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ASSOCIAZIONE NAZIONALE MECCANICA
TEORICA E APPLICATA
RACCOLTA DEI SOMMARI
XX Convegno Nazionale di Meccanica Computazionale
VII Riunione del Gruppo Materiali AIMETA
a cura di Elio Sacco - Sonia Marfia
UNIVERSITA’ DEGLI STUDI DI CASSINO
E DEL LAZIO MERIDIONALE
DIPARTIMENTO DI INGEGNERIA
CIVILE E MECCANICA
Cassino 11 – 13 giugno 2014
www.gimc-gma2014.dicam.unibo.it
Indice dei sommari
GIMC
Error Sensitivity to Refinement: a criterion for optimal grid adaptation
Paolo Luchini, Flavio Giannetti
3
Isogeometric treatment of large deformation contact and debonding problems with
NURBS and T-Splines
Rossana Dimitri
5
Pseudopotentials and thermomechanical response of materials and structures: a convex
analysis approach
Michele Marino
7
Multiphase modeling of porous media: from concrete to tumor growth
Giuseppe Sciumè
9
On the accuracy of the nodal elastic stress of zero thickness interface elements
Giovanni Castellazzi, Daniela Ciancio, Francesco Ubertini
11
The strong formulation finite element method: stability and accuracy
Francesco Tornabene, Nicholas Fantuzzi, Michele Bacciocchi
12
Mixed methods for viscoelastodynamics and topology optimization
Giacomo Maurelli, Nadia Maini, Paolo Venini,
13
Dissipation-based integration algorithm for SMA constitutive models
Edoardo Artioli, Paolo Bisegna
14
Parallel programming techniques for the computation of basins of attraction
Pierpaolo Belardinelli, Stefano Lenci
16
Limit analysis on FRP-strengthened RC members
Dario De Domenico, Aurora A. Pisano, Paolo Fuschi
18
Integrated structure for a resonant micro-gyroscope and accelerometer
Valentina Zega, Claudia Comi, Alberto Corigliano, Carlo Valzasina
20
Numerical analyses in the nonlinear dynamics and control of microcantilevers in atomic
force microscopy
Valeria Settimi, Giuseppe Rega
22
Buckling analysis using a generalized beam model including section distortions
Andrea Genoese, Alessandra Genoese, Antonio Bilotta, Giovanni Garcea
24
Shakedown analysis of 3D frames with an effective evaluation of the elastic domain and
of the load combinations
Leonardo Leonetti, Antonio Bilotta, Giovanni Garcea, Raffaele Casciaro
26
A simple beam model to assess the strength of adhesively bonded tile floorings
Stefano de Miranda, Antonio Palermo, Francesco Ubertini
28
Concrete mechanics at early age
Giuseppe Sciumè, Farid Benboudjema, Giorgio Zavarise
30
Rigid wedge-shaped hull impacting a free surface: a lattice Boltzmann-immersed
boundary study
C. Burrafato, S. de Miranda, A. De Rosis, F. Ubertini
32
i
Analytical evaluation of displacement and stress fields induced in elastic half-spaces by
linear distributions of pressure on the surface
Francesco Marmo, Luciano Rosati
33
How to refine the Sardinia Radio Telescope finite element model
Antonio Cazzani, Flavio Stochino, Emilio Turco
35
A GBT finite element based on elastic solution
S. de Miranda, A. Madeo, D. Melchionda, F. Ubertini
37
Ceramic sanitary wares: reverse engineering strategy for mould prototyping
S. de Miranda, L. Patruno, M. Ricci, R. Saponelli, F. Ubertini
38
Computational modeling of fiber recruitment for statistical distributed biological tissues
Alessio Gizzi, Marcello Vasta, Anna Pandolfi
40
A method of cells-type kinematic limit analysis approach for the evaluation of the
macroscopic strength domain of in-plane loaded periodic masonry
Gabriele Milani, Alberto Taliercio
42
A simple FEM model to predict the mechanical behaviour of an equiatomic NiTi SMA
alloy
Vittorio Di Cocco, Francesco Iacoviello, Alessandra Rossi
43
Evaluation of performance of cold-formed steel structures using Koiter asymptotic
approach
A. Madeo, R. Casciaro, G. Zagari, R. Zinno, G. Zucco
45
A finite-element approach for the analysis of pin-bearing failure of composite laminates
Michele Marino, Francesca Nerilli, Giuseppe Vairo
46
Advanced numerical simulations in biomechanics: patient-specific finite element analysis
of transcatheter aortic valve implantation
S. Morganti, M. Conti, M. Aiello, A. Reali, F. Auricchio
48
A numerically efficient implicit integration algorithm for the Matsuoka-Nakai failure
criterion
Andrea Panteghini, Rocco Lagioia
50
Selective mass scaling for thin structures discretized with multi-layered, solid-shell
elements
Federica Confalonieri, Umberto Perego, Aldo Ghisi
51
A generalized time-domain approach for motion-related wind loads on long-span bridges
S. de Miranda, L. Patruno, F. Ubertini, G. Vairo
53
A new flexible approach for shape memory alloy constitutive modeling
Ferdinando Auricchio, Elena Bonetti, Giulia Scalet, Francesco Ubertini
55
Damage modelling in concrete subject to sulfate attack
Nicola Cefis, Claudia Comi
57
A 3D mixed frame element with multi-axial coupling for thin-walled structures with
damage
Daniela Addessi, Paolo Di Re
59
A basic introduction to isogeometric collocation methods with some applications
Alessandro Reali, Ferdinando Auricchio, Lourenco Beirão da Veiga, Hector Gomez,
Thomas JR Hughes, Giancarlo Sangalli
60
ii
On the state update for isotropic elasto-plastic hardening materials: a dissipation-based
algorithm
Nicola A. Nodargi, Edoardo Artioli, Federica Caselli, Paolo Bisegna
62
A Lagrangian finite element approach for the numerical simulation of landslide runouts
Massimiliano Cremonesi, Francesco Ferri, Umberto Perego
64
Geometry of elastoplasticity in the nonlinear range
Giovanni Romano, Raffaele Barretta, Marina Diaco
66
FE-Meshless multiscale non-linear analysis of masonry structures
Giuseppe Giambanco , Emma La Malfa Ribolla, Antonino Spada
67
Non-linear analysis of 3D elastoplastic framed structures
Valerio Carollo, Giuseppe Giambanco, Antonino Spada
69
Interface poroelastic laws to model fluid-induced damage in oil wells
Carlo Callari, Valentina Fasano
71
Formulation of rate-dependent cohesive-zone models
Giulio Alfano, Marco Musto
72
Porous shape memory alloys: a micromechanical analysis
V. Sepe , F. Auricchio, S. Marfia, E. Sacco
74
A corotational tetrahedral element with rotational degrees of freedom for largedisplacement analysis of inelastic structures
Paolo Bisegna, Federica Caselli, Edoardo Artioli, Nicola A. Nodargi
76
A consistency study of cohesive zone models for mixed-mode debonding problems
Rossana Dimitri, Marco Trullo, Laura De Lorenzis, Giorgio Zavarise
78
A multilevel finite element approach for piezoelectric textiles made of polymeric
nanofibers
Claudio Maruccio, Laura De Lorenzis
80
Computational issues on multiscale FE analysis
Francesco Parrinello, Guido Borino
82
An efficient Bouc & Wen approach for seismic analysis of masonry tower
Luca Facchini, Michele Betti
84
Analysis of masonry arches: a NURBS based simple applicative program
Andrea Chiozzi, Marcello Malagù, Antonio Tralli, Antonio Cazzani
85
Isogeometric collocation for large-deformation frictional contact
Laura De Lorenzis, Roland Kruse, Nhon Nguyen-Thanh
87
An adaptive multiscale approach for the failure analysis of fiber-reinforced composite
materials
Domenico Bruno, Fabrizio Greco, Lorenzo Leonetti, Stefania Lo Feudo, Paolo Lonetti
89
Consistent tangent operator for an exact Kirchhoff rod model
Leopoldo Greco, Massimo Cuomo
91
An implicit G1-continuity interpolation for Kirchhoff plate elements
Leopoldo Greco, Massimo Cuomo
92
Pull-out strength of chemical anchors in natural stone
Loredana Contrafatto, Renato Cosenza
93
Strain gradient elasticity within the symmetric BEM formulation
S. Terravecchia, T. Panzeca, C. Polizzotto
95
iii
Multidomain Symmetric Galerkin BEM for non-linear analysis of masonries in-plane
loaded
L. Zito, S. Terravecchia, T. Panzeca
97
GMA
Sorption of low molecular weight compounds in polymers: thermodynamic issues and
plasticization effects
Giuseppe Mensitieri, Giuseppe Scherillo, Pellegrino Musto
101
Propagation of elastic waves and generation of band-gaps in diffusively damaged
structures
Giorgio Carta, Michele Brun, Alexander B. Movchan
103
On the compressive strength of glass-microballoons/thermoset-matrix syntactic foams
Lorenzo Bardella, Andrea Panteghini
105
Elastically deformable scale through configurational forces
Francesco Dal Corso, Davide Bigoni, Federico Bosi, Diego Misseroni
106
Flaw-tolerance of nonlocal discrete systems and interpretation according to network
theory
Andrea Infuso, Marco Paggi
107
A model to interpret the wedge-shaped spalling in pull-out tests of FRP from concrete
Roberto Ballarini, Annalisa Franco, Gianni Royer Carfagni
108
Morphoelastic rods
Alessandro Tiero, Giuseppe Tomassetti
110
Bending of shape-memory alloys’ beams: constitutive modeling and structural response
Silvia Di Caprera, Michele Marino, Giuseppe Vairo
111
Pre-buckling behavior of composite beams: an innovative approach
Francesco Ascione, Geminiano Mancusi, Marco Lamberti
113
Effective modeling of multilayered composites with cohesive and imperfect interfaces
Roberta Massabò, Francesca Campi
115
Micropolar and second-gradient homogenization of chiral cellular solids
Andrea Bacigalupo, Luigi Gambarotta
117
TWSME of NiTi strips in free bending conditions: experimental and theoretical approach
A. Fortini, M. Merlin, R. Rizzoni, S. Marfia
119
Discrete-to-continuum approaches for complex materials as ‘Non–Simple’ continua
Patrizia Trovalusci
121
Constitutive Behavior of FRCM Materials for Structural Plating: an experimental study
Luigi Ascione, Anna D’Aponte, Geminiano Mancusi
123
A new auxetic lattice model
Luigi Cabras, Michele Brun
125
Cloaking in flexural waves
D. Colquitt, M. Brun, M. Gei, A.B. Movchan, N.V. Mochan, I.S. Jones
126
A contact problem in couple-stress thermoelasticity
Thanasis Zisis, Francesco Dal Corso
127
Flutter analysis of piezoelectric laminate beams in MEMS
Raffaele Ardito, Rocco Musci
128
iv
Variational approach to damage mechanics with plasticity and nucleation of cohesive
cracks
Roberto Alessi, Achille Paolone, Stefano Vidoli
130
Geometric numerical integrators based on the magnus expansion in bifurcation problems
for non-linear elastic solids
Anna Castellano, Pilade Foti, Aguinaldo Fraddosio, Salvatore Marzano, Mario Daniele
Piccioni
132
Experimental and numerical approaches for the ultrasonic characterization of composite
materials
Anna Castellano, Pilade Foti, Aguinaldo Fraddosio, Salvatore Marzano, Mario Daniele
Piccioni
134
A micromechanical four-phase model to predict the compressive failure surface of
cement concrete
Andrea Caporale, Raimondo Luciano
136
Multiscale analyses of a three layers osteochondral scaffold
G. Parisi, S. Bignozzi, E. Kon, P. Vena
138
Damage propagation modeling of masonry structures subjected to dynamic loading
Jessica Toti, Vincenzo Gattulli, Elio Sacco
140
A micromechanical approach for the micropolar modeling of heterogeneous periodic
media
Maria Laura De Bellis, Daniela Addessi, Elio Sacco
142
An experimental investigation on the axial and rotational behavior of web-flange
junctions of open-web pultruded glass fibre-reinforced profiles
Luciano Feo, Ayman S. Mosallam, Rosa Penna
144
Development of biodegradable magnesium alloy stents with coating: the peeling problem
Lorenza Petrini,Wei Wu, Lina Altomare, Barbara Previtali, Maurizio Vedani, Francesco
Migliavacca
146
Interface constitutive relation derived from a representative adhesive layer
Guido Borino , Francesco Parrinello
148
A cohesive-zone model simulating damage, friction and interlocking
Roberto Serpieri, Elio Sacco, Giulio Alfano
150
Crack detection in beam-like structures by nonlinear harmonic identification
Paolo Casini, Oliviero Giannini, Fabrizio Vestroni
152
A data fusion based approach for damage detection in linear systems
Ernesto Grande, Maura Imbimbo
154
Superelastic and Shape Memory effects in shape memory alloy beams
Sara Malagisi, Sonia Marfia, Elio Sacco
156
Anisotropic Swelling in fibrous materials
Paola Nardinocchi, Matteo Pezzulla, Luciano Teresi
158
v
vi
SOMMARI
GIMC
Gruppo Italiano di Meccanica
Computazionale
AIMETA
Congresso GIMC-GMA-2014
11-13 giugno 2014
1
2
Invited Lecture
Error Sensitivity to Refinement: a criterion for optimal grid adaptation
Paolo Luchini1,a *, Flavio Giannetti1,b
1
DIIN, Universita’ di Salerno, Via Giovanni Poalo II, 84084 Fisciano (SA) , Italia
a
[email protected], [email protected]
Keywords: Grid adaptation, error estimation, adjoint, sensitivity
Computational fluid dynamics has become a key technology in the development of new products in
the aeronautical industry. CFD codes are in fact routinely employed to test and optimize different
aerodynamics configurations and offer in many cases a valid alternative to expensive wind-tunnel
experiments. However, despite the progress made in the last decade in terms of code efficiency and
computational resources, large aerodynamic simulations of viscous flows around complex
configurations are still very expensive. The limiting factor of the applicability of CFD as an effective
design tool resides, in fact, in the large number of degrees of freedom needed to accurately predict the
characteristics of complex flow fields. A well known strategy to minimize the computational cost is
automatic mesh adaptation (see for example [1]), i.e. the technique of increasing or decreasing the
number of computational nodes in certain regions of the flow field according to the local features of
the solution. In this way one can achieve substantial savings in memory and computation time while
maintaining a given level of accuracy. A strategy which is often used is based on refining the grid in
certain regions of the flow where some local properties or indicators exceed predetermined values.
Examples are criteria based on the gradient of certain flow quantities such as the velocity or the
vorticity [1, 2]. However, efficient strategies for grid adaptation and grid refinement require a reliable
indicator not only of discretization error, but of the concrete advantage that can be gained by
decreasing it. Although this approach is simple and easy to implement, without a suitable indicator it
is not optimal, in the sense that it does not necessarily guarantee a reduction of the global solution
error and more accurate results. Another approach which is often used in engineering applications is to
assess the error made in predicting important integral quantities such as lift or drag, rather then
focusing on the global error. These strategies are usually based on the properties of the adjoint
equations, which have the ability to relate the error in the required integral quantity to the residual
error of the discretization [3, 4]. Adjoint-based techniques have found widespread use in aerodynamic
calculations to estimate the error of the required output, or as indicators for local grid refinement (see
[5] for a review).
In this work we use an adjoint-based approach to derive a new optimal criterion for an effective mesh
refinement strategy which aims at minimizing the global solution error. Such criterion is derived by
using the properties of the adjoint operators and is based on the sensitivity of the error (or its estimate)
to a local mesh refinement. This sensitivity is derived from the knowledge of two numerical solutions,
one calculated on a coarse and one on a fine mesh. A system of forced adjoint equations is then
derived from a minimization problem in which the objective function is an estimate of the L2 error
norm. By combining the adjoint variables with the local values of the coarse-grid residual we obtain a
spatial map representing the sensitivity of the error to a local refinement of the mesh. As an example
figures (a), (b) and (c) display the error sensitivity to small variations in the residual of the momentum
(horizontal and vertical component) and continuity equations for the Kovasznay flow [6] depicted in
figure (d). By inspecting the spatial structures of these sensitivity maps, it is possible to determine the
regions of the flow where a local refinement of the mesh would be most effective in terms of accuracy
gain per computational effort, and to unveil information on the error propagation properties in the
system.
3
Figures: Sensitivity of the L2 error to local variations in the residual of: (a) horizontal momentum equation, (b) vertical
momentum equation and (c) continuity equation. Results refer to computations performed on a 50 × 50 uniform grid for
the Kovasznay flow at Re = 40. (d) Streamlines for the Kovasznay flow at Re = 40
The error sensitivity so derived can be used as an effective indicator to implement an optimal strategy
of adaptive local refinement. The proposed approach has been tested with a second order finitedifference Navier-Stokes code. Sensitivity maps for different benchmark problems will be presented
for both staggered and colocated discretization.
References
[1] T.J.Baker,“Meshadaptationstrategiesforproblemsinfluiddynamics",Finite Elements Anal. Design
25,243 (1997).
[2] G. P.Warren,W. K. Anderson, J. T. Thomas, and S. L. Krist, “Grid Convergence for Adaptive
Methods", AIAA Pap. 91-1592 (1991)
[3] R. Becker and R. Rannacher, “Weighted A Posteriori Error Control in Finite Element Methods",
Technical report, preprint No. 96-1, Universitat Heidelberg (1994).
[4] M. B. Giles and N. A. Pierce “Adjoint Equations in CFD: Duality, Boundary Conditions and
Solution Behavior", AIAA Pap. 97-1850 (1997).
[5] K. J. Fidkowski and D. L. Darmofal “Review of Output-Based Error Estimation and Mesh
Adaptation in Computational Fluid Dynamics", AIAA Journal 49 No. 4 (2011).
[6] L.I.G.Kovasznay“Laminarflowbehindatwo-dimensionalgrid",Proc.CambridgePhil.Soc
44,58Ð62(1948).
4
Tesi di dottorato selezionata per il premio ECCOMAS
Isogeometric treatment of large deformation contact and debonding
problems with NURBS and T-Splines
Rossana Dimitri
Dipartimento di Ingegneria dell'Innovazione, Università del Salento
Via per Monteroni, 73100, Lecce, [email protected]
Keywords: Bimaterial peel test, T-splines interpolations, cohesive interface.
Within a setting where isogeometric analysis (IGA) has been successful at bringing two different
research fields together, i.e. Computer Aided Geometric Design (CAGD) and numerical analysis, Tspline-based IGA is applied in this work to frictionless contact and debonding problems between
deformable bodies in the context of large deformations. The key feature of IGA is the exact
description of the geometry with a tailorable degree of continuity at the element boundaries, in
addition to the advantageous features of variation diminishing, convex hull properties, and nonnegativeness of the basis functions [1]. The first investigations on contact problems with NURBSbased isogeometric discretizations [2,3] have already shown significant advantages in terms of
robustness and accuracy of contact formulations over conventional finite element descriptions.
However, as a design tool NURBS surfaces are limited by their rigid tensor-product structure and
four-sided nature. NURBS-based design deficiencies can be overcome by using T-splines, which
allow for local refinement through the introduction of T-junctions and extraordinary points [4].
The continuum is here discretized with cubic T-splines and NURBS, which are incorporated into an
existing finite element framework by using Bézier extraction, i.e. a linear operator which maps the
Bernstein polynomial basis on Bézier elements to the global NURBS or T-spline basis. A Gausspoint-to-surface (GPTS) formulation is adopted for the enforcement of the contact constraints,
whereby a desired number of quadrature points is located on the contact surface and the contact
constraints are enforced independently at each quadrature point [5]. Some numerical examples
demonstrate the potential of T-spline IGA to solve challenging contact problems in 2D and 3D. More
specifically, the Hertz problem is used as benchmark to compare the performance of cubic T-spline
discretizations with NURBS of equal order from the standpoint of spatial convergence, characterized
by uniform (Nu) and non-uniform (Nnu) patterns. The convergence study shows a very similar order of
convergence, due to the equal polynomial degree and contact formulation and to the absence of error
estimation criteria in performing the local T-spline refinement. However, the T-spline error curve is
shown to lie below all the NURBS curves, thus demonstrating the superior accuracy of T-splines for a
given number of degrees of freedom (DOFs) D0 (Fig. 1).
Figure 1: L2 error norm of the contact pressure. Penalty parameter N=103.
The purely geometric enforcement of the non-penetration condition in compression is then generalized
to encompass both contact and mode-I debonding of interfaces, which is here approached through
5
cohesive zone (CZ) modeling [6]. Depending on the contact status, an automatic switching procedure
is used to choose between cohesive and contact models. A challenge in the numerical computation of
debonding problems by applying CZ models is to capture correctly the strain field around the crack
front during its propagation. Unless a sufficiently fine mesh is provided in the process zone, the
computed load-deflection response is usually non-smooth and may exhibit artificial snap-throughs and
snap-backs. Within the isogeometric context, however, NURBS and T-spline discretizations feature
higher inter-element continuity. This is the primary reason why their use proves to be a
computationally accurate and efficient technology for the solution of interface problems.
Results for the double cantilever beam (DCB) test and for the bimaterial peel test with varying
resolutions of the process zone and number of Gauss points used for the enforcement of the contact
constraints are presented and compared. The superior accuracy of T-splines interpolations with respect
to the NURBS and Lagrange ones for a given number of DOFs is verified. Fig. 2 illustrates the main
results obtained for a peel test. A bilinear cohesive law is adopted with cohesive strength pNmax=6
MPa, fracture energy GIC=0.1 N/mm, and ratio between the ultimate and maximum opening
displacements gNu/gNmax=10.
6
L3
N3
T
5
3
L oad [N]
L oad [N]
4
2
1
0
0
2
4
6
8
Displacement [mm]
10
1.8
1.75
1.7
1.65
1.6
1.55
1.5
1.45
1.4
L3
N3
T
4
4.5
5
5.5
Displacement [mm]
6
Figure 2: Load-displacement response for a peel test problem.
Acknowledgements: The European Research Council provided funding for this research under the
EU’s FP7/2007-2013, ERC Starting Researcher Grant “INTERFACES”, G.A. n° 279439.
References
[1] T.J.R. Hughes, J.A. Cottrell, Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS,
exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg. 194 (2005) 4135-4195.
[2] L. De Lorenzis, I. Temizer, P. Wriggers, G. Zavarise. A large deformation frictional contact
formulation using NURBS-based isogeometric analysis, Int. J. Numer. Meth. Eng. 87(13) (2011)
1278-1300.
[3] L. De Lorenzis, P. Wriggers, G. Zavarise. A mortar formulation for 3D large deformation contact
using NURBS-based isogeometric analysis and the augmented Lagrangian method, Comp. Mech.
49(1) (2012) 1-20.
[4] M.A. Scott, X. Li, T.W. Sederberg, T.J.R. Hughes. Local refinement of analysis-suitable T-splines,
Comput. Methods Appl. Mech. Engrg. 213-216 (2012) 206-222.
[5] R. Dimitri, L. De Lorenzis, M. Scott, P.Wriggers, R.L. Taylor, G. Zavarise. Isogeometric large
deformation frictionless contact using T-splines, Comput. Methods Appl. Mech. Engrg. 269 (2014)
394-414.
[6] R. Dimitri, L. De Lorenzis, P. Wriggers, G. Zavarise. NURBS- and T-spline-based isogeometric
cohesive zone modeling of interface debonding, Comput. Mech. DOI 10.1007/s00466-014-0991-7.
6
Tesi di dottorato che ha partecipato alla selezione per il premio ECCOMAS
Pseudopotentials and thermomechanical
response of materials and structures: a convex analysis approach
Michele Marino
Department of Civil Engineering and Computer Science,
Università degli Studi di Roma “Tor Vergata”, Via del Politecnico 1, 00133 Rome, Italy
[email protected]
Keywords: Constitutive modeling, damage modeling, thermomechanical structural response.
The thermomechanical response of materials and structures in many engineering applications is
characterized by dissipative mechanisms highly affecting their functioning behavior. Dissipation can
be associated with a change of physical/chemical properties (that is, phase change) or with damage.
Dissipation characterizes also the response of structures undergoing a fast and transient change in
boundary conditions as, for instance, in collisions. Moreover, the behavior of unilateral constraints can
be modeled recurring to different dissipative behaviors depending on the values of state quantities’
evolution. The thermomechanical evolution of systems characterized by dissipation is highly nonlinear and the modeling of such behavior represents an open research issue.
In present thesis, the response of a number of materials and structures under dissipative conditions are
addressed by means of a unified approach based on: the choice of the state quantities describing the
physical phenomena under investigation, the formulation of equilibrium conditions from the
application of the Principle of Virtual Power, and the definition of suitable constitutive laws allowing
thermodynamical prescriptions to be a-priori satisfied. The constitutive laws are defined by
introducing the free-energy and the pseudo-potential of dissipation, as proposed by Jean-Jacques
Moreau. Accordingly, the convexity of the dissipative pseudo-potential allows the fulfillment of the
second law of thermodynamics. Moreover, employing arguments from convex analysis, physical
restrictions either on the value of state quantities or on their evolution are imposed within a variational
framework as internal constraints on the free-energy or on the pseudo-potential of dissipation.
Applications will be devoted to shape-memory alloys (undergoing thermal-induced and/or stressinduced phase-change), biological tissues (including possible damage of collagenous biostructures)
and tensegrity structures (where the unilateral response of cables is reproduced by means of
dissipative models). The mechanisms under investigation are modeled under the hypothesis of
smooth-evolution of the systems, apart when phase-change of shape-memory alloys due to collisions
is investigated and modeled as an instantaneous phenomenon. In this case, since the response is
assumed to be discontinuous in time, the system is characterized by a non-smooth evolution, and the
consistency of the employed thermomechanical framework is shown.
Addressing shape-memory alloys under smooth evolution, the model proposed by Michel Frémond is
generalized in order to improve model capabilities in capturing their macroscopic behavior: pseudoelasticity, shape-memory effects, as well as the complex thermomechanical coupling are taken into
account. The model is validated by means of comparisons with available experimental data on the
isothermal uniaxial traction response, on the thermal behavior as obtained by differential scanning
calorimetry, and on the shape-memory effect. Moreover, sensitivity on model parameters without a
clear physical meaning is analyzed and analytical relationships for an effective identification of
parameters’ values are presented. The model is also applied for showing its capabilities in reproducing
the dependence of the uniaxial response on temperature, the occurrence of residual strain, materialtraining effects, and biaxial responses. It is also shown that the model predicts a strain-rate
independent mechanical response under isothermal conditions. Nevertheless, when the non-isothermal
case is addressed, the model effectively reproduces well-established experimental evidence depending
on thermomechanical-coupling-effects. In fact, material’s mechanical response highly depends on the
conductive and convective properties of the surrounding medium in which the experiment is carried
7
out, as well as on the strain-rate when the latent heat released during martensitic transformation
determines a significant increase of specimen temperature.
Moreover, phenomena occurring when a solid collides against an object made up of a shape-memory
alloy are modeled, proving that the proposed impact theory predicts that temperature increases, that
austenitic phase appears when the collision is violent and that post-impact velocity starts the motion
after the collision. The resulting equations involve standard mathematical operators and can be solved
recurring to finite element discretizations. Parametric and sensitivity analyses are presented and the
results of a campaign of numerical simulations are shown as representative of the process behind the
design of damping or energy-absorbing devices.
When damage in biological tissues is addressed, the starting point is the well-established evidence,
numerically obtained by molecular dynamical simulations, that inelastic mechanisms at tissue
macroscale are related at the fibril level to the rupture of intra-molecular and inter-molecular covalent
bonds, as well as to slip-pulse mechanisms associated with inter-molecular weak interactions.
Therefore, the mechanical behavior of collagenous fibrils at nano/microscale is herein modeled
consistently up-scaling nanoscale molecular and inter-molecular behavior by means of a multiscale
homogenization technique allowing to obtain the elasto-damage response of collagenous fibrils under
uniaxial traction. Moreover, planar collagenous fibers with a curvilinear centerline, as found in in-vivo
biological tissues, are treated following the same rationale in order to formulate the equations
governing the elasto-damage response of such microscale structures based on nanoscale quantities.
Several numerical applications are presented and discussed, aiming to highlight soundness and
effectiveness of the present approach, and recovering a number of well-established experimental
evidences. Present model opens to the possibility of correlating structure/arrangement of tissue
constituents with their mechanical function, in the way of an effective integration of mechanics,
biochemical surrounding and histology at different scales of investigation.
Finally, addressing tensegrity structures, several classical results are consistently recovered in a novel
variational framework addressing ideal and non-ideal models for structural members. Basic results are
coupled with energy-based non-conventional physical interpretations, even in the case of ideal or
mixed-type structures wherein application of differentiable energy-based arguments fails. To provide
an efficient structural design tool for tensegrities, unilateral conditions on infinitesimal motions and
reaction forces are embedded into a quadratic minimization problem under equality and inequality
linear constraints. Moreover, in order to investigate on the stability of mixed-type tensegrities, an
operative stability criterion based on a new necessary and sufficient condition is established. Both
results show as convex optimization arguments properly apply for the development of an efficient
specific algorithm devoted to the design of tensegrity structures in civil and mechanical applications.
8
Tesi di dottorato che ha partecipato alla selezione per il premio ECCOMAS
Multiphase modeling of porous media: from concrete to tumor growth
Giuseppe Sciumè
Department of Innovation Engineering, Università del Salento, Lecce, Italy
[email protected]
Keywords: porous media mechanics, multiphase flow, TCAT.
Porous media mechanics (PMM) has been ordinarily used in the past for geomechanical problems at
large, but nowadays it is also currently applied to model biomechanical and biomedical ones: teeth
and bones decalcification, herniation of intervertebral discs, glaucoma and tumor growth are examples
of clinical pathologies which can be modeled using mathematical approaches based on it.
To highlight the versatility of PPM, two very different applications are presented: a multi-physics
model for concrete at early age [1-2], and a model for tumor growth [3-4]. Both are multiphase
models governed by macroscopic balance equations derived via the Thermodynamically Constrained
Averaging Theory (TCAT) [5] which gives consistency across scales.
Figure 1: Volume fractions of TCs, HCs and mass fraction of oxygen for a multicellular tumor spheroid (MTS)
growing in a host tissue.
After an overview of the formal analogies between these two models and of their major features, the
attention is focused on the second one, that of tumor growth.
Tumor is modeled as a four-phase system which consists of a solid phase, the extracellular matrix
(ECM), and three immiscible fluid phases. The fluid phases are the interstitial fluid (IF), tumor cells
(TC) and healthy cells (HC), with the latter two phases modeled as adhesive fluids. Being the tumor
growth strongly influenced by nutrients availability, the diffusion of oxygen coming from the nearby
existing vessels is also considered.
In the previous version of the model a unique pressure was considered for both cell populations (pTC =
pHC) [3]. Nowadays, appropriate constitutive relationships for the pressure difference among each pair
of fluid phases are introduced, allowing for different pressures in the three fluid phases [4]. These
9
relationships respect the relative wettability of fluids and take into account explicitly fluid–fluid
interfacial tensions, resulting in a more realistic modeling of cell adhesion and invasion.
High interfacial tension at the TC–HC interface support a rapid growth of the malignant mass, with a
relevant amount of HC which cannot be pushed out by TC and remains in place; conversely, a lower
TC–HC interfacial tension tends to originate a more compact and dense tumor mass with a slower
growth rate of the overall size. This enhancement together with the recent relaxation of the assumption
of a rigid ECM [6], generalize the model and allow to properly take into account the physical
properties of the host tissue in which tumor grows and evolves.
Acknowledgements
This work has been carried out within a PhD co-tutelage program between the Department of Civil,
Environmental and Architectural Engineering of Padua and the Laboratoire de Mécanique et
Technologie of ENS-Cachan. In this context the scientific support and contribution of Bernhard
Schrefler, Professor at University of Padua, Yves Berthaud, Professor at University Pierre et Marie
Curie (UPMC), Farid Benboudjema and Caroline De Sa, Professors at ENS-Cachan, are gratefully
acknowledged.
References
[1] G. Sciumè, F. Benboudjema, C. de Sa, F. Pesavento, Y. Berthaud, B.A. Schrefler, A multiphysics
model for concrete at early age applied to repairs problems, Engineering Structures 57 (2013) 374387.
[2] G. Sciumè, F. Pesavento, B.A. Schrefler, Thermo-hygro-chemo-mechanical modeling of the
behavior of a massive beam with restrained shrinkage. Proceedings of RILEM-JCI international
workshop on crack control of mass concrete and related issues concerning early-age of concrete
structures, (2012) 133-144.
[3] G. Sciumè, S.E. Shelton , W.G. Gray, C.T. Miller, F. Hussain, M. Ferrari, P. Decuzzi, B.A.
Schrefler, A multiphase model for three dimensional tumor growth, New Journal of Physics, 15
(2013) 015005.
[4] G. Sciumè, W.G. Gray, F. Hussain, M. Ferrari, P. Decuzzi and, B.A. Schrefler, Three phase flow
dynamics in tumor growth, Computational Mechanics, 53(3) (2014) 465-484.
[5] W.G. Gray, C.T. Miller, Thermodynamically constrained averaging theory approach for modeling
flow and transport phenomena in porous medium systems: 1. Motivation and overview. Advances in
Water Resources, 28 (2005) 161–180.
[6] G. Sciumè, R. Santagiuliana, M. Ferrari, P. Decuzzi and, B.A. Schrefler, A tumor growth model
with deformable ECM, submitted to Physical Biology (2014).
10
On the accuracy of the nodal elastic stress of zero thickness interface
elements
Giovanni Castellazzi1,a *, Daniela Ciancio2,b, Francesco Ubertini1,c
1
DICAM – School of Engineering and Architecture, Viale del Risorgimento, 2 – Bologna, Italy
The University of Western Australia, 35 Stirling Highway, CRAWLEY WA 6009, Australia
a
[email protected], [email protected], [email protected]
1
Keywords: Interface Elements, Stone Masonry Walls, Penalty Stiffness Factors, Sequential Linear
Analysis.
A recent study [1] has discussed the accuracy of the nodal elastic stresses of zero thickness interface
elements when fictitious elastic parameters (often called penalty stiffness factors) are used. These
stresses, commonly used to calculate the triggering conditions of the opening of the interface
simulating a discrete crack in quasi-brittle materials, are affected by non-negligible errors if the
interfaces are embedded in unstructured/irregular meshes. A procedure to avoid these errors has been
proposed for homogeneous materials [1] and bi-material interfaces [2]. Accurate nodal stresses are
recovered if certain geometry-dependent pre-processed coefficients are used. In this paper, this
method is proposed in the analysis of Historical Stone Masonry Walls: zero-thickness interface
elements represent the interaction between stones of irregular shape, similarly to other studies [3, 4]
currently available in the literature. The crack opening triggering conditions are calculated for two
identical meshes with rigid continuous elements representing the stones and fictitiously elastic zerothickness interfaces representing the mortar layers; one mesh contains the coefficients proposed in [1]
and the other doesn’t. The precision of the method and a mesh-dependency study is presented for
several numerical examples. The implementation of this procedure within the framework of the
Sequential Linear Analysis (SLA) is also discussed.
References
[1] D. Ciancio, I. Carol, G. Castellazzi, Optimal penalty stiffness values of concurrent 2D elastic
interface elements leading to accurate stress tractions, Int. J. Num. Meth. Engng., to appear, (2014).
[2] D. Ciancio, G. Castellazzi, Fictitious elastic stiffness parameters of zero-thickness finite elements
at bi-material interfaces, Applied Mechanics and Materials, to appear, (2014).
[3] R. Senthivel, P.B. Lourenço, Finite element modelling of deformation characteristics of historical
stone masonry shear walls, Engineering Structures 31 (2009) 1930-1943.
[4] B. Villemus, J.C. Morel and C. Boutin, Experimental assessment of dry stone retaining wall
stability on a rigid foundation, Engineering Structures, 29 (2007) 2124-2132.
11
The strong formulation finite element method: stability and accuracy
Francesco Tornabene1,a *, Nicholas Fantuzzi1,b , Michele Bacciocchi1,c
1
DICAM Department, Viale del Risorgimento 2, 40136 Bologna, Italy.
[email protected], [email protected], [email protected]
* corresponding author
a
Keywords: Strong Formulation Finite Element Method, Differential Quadrature Method, Finite
Element Method, Free vibration analysis, Static Analysis, Numerical Stability.
The Strong Formulation Finite Element Method (SFEM) is a numerical solution technique for solving
arbitrarily shaped structural systems. This method uses a hybrid scheme given by the Differential
Quadrature Method (DQM) and the Finite Element Method (FEM). The former is used for solving the
differential equations inside each element and the latter employs the mapping technique to study
domains of general shape. A general brief review on the current methodology has been reported in the
book [1] and recalled in the works [2,3], where a stress and strain recovery procedure was
implemented. The aim of this manuscript is to present a general view of the static and dynamic
behaviors of one- and two-dimensional structural components solved by using SFEM. It must be
pointed out that SFEM is a generalization of the so-called Generalized Differential Quadrature Finite
Element Method (GDQFEM) presented by the authors in some previous papers [4-8]. Particular
interest is given to the accuracy, stability and reliability of the SFEM when it is applied to simple
problems. Since numerical solutions - of any kind - are always an approximation of physical systems,
all the numerical applications are compared to well-known analytical and semi-analytical solutions of
one- and two-dimensional systems. Ultimately, this work presents typical aspects of an innovative
domain decomposition approach that should be of wide interest to the computational mechanics
community.
References
[1] F. Tornabene, N. Fantuzzi, Mechanics of Laminated Composite Doubly-Curved Shell Structures,
Esculapio, Bologna, 2014.
[2] N. Fantuzzi, F. Tornabene, Strong Formulation Finite Element Method for Arbitrarily Shaped
Laminated Plates – I. Theoretical Analysis, Adv. Aircraft Space. Sci. 1 (2014) 124-142.
[3] N. Fantuzzi, F. Tornabene, Strong Formulation Finite Element Method for Arbitrarily Shaped
Laminated Plates – II. Numerical Analysis, Adv. Aircraft Space. Sci. 1 (2014) 143-173.
[4] N. Fantuzzi, F. Tornabene, E. Viola, Generalized Differential Quadrature Finite Element Method
for Vibration Analysis of Arbitrarily Shaped Membranes, Int. J. Mech. Sci. 79 (2014) 216-251.
[5] E. Viola, F. Tornabene, E. Ferretti, N. Fantuzzi, On Static Analysis of Composite Plane State
Structures via GDQFEM and Cell Method, CMES 94 (2013) 421-458.
[6] E. Viola, F. Tornabene, E. Ferretti, N. Fantuzzi, GDQFEM Numerical Simulations of Continuous
Media with Cracks and Discontinuities, CMES 94 (2013) 331-369.
[7] E. Viola, F. Tornabene, E. Ferretti, N. Fantuzzi, Soft Core Plane State Structures Under Static
Loads Using GDQFEM and Cell Method, CMES 94 (2013) 301-329.
[8] E. Viola, F. Tornabene, N. Fantuzzi, Generalized Differential Quadrature Finite Element Method
for Cracked Composite Structures of Arbitrary Shape, Compos. Struct. 106 (2013) 815-834.
12
Mixed methods for viscoelastodynamics and topology optimization
Giacomo Maurelli1, a *, Nadia Maini1,b And Paolo Venini1,c,*
1
Department of Civil Engineering and Architecture, University of Pavia, Italy
[email protected], [email protected], [email protected]
a
Keywords: Mixed methods, viscoelasticity, dynamics.
We present an innovative method for the analysis of viscoelastic plane systems based on a truly-mixed
Hellinger-Reissner variational principle, wherein stresses and velocities are the main variables and
Lagrange multipliers, respectively. Our discretization adopts the Arnold-Winther element [1] as to the
stress variables along with usual element wise-linear displacements. Our approach is an original
variation of that presented in [2] wherein the symmetry of the stress tensor is imposed weakly as
opposed to the strong stress-symmetry approach that is assumed in this paper with a considerable
reduction of the computational burden. In the second part of the paper, a novel topology optimization
approach is proposed for viscoelastic structures focusing on eigenvalue-based objective functions [3].
For a given representative problem, the viscoelastic optimal solutions are presented and compared to
more classical elastic solutions, see Figure 1, that neglect the effect of viscosity that is however known
to be crucial when dissipative or isolation devices are under investigation.
Figure 1: Elastic design to be confronted with the new viscoelastic solution.
References
[1] D. Arnold, R. Winther, Mixed finite elements for elasticity, Numer. Math., 92 (2002) 401-419.
[2] M.E. Rognes, R. Winther, Mixed finite elements for linear viscoelasticity using weak symmetry,
Math. Models & Methods in Applied Sciences, 20(6) (2010) 955-985.
[3] M. Bruggi, P. Venini, Eigenvalue-based optimization of incompressible media using mixed finite
elements with application to isolation devices, Comp. Meth. Appl. Mech. Engng., 197(13-16) 12621279.
13
Dissipation-based integration algorithm for SMA constitutive models
Edoardo Artioli1,a* and Paolo Bisegna1,b
1
DICII - University of Rome Tor Vergata, via del Politecnico 1, 00133 Rome, Italy
a
[email protected], [email protected]
Keywords: Shape memory alloy, Constitutive model, Integration algorithm, Dissipation.
Shape memory alloys (SMA) are materials which, after being subjected to a severe apparently plastic
deformation, can recover their original shape if subjected to an appropriate temperature increase. Such
unique mechanical behavior is associated with stress-induced solid phase transformations from
twinned to detwinned martensite (during the “plastic” deformation) and from detwinned martensite to
austenite (during shape recovery). The former characteristic is referred to as pseudoelasticity, the
latter as shape memory effect. Representative materials exhibiting shape memory include NiTi,
CuZnAl, CuAlNi and AuCd alloys. Shape memory alloys represent a class of materials that can be
engineered and are found in several engineering applications.
One of the most efficient approaches for the thermo-mechanical modeling of SMA is based on
thermodynamics with internal variables [1]. Constitutive models developed within this approach are
said to be dissipative, inasmuch they are consistent with the fundamental laws of thermodynamics. In
particular, the material state is defined through an appropriate energy potential   ε, e tr ,T 
(Helmholtz free energy in the present case) depending on internal variables, usually an inelastic
macroscopic strain tensor e tr (transformation strain), and a set of observable control variables, usually
the strain tensor ε and the temperature T . Typically, internal free energy is additively split and
contains at least an elastic term and a chemical energy term. Constitutive equations are derived writing
state equations, which define entities conjugate to control variables and to internal variables,
supplemented by a rate equation for the transformation strain. The latter is a flow rule associated to
some plasticity-like yield function f  X  , or transformation-function, separating the elastic domain
from the evolution activation threshold, in terms of the stress measure X , conjugate to e tr . In the
format originally proposed by Souza et al. [4] and subsequently developed by Auricchio and Petrini
[5], the transformation function is taken in von-Mises or Prager-Lode form, respectively. The solution
procedure applies a backward Euler scheme for the integration of the evolution equation; the update of
material state is carried out using a return map algorithm and a branch detection scheme,
distinguishing between three different material phases. A regularized transformation strain is applied
in case e tr  0 , for which chemical energy is non differentiable.
In the present context, the constitutive model is formulated as a minimum principle for the total rate of
energy and dissipation[3,4]:


inftr   ε, etr ,T   D  e tr 
e 
(1)
resulting in the following non-smooth nonlinear differential inclusion:
0   ε, e tr ,T   D  e tr 
(2)
where D  e tr  is the dissipation potential, a degree-one positively homogeneous function of the
transformation strain rate, non differentiable at e tr  0 . The proposed solution strategy implies time
discretization of rate equation adopting backward Euler method, and is based on efficient computation
of the dissipation function and its derivatives, using the so-called Haigh-Westergaard invariants. This
permits to solve for the transformation strain increment through Newton method in quite a neat
fashion. The present work aims at providing the following results:
14

A robust integration algorithm capable of resolving all the singularity points relative to
variational formulation (1);

A generalization to a wide class of deviatoric isotropic transformation function forms
expressed in terms of Haigh-Westergaard stress invariants;
 An efficient user material model subroutine for FEM implementation.
Numerical tests on a single integration point are provided to prove such points.
References
[1] B.D. Coleman and M.E. Gurtin. Error estimates for discretizations of a rate-independent
variational inequality. J. Chem. Phys. 47 (1967) 597-613.
[2] Q. Yang, L. Stainier and M. Ortiz. A variational formulation of the coupled thermo-mechanical
boundary-value problem for general dissipative solids. J. Mech. Phys. Sol. 54 (2006) 401-424.
[3] K. Hackl and F.D. Fischer. On the relation between the principle of maximum dissipation and
inelastic evolution given by dissipation potentials. Proceedings of the Royal Society, Series A. 464
(2008) 117-132.
[4] A.C. Souza, E.M. Mamiya and N. Zouain. Three-dimensional model for solids undergoing stressinduced phase transformations. Eur. J. Mech. A/Solids. 17 (1998) 789-806.
[5] F. Auricchio and L. Petrini. A three-dimensional model describing stress-temperature induced
solid phase transformations: solution algorithm and boundary value problems. Int. J. Numer. Meth.
Eng. 61 (2004) 807-836.
15
Parallel programming techniques for the computation of basins of
attraction
Pierpaolo Belardinelli1,a*, Stefano Lenci1,b
1
DICEA, Polytechnic University of Marche, 60131, Ancona, Italy
a
[email protected], [email protected]
Keywords: Parallel programming, computing performance, basins of attraction, MPI
The analysis of nonlinear systems of differential equations is an essential task for scientists and
engineers of many disciplines. In particular the study of dynamic attractors and their basins of
attraction represents a key point to get an overall description of the problem and to predict the
behaviour in several conditions [1]. Here we undertake the computation of basins of attraction, by
addressing first the computation itself, looking to develop an efficient algorithm. The powerful tool
we want to apply, by taking a conscious look at the applicability and the performances, is the parallel
programming. Since the parallel programming is strictly correlated to the hardware, it is a little bit
involved: several techniques must be implemented to take advantage of the newer computers
architecture [2]. In the past, we have assisted to an ever increasing performance improvement of
computers. Nowadays, the core frequency and performances will not grow following the Moore’s law
any longer [3], thus, in order to maintain the architectures evolution, chip manufactures are increasing
the number of cores.
We exploit the MPI programming interface [4] to develop a parallel code for the computation of
basins of attraction. The mere calculus is done by the computing tasks, a master process denominated
master of the initial conditions distributes the work and collects the results, finally another master,
namely the master of the attractors, picks up the information about the attractors and acts a sort of
coherence operation (see Figure1).
Figure 1: Organization of the tasks for a multi-cores environments.
We present two schemes of implementation based on a real-time synchronization between the
computing nodes or with a posteriori processing. As a first benchmark of our codes we chose the
Duffing equation since it can describe many nonlinear systems [5]. The code with a posteriori
coherency presents a good scalability and, by increasing the number of cores, the computational time
decreases. The behaviour is similar for all the grid dimensions but we get a greater advantage with a
larger grid dimension (see Figure 2). The curves obtained by means of the code with a real-time
synchronization present an U-shape with a minimum. Increasing the number of cores we have first a
better response but then we assist to worst performances. This trend is due to the updating of all the
nodes with the attractors informations that, for large amount of computing nodes, became too heavy.
The position of the minimum, as expected, is related to the problem dimension and indicates the
saturation of the communication with respect to the work load. For a few-cores application the
performance of this code are better, and increasing the dimension of the grid the advantage became
more visible, e.g. up to about 30 cores for a dimensions of 30002 this code perform the computation of
the basin in less time. Thus, the results show that for large scale problems, only for a low number of
cores an instantaneous synchronism between the nodes is preferred.
16
Figure 2: Computational time as function of the number of cores. The figure show the results for three different
discretization grids (30002 -> circle, 20002 -> triangle, 10002 -> square). The black solid lines refer to the code with postupdating while the results of the code with the real-time synchronization are reported with red dashed lines.
The performance of the codes have been tested in an heterogeneous cluster in order to verify the time
execution balance on the nodes. The excellent balance justifies the choice of a master scheduler and
the resulting time differences between the nodes is only of few second and only caused by the MPI
initialization and finalization [6].
We believe that our approach can be consider only the first step in the application of parallel
programming in the study of the dynamical systems. The work wants to give some skill, rules and
results to better deal with large scale problem characterized by a deep seriality. We paid attention at
the optimization of the computing time as well as the work time load on each node in order to develop
a performing and portable code. By performing a comparison with the serial software is demonstrated
the capabilities of the parallelism in the elaborations of basins with a large set of initial conditions.
The computational time as function of the number of cores show that the problem take advantage by
the use of the parallel programming and we can obtain good applicability also in small clusters.
Further implementation will regard the use of an hybrid MPI-openMP infrastructure; moreover to take
advantage of GPU and accelerators in the critical sections of the software.
References
[1] G. Rega and S. Lenci. Identifying, evaluating, and controlling dynamical integrity measures in
non-linear mechanical oscillators. Nonlinear Analysis: Theory, Methods & Applications, 63(57):902-914, 2005.
[2] H. Sutter and J. Larus. Software and the concurrency revolution. Queue, 3(7):54-62, 2005.
[3] G. Moore. Cramming more components onto integrated circuits. Electronics Magazine, 38(8),
1965.
[4] MPI: A Message-Passing Interface Standard, Version 3.0. High Performance Computing Center
Stuttgart, 2012.
[5] I. Kovacic and M.J. Brennan. The Duffing Equation: Nonlinear Oscillators and their Behaviour.
Wiley, 2011.
[6] MPI: The Complete Reference. MIT Press, 1996.
17
Limit analysis on FRP-strengthened RC members
Dario De Domenico1, a *, Aurora A. Pisano1,b and Paolo Fuschi1,c
1
University Mediterranea of Reggio Calabria, Dept. PAU
via Melissari, I-89124 Reggio Calabria, Italy
a
[email protected], b [email protected], c [email protected]
Keywords: FE-based limit analysis, multicriteria approach, reinforced concrete elements, FRPstrengthening systems.
Many existing steel-reinforced concrete (RC) structures, including decks and beams in highway
bridges as well as beams, slabs and columns in buildings, are being assessed as having insufficient
load carrying capacity due to their deterioration, ageing, poor initial design and/or construction, lack
of maintenance, corrosion of steel reinforcement or underestimated design loads. In other cases they
no longer comply with the current standards and requirements because of changed load conditions or
modification of structural system for some reason. It is both economically and environmentally
preferable to upgrade these structures rather than replace/rebuild them, even more if rapid, simple and
effective strengthening techniques are employed. In this context, flexural and/or shear repair and
rehabilitation of RC structures with externally bonded fiber reinforced polymer (FRP) sheets, strips
and fabrics is generally viewed as a valid and viable solution. Moreover, this technique can be carried
out while the structure is still in use with relative ease of application and it can also be targeted at
where the structural deficiency is more marked [1,2][1].
Experimental investigations confirm that a significant increase in flexural/shear capacity of the RC
elements (up to about 125%) is achieved after the application of such FRP techniques [3].
Experiments also show the enhanced concrete confinement due to the FRP laminates, resulting in
shifting the failure mode from brittle concrete crushing to more ductile steel yielding and/or FRP
rupture [4]. In fact, the FRP strengthening system mitigates crack development and, as a result,
increases ductility of the RC element as a whole.
On the other hand, to estimate the actual efficacy of the strengthening system, without performing
expensive laboratory tests, as well as to design the proper repair interventions, to reach a given gain in
load carrying capacity, analytical tools and predictive models are highly needed. In this contribution a
numerical methodology, based on the theory of limit analysis, is adopted to predict the peak load of
FRP-strengthened RC elements. The above considerations make indeed a limit analysis approach, as
the one here proposed, both applicable and effective, especially when primary interest is in
determining the limit (peak) load at collapse of the RC strengthened element.
The numerical methodology here followed, already used by the authors to predict the limit-state
solution of RC elements (see e.g. [5]) and of pinned-joint orthotropic composite laminates (see e.g.
[6]), is quite versatile and does not require any specialist program employing conventional finite
element (FE) iterative analyses. A more general multicriteria formulation of the above-mentioned
limit analysis methodology is here presented to appropriately describe the behaviour at collapse of
structural elements of engineering interest strengthened by FRP techniques. Precisely, to simulate the
constitutive behaviour of the three constituent materials, concrete is described by a Menetréy-Willamtype yield criterion endowed with cap in compression and formulated in terms of the Haigh–
Westergaard coordinates; steel reinforcement bars (re-bars) are handled by a von Mises yield
criterion; FRP strengthening laminates are governed by a Tsai–Wu-type criterion particularized in the
case of an orthotropic lamina under plane stress conditions.
Operationally the iterative linear FE analyses are carried out on a structure with spatially varying
moduli. The elastic parameters of the various FEs are iteratively adjusted in such a way as to simulate,
with reference to the assumed yield criteria, a collapse mechanism and an admissible stress field for
the given structure so as to apply the kinematic and the static approach of limit analysis, respectively.
18
On taking into account the nonstandard nature of the constitutive behaviour, the peak load value of the
analyzed specimens is in facts numerically detected by an upper and a lower bound to it.
To demonstrate the actual capabilities of the proposed numerical procedure to deal with practical
problems, large-scale prototypes of a few FRP-strengthened RC beams and slabs, experimentally
tested up to collapse, are numerically investigated. The obtained results correlate quite well with the
corresponding experimental findings taken from the relevant literature [4,7,8].
References
[1] FIB Bulletin 14. Externally bonded FRP reinforcement for RC structures, Task group 9.3,
International Federation of Structural Concrete, (2001).
[2] American Concrete Institute ACI 440. Guide for the design and construction of externally bonded
FRP systems for strengthening concrete structures, ACI 440.2R-08 (2008).
[3] J. Dong, Q. Wang, Z. Guan, Structural behaviour of RC beams with external flexural and
flexural–shear strengthening by FRP sheets, Composites: Part B 44 (2013) 604–612.
[4] M.A. Shahawy, M. Arockiasamy, T. Beitelmant, R. Sowrirajan, Reinforced concrete rectangular
beams strengthened with CFRP laminates, Composites Part B 27B (1996) 225–233.
[5] A.A. Pisano, P. Fuschi, D. De Domenico, Peak loads and failure modes of steel-reinforced
concrete beams: predictions by limit analysis, Engineering Structures 56 (2013) 477–488.
[6] A.A. Pisano, P. Fuschi, D. De Domenico, A layered limit analysis of pinned-joints composite
laminates: Numerical versus experimental findings, Composites: Part B 43 (2012) 940–952.
[7] R. Al-Rousan, M. Issa, H. Shabila, Performance of reinforced concrete slabs strengthened with
different types and configurations of CFRP, Composites: Part B 43 (2012) 510–521.
[8] D. Kachlakev, T. Miller, S. Yim, K. Chansawat, T. Potisuk, Finite Element Modeling of
Reinforced Concrete Structures Strengthened with FRP Laminates, Final Report SPR 316 (2001),
Oregon Department of Transportation Research Group, USA, May 2001.
19
Integrated structure for a resonant micro-gyroscope and accelerometer
Valentina Zega1,a* , Claudia Comi1,b, Alberto Corigliano1,c, Carlo Valzasina2,d
1
Department of Civil and Environmental Engineering – Politecnico di Milano
Piazza Leonardo da Vinci 32, 20133 Milano, Italy)
2
AMS Division, STMicroelectronics, via Tolomeo 1, 20010 Cornaredo, (Milano), Italy
a
[email protected], [email protected], [email protected],
c
[email protected],
Keywords: MEMS, gyroscope, accelerometer, resonators.
The present paper presents the study of the mechanical behavior of a microstructure designed to detect
acceleration and angular velocity simultaneously. MEMS accelerometers and gyroscopes have been
proposed and are used, thanks to their high compactness, their reduced levels of consumption, and
their good electrical performance, in a wide range of contexts of application, for example in the field
of portable electronic apparatuses.
In this work a new resonant micro-sensor is proposed, made with the Thelma surface-micromachining
technique, which bases detection of two components of external acceleration (one in-plane component
and one out-of plane component) and two components of angular velocity (roll and yaw) on the
variation of frequency of several elements set in resonance. Resonant detection, as compared to other
measuring techniques, has the advantage of affording a direct frequency output, of a quasi-digital
type, high sensitivity and a wide dynamic range. While several resonant accelerometers have been
proposed in the literature [1-5] there exist a few examples of micro-gyroscopes with resonant
detection [6-7].
In the proposed integrated detection structure, schematically shown in Figure 1, both bending
resonators and torsional resonators are included. The variation of the resonance frequency in the
flexural resonators (labelled I, II, III and IV in Figure 1) is induced by the presence, upon
displacement of the inertial mass, of axial stresses while in the torsional resonators (labelled 1, 2, 3
and 4 in Figure 1) it is induced by variations of the so-called “electrical stiffness” to which the
resonator mass is subjected. The simultaneous use of these two different type of resonators allows
realization of a four-axis sensor with reduced dimensions.
a
a
torsional resonators
IV
II
1
2
4
3
I
sensing plates
III
proof mass
y
beam resonators
a
driving plate
springs
a
anchors
x
Figura 1: Schematic plan view of the structure, for detection of acceleration and angular velocity.
20
By means of the flexural resonator elements, the integrated detection structure enables differential
detection of an angular velocity acting about a direction out of the horizontal plane xy, the so-called
yaw angular velocity Ωz, and of a linear acceleration ay along the second axis y. In addition, by means
of the torsional resonator elements, the integrated detection structure enables differential detection of
an angular velocity acting about the second axis y, the so-called roll angular velocity Ωr, and of a
linear out-of-plane acceleration az.
The two proof masses, in grey in Fig 1, are kept in resonance according to the third natural mode of
motion (i.e., the translation along the first axis x), by means of electrostatic driving implemented by
the respective driving electrodes. When an external angular velocity is applied, Coriolis forces, having
opposite signs, originate on the two inertial masses, see Figure 2, while when an external acceleration
is applied, inertial forces originate on the two inertial masses having equal directions. By properly
adding or subtracting the frequency variation in the four resonators, one can separate the signal
coming from acceleration or angular velocity and obtain a differential four-axis sensor.
y
a
a
Fc
Fc
1
2
4
3
driving
proof mass
y
a
x
z
a
Fc
Fc
1
2
3
4
substrate
x
Figura 2: Plan view and lateral section of the structure in an operating condition of detection of an angular velocity of roll
References
[1] R. Zhu, G. Zhang, G. Chen “A novel resonant accelerometer based on nanoelectromechanical
oscillator”, Proc. MEMS 2010, Hong Kong, 440-443, (2010).
[2] C. Comi, A. Corigliano, G. Langfelder, A. Longoni, A. Tocchio, B. Simoni, A resonant microaccelerometer with high sensitivity operating in an oscillating circuit, Journal of
Microelectromechanical Systems, 19, 1140 – 1152, (2010).
[3] B. Lee, C. Oh, S. Lee, Y. Oh, K. Chun, “A vacuum packaged differential resonant accelerometer
using gap sensitive electrostatic stiffness changing effect”, Proc. MEMS (2000), 352-357.
[4] H.C. Kim, S. Seok, I. Kim, S-D. Choi, K. Chun, Inertial-grade out-of-plane and in-plane
differential resonant silicon accelerometers (DRXLs), Proc. Transducers05, Seoul, 172-175, (2005).
[5] C. Comi, A. Corigliano, A. Ghisi, S. Zerbini, A resonant micro accelerometer based on
electrostatic stiffness variation, Meccanica 48, 1893–1900, (2013).
[6] A.A. Seshia, R.T. Howe, S. Montague, “An integrated microelectromechanical resonant output
gyroscope”, Proc. MEMS2002, 722-726 (2002).
[7] J. Li, J. Fang, H. Dong, Y. Tao, Structure design and fabrication of a novel dual-mass resonant
output micromechanical gyroscope, Microsyst. Technology, 16, 4, 543- 552, (2010).
21
Numerical analyses in the nonlinear dynamics and control of
microcantilevers in atomic force microscopy
Valeria Settimi1, a *, Giuseppe Rega1,b
1
Dipartimento di Ingegneria Strutturale e Geotecnica, Sapienza University of Rome, Rome, Italy
a
[email protected], [email protected]
Keywords: Noncontact AFM, Bifurcation Scenarios, Response charts, Dynamical integrity, External
Feedback Control.
It is well known in the literature that AFMs operating in dynamic mode can exhibit several nonlinear
phenomena, such as bifurcations, in-well instability regions and eventually chaotic motion, that are
common to many other dynamical systems and represent an undesirable behavior which restricts the
operating range of many electronic and mechanical devices. The deep investigation of their dynamical
bifurcation behavior as a function of the main system parameters is thus a topic of great theoretical
and practical importance, not only to frame such systems in the literature scene, but also because its
potentiality in enhancing performance, effectiveness, reliability and safety of systems is crucial to the
aim of developing novel design criteria.
In this perspective, the nonlinear response of a single-mode model of noncontact AFM [1] has been
analyzed by making use of several computational tools, in order to investigate the evolution of the
main system periodic solutions and relevant basins of attraction under variations of the most
significant system parameters [2]. Different numerical simulations and continuation techniques have
been employed (using Dynamics software and AUTO software) taking into account the presence of
the horizontal parametric excitation and of the vertical external one, separately. Several bifurcation
diagrams have been obtained in a large range of forcing frequencies which includes the fundamental
(primary) (ωu (ωv) ≈ ω1) and principal (subharmonic) (ωu (ωv) ≈ 2ω1) parametric (external)
resonances, whereby the main periodic solutions and local bifurcations have been detected thanks to
the Floquet multipliers computation. The local bifurcation loci have been summarized in behavior
charts, which report also the system stability threshold obtained as the envelope of local bifurcation
escape thresholds in different parameter ranges (Figure 1a).
Moreover, erosion process of the basins of attraction of the various solutions, which is indeed a
critical issue corresponding to system impending escape (corresponding to the unwanted jump-tocontact) and thus governing its practical safety, is investigated by applying the dynamical integrity
Figure 1: Local bifurcations map and overall escape threshold in the frequency/amplitude space of parametric
excitation. Gray area: region of stable reference response; SN1H: saddle-node bifurcation of the P1H solution;
SN1L: saddle-node bifurcation of the P1L solution; SN2: saddle-node bifurcation of the P2 solution; SpPD1:
supercritical period doubling of the P1 solution; SpPD2: supercritical period doubling of the P2 solution; SbPD1:
subcritical period doubling of the P1 solution (a). Iso-integrity curves obtained by expressing the erosion profiles in
terms of remaining safe basin percentage (b).
22
concepts [2,3]. Thanks to the analysis of basins of attraction evolution, and making use of specific
computational tools such as the evaluation of different integrity measures (GIM and IF), several
erosion profiles have been obtained as a function of the increasing excitation amplitude (reported in
Figure 1b), with the aim to detect thresholds of residual integrity able to ensure acceptable safety
targets established a priori according to the required system performances.
The topic of controlling undesirable system dynamical responses is then addressed through the
insertion in the AFM model of an external feedback control technique [4], with the aim to take the
system response to a selected reference one. The periodic motion used as reference in the control
procedure is chosen to be the response of the corresponding uncontrolled system, for which the
previous analyses have already allowed to detect the main stability regions in various parameters
planes. Upon checking the effectiveness of the procedure in the weakly nonlinear regime via a
perturbation approach, the description of bifurcation/response scenarios of the controlled system under
scan excitation up to the strongly nonlinear regime, and the critical comparison with the results related
to the uncontrolled system permit to highlight the influence of the applied control on the overall
dynamical behavior of the AFM system, and provide indications to refer to in practical applications
(Figure2a) [5].
Ongoing studies are focusing on the effect of feedback control on the evolution of global dynamics
[2], as well as on possibly controlling it through the shift of the homoclinic bifurcation threshold
triggering erosion and escape, obtainable by optimally modifying the harmonic shape of the excitation
via the addition of controlling superharmonics (Figure 2b) [3].
References
[1] Hornstein S., Gottlieb O., Nonlinear dynamics, stability and control of the scan process in
noncontacting atomic force microscopy, Nonlinear Dyn. 54 (2008) 93-122.
[2] Rega G., Settimi V., Bifurcation, response scenarios and dynamic integrity in a single-mode model
of noncontact atomic force microscopy, Nonlinear Dyn. 73 (2013) 101-123.
[3] Rega G., Lenci S., Dynamical integrity and control of nonlinear mechanical oscillators, J. Vib.
Control 14 (2008) 159-179.
[4] Yagasaki K., New control methodology of microcantilevers in atomic force microscopy, Phys.
Lett. A 375 (2010) 23-28.
[5] Settimi V., Rega G., Bifurcation and escape scenario of noncontact AFM with external feedback
control, submitted to Commun Nonlinear Sci Numer Simul (2014).
Figure 2: Behavior chart in the ωu-U plane with detection of the overall stability thresholds and stability regions for the
controlled (orange line, dark gray area) and uncontrolled (black line, light gray area) systems under parametric
excitation (a). Comparison between homoclinic bifurcation thresholds for the system with harmonic (black) and optimal
control (blue) excitations (N = 2) (b).
23
Buckling analysis using a generalized beam model including section
distortions
Andrea Genoesea, Alessandra Genoeseb, Antonio Bilottac , Giovanni Garcead*
Laboratorio di Meccanica Computazionale (DIMES), Univ. della Calabria, 87036 Rende (CS), Italy
a
[email protected], b [email protected],
c
[email protected], d [email protected]
Keywords: laminated beams, 3D stress field, mixed formulation, corotational strategy, buckling
analysis
A geometrically nonlinear beam model suitable to describe complex 3D effects due to non-uniform
warpings including non-standard in-plane distortions of the cross-section or to the anisotropy and
heterogeneity of the material is presented.
The basic idea of the proposal is that of generalizing advanced linear formulations for beams as those
presented in [1, 2] to the case of large displacements but small strains through the Implicit
Corotational Method (ICM) proposed in [3]. ICM extends the corotational description at the
continuum level by introducing a corotational reference system for each cross-section. In this system,
following a mixed approach, the linear stress tensor is shown to be a good approximation of the Biot
nonlinear one, while a quadratic approximation of the strain is easily obtained from the symmetric
and the skew-symmetric parts of the displacement gradient of the parent linear solution. The two
fields so defined are introduced in the Hellinger-Reissner functional to describe the beam behaviour in
terms of generalized static and kinematic quantities only, while change of observer algebra are used to
complete the framework. The nonlinear model maintains all the information of its linear counterpart,
but is objective and accurate up to the required order. This condition makes it suitable to be used
within both a standard incremental iterative approach or FEM implementations of the Koiter
asymptotic method. Readers are referred to [4] for its first application to the Saint-Venànt (SV) and
the Kirchhoff solutions for beams and plates. Reference [5] presents an extension to homogeneous and
isotropic beams subjected to variable shear/torsion warping deformations.
The linear formulations used in [1, 2] have been proved to be very effective for modeling beams made
by isotropic and homogeneous material or by composites, also when important warping effects
including non-standard in-plane distortions of the cross-section arise (see [1] in particular). These
models are defined exploiting a semi-analytical solution of the Cauchy continuum problem for beamlike bodies under the usual SV loading conditions, based on a FEM discretization of the cross-section
(see also [6] for details). The stress field considered in this way is potentially fully 3D, allowing to
recover the SV solution for standard materials (see [5] for instance) or to generalize it to
inhomogeneous and anisotropic cross-sections. Furthermore some additional relevant strain modes
(generalized warpings) of the cross-section can be defined in a coherent and effective way. On the
basis of these information, the 1D linear model is described in a mixed format as required by the ICM
framework. In particular, as in [5,7], the displacement field is approximated in terms of a rigid section
motion and other relevant generalized warping modes independently amplified along the beam axial
direction. The stress field coincides with that provided by the generalized SV solution plus the
contributions due to all the other effects considered.
With respect to the nonlinear beam model, a mixed finite element suitable to interpolate both the
kinematic and static generalized unknowns is proposed. A pseudo-compatible solution scheme is used
to improve the computational efficiency of the numerical procedures. The element is then exploited
inside a Koiter-like asymptotic algorithm for the buckling analysis of isotropic and composite beams.
In order to validate the new proposal, some numerical tests are carried out and results are compared
with those obtained on the bases of solid or shell finite elements which represent a reference solution
but computationally expensive in cases of complex tests.
24
References
[1] A. Genoese, A. Genoese, A. Bilotta, G. Garcea, A generalized model for heterogeneous and
anisotropic beams including section distortions, Thin Wall. Struct., 74 (2014), 85–103.
[2] A. Genoese, A. Genoese, A. Bilotta, G. Garcea, A composite beam model including variable
warping effects derived from a generalized Saint Venant solution, Compos. Struct., 110 (2014), 140–
51.
[3] G. Garcea, A. Madeo, R. Casciaro, The Implicit Corotational Method and its use in the derivation
of nonlinear structural models for beam and plates, J. Mech. Mater. Struct., 7 (2012), 509–539.
[4] G. Garcea, A. Madeo, R. Casciaro, Nonlinear FEM analysis for beams and plate assemblages
based on the Implicit Corotational Method, J. Mech. Mater. Struct., 7 (2012), 539–574.
[5] A. Genoese, A. Genoese, A. Bilotta, G. Garcea, A mixed beam model with non-uniform warpings
derived from the Saint Venant rod, Comp. Struct., 121, 87-98, 2013.
[6] M. Morandini, M. Chierichetti, P. Mantegazza, “Characteristic behavior of prismatic anisotropic
beam via generalized eigenvectors”, Int. J. Solids Struct., 47 (2010), 1327–1337.
[7] A. Genoese, A. Genoese, A. Bilotta, G. Garcea, A geometrically exact beam model with nonuniform warping coherently derived from the Saint Venant rod, Eng. Struct., 68 (2014), 33–46.
25
Shakedown analysis of 3D frames with an effective evaluation of the
elastic domain and of the load combinations
Leonardo Leonettia*, Antonio Bilottab,Giovanni Garceac,Raffaele Casciarod
Dipartimento di Informatica, Modellistica, Elettronica e Sistemistica (DIMES) - Università della
Calabria - 87030 Rende (CS) - Italy
a
[email protected], b [email protected], c [email protected],
d
[email protected]
Keywords: Shakedown, 3D steel and reinforced concrete frames, direct methods.
The possibility, offered by many National Codes, to design structures taking into account their
nonlinear behavior has given great importance to elasto-plastic analysis in civil engineering practice.
If the external load can be assumed as monotonically increasing, the safety factor of elasto-plastic
structures can be evaluated efficiently by numerical implementations of the classical theorems of limit
analysis. However, structures are generally subjected to variable cyclic actions, often only defined
through their overall envelope domain. In these cases limit analysis does not provide a reliable safety
evaluation, even if each possible combination of basic loads contained in the load domain is
considered separately. In fact a continuous increase in plastic deformations, along successive
plastically admissible load cycles, could lead to excessive displacements or produce collapse due to
fatigue, also for values of the load multiplier lower than the theoretical one given by limit analysis. On
the contrary a further requirement has to be met in this case: the rising of plastic deformation needs to
be confined to the initial phase, after which the structure behaves elastically.
The correct formulation of this kind of problem is well established inside shakedown theory through
the Melan's and Koiter's theorems. On the basis of these theorems, the so-called direct methods are
capable to evaluate the safety factor for an elasto-plastic structures subjected to generic variable loads.
If real structures, discretized by means of finite elements, are considered, the resulting non-linear
convex optimization problem usually involves hundreds of thousands of unknowns and constraints, a
very demanding problem for any kind of solver.
In the last two decades, the interest in direct methods to be applied to limit and shakedown analysis
has rapidly increased, principally due to the availability of highly efficient new optimization
algorithms. Starting from the Karmarkar proposal the Interior Point revolution has completely
changed the way of solving convex programming and, today, these algorithms are efficiently
employed for very large non-linear problems. Nowadays, the best performance of the Interior Point
Method (IPM) is obtained when the problem is formulated as conic programming including, as
particular cases, linear, semidefinite and second-order cone programming (SOCP) and the solution is
obtained using primal–-dual formulations [1,2]. A great number of yield constraints can be described
as second-order cones allowing the proposal of efficient interior point algorithms for limit and
shakedown analysis.
Alternatively to standard mathematical programming formulations a specialized direct method, named
pseudo elastoplastic analysis, can be used to evaluate the limit and shakedown safety factors. This
approach, see [3,4], is based on a strain-driven strategy of analysis hinged on closest point projection
return mapping schemes and Riks arc-length solution techniques. This method can be seen as the
application of the proximal point algorithm to the static shakedown or limit analysis theorem and the
solution of the resulting problem is performed by means of the dual decomposition strategy.
Independently from how the shakedown problem is solved, its practical application to the analysis of
reinforced concrete (RC) 3D frames requires a fine tuning o two important aspects: i) an accurate and
simple definition of the section yield function; ii) the limitation of the number of load combination to
be considered.
26
The yield function of RC 3D frame is usually evaluated only considering flexural failures, i.e.
implicitly assuming that non-flexural types of failure are prevented through appropriate transverse
reinforcements. In spite of this simplifying assumption, computing accurate yield surfaces with
combined axial force and bending moments is not an easy task and has received increasing attention in
the literature [2,5].
A piecewise linearization often requires a large number of polyhedral facets to obtain a sufficiently
accurate approximation, which can have an important effect on the quality of the estimated bounds [5]
but also on the efficiency of the algorithm. Since the yield criterion has to be verified for a large
number of points throughout the whole structure, a compromise between accuracy and computational
efficiency is required in the case of large-scale problems. These aspect becomes also more important
for shakedown analysis where the number of constraints depends on the number of basic loads
exponentially[3,4].
Recently a strategy for treating 3D frames describing them through nonlinear yield surfaces was
suggested in [2] in the context of limit analysis, where the real yield surface is approximated by using
ellipsoids. In this way the arising optimization problem becomes a SOCP problem which can be
efficiently solved also with a commercial code such as MOSEK. A similar approach is also adopted in
the present work allowing to accurately describe the material elastic domain and to use only few
analytical yield functions. However due to the several load conditions to be considered, the number of
constraints required for the shakedown analysis of 3D frame can be very large. In the more simple
case of load domain defined by means of a sum of n basic loads varying between a minimum and a
maximum value we have to verify the plastic admissibility for each of the resulting 2n possible
vertexes of the convex polytope defined by the elastic stresses associated to each load, that is for each
vertex of the so called elastic envelope polytope.
The number of vertexes of the elastic envelope heavily affects the computational performance of the
analysis whatever method is employed, a standard direct formulation, such as the interior point
algorithm used in [1] or, also with a minor impact, the pseudo elastic-plastic formulation.
In the paper we propose an efficient and effective strategy to select for each finite element or Gauss
point, i.e. where the plastic admissibility condition has to be tested, only a subset of vertexes without
affecting the accuracy of the shakedown analysis.
References
[1] J.-W. Simon,, D. Weichert, Shakedown analysis with multidimensional loading spaces,
Computational Mechanics, 49 (2012), 477-485.
[2] J. Bleyer, P. de Buhan, Yield surface approximation for lower and upper bound yield design of 3D
composite frame structures, Computers and Structures, 129 (2013), 86-98.
[3] R. Casciaro, G. Garcea, An iterative method for shakedown analysis, Computer Methods in
Applied Mechanics and Engineering, 191 (2002), 5761-5792.
[4] G. Garcea, L. Leonetti, A unified mathematical programming formulation of strain driven and
interior point algorithms for shakedown and limit analysis, Int. J. Numer. Methods Eng., 88 (2011),
1085-1111.
[5] M. Malena, R. Casciaro, Finite element shakedown analysis of reinforced concrete 3D frames,
Computers and Structures, 86 (2008), 1176–1188.
27
A simple beam model to assess the strength of adhesively bonded tile
floorings
Stefano de Miranda1,a, Antonio Palermo1,b* and Francesco Ubertini1,c
1
DICAM Department, Viale del Risorgimento 2-40136 Bologna (Italy)
[email protected], [email protected], [email protected]
a
Keywords: Adhesive joints, Tiles, Flooring, Kerr foundation
Tile floorings are extensively used in residential and industrial buildings. A typical tiled floor consists
of an upper tile layer attached via an adhesive stratum to a lower cementitious substrate. Substrate
shrinkage, either due to thermal gradients or residual maturation of cementitious substrate, may induce
eccentric compression in tiles transmitted by a shearing mechanism through the adhesive. This stress
state may cause debonding of tile with a typical Mode I failure of the adhesive layer. Several types of
adhesives are available in the market in order to satisfy different performances in terms of strength
and deformations. Therefore, for a cost-effective design it is desirable to accurately evaluate the
required adhesive strength.
In recent years, the assessment of adhesive strength has been the object of different studies carried out
by modelling the adhesive layer as a fracturing interface. Analytical models have been developed for
describing the interface decohesion in laminated beam (see e.g. [1,2]) and modelling peeling tests.
Finite element models have been widely used to simulate the debonding of adhesive lap joints (see
e.g. [3,4]). The debonding of tiles in external wall claddings has been investigated by
Mahaboonpachai et al. [5] with the formulation of a two dimensional cohesive interface element.
With reference to the case of tile flooring, of particular interest is the work by Perego et al. [6]. The
authors modelled a tile bonded to a rigid substrate through an elastic adhesive as an eccentrically
compressed beam on a Pasternak foundation. The eccentricity of compression is promoted by the
presence of an out-of-plane workmanship defect leading to a Mode I failure of the adhesive. This
approach leads to closed-form estimation of the ultimate strength of tile-substrate adhesive joint.
In this work, the model by Perego et al. [6] is refined by considering the flexibility of the substrate. A
tile bonded to a flexible substrate by means of an elastic adhesive is modelled as a beam on a
Pasternak foundation connected to a second layer of elastic springs (see Figure 1). Substrate shear
deformability is taken into account by means of an elastic shear layer. This model recalls some
features of a Kerr type foundation [7].
An ad-hoc finite element is developed in order to solve the governing differential equations of the tileadhesive-substrate system. The presence of inter-tile grouting is modelled through a rotational/
translational spring. Eccentricity induced by workmanship defect is taken into account by means of
the same rotational/translational spring collocated in eccentric position. Sequence Linear Analyses are
carried out to determine the elastic limit and Mode I failure condition of the adhesive layer. Results
provided by the mechanical model are compared with those obtained through a fully 2D FE model
developed with the commercial software Abaqus.
28
Figure 1: Mechanical model of the tile-adhesive-substrate system.
References
[1] A. Carpinteri, M. Paggi, G. Zavarise, The effect of contact on the decohesion of laminated beams
with multiple microcracks, International Journal of Solids and Structures 45 (2008), pp. 129-143.
[2] J. Williams, H. Hadavinia, Analytical solution for cohesive zone models, Journal of the
Mechanics and Physics of Solids 50 (2002), pp. 809-825.
[3] P. Schmidt, U. Edlund, Analysis of adhesively bonded joints: a finite element method and a
material model with damage, International Journal for Numerical Methods in Engineering 66 (2006),
pp. 1271-1308.
[4] J. P. M Goncalves, M. F. S. F. de Moura, P. M. S. T de Castro, A three-dimensional finite
element model for stress analysis of adhesive joints, International Journal of Adhesion & Adhesives,
22(2002), pp. 357–365.
[5] T. Mahaboonpachai, T. Matsumoto, Y. Inaba, Investigation of interfacial fracture toughness
between concrete and adhesive mortar in an external wall tile structure, International Journal of
Adhesion and Adhesives 30 (2010), pp. 1–9.
[6] G. Cocchetti, C. Comi, U. Perego, Strength assessment of adhesively bonded tile claddings,
International Journal of Solids and Structures, 48 (2011), pp. 2048-2059.
[7] I. E. Avramidis, K. Morfidis, Bending of beams on three-parameter elastic foundation,
International Journal of Solids and Structures 43 (2006), pp. 357-375.
29
Concrete mechanics at early age
Giuseppe Sciumè1,a *, Farid Benboudjema2,b and Giorgio Zavarise1,c
1
2
Department of Innovation Engineering, Università del Salento, Lecce, Italy
Laboratoire de Mécanique et Technologie, Ecole Normale Supérieure de Cachan, France
a
[email protected], [email protected],
c
[email protected]
* corresponding author
Keywords: hydration, damage, cracking, shrinkage, creep.
Prediction of concrete strain at early age may be a critical point on the design of some classes of civil
engineering structures. Among these are massive structures like concrete dams, reactor’s containments
in nuclear power plants, tunnels, etc., in which hydration is accompanied by an important increase of
temperature (see [1]). Cement hydration is a thermo-activated reaction and therefore the rise of
temperature, not well dissipated in mass concrete, increases the rate of reaction which may become
very important inducing a ΔT of the order of 40-60 °C (see Figure 1).
Figure 1: Temperature in the massive wall at 2 days after the casting (a); relative humidity at 2 and 10 years in the
proximity of point A (b). Numerical results from the casting to 7 days for the points A, B, C and D: temperature (c); degree
of reaction (d).
The positive thermal strain associated with heating is very often restrained by an existing substrate, or
self-restrained due to the cast geometry, leading to compressive stress. Then, during the subsequent
cooling phase the volume of concrete decreases progressively and also this compressive stress
decreases; however, due to the greater stiffness of the material (Young’s modulus changes a lot with
hydration) the stress in certain parts of the structure becomes of traction and may induce diffuse
cracking or traversing localized cracks. Furthermore, thermal strain is coupled with hygral strain
(autogenous and drying shrinkage) and creep strain (basic and drying creep) and this makes modeling
the mechanical behavior of concrete at early age a very demanding task.
30
To deal with such problems, a multiphysics model for concrete at early age is here presented.
Concrete is modeled as a multiphase system in non-isothermal condition, consisting of three phases: a
solid phase, s, a liquid phase, l, and a gaseous phase, g. The solid phase contains several species:
anhydrous grains of cement, aggregates, and hydrates (CSH, etringite, etc.). The liquid phase is liquid
water and the gaseous phase is modeled as an ideal binary gas mixture of dry air and water vapour.
The mathematical model shares the general conservation equations of mass, energy and linear
momentum of Gawin et al. [2], whereas several originalities have been introduced at the constitutive
level. For instance, the analytical equation used for the desorption isotherm has been properly
modified introducing the dependency on the hydration degree of concrete. Together with the adoption
of a hydration-dependent Biot’s coefficient this gives autogenous shrinkage mechanically, in a unified
way with drying shrinkage, so without a dedicated constitutive equation. Furthermore, this allows to
properly compute its time-dependent part (i.e., the viscous autogenous shrinkage) which in high and
ultra-high performance concretes may be very large. Other novelties are mechanical damage, added as
internal variable, and the 3D implementation in Cast3M (FE code of the French Atomic Agency),
which allows to model suitably reinforced concrete structures [3, 4].
After an outline of the mathematical model, the attention is focused on mechanical features, damage
and creep specially. For details on the governing and constitutive equations, and on computational
aspects refer to [3].
As introduced before, during hydration the changes of mechanical properties, i.e. Young’s modulus,
E, tensile strength, ft, and fracture energy, Gft, have a key impact on the mechanical response of the
material and its susceptibility to cracking. The evolution of mechanical properties is taken into
account in the model using the equations proposed by De Schutter et al. [5, 6], which relate them to
the degree of reaction of cement hydration and fit very well experimental data.
The adopted damage model is that of Mazars [7], adapted for young concrete and coupled with creep.
Thus, within this context we will show how the more or less rapidity of gain stiffness and strength
may be the discriminant between cracking or not. Finally, some practical suggestions will be given, to
reduce cracking susceptibility at early age. For some of the presented numerical results, e.g. those
concerning loading – unloading – reloading cycles in concrete at early ages, the validation is still
pending but experiments are under design at LMT Cachan.
References
[1] F. Benboudjema, J.M. Torrenti, Early-age behaviour of concrete nuclear containments, Nuclear
Engineering and Design 238(10) (2008) 2495-2506.
[2] D. Gawin, F. Pesavento, B.A. Schrefler, Hygro-thermo-chemo-mechanical modelling of concrete
at early ages and beyond. Part I: Hydration and hygro-thermal phenomena, International Journal for
Numerical Method in Engineering, 67(3) (2006) 299-331.
[3] G. Sciumè, F. Benboudjema, C. de Sa, F. Pesavento, Y. Berthaud, B.A. Schrefler, A multiphysics
model for concrete at early age applied to repairs problems, Engineering Structures 57 (2013) 374387.
[4] G. Sciumè, F. Pesavento, B.A. Schrefler, Thermo-hygro-chemo-mechanical modeling of the
behavior of a massive beam with restrained shrinkage. Proceedings of RILEM-JCI international
workshop on crack control of mass concrete and related issues concerning early-age of concrete
structures, (2012) 133-144.
[5] G. De Schutter, L. Taerwe, Degree of hydration based description of mechanical properties of
early-age concrete, Materials and Structures 29(6) (1996) 335-344.
[6] G. De Schutter, L. Taerwe, Fracture energy of concrete at early ages, Materials and Structures 30
67-71.
[7] J. Mazars, A description of micro and macroscale damage of concrete structures Engineering
Fracture Mechanics 25(5-6) (1986) 729-737.
31
Rigid wedge-shaped hull impacting a free surface: a lattice Boltzmannimmersed boundary study
C. Burrafato1,a, S. de Miranda1,b , A. De Rosis2,c* and F. Ubertini 1,d
1
Department of Civil, Environmental, Chemical and Materials Engineering (DICAM), Viale del
Risorgimento 2, 40136 Bologna, Italy; [email protected]
2
Department of Agricultural Sciences (DIPSA), Viale Giuseppe Fanin 48, 40127 Bologna, Italy
b
[email protected], [email protected], [email protected]
Keywords: hull slamming, lattice Boltzmann method, immersed boundary method.
Violent impacts between a water surface and the hull of a ship induce large impulsive loads. Such
forces are characterized by a short duration and high localized peaks of pressure. As a consequence,
deleterious vibrations and even local structural damages can arise, especially due to stress
concentration and fatigue. Thus, an accurate prediction of forces acting upon the ships during the hull
slamming phenomena plays a crucial role in the design of marine structures. Here, we predict the
forces acting on a rigid wedge as it impacts a free water surface by using the lattice Boltzmann
method (LBM). The LBM [1] is a relatively new simulation technique for fluids which, unlike the
traditional numerical methods, which solve the conservation equations of macroscopic properties,
consists of a set of particles streaming and colliding in a discrete space-time universe moving on a
lattice mesh through fixed velocity vectors. Due to its intrinsically mesoscopic nature, the LBM has
several advantages over other conventional methods, especially in dealing with complex boundaries
and geometry. In order to account for the presence of the wedge in the lattice fluid background, the
immersed boundary method (IBM) [2] is adopted. The choice oft he IBM over standard and wellconsolidated bounce-back rules is motivated by its superior properties in terms of stability and
involved computational effort [3]. Moreover, the combination of these two methods leads to an
algorithm that is quite general and independent from the shape of the immersed body [4]. Slamming
forces predicted using the LBM method are compared to the ones given by classical Wagner’s
solution [5], which is derived by assuming the flow to be incompressible and irrotational and the fluid
to be inviscid. In our numerical simulations, the hull penetrates the water surface with a constant
velocity. Scenarios characterized by different deadrise angles and penetration velocity are
investigated. Findings are presented in terms of time history of the forces acting upon the wedge,
together with the pressure distribution at different time instants. In particular, we find that for
progressively higher values of the Reynolds number and lower values of the Mach number our
estimations about slamming forces get closer to Wagner’s solution. Therefore, it is possible to assess
that the present strategy is promising for simulating hull slamming phenomena.
References
[1]
S. Succi, The lattice Boltzmann equation: for fluid dynamics and beyond. Oxford University
Press. (2001)
[2] C.S. Peskin, The immersed boundary method. Acta Numerica 11. (2002)
[3] A. De Rosis, S. Ubertini, F. Ubertini, A Comparison Between the Interpolated Bounce-Back
Scheme and the Immersed Boundary Method to Treat Solid Boundary Conditions for Laminar Flows
in the Lattice Boltzmann Framework. Journal of Scientific Computing 1-13. (2014)
[4] A. De Rosis, S. Ubertini, F. Ubertini. A partitioned approach for two-dimensional fluid–structure
interaction problems by a coupled lattice Boltzmann-finite element method with immersed boundary.
Journal of Fluids and Structures 45:202-215. (2014)
[5] A. De Rosis, G. Falcucci, M. Porfiri, F. Ubertini, S. Ubertini, Hydroelastic analysis of hull
slamming coupling lattice Boltzmann and finite element methods. Computers & Structures 138: 2435. (2014)
32
Analytical evaluation of displacement and stress fields induced in elastic
half-spaces by linear distributions of pressure on the surface
Francesco Marmo1,a* and Luciano Rosati1,b
1
Department of Structures for Engineering and Architecture, University of Naples Federico II
a
[email protected], b [email protected]
* corresponding author
Keywords: potential theory, elastic half-space, foundations.
The classical approach to the evaluation of stresses and displacements in an elastic, homogeneous and
isotropic half-space due to surface loads has been first developed by Boussinesq [1] who provided the
solution for a point load by making use of the potential theory. Applications of Boussinesq’s solution
range from contact mechanics to rock mechanics, geodesy and geomechanics. For instance, in the
former case [2] classical Hertz theory assumes that for the purpose of calculating the local
deformations of two bodies in contact each body can be regarded as an elastic half-space loaded over
a small region of its surface. In rock mechanics [3] the magnitude and distribution of the
displacements and stresses are predicted by using solutions that model rock as a linearly elastic,
homogeneous and isotropic continuum. In space geodesy [4] the Boussinesq solution is applied to
predict the seasonal fluctuations of the Earth’s surface due to changes of crustal loads induced by
shifting masses of water, snow and ice. Clearly the most useful field of application of the results
contributed by Boussinesq is in geomechanics in order to predict stress and displacements induced in
soil [5] by foundation loads.
For this reason a generalization of the Boussinesq solution to cases in which the load is distributed on
a region of the half-space surface is of great interest in practical applications. The case of uniform
pressure applied on a circular domain was addressed by Lamb [6] while Love [7] considered the case
of isotropic uniform pressure applied to a rectangular region. Recently, by adopting the Gauss
theorem, D’Urso and Marmo obtained closed-form expressions for the evaluation of the vertical
stresses induced by pressure distribution of polynomial type [8] and the displacements due to uniform
pressures [9]. These results can be further generalized to analytically derive the stress and
displacement fields induced by linearly distributed pressures applied over polygonal regions of the
half-space.
Numerical examples and comparisons with alternative results in the literature [10, 11] show the
correctness of the derived formulas and their applicability to a vast range of practical cases thanks to
the versatility of the polygonal representation of the loaded region.
References
[1] J. Boussinesq, Applications des Potentials a l’Etude de l’Equilibre et Mouvement des Solides
Elastiques. Gauthier-Villard, Paris (1885).
[2] K. L. Johnson, Contact Mechanics. Cambridge University Press, New York (1985).
[3] C. D. Wang, C. S. Tzeng, E. Pan, and J. J. Liao, Displacements and stresses due to a vertical point
load in an inhomogenous transversely isotropic half-space, Int. J. Rock Mech. Mining Sci. 40 (2003),
667-685.
[4] D. Dong, P. Fang, Y. Bock, M. Cheng, and S. Miyazaki, Anatomy of apparent seasonal variations
from GPS-derived site position time series, J. Geophys. Res. 107, ETG, (2002) 9.1-9.18.
[5] J. E. Bowles, Foundation Analysis and Design. McGraw-Hill, New York, (1996), 285-300.
[6] H. Lamb, On Boussinesq’s problem, Proc. Lond. Math. Soc. 34, (1902) 276-284.
33
[7] A. E.H. Love, The stress produced in a semi-infinite solid by pressure on part of the boundary,
Philos. Trans. R. Soc. Lond. A 667 (1929), 377-420.
[8] M.G. D’Urso, F. Marmo, Vertical stress distribution in isotropic half-spaces due to surface vertical
loadings acting over polygonal domains, Zeit. Angew. Math. Mech. (2013),
doi:10.1002/zamm.20130003.
[9] M.G. D’Urso, F. Marmo, On a generalized Love’s problem, Comput. Geosc. 61 (2013), 144-151.
[10] J.R. Dydo, H.R. Busby, Elasticity solutions for constant and linearly varying load applied to a
rectangular surface patch on the elastic half--space. Journal of Elasticity, 38 (1995) 153-163.
[11] J. Li, E.J. Berger, A Boussinesq-Cerruti solution set for constant and linear distribution of normal
and tangential load over triangular area. Journal of Elasticity, 63 (2001) 137-151.
34
How to refine the Sardinia Radio Telescope finite element model
Antonio Cazzani1, a , Flavio Stochino2,b* and Emilio Turco2,c
1
2
DICAAR, Università degli Studi di Cagliari, via Marengo 2 09122, Italy,
DADU, Università degli Studi di Sassari, Asilo Sella, via Garibaldi 35, 07041 Alghero (SS), Italy
a
[email protected] , b [email protected], [email protected]
* corresponding author
Keywords: Sardinia Radio Telescope, Finite Element Model Updating, Structural Modelling, Huge
Structures, Active Structures.
The Sardinia Radio Telescope (SRT), located near Cagliari (Italy), is the world’s second largest fully
steerable radio telescope with an active surface. It is shown in Figure 1 and it has a gregorian
configuration with a primary and secondary mirror diameters of 64 m and 8 m respectively.
Figure 1: front view of SRT.
Among its peculiarities there is the capability of modifying the configuration of the primary mirror
surface by means of electro-mechanical actuators. This capability ensures, within a fixed range, the
balancing of the deformation caused, for example, by external loads (self-weight, thermal load, wind
pressure). This balancing allows to reduce the difference between the ideal shape of the mirror (which
maximizes its efficiency) and the actual surface.
Actually the theoretical shape is accurately designed to optimize the SRT antenna gain and it is strictly
linked to the accuracy of the radio telescope measurements. In reference [1] is shown that to achieve
an antenna efficiency of 67% in recording signals at 100 GHz, it is necessary to restrict to 0.150 mm
the Root Mean Square Deviation (RMSD) of the above mentioned difference.
A thorough photogrammetric survey has been developed, during the primary and secondary mirror
panels alignment, in order to estimate the actual surface deformation and to determine what would be
the optimal actuators displacements for a given set of SRT configurations.
The finite element method (FEM) has been exploited for modeling huge radio telescopes. In [2], [3]
and [4] is described the FEM analysis respectively for the RT 70m radio telescope (located in
Yevpatoria, Crimea) and for the IRAM 30m radio telescope (located in the Spanish Sierra Nevada).
In [5] and [6] is shown the numerical model of SRT and its effectiveness in estimating its actuators
displacements. After having accurately described the SRT, the experimental data provided by
photogrammetric techniques and the adopted finite element model in [5]. Then, in reference [6], an
35
approach to improve the accuracy of the finite element model is developed. The comparison with field
recordings showed a significant reduction of the differences between photogrammetric measurements
and the results of the numerical model, which emphasizes the effectiveness of the updating procedure.
Unfortunately the given photogrammetric data are referred to a few configurations of the SRT. For
this reasons it is necessary to extend the procedure developed in [6] in order to consider all the
possible configurations.
In this work an effective estimation of the SRT structural behavior and of its actuators displacements,
based on an updated FE model, is developed regardless of the given set of configurations considered
for the photogrammetric survey.
In addition the whole updating procedure is enhanced with a more refined sub-structuring which
allows to reach higher accuracy than the ones obtained in the previous papers. Thus the enhanced FE
model can be an effective part of the SRT control loop. Especially if associated with a real time
survey of climatic conditions (thermal load, wind pressure) that can be developed by means of a set of
sensors located on the main reflector surface, it will be able to precisely estimate on the fly the
actuators displacements for any radio telescope configuration.
References
[1]
T. Pisanu, F. Buffa, M. Morsiani, M. Natalini, C. Pernechele, G. Vargiu, How to improve the
high-frequency capabilities of the SRT, Mem. S. A. It. Suppl., 10 (2006) 136–140.
[2] A.I. Borovkov, D.V. Shevchenko, A.V. Gaev, A.S. Nemov, Finite element modeling and thermal
analysis of the RT-70 radio telescope main reflector, Int. Conf. on Antenna Theory and Techniques,
(2004) 651-654.
[3] A.I. Borovkov, D.V. Shevchenko, A.V. Gaev, A.S. Nemov, 3D finite element thermal and
structural analysis of the RT-70 full-circle radio telescope., Proceedings of International ANSYS
Conference, (2004).
[4] A. Greve, M. Bremer, J. Penalver, P. Raffin, D.Morris , Improvement of the IRAM 30-m
telescope from temperature measurements and finite-element calculations, IEEE Transactions on
Antennas and Propagation, 53(2) (2005) 851-860.
[5] A. Cazzani , F. Stochino, E. Turco, Finite element model updating of the Sardinia Radio
Telescope. Part I: field recordings, submitted to J. Struct. Eng. 2014.
[6] F. Stochino , E. Turco, A. Cazzani, Finite element model updating of the Sardinia Radio
Telescope. Part II: fine tuning, submitted to J. Struct. Eng. (2014).
36
A GBT finite element based on elastic solution
S. de Miranda1,a, A. Madeo2,b , D. Melchionda3,c* and F. Ubertini4,d
1,3,4
DICAM, University of Bologna, V. Risorgimento 2, 40136 Bologna, Italy
2
DIMES, University of Calabria, P. Bucci, 87036 Rende, Italy
a
[email protected], [email protected],
c
[email protected], [email protected]
Keywords: Generalized Beam Theory (GBT), finite element, elastic analytical solution.
The Generalized Beam Theory (GBT) allows to consistently account for cross-section distortion along
with the more “classical” kinematics of axial displacement, bending and torsional rotation in a
comprehensive fashion. Since the initial work by Schardt [1], during the years, the research on the
GBT led to its extensive use for the design and evaluation of the performance of steel members. The
report by Camotim and Basaglia [2], offers a survey on the state-of-art in the use of GBT for buckling
analysis, while the recent application to post-buckling are discussed in [3]. The material non-linearity
and the dynamic analysis cases are discussed in [4, 5]. Recently, applications to composite materials
have been investigated [6] and an effective procedure to recover the three-dimensional stresses has
been proposed [7]. The effort to employ GBT in technical/real contexts and large-scale analyses
clearly requires an optimization of the analysis, especially when the GBT members are assembled to
form space frame structures. In this sense, the development of new, high performance GBT finite
elements is becoming more and more important.
In this work, starting from the shear deformable GBT formulation presented by de Miranda et. al [8],
an analytical solution, in the case of zero distributed load, was recovered. The solution can be divided
into two parts. The first one coincides with the classical solution of the Vlasov’s theory. The second
part corresponds to the effects of cross-section distortions along the length of the beam. Reusing this
analytical solution for the representation of the independent fields a new finite element was obtained.
The kinematics of the element is ruled by parameters associated to the ends of the beam, as in
standard elements. The results for various steel thin-walled beams with different cross-sections are
discussed and convergence studies are reported. The new element shows, as expected, great accuracy
also for very rough meshes.
References
[1] R. Schardt, The generalized beam theory, U. of Manchester (Ed.), Proocedinds of the M.R. Horne
Conference (1983) 469-475.
[2] D. Camotim, C. Basaglia, Buckling analysis of thin-walled steel structures using generalized beam
theory (GBT): state-of-the-art report, Steel Construction, 6 (2) (2013) 117-131.
[3] C. Basaglia, D. Camotim, N. Silvestre, Post-buckling analysis of thin-walled steel frames using
generalized beam theory (GBT), Thin-Walled Structures, 62 (2013) 229-242.
[4] R. Goncalves, D. Camotim, Geometrically non-linear generalized beam theory for elastoplastic
thin-walled metal members, Thin-Walled Structures, 51 (2012) 121-129.
[5] R. Rebiano, D. Camotim, N. Silvestre, Dynamics analysis of thin-walled members using
Generalized Beam Theory (GBT) ), Thin-Walled Structures, 72 (2013) 188-205.
[6] N. Silvestre, D. Camotim, Shear Deformable generalized beam Theory for the Analysis of ThinWalled Composite Members, Journal of Engineering Mechanics, 139 (2012) 1010-1024.
[7] S. de Miranda, A. Gutierrez, R. Miletta, Equilibrium-based reconstruction of three-dimensional
stresses in GBT, Thin-Walled Structures, 74 (2014) 146-154.
[8] S. de Miranda, A. Gutierrez, R. Miletta, F. Ubertini, A generalized beam theory with shear
deformation, Thin-Walled Structures, 67 (2013) 88-100.
37
Ceramic sanitary wares: reverse engineering strategy for mould
prototyping
S. de Miranda1, L. Patruno1, M. Ricci1* a, R. Saponelli2, F. Ubertini1
1
DICAM, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy
2
SACMI, Via Selice Provinciale 17/A, 40026 Imola (BO), Italy
[email protected]
* Corresponding author
Keywords: Digital manufacturing, Ceramic materials, Mould prediction, Sanitary ware, Reverse
engineering
Ceramic materials are nowadays widely used in several industrial applications, ranging from the
production of sanitary wares to high-performance mechanical components. During the production
process, the mechanical behavior of the material is extremely non-linear and prone to plastic and creep
deformations, while displacements can reach up to 20% of the product characteristic dimension. A
common problem for the industries operating in such sector is predicting the initial shape to be given
to the mould in order to obtain a given design at the end of the production process. Considering that
tolerances with respect to the target shape are typically 0.5% of the product dimensions, the mould
prediction for sanitary wares appears to be an extremely challenging engineering problem.
The production process of sanitary wares can be subdivided into three main phases, namely: forming,
drying and firing. During the forming phase, a mixture of ceramic powders is poured into a porous
mould. Water filtration allows the deposition of a thin layer of clay in adherence to the mould walls.
Once the design thickness is reached, the mould is removed. At this stage, the product is called green
body. Immediately after the mould removal, the green body behaves as an unsaturated soil [1,2] and
deforms under its self weight in an irreversible way.
During the drying phase, pieces are exposed to the environmental humidity and temperature. Moisture
gradients between the clay layers and the environment lead to water migration and, therefore, to the
piece drying. In such phase, the material mechanical properties improve due to suction as the dying
process advances [3]. Moreover, in this phase, additional plastic deformations and volumetric
shrinkage deform pieces significantly.
Once pieces are completely dried, their shape is further deformed in the firing phase when sintering
and pyroplasticity cause relevant creep deformation together with additional shrinkage [4, 5].
Due to the deformations accumulated in the production process, the final shape significantly differs
from the mould that produced it. This problem has been traditionally dealt with by using a trial and
error approach, strongly relying on the technician experience.
However, such strategy appears to be highly inefficient leading to high costs and long product
development cycles. As a consequence, the mould prediction using numerical tools receives a strong
interest from the industries operating in such sector.
In this paper, a numerical model aimed at simulating the whole sanitary ware production process is
proposed. Then, a reversal algorithm is described in order to obtain the mould shape starting from the
desired final product geometry. The topics presented in the paper have been addressed in collaboration
with SACMI SC, a world leading company in the production of machineries for the ceramic industry.
38
References
[1] D.Sun, W. Sun, L. Xiang, Effect of degree of saturaion on mechanical behaviour of unsaturated
sols and its elastoplatic simulation, Computer and Geotechnics 37 (2010) 678 -688.
[2] E. Alonso, E. Gens, A. Josa, A constituive model for partially saturated soils, Géotechnique 40
(1990) 405-430.
[3] J. Amoròs, E. Sànchez, V. Cantavella, J. Jarque, Evolution of the mechanical strength of
industrially dried ceramics tiles during storage, Journal of the European Ceramic Society 23 (2003)
1839-1845.
[4] M. D. Noirot, W. M. Carty, Dynamic pyroplatic deformation study: digital time-lapse photography
of porcelain firing, Ceramic Engineering and Science Proceedings 24 (2003) 133-147.
[5] D. Tuncel, E. Ozel, Evaluation of pyroplatic deformation in sanitaryware porcelain bodies,
Ceramic International 38 (2012) 1399-1407.
39
Computational modeling of fiber recruitment for statistical distributed
biological tissues
Alessio Gizzi1,a*, Marcello Vasta2,b and Anna Pandolfi3,c
1
University Campus Bio-Medico of Rome, Department of Engineering,
Via A. del Portillo 21, 00128 Rome, Italy
2
Università di Chieti-Pescara, Dipartimento INGEO, Viale Pindaro 42, Pescara, Italy
3
Politecnico di Milano, Dipartimento di Ingegneria Civile ed Ambientale,
Piazza Leonardo da Vinci 32, Milano, Italy
a
[email protected], [email protected], [email protected]
Keywords: Statistical fiber distribution, Fiber recruitment, Hyperelasticity, Computational models.
Constitutive theories for biological tissues accounting for the distribution of collagen fiber orientation,
starting from the seminal work of Lanir on connective tissues [1], have been extensively analyzed in
two- and three-dimensional settings and extended to include the mechanism of fiber recruitment.
Finite kinematics hyperelastic theories accounting for the anisotropic behavior of biological media
with distributed fibers are, in general, based on the additive decomposition of the Helmholtz free
energy to neatly separate isotropic and anisotropic contributions. In the presence of inelastic
processes, it is customary to decompose the deformation gradient multiplicatively into an elastic part,
associated to the stress, and inelastic part, associated to the non-compatible inelastic phenomenon [2].
Approaches trying to extend this framework to statistical distributions of fiber orientations may
consider generalized structure tensors or proceed directly with the numerical integration of the
orientation distributions. Both approaches, though, suffer of well-known shortcomings, including
large errors on the stress for fiber distributions with reduced alignment and unaffordable
computational costs. An acceptable solution recently explored is to adopt hyperelastic approaches that
account for second order statistical parameters (variance) of the fiber orientation distribution [3]. Such
an approach can be easily extended to include inelasticity, including fiber recruitment mechanisms.
Experimental results and finite kinematics models of fiber recruitment have been discussed in [4], see
Figure 1. The fiber recruitment mechanism was included in the material model by means of a
probability distribution function (PDF), accounting for the first order (mean) statistical parameter of
the fiber orientation distribution.
Figure 1: Multi-photon images of collagen at stretches of 1.4 (left) and 2.0 (right). White bars = 50 m [4].
A fiber recruitment mechanism based on the second order approximation (i.e., including mean and
variance) of the fiber distribution has been proposed recently in [5]. The comparison with the first
order approximation and the exact integration of the statistical distribution, in both two- and threedimensional setting, testifies a clear superiority of the second order approximation with respect to the
first order approximation in terms of accuracy, efficiency and robustness.
In the present study, the features of the anisotropic material model characterized by a recruitment
mechanism introduced in [5] are discussed in detail. The fiber reinforced material model is
40
characterized by a von Mises statistical distribution of the fiber orientation. The anisotropic term of
the Helmholtz free energy density related to the fiber contribution is assumed to be an exponential
function of the mean and the variance of the fiber distribution, through the isochoric pseudo-invariant
I4. The finite kinematics is described by means of a multiplicative decomposition of the deformation
gradient F = Fe Fa in elastic and inelastic (recruitment) parts, see Figure 2.
Figure 2: Schematic representation of the multiplicative decomposition of the deformation gradient in the presence of fiber
recruitment [5].
The explicit expression of the stress and tangent stiffness tensors are derived, and the proposed
material model is validated against experimental behaviors in terms of uniaxial, biaxial and shear
tests. Alternative PDFs for recruitment are analyzed, and their influence on the material response is
quantified. Planar and three-dimensional fiber distributions are compared with reference to
experimental data taken from the literature. Specializations of the model are considered in view of
applying the recruitment mechanism to biological tissues showing evidence of recruitment, e.g.,
arterial wall, skin, and tendon; and generalization of the model in view of extensions to active biomaterials, such as muscle, heart, and intestine.
References
[1]
Y. Lanir, Constitutive equations for fibrous connective tissues, J. Biomech. 16 (1983) 1-12.
[2]
V.A. Lubarda, Constitutive theories based on the multiplicative decomposition of deformation
gradient: Thermoelasticity, elastoplasticity, and biomechanics, Appl. Mech. Rev. 57 (2004) 95-108.
[3]
A. Pandolfi, M. Vasta, Fiber distributed hyperelastic modeling of biological tissues. Mech.
Mat. 44 (2012) 151-162.
[4]
M.R. Hill, X. Duan, G.A. Gibson, S. Watkins, A.M. Robertson, A theoretical and nondestructive experimental approach for direct inclusion of measured collagen orientation and
recruitment into mechanical models of artery wall. J. Biomech. 45 (2012) 762-771.
[5]
A. Gizzi, M. Vasta, A. Pandolfi, Modeling collagen recruitment in hyperelastic bio-material
models with statistical distribution of the fiber orientation, Int. J. Eng. Sci. 78 (2014) 48-60.
41
A method of cells-type kinematic limit analysis approach for the
evaluation of the macroscopic strength domain of in-plane loaded
periodic masonry
Gabriele Milani1, a *, Alberto Taliercio2,b
1
Department of Architecture, Built Environment and Construction Engineering (A.B.C.), Technical
University of Milan, Piazza Leonardo da Vinci 32, 20133, Milan, Italy
2
Departiment of Civil and Environmental Engineering (DICA), Technical University of Milan, Piazza
Leonardo da Vinci 32, 20133, Milan, Italy
a
[email protected], [email protected]
*corresponding author
Keywords: Masonry, Homogenization, Limit analysis, Cell method, Joint thickness.
A simple model based on the so-called "method-of-cells" for the determination of macroscopic inplane loaded masonry strength domain is presented.
The approach proposed subdivides the elementary cell into six rectangular sub-domains, where a
piecewise differentiable strain-periodic velocity field is a-priori assumed, which approximates more
accurate FE solutions reasonably well.
In the framework of the upper bound theorem of limit analysis, a simple linear programming
optimization problem is derived to estimate an outer bound to the homogenized in-plane strength
domain of periodic brickwork (e.g. with running bond or header bond texture).
The main advantages related to the proposed approach are the following: (1) the homogenized failure
surface can be directly estimated, without the need of performing expensive step-by-step elasto-plastic
non-linear FE analyses; (2) as the linear programming problem involves very few variables, it is
intrinsically very robust and, at the same time, allows the failure surface to be quickly estimated; (3)
there is no need to reduce mortar joints to interfaces, so that the actual thickness of the joints and the
failure mechanisms taking place therein can be accounted for in the model.
Several comparisons are provided, showing the match between the homogenized strength domain
surfaces computed both with computationally expensive FE procedures and the proposed limit
analysis approach.
References
[1] G. Milani, A. Taliercio. Kinematic limit analysis with embedded cell-method of brickwork
masonry in-plane loaded. Under review.
[2] A. Taliercio. Closed-form expressions for the macroscopic in-plane elastic and creep coefficients
of brick masonry. International Journal of Solids and Structures (2014), in press.
[3] G. Milani G. Simple lower bound limit analysis homogenization model for in- and out-of-plane
loaded masonry walls. Construction & Building Materials 25 (2011), 4426–4443.
[4] G. Milani, P.B. Lourenco, A. Tralli. Homogenised limit analysis of masonry walls, Part I: failure
surfaces. Computers and Structures 84 (2006), 166-180.
[5] G. Milani, P.B. Lourenco, A. Tralli. Homogenised limit analysis of masonry walls, Part II: failure
surfaces. Computers and Structures 84 (2006), 181-195.
42
A simple FEM model to predict the mechanical behaviour of an
equiatomic NiTi SMA alloy
Vittorio Di Cocco1, a *, Francesco Iacoviello1,b, Alessandra Rossi1,c
1
DICeM – University of Cassino and Souther Lazio, Via G. Di Biasio 43, 03043 Cassino (FR) Italy
a
[email protected], [email protected], [email protected]
Keywords: Shape Memory Alloy, NiTi, Martensite, Austenite.
Shape memory alloys (SMA) are able to recover their original shape also after high mechanical
deformations, by heating up to a characteristic temperature (SME) or by removing the mechanical
load (PE). This particular behaviour is due to a reversible solid state austenite-martensite
microstructural transitions, which can be activated by mechanical and/or thermal actions.
Some examples per SMA are:
1) CuZnAl alloys;
2) NiTi alloys.
Cu-Zn-Al alloys are characterized by good shape memory properties due to a bcc disordered structure
stable at high temperature called β-phase, which is able to change by means of a reversible transition
to a B2 structure after appropriate cooling, and reversible transition from B2 secondary to DO3 order,
under other types of cooling. In β-Cu-Zn-Al shape memory alloys, the martensitic transformation is
not in equilibrium at room temperature. It is therefore often necessary to obtain the martensitic
structure, using a thermal treatment at high temperature followed by quenching. The martensitic
phases can be either thermally-induced spontaneous transformation, or stress-induced, or cooling, or
stressing the β-phase. Direct quenching from high temperatures to the martensite phase is the most
effective because of the non-diffusive character of the transformation. The martensite inherits the
atomic order from the β-phase.
NiTi SMA’s phase transitions of near equiatomic NiTi systems at T<900°C are not well specified. In
the last years a triple transitions has been accepted, from an austenitic B2 phase for slowly cooling a
B19 orthorhomic phase transformation occurs, but for long time at 500°C (about 120 hours) an
monoclinic B19’ phase is obtained. But not all the transformation are possible when changing the Ni
content; in particular when increasing the Ni content last transformation (B19’) can take place only
under environmental temperature or under the zero absolute.
The near equiatomic NiTi system is capable of two successive martensitic phase transformations
during cooling from its high temperature austenitic phase. In Ti rich NiTi SMAs, the first phase
transformation during cooling is observed just above room temperature and results in the R-phase, the
second one occurs around room temperature and results in M-phase (monoclinic structure), often with
a fine lath morphology. These transformations give rise to thermo-elasticity and twin deformations in
NiTi alloy facilitating shape memory effect (SME). The phase transition can be activated by a
temperature change (TIM, Thermally Induced Martensite), between the characteristic phase transition
temperature, or by external mechanical loads (SIM, Stress Induced Martensite) [1-4]. Among these
alloys, the near equiatomic NiTi binary system shows the most exploitable characteristics and it is
currently used in an increasing number of applications in many fields of engineering [1-6], for the
realization of smart sensors and actuators, joining devices, hydraulic and pneumatic valves,
release/separation systems, consumer applications and commercial gadgets. Due to their good
mechanical properties and biocompatibility the most important applications of NiTi alloys are in the
field of medicine, where pseudoelasticity is mainly exploited for the realization of several
components, such as cardiovascular stent, embolic protection filters, orthopedic components,
orthodontic wires, micro surgical and endoscopic devices [7].
43
In any case, due to their interesting features and the efforts o many researcher the use of NiTi alloys is
expected to rise considerably in the near future, even in low cost applications, due to a continuos
improvement in product quality and cost reduction [8].
In this work a commercial pseudo-elastic NiTi alloy (Type S, Memry, USA), with nominal chemical
composition of 50.8at.% Ni - 49.2 at.% Ti, was investigated.
The evolution of the microstructure during uniaxial deformation was analyzed using miniature dogbone shape specimens and a customized testing machine which allows in-situ X-Ray diffraction
(XRD) analyses. A removable loading frame allows X-Ray analyses at fixed values of applied load
and/or deformations. The specimens were machined from commercial NiTi sheets with thickness of
the 0.5 mm, by wire electro discharge machining, due to the poor workability of this class of materials
by conventional machining processes as well as to reduce the formation of thermo-mechanical
affected zone.
Finally a simple stress strain model has been proposed in order to predict both the mechanical
behaviour and the structure transitions in an equiatomic NiTi SMA under loading conditions.
The stress-strain model is based on the maximum austenite strains that implies the austenitemartensite transitions under loading condition. Model has been implemented in an commercial FEM
code and results were compared with experimental results both in mechanical and in diffraction
results.
References
[1] Y. Liu, G.S. Tan., Intermetallics (2000) 8.
[2] Y. Liu,D. Favier, Acta Mater (2000) 48.
[3] S. Miyazaki, M. Kimura In: Otsuka K, Fukai Y, editors. Advance materials _93, V/B: Shape
memory materials hydrides. Amsterdam: Elsevier (1994) 1101.
[4] Sato, E. Chishima, K. Soma, T. Mori, Acta Metall (1982);30:1177.
[5] K. Otsuka Shimizu, Scripta Metall 1977;11:757.
[6] Otsuka, K., Ren, X. (2005) Progress in Materials Science 511.
[7] Dong, Y., Boming, Z., Jun, L. (2008) 485 Materials Science and Engineering A, 243–250.
[8] K.C. Russel, Phase transformation, Ohio, ASM (1969) 1219.
44
Evaluation of performance of cold-formed steel structures using Koiter
asymptotic approach
A. Madeoa*, R. Casciarob, G. Zagaric, R. Zinnod, G. Zuccoe
DIMES, University of Calabria. Ponte P. Bucci, 87036 Rende (CS) Italy
a*
b
c
[email protected], [email protected], [email protected]
d
e
[email protected], [email protected]
Keywords: Cold-formed steel structures, Koiter asymptotic approach, buckling and post-buckling,
imperfection sensitivity analysis.
Cold-formed steel structures are recently widely used in framing, metal building and racks. The
review by Schafer [1] offers a complete overview on their applications and on the methodologies
currently employed for the evaluation of their performance. Generally, due to the small thickness of
cross section and the high span, the structures suffer of instability phenomena worsen by the presence
of perforated cross-sections, residual stress and geometrical imperfections [2, 3] and modal buckling
interaction [4].
In this context, the asymptotic approach, initially proposed by Koiter [5], is very attractive for its
advantages respect to path-following analysis. These consist in an accurate evaluation of performance
in post-buckling range, with low computational cost, especially in the case of modal interaction and in
very efficient and robust imperfection sensitivity analysis. The main difficulties arise in the
availability of geometrically coherent (almost until fourth order) structural model and in an accurate
evaluation of their high order energy variations. The use of corotational formulation, within a mixed
formulation, allows to have a general finite element implementation of Koiter analysis [6, 7].
Our recent developments in terms of numerical implementation [8, 9] are here applied for the large
scale analysis and performance evaluation of cold-formed structures. Several results are presented and
discussed, highlighting in particular those about imperfection sensitivity analysis. Numerical testing
confirms the reliability, robustness and low computational cost of Koiter approach.
References
[1] B.J. Schafer, Cold-formed steel structures around the world (A review of recent advances in
applications, analysis and design), Steel Construction, 4(3) (2011).
[2] A. Crisan, V. Ungureanu, D. Dubina, Behaviour of cold-formed steel perforated sections in
compression. Part-1-Experimental investigations, Thin Wall Struct, 61 (2012) 86-96.
[3] A. Crisan, V. Ungureanu, D. Dubina, Behaviour of cold-formed steel perforated sections in
compression. Part-2-Numerical investigations and design considerations, Thin Wall Struct, 61 (2012)
97-105.
[4] D. Dubina, V. Ungureanu, Instability mode interaction: From Van der Neut model to ECBL
approach, Thin-Walled Structures, doi: http://dx.doi.org/10.1016/j.tws.2012.07.013.
[5] W.T. Koiter. 'On the stability of elastic equilibrium'. Thesis, Delft, 1945. English transl. NASA
TT-F10, 883 (1967) and AFFDL\TR70-25 (1970).
[6] G. Garcea, A. Madeo, R. Casciaro, The implicit corotational method and its use in the derivation
of nonlinear structural models for beams and plates, J Mech Mater Struct; 7 (6) (2012) 509-538.
[7] G. Garcea, A. Madeo, R. Casciaro, Nonlinear FEM analysis for beams and plate assemblages
based on the implicit corotational method. J Mech Mater Struct, 7 (6) (2012) 539-574.
[8] G. Garcea, A. Madeo, G. Zagari, R. Casciaro. Asymptotic post-buckling FEM analysis using
corotational formulation. Int J Solids Struct, 46 (2009) 377-397.
[9] G. Zagari, A. Madeo, R. Casciaro, S. de Miranda, F. Ubertini, Koiter analysis of folded structures
using a corotational approach. Int J Solids Struct, 50(1) (2013) 755-765.
45
A finite-element approach for the analysis of pin-bearing failure of
composite laminates
Michele Marinoa , Francesca Nerillib,* and Giuseppe Vairoc
Department of Civil Engineering and Computer Science (DICII)
University of Rome “Tor Vergata”, via del Politecnico 1, 00133 Roma, Italy
a
[email protected], [email protected], [email protected]
* corresponding author
Keywords: FRP composite laminates, bolted joints, progressive damaging, pin-bearing failure.
Fiber reinforced polymer (FRP) composites are characterized by interesting mechanical properties, as
for instance high values of specific stiffness, specific strength and high corrosion resistance, that make
these materials appealing for advanced engineering applications. In this context, an open issue is
surely related to the design and the analysis of structural joints between composite structural
members. Referring to bolted laminates, they develop local failures or exhibit local damage such as
matrix cracks, fiber failures, fiber-matrix shear-outs and delamination. In detail, pin-bearing failure
mode of bolted FRP joints, locally associated with matrix cracks, is an important design problem that
has attracted the interest of the international scientific community, as confirmed by the great number
of researches carried out in the last years (e.g., [1–10]). Accordingly, a computational model able to
give parametric indications on the mechanical performance of bolted FRP joints, as well as able to
predict their failure mechanisms, would be a powerful and useful design tool for both civil and
mechanical advanced applications.
This paper presents a numerical model based on a non-linear finite-element formulation for the
analysis of the progressive damaging and the failure modes in bolted joints between fiber-reinforced
composite laminates.
In order to describe the damage evolution, the model implements several failure criteria available in
the literature and involving different stress-strain measures at different material scales [11]. The
proposed numerical formulation has been preliminarily applied to a pin-plate system, by adopting a
bidimensional model under plane-stress assumptions and by considering an incremental displacementbased approach driven by the pin position. Neglecting friction, the unilateral contact at the pin-plate
interface has been treated through a surface-to-surface penalty method. Numerical analyses have been
carried out by means of a Matlab home-made code employing, for finite-element computations and
for managing the non-linear contact problem, the libraries encoded in the commercial solver Comsol
Multyphysics.
Proposed numerical results predict a bearing failure mechanism fully in agreement with the
experimental evidence discussed in [10], contributing also to clarify the influence of the failure
criterion on the predicted failure strength and failure mode. In particular, proposed results clearly
suggest that failure criteria not accounting for micro-structural stress-strain localization mechanisms
generally can be not able to describe suitably the global failure of bolted joints between FRP
laminates.
References
[1] G. Kelly, S. Hallström, Pin-bearing strength of carbon fibre/epoxy laminates: effects of bolt-hole
clearance, Compos. Part B-Eng. 35 (2004) 331–343.
[2] W.A. Counts, W.S. Johnson, Bolt pin-bearing fatigue of polymer matrix composites at elevated
temperature, Int. J. Fatigue 24 (2002) 197–204.
46
[3] Y. Xiao, T. Ishikawa, Bearing strength and failure behaviour of bolted composite joints (part I:
experimental investigation), Compos. Sci. Technol.: Eng. 65 (2005) 1022–1031.
[4] Y. Xiao, T. Ishikawa, Bearing strength and failure behaviour of bolted composite joints (part II:
modelling and simulation), Compos. Sci. Technol.: Eng. 65 (2005) 1032–1043.
[5] B. Vangrimde, R. Boukhili, Pin-bearing stiffness of glass fibre-reinforced polyester: influence of
coupon geometry and laminate properties, Compos. Struct. 58 (2002) 57–73.
[6] B. Vangrimde, R. Boukhili, Descriptive relationships between pin-bearing response and
macroscopic damage in GFRP bolted joints, Compos. Part B-Eng. 34 (2003) 593–605.
[7] B. Vangrimde, R. Boukhili, Analysis of the pin-bearing response test for polymer matrix
composite laminates: pin-bearing stiffness measurements and simulation, Compos. Struct. 56 (2002)
359–374.
[8] R. Li, D. Kelly, A. Crosky, Strength improvement by fibre steering around a pin loaded hole”,
Compos. Struct. 57 (2002) 337–383.
[9] F. Ascione, L. Feo, F. Maceri, An experimental investigation on the pin-bearing failure load of
glass fibre/epoxy laminates, Compos. Part B-Eng. 40 (2009) 197–205.
[10] F. Ascione, L. Feo, F. Maceri, On the pin-bearing failure load of GFRP bolted laminates: An
experimental analysis on the influence of bolt diameter, Compos. Part B-Eng. 41 (2010) 482–490.
[11] M.J. Hinton, A.S. Kaddour, P.D. Soden, A comparison of the predictive capabilities of current
failure theories for composite laminates, judged against experimental evidence, Compos. Sci.
Technol.: 62 (2002) 1725–1797.
47
Advanced numerical simulations in biomechanics: patient-specific finite
element analysis of transcatheter aortic valve implantation
S. Morganti1,a*, M. Conti2,b, M. Aiello3,c, A. Reali2,d and F. Auricchio2,e
1
Dip. Ing. Industriale e dell’Informazione, Università di Pavia, Via Ferrata 3, 27100 Pavia, Italia
2
Dip. Ing. Civile e Architettura, Università di Pavia, via Ferrata 3, 27100 Pavia, Italia
3
Dip. Cardiotoracovascolare, IRCCS Policlinico San Matteo, Viale Golgi 19, 27100 Pavia, Italia
a
[email protected], [email protected], [email protected],
d
[email protected], [email protected]
Keywords: aortic valve, finite element analysis, patient-specific modelling, TAVI.
Introduction. The first percutaneous transcatheter implantation of an aortic valve prosthesis in
humans was described more than 10 years ago, in 2002, by Cribier [1]. Since then, such a minimallyinvasive procedure to restore valve functionality in case of calcific stenosis has become a routine
approach for high-risk or even inoperable patients [2]. However, a high percentage of treated patients
have shown moderate to severe perivalvular aortic regurgitation which is one of the most frequent
complications associated with TAVI which correlates with an increased rate of mortality [3].
Incomplete prosthesis apposition due to calcifications or annular eccentricity, undersizing of the
device, and malpositioning of the valve are the most common determinants of paravalvular leakage
[3]. As a direct consequence, appropriate annular measurements, a correct evaluation of calcifications
and of how they can affect prosthesis placement, as well as the optimal selection of prosthetic valve
size are “of utmost importance” [4]. Given such considerations, advanced computational tools
integrating patient-specific information and accurate device data can be used to support pre-operative
planning.
Materials and Methods. Two patients were recruited for the present study, both with severe
symptomatic aortic stenosis. For both patients the Edwards SAPIEN XT size 26 was selected by
physicians as the optimal device for implantation. The overall strategy we developed to obtain
predictive outcomes of transcatheter aortic valve implantation through advanced computational tools
can be roughly divided into three main steps.
1. Processing of medical images
The native aortic valve geometry, including aortic sinuses and leaflets, as well as positions and
dimensions of calcifications are extracted as STL representations from Computed Tomography (CT)
images using OsiriX.
2. Creation of analysis-suitable models
Elaboration of medical images leads to very accurate patient-specific meshed geometries of the native
aortic valve, which are modelled using the nearly-incompressible Yeoh model calibrated on human
experimental data [5]. Calcium is also included whose properties are defined according to [6]. The
device model is obtained from high-resolution micro-CT scan of a real device sample; a von MisesHill plasticity model with isotropic hardening is used to reproduce the CoCr alloy behavior of the stent
while an isotropic hyperelastic model is used for the prosthetic leaflets made of bovine pericardium
[7].
3. Analyses to reproduce the clinical procedure
TAVI is a complex intervention composed of several steps; to realistically reproduce the whole
procedure, we set-up a simulation strategy consisting in the following two main stages [8]: (i) stent
crimping and deployment and (ii) valve mapping and closure. In Figure1 the principal phases of the
TAVI simulation strategy are shown. All the numerical analyses are non-linear problems involving
large deformation and contact performed using Abaqus Explicit solver v6.10.
Results. On one side, from the simulation of stent expansion we can evaluate the impact of the
metallic frame of the stent on the native calcified aortic valve; in particular, we can measure the
48
stress/strain patterns induced from the stent to the native aortic valve. On the other side, after
performing prosthetic valve mapping, from the simulation of valve closure we can predict the postoperative device performance in terms of paravalvular leackage (Figure 2a) and coaptation area. We
finally try to compare the obtained results with post-operative in vivo measurements (Figure 2b).
Figure 1. Procedural steps of TAVI reproduced through a computer-based simulation strategy.
(a)
(b)
Fig 2. Paravalvular leackage; evaluation of the degree of apposition between the prosthesis stent and the patient-specific
aortic root: (a) contour plot of the distance [mm] between the aortic wall and the prosthetic stent; (b) correspondent
postoperative ultrasound images highlighting the presence of a retrograde blood flow for the considered patient.
References
[1] A. Cribier, Percutaneous transcatheter implantation of an aortic valve prosthesis for calcific aortic
stenosis: First human case description. Circulation (2002), vol. 106, pp. 3006–3008.
[2] C.R. Smith et al., Transcatheter versus surgical aortic-valve replacement in high-risk patients,
New England Journal of Medicine (2011), vol. 364, pp. 2187-2198.
[3] P. Generaux et al., Paravalvular leak after transcatheter aortic valve replacement. JACC (2013),
vol. 61, p1125-1136
[4] R. Gurvitch et al., Transcatheteraortic valve implantation: Lessons from the learning curve of the
first 270 high-risk patients, Catheterization and Cardiovascular Interventions (2011), vol. 78, pp.
977-984
[5] C. Martin et al., Significant differences in the material properties between aged human and porcine
aortic tissues, EJCTS (2011), vol. 40, pp. 28-34.
[6] C. Capelli et al., Patient-specific simulations of transcatheter aortic valve stent implantation,
Medical and Biological Engineering and Computing (2012), vol. 368, pp. 183-192.
[7] F.L. Xiong et al., Finite element investigation of stentless pericardial aortic valves: Relevance of
leaflet geometry. Annals of Biomedical Engineering (2010), 38, pp. 1908-1918.
[8] F. Auricchio et al., Simulation of transcatheter aortic valve im-plantation: a patient-specific finite
element approach, Computer Meth. in Biomech. and Biomed. Eng. (2013), DOI:
10.1080/10255842.2012.746676.
49
A numerically efficient implicit integration algorithm for the MatsuokaNakai failure criterion
Andrea Panteghini1, a *, Rocco Lagioia2,a
1
University of Brescia – DICATAM, via Branze, 43 – 25123 Brescia (Italy)
a
[email protected], [email protected]
Keywords: Finite Elements, Implicit integration, Matsuoka-Nakai failure criterion, Mohr-Coulomb
failure criterion
We present a reformulation of the original Matsuoka–Nakai criterion for overcoming the limitations
which make its use in a stress point algorithm problematic. In fact, its graphical representation in the
principal stress space is not convex, and it comprises physically meaningless branches, plotting also in
negative octants. Moreover it does not increase monotonically as the distance of the stress point from
the failure surface rises. We propose an exact mathematical reproduction of the only significant
branch of the original criterion that plots as a single, convex surface, which entirely lies in the positive
octant of the stress space. The new formulation monotonically increases as the stress point moves
away from the failure surface. It is also suitable for shaping in the deviatoric plane the yield and
plastic potential surfaces of complex constitutive models. A very efficient numerical algorithm is also
provided for the implementation of a constitutive model based on this expression. An innovative
implicit integration scheme is formulated, which can be easily adapted for other models. Although the
yield and plastic potential surfaces described by the proposed expression is formulated in terms of
three invariants, a single scalar equation is finally obtained for the determination of the stress at the
end of each increment. This can be used both in associated and non-associated plasticity, and the
singularity at the apex of the surfaces is exactly handled during the numerical integration. It is shown
that all this results in extremely fast solutions of boundary value problems.
References
[1] A. Panteghini, R. Lagioia, A single numerically efficient equation for approximating the Mohr–
Coulomb and the Matsuoka–Nakai failure criteria with rounded edges and apex, Int J Numer Anal
Met. 38, 4 (2014) 349-369.
[2] A. Panteghini, R. Lagioia, A fully convex reformulation of the original Matsuoka–Nakai failure
criterion and its implicit numerically efficient integration algorithm, Int J Numer Anal Met. 38, 6
(2014) 593-614.
50
Selective mass scaling for thin structures discretized with multilayered, solid-shell elements
Federica Confalonieria *, Umberto Peregob and Aldo Ghisic
Dipartimento di Ingegneria Civile ed Ambientale, Politecnico di Milano,
piazza Leonardo da Vinci 32, Milano, Italy
a
[email protected], [email protected], [email protected]
* corresponding author
Keywords: mass scaling, solid-shell elements, thin structures.
It is well known that explicit time integration algorithms for structural dynamics are conditionally
stable according to the Courant-Friedrichs-Lewy (CFL) condition [1], stating that the critical time step
coincides with the so called “traversal time”, i.e. the time required by a dilatational stress wave to run
across the shortest element dimension. Thin structures discretized with solid-shell elements [2,3] are,
therefore, computationally expensive, because of their intrinsic small dimension in the thickness
direction. A possible solution, that allows to enlarge the critical time step and, hence, to improve the
computational efficiency without affecting the dynamical response, is to introduce a selective mass
scaling procedure. The basic idea is to locally modify the solid-shell element mass matricx, artificially
increasing the coefficients related to thickness eigenmodes, while those related to the translational
rigid body motions are left unchanged. In this manner, the highest structural eigenfrequencies are
reduced, without significantly alter the lowest ones. In [4], a mass scaling technique preserving the
mass lumping, and based on a simple, computationally inexpensive, linear transformation of the
element degrees of freedom, has been proposed for a single layer (solid-shell element) thin structure.
In the present contribution, the approach is extended to a multi-layered thin structure (Figure 2).
Figure 1: solid-shell element.
As in [4], the starting point is the definition of the average and difference accelerations of each
element as a function of the corresponding upper or lower nodal degrees of freedom:
and
,
i = 1,2, 3,4
(1)
where i is the node number for the upper or lower face of the solid-shell element (see Figure 1). Since
the average degrees of freedom are related to the translational rigid body modes, the element
maximum frequency can be reduced by scaling only the mass coefficients related to the difference
accelerations. Thus, the elements mass matrix becomes:
0
(2)
0
51
Once the e is retrieved for each element, it is possible to increase the time step by scaling down the
highest eigenfrequencies related to the rotational degrees of freedom, since the CFL condition states
that the time step must be t  2 /max (with max the maximum eigenfrequency of the system), and
max  e (maximum element eigenfrequency)
Figure 2: reference scheme for the discretization of a multi-layered thin structure with solid-shell elements.
It can be shown that, when a multi-layered structured is considered, the global mass matrix becomes a
diagonal block matrix, each block corresponding to the degrees of freedom of a single fiber b. The
overall solution can be computed simply solving a set of subsystems in the form:
(3)
∑
being
a tridiagonal matrix, built by assembling the mass matrices arising for each
layer l along the fiber, namely
1
1
1
1
The resulting mass matrix is not diagonal. However, it is block diagonal, each block having a tridiagonal structure and dimensions directly related to the number of layers, which are in general in a
limited number through the shell thickness. Even though accelerations cannot be computed explicitly,
the solution of the small linear system providing accelerations of nodes belonging to the same fiber is
inexpensive and the small additional burden is by far compensated by the largest stable time step
which can be used in the computation. This turns out to be a critical issue, as several other selective
mass scaling techniques proposed in the literature provide a theoretically rigorous scaling, though at
the cost of densely populated mass matrices, requiring an iterative solver for acceleration computation
at each time step.
The accuracy of the mass scaling procedure and the computational gain are checked with the aid of
numerical examples.
References
[1] R. Courant, K. Friedrichs, H. Lewy, On the Partial Difference Equations of Mathematical
Physics. IBM J. 11 (1967) 215-234.
[2] S. Reese, A large deformation solid-shell concept based on reduced integration with hourglass
stabilization, Int. J. Numer. Meth. Eng. 69 (2007) 1671-1716.
[3] M. Pagani, S. Reese, U. Perego, Computationally efficient explicit nonlinear analyses using
reduced integration-bsed solid-shell finite elements, Comp. Meth. Appl. Mech. Eng. 268 (2014) 141159.
[4] G. Cocchetti, M. Pagani, U. Perego, Selective mass scaling and critical time-step estimate for
explicit dynamics analyses with solid-shell elements, Comput. & Struct. 127 (2013) 39-52.
52
A generalized time-domain approach for motion-related wind loads on
long-span bridges
S. de Miranda1,a, L. Patruno1,b, F. Ubertini1,c and G. Vairo2,d *
1
2
DICAM, University of Bologna, viale Risorgimento 2, 40136 Bologna, Italy
DICII University of Rome “Tor Vergata”, via del Politecnico 1, 00133 Roma, Italy
a
[email protected], [email protected],
c
[email protected], [email protected]
* corresponding author
Keywords: bridge aeroelasticity, indicial functions, flutter derivatives.
Long-span bridges are slender, light, and flexible large-scale line-like structures, highly sensitive to
unsteady wind effects. As a result of the aeroelastic interaction between wind and structure, dynamic
and static instabilities can occur at current wind speeds. In this context, the major concern is surely
represented by flutter instability which, in case of unfavorable aerodynamic properties of the deck, can
lead to unbounded growing bridge oscillations even at relatively mild wind speed.
Available procedures for designing bridges against flutter are mainly based on the frequency-domain
approach introduced by Scanlan [1-3], that has proved to be extremely effective in synthetically
representing the aeroelastic response and wind-structure interaction mechanisms in long-span bridges,
allowing to straight estimate critical states for flutter onset (e.g., [3-6]). Nevertheless, the Scanlan's
approach needs the preliminary evaluation of frequency-response functions, namely the flutter
derivatives, that linearly relate the aeroelastic forces acting on the vibrating structure to its motion,
leading to the identification of equivalent aerodynamic stiffness and damping that couple with the
structural dynamical features.
Attempts to define effective time-domain formulations, based on the definition of suitable indicial
functions (describing the time evolution of the aerodynamic forces induced by a step variation in the
effective angle of attack), can be also found in recent literature (e.g., [2, 4, 7]). Nevertheless, the
characterization of the aerodynamic response to step-like body movements is generally tough and it
poses a number of technical drawbacks. Therefore, indicial functions necessary to define time-domain
loading models are usually extracted a posteriori from flutter derivatives [7], leading to possible
theoretical inconsistencies [8].
Starting from a critical and unified review of the state of the art in modeling of wind loads acting upon
bridge decks, actual approaches and open issues will be outlined. Moreover, a novel time-domain
description of the aeroelastic loads will be proposed, aiming to consistently extend the framework of
the thin airfoil theory to mildly bluff sections, such as those usually employed for decks of modern
long-span bridges. Circulatory and non-circulatory contributions are separately described and
superimposed, and generalized downwash-related terms are introduced. The strong duality between
time-domain and frequency-domain representations is focused, and direct relationships between
proper Wagner-like [9] indicial functions and Theodorsen-like [10] circulatory functions are deduced.
Thereby, following the Scanlan formulation for bridge deck sections, flutter derivatives are
represented by superimposing circulatory and non-circulatory effects, resulting in a frequency-domain
description fully consistent with the Theodorsen’s theory.
The model is based on few parameters that can be estimated by simplified strategies and by
asymptotic relationships. An identification procedure involving few experiments or numerical
simulations is proposed and numerically implemented. Several simulation results will be presented,
highlighting effectiveness and soundness of the presented identification strategy.
53
References
[1]
R.H. Scanlan, Motion-related body-force functions in two-dimensional low-speed flow, J.
Fluid. Struct. 14 (2000) 49-63.
[2]
R.H. Scanlan, Reexamination of sectional aerodynamic force functions for bridges, J. Wind
Eng. Ind. Aerod. 89 (2001) 1257-1266.
[3] E. Simiu, R.H. Scanlan, Wind Effects on Structures, John Wiley and Sons Inc., New York, 1996.
[4]
X. Chen, A. Kareem, Advances in modeling aerodynamic forces on bridge decks, J. Eng.
Mech. ASCE 128 (2002) 1193-1205.
[5]
G. Vairo, A numerical model for wind loads simulation on long-span bridges, Simul. Model.
Pract. Th. 11 (2003) 315-351.
[6]
G. Vairo, A simple analytical approach to the aeroelastic stability problem of long-span cablestayed bridges, Int. J. Comput. Meth. Eng. Sci. Mech. 11 (2010) 1-19.
[7]
L. Caracoglia, N. Jones N, Time domain vs. frequency domain characterization of aeroelastic
forces for bridge deck sections, J. Wind Eng. Ind. Aerod. 91 (2003) 371-402.
[8]
S. De Miranda, L. Patruno, F. Ubertini, G. Vairo, Indicial functions and flutter derivatives: a
generalized approach to the motion-related wind loads, J. Fluid. Struct. 42(2013) 466-487.
[9]
H. Wagner, Űber die entstehung des dynamischen auftriebes von tragflügeln (in German),
ZAMM 5 (1925) 17-35.
[10] T. Theodorsen, General theory of aerodynamic instability and the mechanism of flutter, NACA
Rep. n. 496 (1935).
54
A new flexible approach for shape memory alloy constitutive modeling
Ferdinando Auricchio1,a *, Elena Bonetti2,b, Giulia Scalet1,c and Francesco Ubertini3,d
1
Dipartimento di Ingegneria Civile e Architettura, Università di Pavia,
via Ferrata 3, 27100 Pavia, Italy
2
Dipartimento di Matematica, Università di Pavia, via Ferrata 1, 27100 Pavia, Italy
3
Dipartimento di Ingegneria Civile, Chimica, Ambientale e dei Materiali, Università di Bologna,
viale Risorgimento 2, 40136 Bologna, Italy
a
b
[email protected], [email protected], [email protected], [email protected]
Keywords: Shape memory alloys, constitutive modeling, phase transformation, reorientation, FischerBurmeister function.
Among the broad class of smart materials, shape memory alloys (SMAs) have unique features due to
their ability to undergo large deformations (up to 8-10%) and to regain the original shape either during
unloading (superelasticity) or through a thermal cycle (one-way and two-way shape memory effects).
Since such effects are not present in standard alloys, SMAs are exploited in innovative applications,
within the mechanical, aeronautical, biomedical, structural engineering fields.
In the past three decades SMAs have been deeply investigated from the point of view of modeling,
analysis, and computation with the aim of developing flexible and reliable constitutive models to be
used as design tools for SMA devices. Thanks to their simple numerical implementation, less timeconsuming calculations and the possibility to be adjusted for a particular type of material easily,
macroscopic models appear to be a powerful tool for SMA behavior simulation. As a consequence,
many phenomenological models have been proposed in the literature. However, some of the well
performing models (e.g., see [1]) have still limitations, since they are not able to properly describe
material secondary effects, which however corresponds to the most adopted conditions in several
applications.
Accordingly, the present work aims to formulate a general, complete and flexible theoretical
framework that can predict the complex behavior of SMAs and is based on a physical interpretation of
material parameters as well as to offer a robust numerical framework to be then used for the
simulation of real devices.
To this purpose, the work develops a more refined and general 3D constitutive model, along the lines
of what recently proposed [2]. The new flexible model is capable of realistic simulations of several
physical phenomena, involving phase transformations between austenite, twinned and detwinned
martensites, martensite reorientation, different kinetics between forward/reverse PTs and
transformation-dependent elastic properties.
The model is then treated numerically through an effective and efficient procedure based on the
Fischer-Burmeister complementarity function [3], since standard predictor-corrector methods are no
more suitable due to the increased complexity of the governing equations.
Finally, the work presents several numerical simulations, ranging from proportional and nonproportional loading conditions to boundary-value problems of industrial interest (see Figure 1), in
order to assess the reliability of the proposed model as well as the robustness of its numerical
counterpart.
55
Figure 1. Pseudoelastic test of a SMA helical spring actuator. (Top) Spring initial geometry and scaled deformed shape
under the maximum force. (Bottom) Force vs. vertical displacement of the loaded end (markers, large load increments;
solid line, small load increments).
References
[1] A.C. Souza, E.N. Mamiya, N. Zouain, Three-dimensional model for solids undergoing stressinduced phase transformations. Eur. J. Mech. A-Solids. 17 (1998) 789-806.
[2] F. Auricchio, E. Bonetti, A new 3D macroscopic model for shape memory alloys describing
martensite reorientation. Discret. Contin. Dyn. Syst. Ser. S. 6(2) (2013) 277-291.
[3] A. Fischer, A special Newton-type optimization method. Optim. 24 (1992) 269-284.
56
Damage modelling in concrete subject to sulfate attack
Nicola Cefis1,a* and Claudia Comi1,b
1
Department of Civil and Environmental Engineering, Politecnico di Milano,
P.zza Leonardo da Vinci 32, 20133 Milano (Italy)
a
[email protected], [email protected]
Keywords: Concrete, Sulfate attack, Damage, Delayed ettringite formation
Under particular environmental conditions, some kinds of concrete may be subject to deleterious
chemical reactions that cause swelling and micro-cracking, alter the mechanical properties and affect
the durability of concrete structures. The present work is devoted to the modeling and numerical
simulation of the degradation in concrete subject to sulfate attack.
There are two kinds of sulfate attack: the internal sulfate attack and external sulfate attack. In the first
case, the sulfate ions are already present within the material because of the thermal depletion of
primary ettringite due to curing at high temperature or to the excessive heat of hydration developed in
massive structures, see e.g. [1]. In the second case, the sulfate is present in the environment and
diffuses within the material through the porous microstructure; this happens e.g. in foundations,
galleries, stores of radioactive waste in contact with sulfate-rich soils, [2]. In both cases, the reaction
between the sulfate and hydrated products of the cement leads to the formation of gypsum and of
secondary ettringite, [3, 4]. The product formed in the hardened paste exerts an internal pressure
resulting in the appearance of micro-cracks and material degradation.
The kinetics of the reactions and, consequently, the severity of the damage depends on environmental
factors (species and concentration of sulfate, pH of the solution, humidity, temperature) and intrinsic
material properties (chemical composition of the cement paste, in particular aluminates content, pore
distribution, diffusivity properties).
The numerical description of these phenomena requires a proper diffusion-reaction model, for the
computation of the amount of reaction expansive products and a mechanical model for the prediction
of swelling and material damage. In this work we use the coupled model developed in [3] which
allows to compute the sulfate molar concentration s(x,t) from a diffusive-reaction equation, taking into
account the aluminate depletion due to ettringite formation. The governing system reads ( ceq being the
aluminate molar concentration, Ds the diffusion parameter, k the rate of take-up of sulfates and q the
average stoichiometric parameter):
 s
 t  div ( Ds (grad s))  k ceq s
 c
 eq   k ceq s
 t
q
(1)
From the molar concentration of aluminates and sulfate one obtains the distribution of ettringite and
the volume change related to the difference between the molar volumes of reactants and reaction
products.
The mechanical response of the material to this expansion is then computed by a new poroelasticdamage model. Within the framework of the Biot’s theory [5], the concrete is described as a twophase material: the homogenized skeleton phase, including cement paste and aggregates, and the
expansive phase of the products of the reaction. The total stress  is the sum of the effective stress
acting on the solid skeleton σ and of the stress on the reaction products phase, p being the pressure
at the microscale:
σ  σ   bp1
(2)
57
The effective stress on the concrete skeleton is related to the total strain by an elastic law, with
isotropic damage D. The pressure depends on the volumetric deformation  v and on the volumetric
expansion due to ettringite formation, M and b being the Biot’s parameters
σ   (1  D)d : ε
(3)
p  1  D  bM v  M 
(4)
The expansion term  can be expressed as
 
K  Mb 2
 (ceq0  ceq )  f 0
Mb
(5)
where ceq0  ceq is the amount of calcium aluminate,  accounts for the volume change involved in
the reaction of ettringite formation, f 0 if a fraction of the initial porosity and  denotes the
positive part of  .
The damage evolution is governed by the loading-unloading conditions proposed in [6], expressed in
term of the inelastic effective stress σ   σ   p1 (   b) .
The numerical solution of the diffusion-reaction problem and of the subsequent chemo-damage
problem are obtained by an ad-hoc developed finite element code.
The presented model is employed to simulate the expansion of mortar samples subject external sulfate
attack (experimental tests reported in [7]) and the influence of restraint on the expansion of mortar
specimens affected by internal sulfate attack (experimental tests reported in [8]). A reasonably good
agreement is obtained in both cases.
References
[1] M. Al Shamaa, S. Lavaud, L. Divet, G. Nahas, J.M. Torrenti. Coupling between mechanical and
transfer properties and expansion due to DEF in a concrete of a nuclear plant. Nuclear Engineering
and Design 266 (2014) 70-77.
[2] M. Lei, L. Peng, C. Shi, S. Wang. Experimental study on the damage mechanism of tunnel
structure suffering from sulfate attack. Tunnelling and underground space technology 36 (2013) 5-13.
[3] J. Skalny, J. Marchand, I. Odler, Sulfate Attack on Concrete, Spon Press, 2002
[4] R. Tixier, B. Mobasher, Modeling of Damage in Cement-Based Materials Subjected to External
Sulfate Attack, J. Mater. Civ. Eng, 15 (2003) 305-310.
[5] O. Coussy, Poromechanics, John Wiley & Sons, 2004.
[6] C. Comi, R. Fedele, U. Perego, A chemo-thermo-damage model for the analysis of concrete dams
affected by alkali-silica reaction, Mechanics of Materials, 41 (2009) 210-230.
[7] A.E. Idiart, Coupled analysis of degradation processes in concrete specimens at the meso-level,
Doctoral Thesis, Universitat Politècnica De Catalunya, (2009).
[8] H. Bouzabata, S. Multon, A. Sellier, H. Houari, Effects of restraint on expansion due to delayed
ettringite formation, Cement and Concrete Research, 42 (2012) 1024-1031.
58
A 3D mixed frame element with multi-axial coupling for thin-walled
structures with damage
Daniela Addessi1, a * and Paolo Di Re1,b
1
Dip. di Ingegneria Strutturale e Geotecnica, Università di Roma “Sapienza”,
Via Eudossiana 18, 00184, Rome, Italy
a
[email protected], [email protected]
Keywords: Thin-walled structures, Mixed beam formulation, Warping, Damage, Softening,
Regularization.
The development of accurate and efficient finite element (FE) codes is a significant challenge in many
engineering fields and, in particular, in structural engineering. In fact, in today’s professional
structural applications it is often required to analyze large scale structures with irregular geometry,
made of innovative composite materials, under severe loading conditions. Thus, in order to accurately
describe the global nonlinear response and the local distributions of stresses and damaging paths, it’s
of great interest to formulate enhanced FEs taking into account nonlinear geometric and constitutive
behavior.
In this work, a 2-node frame element, derived on the basis of the Hu-Washizu variational potential, is
presented, as proposed in [1]. It can describe the warping of the cross sections, adding a variable
number of degrees of freedom to the standard ones defined in a 3D beam FE. The evolution of the
warping displacements is defined through the interpolation of these degrees of freedom, that is
developed at two independent levels: along the element axis and on the cross section, both using
Lagrange polynomials. Thanks to the adopted enriched kinematic description, the coupling of the
effects produced by shear and torsion with those produced by axial and flexural stresses is captured.
With the purpose of modeling the complex behavior of the cross section, that arises when considering
nonlinear material behaviors, a fiber discretization is introduced. Hence, stress and strain variables are
determined at each integration point of the section and integrated over the area to obtain the
generalized ones. The stress-strain relationship adopted is aimed at the description of damaging
mechanisms. It is based on the definition of two scalar damage variables, evolving according to two
different laws: one related to the damage produced in tension and the other to the damage produced in
compression, as proposed in [2]. The irreversibility of the degrading processes is considered.
The proposed frame element is implemented in the FE analysis program FEAP, which is used to
perform all the numerical analyses. In particular, different shapes of the beam cross section are
considered. Thus, the effects of the warping on the damage evolution are investigated, comparing the
responses obtained with the proposed element to those obtained with standard ones, as well as with
experimental outcomes. Furthermore, the localization problems due to the considered strain-softening
material behavior are analyzed and a regularization methodology [3] is introduced to overcome the
related numerical drawbacks.
References
[1]
V. Le Corvec, Nonlinear 3d frame element with multi-axial coupling under consideration of
local effects, UC Berkeley Electronic Theses and Dissertations Degree: Ph.D, Civil and
Environmental Engineering UC Berkeley, 2012.
[2]
J. Mazars, P. Kotronis, F. Ragueneau, G. Casaux, Using multifiber beams to account for shear
and torsion. Application to concrete structural elements, Comp. Methods Appl. Mech. Engrg. 195
(2006) 7264-7281.
[3]
D. Addessi, V. Ciampi, A regularized force-based beam element with a damage–plastic section
constitutive law, Int. J. Numer. Meth. Eng. 70 (2007) 610-629.
59
A basic introduction to isogeometric collocation methods with some
applications
Alessandro Reali1,a*, Ferdinando Auricchio1,b, Lourenco Beirão da Veiga2,c, Hector
Gomez3,d, Thomas JR Hughes4,e, Giancarlo Sangalli1,f
1
University of Pavia, Italy; 2University of Milan, Italy; 3
University of A Coruña, Spain; 4 University of Texas at Austin, USA;
a
[email protected], [email protected], [email protected], [email protected],
e
[email protected], [email protected]
Keywords: Isogeometric analysis, collocation methods, elasticity, explicit dynamics, Cahn-Hilliard
equations, phase field modeling.
Isogeometric Analysis (IGA) is a recent idea (see [1,2]) introduced to bridge the gap between
Computational Mechanics and Computer Aided Design (CAD). The key feature of IGA is to extend
the finite element method representing the geometry by functions - such as NURBS - typically used
by CAD systems, and then invoking the isoparametric concept to define field variables. Thus, the
computational domain exactly reproduces the NURBS description of the physical domain, and, also
thanks to the high regularity properties of the employed functions, numerical testing in different
situations has shown a substantial increase, with respect to standard finite elements, of the ratio
between accuracy and number of degrees-of-freedom.
In the framework of NURBS-based IGA, collocation methods have been recently proposed as a viable
and interesting low-cost alternative to standard isogeometric Galerkin approaches, where, within the
isoparametric paradigm, PDEs are collocated in strong form at suitable points (cf. [3,4]). In this work,
we introduce the basics of such methods and focus on some relevant applications, ranging from
elastostatics and explicit elastodynamics (see [5] and Figure 1) to the solution of the Cahn-Hilliard
equation (see [6] and Figure 2), for which isogeometric collocation represents an accurate, efficient,
and geometrically flexible option.
Figure 1. Numerical simulation of a pressurized thick-walled elastic cylinder as described in [5].
60
Figure 2. Numerical simulation snapshot of the spinodal decomposition of two immiscible fluids as described in [6].
References
[1] Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y., Isogeometric analysis: CAD, finite elements,
NURBS, exact geometry, and mesh refinement. Computer Methods in Applied Mechanics and
Engineering, vol. 194, pp. 4135-4195 (2005).
[2] Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y., Isogeometric Analysis. Towards integration of
CAD and FEA. Wiley, (2009).
[3] Auricchio, F., Beirão da Veiga, L., Hughes, T.J.R., Reali, A., Sangalli, G., Isogeometric
Collocation Methods. Mathematical Models and Methods in Applied Sciences, vol. 20, pp. 2075-2107
(2010).
[4] Schillinger, D., Evans, J.A., Reali, A., Scott, M.A., Hughes, T.J.R. Isogeometric Collocation: Cost
Comparison with Galerkin Methods and Extension to Adaptive Hierarchical NURBS Discretizations.
Computer Methods in Applied Mechanics and Engineering, vol. 267, pp. 170-232 (2013).
[5] Auricchio, F., Beirão da Veiga, L., Hughes, T.J.R., Reali, A., Sangalli, G., Isogeometric
collocation for elastostatics and explicit dynamics. Computer Methods in Applied Mechanics and
Engineering, vol. 249-252, pp. 2-14 (2012).
[6] Gomez, H., Reali, A., Sangalli, G., Accurate, efficient, and (iso)geometrically flexible collocation
methods for phase-field models. Journal of Computational Physics, vol. 262, pp. 153-171 (2014).
61
On the state update for isotropic elasto-plastic hardening materials: a
dissipation-based algorithm
Nicola A. Nodargi*, Edoardo Artioli, Federica Caselli and Paolo Bisegna
Department of Civil Engineering and Computer Science
University of Rome "Tor Vergata", via del Politecnico 1, 00133 Rome, Italy
[email protected], URL: www.dicii.uniroma2.it
Keywords: Plasticity, Hardening, Incremental problems, Energy minimization, Dissipation,
Integration algorithm
The solution of inelastic structural boundary value problems in a typical finite element
implementation requires the numerical integration of the material constitutive law at each Gauss point.
In the context of hardening elasto-plastic media, the kinematic description of the material state is
provided through the total strain ε, additively decomposed into its elastic εe and plastic εp parts, and
the strain-like internal variables α, i.e. kinematic and isotropic hardening variables. The corresponding
conjugated stress-like variables are the stress σ and the stress-like internal variables q respectively.
In a strain-driven framework, the material state has to be updated for a given total strain increment. A
standard approach for the integration of the constitutive law is the use of elastic-predictor inelasticcorrector return map algorithms. In a first step it is checked whether the state determined with the
assumption of no plastic evolution is plastically admissible. In case not, a solution for plastic
evolution equations, i.e. flow rule, hardening law and plastic consistent condition, is sought for in the
plastic corrector step. To this purpose, a common strategy in practical applications is a backwardEuler-type approximation of the governing equations, leading to the so-called closest-point projection
algorithm (e.g. see [1]).
Under the assumptions of (i) associative flow rule, (ii) convex Helmholtz free energy function ψ(εe , α)
and (iii) convex yield function f (σ, q), the approximated equations governing the plastic evolution can
be recast in a variational framework. In that approach, the minimization of a functional involving the
complementary energy χ (σ, q), i.e. the Fenchel conjugate of the free energy, with respect to
plastically admissible stress σ and stress-like internal variables q is required [2]. However the
constrained character of the return map strategy and the numerical difficulties in convergence when
the yield surface presents singularities or points with large curvature motivate the development of
alternative techniques.
The present work focuses on an equivalent statement of the aforementioned variational formulation
(e.g. see [3]). In a time-discrete framework, this allows to update the material state from time tn to
time tn+1 by solving the infimum problem
inf
Δεp ,Δα
ψ εn+1
p
p
εn
p
Δεp , Δα + D Δεp , Δα
where Δεp = εn+1 εn , Δα = αn+1
variables respectively and
D Δεp , Δα = sup
σ, q
f (σ, q) ≤
,
(1)
αn are the increments of plastic strain and strain-like internal
σ · Δεp + q · Δα
(2)
is the dissipation function, i.e. the support function of the elastic domain.
62
The present work exploits this dissipation-based variational formulation to perform the material state
update. To this purpose, a two-step algorithm is proposed. In the first step, an elastic prediction of the
updated material state is carried out. In case it is not plastically admissible, the infimum problem Eq. 1
is solved adopting the Newton-Raphson method. An efficient strategy to compute the dissipation
function for isotropic yield criteria is proposed. In particular, adopting the Haigh-Westergaard
representation (e.g. see [4]) for the analysis of isotropic yield functions, the supremum problem Eq. 2
is reduced to a non-linear scalar equation. Moreover closed-form expressions for the gradient and the
Hessian of Haigh-Westergaard coordinates and dissipation function are presented. With the aim of
proving the robustness and stability of the proposed algorithm, numerical tests on a single integration
point and FEM simulations are provided. This numerical approach appears to be competitive with the
typical return map strategy, especially when the yield surface presents singularities or points with
large curvature because no difficulty in convergence arises.
References
[1] J.C. Simo, T.J.R. Hughes, Computation inelasticity, Springer Verlag, New York, 1998.
[2] F. Armero, A. Perez-Foguet, On the formulation of closest-point projection algorithms in
elastoplasticity - part I: The variational structure, Int. J. Numer. Meth. Engng., 53 (2002), 297-329.
[3] W. Han, B. Reddy, Plasticity. Mathematical theory and numerical analysis, Springer Verlag,
Berlin, 1999.
[4] G. Nayak, O. Zienkiewic, Convenient forms of stress invariants for plasticity, Proceedings of the
ASCE Journal of the Structural Division, 98(ST4) (1972), 949-953.
63
A Lagrangian finite element approach for the numerical simulation of
landslide runouts
Massimiliano Cremonesia *, Francesco Ferrib and Umberto Peregoc
a
Department of Civil and Environmental Engineering, Politecnico di Milano, Italy
massimiliano.cremonesi@polimi.it, bfrancesco.ferri@polimi.it, cumberto.perego@polimi.it
* corresponding author
Keywords: landslide simulation, Lagrangian finite element approach, slip boundary conditions.
Landslides are extreme natural phenomena frequently occurring in our country and causing casualties
and extensive damage to residential structures, infrastructures and to the historical and cultural
heritage.
The numerical simulation of these events requires capabilities for tracking evolving interfaces and free
surfaces, accounting for the mixing of different constituents, for complex constitutive behaviors,
extremely large deformations and possible multi-physics processes.
Recently, several scientific contributions have treated landslides as viscous fluids in motion (see e.g.
[1]), an approach that has opened the way to new applications of computational methods conceived
for the simulation of fluid problems. While in the case of fluids most approaches are developed in an
Eulerian framework, in the case of landslides, the complex constitutive behavior of the soil and the
rapidly evolving free surface are more appropriately modeled with a Lagrangian approach. The PFEM
(Particle Finite Element Method) [2] is an innovative Lagrangian numerical method, particularly
suited to the solution of problems with interactions between fluids and structures.
In the PFEM, the Navier-Stokes equations, governing the motion of fluids and structures, are
approximated using a material formulation (Lagrangian) in which the mesh nodes move together with
material particles. For this reason, in order to emphasize the material description of motion, the nodes
are named 'particles'. All the physical properties such as density, viscosity, velocity, position, and
other variables such as temperature, are assigned to the particles and are transported during the motion
of the mesh nodes. The problem of elements distortion due to the large deformations of the moving
fluid, is solved with a very efficient and continuous mesh re-triangulation based on the Delaunay
technique [2,3,4] and driven by a fast and effective geometric distortion criterion.
A non-Newtonian constitutive law has been introduced to describe the behavior of the granular
material, which is assumed to be incompressible in the landslide running out regime. The deviatoric
stress is related to the deviatoric strain rate through an apparent viscosity defined as:
tan
| |
2 1
| |
where is the pressure field, is the friction angle and n is a regularization parameter.
To better describe the interaction between the moving landslide and the slope substrate, slip boundary
conditions have been introduced. The velocity component along the slope
is written as:
0
where
is the tangential stress, 0 represents a stress threshold, below which no slip can occur, and
a parameter, having the dimensions of a length over a viscosity, defining the amount of slip. This
condition states that the slip is resisted by a tangential force proportional to the relative velocity. For
0 the no-slip boundary condition is recovered, while → ∞ represents the stress free boundary
condition.
64
The proposed approach has been validated against experimental tests, showing a good agreement with
the expected results. In Figure 1 an example of a tridimensional simulation of a landslide along a slope
is presented.
Figure 1: Propagation of a landslide on a slope: snapshots at different time istants.
References
[1] M. Quecedo, M. Pastor and M.I. Herreros, Numerical modelling of impulse wave generated by
fast landslides, International journal for numerical Method in Engineering, 59(12), pp.1633-1656,
(2004).
[2] E. Oñate, S.R. Idelsohn, F. del Pin and R. Aubry. The Particle Finite Element Method. An
Overview. International Journal Computational Method, 1(2), pp. 267-307, (2004).
[3] M Cremonesi, A. Frangi, A. and U. Perego, A Lagrangian finite element approach for the analysis
of fluid-structure interaction problems. International journal for numerical Method in Engineering
84:pp 610-630 (2010).
[4] M.Cremonesi, A. Frangi, U. Perego, A Lagrangian finite element approach for the simulation of
water-waves induced by landslides, Computers &.Structures, 89(11-12), pp. 1086-103, (2011).
65
Geometry of elastoplasticity in the nonlinear range
Giovanni Romano, Raffaele Barretta and Marina Diaco
Department of Structures for Engineering and Architecture,
University of Naples Federico II, Italy
via Claudio 21 – 80125 - Naples,
romano@unina.it - rabarret@unina.it - diaco@unina.it
Keywords: Material and spatial fields, rate-elastoplaticity, naturality and frame invariance, Liederivatives.
The conceptual and operational revisitation of fundamentals of Continuum Mechanics, carried out in a
series of recent contributions by the authors [1-6], is applied to the formulation of a geometrically
consistent theory of Elasto-Plasticity in the full nonlinear range.
The bias of the presentation is towards physical and geometrical new ideas and notions involved in the
analysis.
Main issues are the natural definition of material and spatial tensor fields and a list of four basic
geometric principles to be addressed in a constitutive theory.
Treatments in literature are revisited to underline the need for the new geometric theory according to
which, in the constitutive framework, only current placements of the body are to be considered and
only relations between material tensor fields, with a common base point in the space-time trajectory,
are involved.
The rate elastic and plastic responses to stress and stress-rate are assumed as additive components of
the total stretching. Reference local manifolds are considered as purely computational tools.
Unphysical notions such as reference and intermediate local configurations are eliminated from the
basic constitutive formulation and referential finite elastic and plastic strains are shown to have a
purely computational role without physical interpretation.
The requirement of invariance under change of observer is treated anew to provide a correct geometric
interpretation of the physical experience.
The outcome is a model of nonlinear Elasto-Plasticity testable by experiments, suitable for
implementation in computational codes and apt to provide a consistent treatment of other complex
constitutive behaviors such as phase transformations in metal alloys or tissue growth in biomechanics.
References
[1] G. Romano, R. Barretta, Covariant hypo-elasticity, Eur. J. Mech. A-Solids 30 (2011) 1012-1023.
[2] G. Romano, R. Barretta, On Euler's Stretching Formula in Continuum Mechanics, Acta Mech. 224
(2013) 211-230.
[3] G. Romano, R. Barretta, Geometric Constitutive Theory and Frame Invariance, Int. J. Non-Linear
Mech. 51 (2013) 75-86.
[4] G. Romano, R. Barretta, M. Diaco, Geometric Continuum Mechanics, Meccanica 49 (1) (2014)
111-133.
[5] G. Romano, R. Barretta, M. Diaco, Rate Formulations in Non-Linear Continuum Mechanics
Acta Mech. online (2013) doi: 10.1007/s00707-013-1002-3
[6] G. Romano, R. Barretta, M. Diaco, The Geometry of Non-Linear Elasticity Acta Mech. online
(2014) doi: 10.1007/s00707-014-1113-5
66
FE-Meshless multiscale non linear analysis of masonry structures
Giuseppe Giambanco 1,a *, Emma La Malfa Ribolla1,b and Antonino Spada1,c
1
Department of Civil, Environmental, Aerospace and Materials Engineering (DICAM)
University of Palermo, Viale delle Scienze – Ed. 8, 90128 Palermo, Italy
a
giuseppe.giambanco@unipa.it, bemmalamalfa@hotmail.it, cantonino.spada@unipa.it
* corresponding author
Keywords: multiscale, mesomodeling, meshless, masonry.
In masonry structures, the most relevant kinematical and mechanical phenomena take place at a scale
which is small if compared to the dimensions of the structure. On the other side, the structure is
governed, in its peculiar overall response, by its global geometrical and morphological configuration.
In literature, two different scales of interest are distinguished, directly linked to as many theoretical
approaches: the mesoscopic approach and the macroscopic approach. The mesoscopic approach
considers the heterogeneous materials and their interfaces individually [1, 2], but many difficulties
arise in the mesh creation and a fine discretization of the structure has to be used, which leads to
prohibitive computational costs. The macroscopic approach considers the structure constituted by a
fictitious homogeneous and continuous material. The multiscale techniques belong to the second
approach and couple different scales of interest by means of apposite transition laws capable to
exchange information between different consecutive scales [3-5].
In this work a multiscale first order computational homogenization technique is applied to simulate
masonry structures. A unit cell (UC) is identified. The UC is assumed constituted by a block
surrounded by mortar joints, which are simulated by zero-thickness interface models. The material of
the block is assumed indefinitely elastic while the interface laws are expressed in the framework of
elastoplasticity.
The scale transition between macroscale and mesoscale is based on the Hill-Mandel principle. By
imposing the equilibrium over the entire structure, the displacement u M and strains ε M fields at the
macroscopic level are derived. The macroscopic stress field σ M associated to the strain field ε M is
instead obtained averaging the UC reactions r over the volume UC . To this end, the response of the
UC is evaluated by solving a boundary value problem, that in this work are assumed to be of TaylorVoigt type:
um  εM x
on UC
(1)
where um are the prescribed displacements for the point of position x located on the boundary UC of
the UC. Once the solution of the boundary value problem is obtained, the macroscopic stress σ M is
calculated according to the following equation:
σM 
1
2 UC
 r  x  x  r  d 
(2)
UC
The interface constitutive laws are developed in the framework of elastoplasticity for non standard
materials. The elastic domain is defined by two convex limit surfaces intersecting in a non-smooth
fashion: the Coulomb bilinear limit surface and a tension cut-off. Non-associative flow rules are
derived to express the displacement discontinuities at the interfaces.
67
The solution of the UC boundary value problem is generally approached in an approximated way
making use of the finite element method. In the present study the numerical solution of the
mesoscopic model is obtained by means of a meshless strategy [6]. The UC is divided in five
integration domains: the first domain corresponds to the volume occupied by the block, the other four
domains are the interfaces. The displacement field inside each sub-domain is obtained from the nodal
displacement values by a Moving Least Square approximation. The influence of a node on a point on
the UC is defined by a weight function depending on the distance between the sampling point and the
node.
The proposed model has been implemented on a research oriented finite element analysis program to
run 2D simulations on masonry structures.
The solution can be separated in an elastic and a plastic phases. To find an elasto-plastic solution the
interface constitutive laws are rewritten in a discrete way and integrated for a given incremental strain
history. The solution over a single time step is obtained employing a Backward Euler integration
scheme, separated in an elastic predictor stage and a plastic corrector stage. The governing equations
for the UC at the elasto-plastic step are finally expressed in matrix form as follows:
 K -G   Um   Fp 
 T 0   R    U 
G

  m
(3)
where K and G are two matrices depending on the UC geometry, Um is the increment of nodal
displacements, R the increment of nodal reactions, Fp the elasto-plastic forces evaluated in the
correction stage, U m the imposed displacements on boundary nodes.
A macroscopic elastic tangent stiffness matrix can be evaluated at each integration point and a
procedure is performed to localize the plastic zones at the macroscale, starting from the results at the
mesoscale.
The FE-Meshless multi-scale computational strategy has been applied to simulate experimental tests
available in literature in plane-stress conditions. The classical finite element analysis is run at the
macroscale, while the meshless procedure is applied at the mesoscale on the UC. Two iterative
Newton-Raphson procedures have been used during the analyses: one for the macroscale finite
element procedure, one for the solution of the UC in the plastic phase. The convergence criterion at
the mesoscale is considered satisfied when the difference between the elasto-plastic forces between
two successive iterations is less than a tolerance value.
References
[1] G. Giambanco, S. Rizzo, R. Spallino, Numerical analysis of masonry structures via interface
models, Comput. Methods Appl. Mech. Eng., 190 (2001) 6493-6511.
[2] A. Spada, G. Giambanco, P. Rizzo, Damage and plasticity at the interfaces in composite materials
and structures, Comput. Methods Appl. Mech. Eng., 198 (2009) 3884-3901.
[3] V. Kouznetsova, M.G. Geers, W.M. Brekelmans, Multi-scale constitutive modelling of
heterogeneous materials with a gradient-enhanced computational homogenization scheme, Int. J.
Numer. Meth. Eng., 54 (2002) 1235-1260.
[4] T.J. Massart, Multi-scale modelling of damage in masonry structures, PhD Thesis, University of
Technology, Eindhoven, 2003.
[5] D. Addessi, E. Sacco, A multi-scale enriched model for the analysis of masonry panel, Int. J.
Solids and Struct, 49 (2012) 865-880.
[6] S.N. Atluri, S. Shen, The meshless method, Tech. Sci. Press, Forsyth, 2002.
68
Non-linear analysis of 3D elastoplastic framed structures
Valerio Carollo1, a, Giuseppe Giambanco1,b*, and Antonino Spada1,c
1
Department of Civil, Environmental, Aerospace, and Materials Engineering
University of Palermo - Viale delle Scienze, Edificio 8, 90128 Palermo – Italy.
a
valeriocarollo@hotmail.it, bgiuseppe.giambanco@unipa.it, cantonino.spada@unipa.it
* corresponding author
Keywords: elastoplastic frames, finite elements, lumped hinge model.
Studies on the post-elastic behavior of materials and structures have permitted the updating of national
and international codes with particular reference to the design of structures in seismic zones. More
severe conditions are also imposed for those regions declared seismic and zones considered nonseismic in the past are today included into these categories. Most of buildings are therefore inadequate
with respect to the prescriptions of actual codes. In the worst case some structures have not been
designed to absorb horizontal actions. The evaluation of the vulnerability of existing structures to
seismic loads is therefore of extreme importance and can be done by performing nonlinear finite
element analyses. In literature, with respect to framed structures, three different finite element models
are utilized to describe the elastoplastic behavior of a beam/column element [1]: lumped models,
distributed nonlinearity models, fiber models. Lumped models consider the constitutive nonlinearity
concentrated at a section level of a frame element, usually employing nonlinear springs at the ends of
beam/column elements [2]. Distributed nonlinearity models average the nonlinearity over a finite
element by considering the possibility to form plastic hinges at different evaluation points of the
element and calculating weighted integrals of the section responses [3]. Fiber models subdivide a
section with a large number of finite elements and nonlinearity is related to the stress-strain
relationship of a single finite element [4].
The present work concentrates on the framework of lumped models. A new three-dimensional twonode Euler-Bernoulli beam/column finite element is proposed with the aim to run nonlinear analyses
on 3D RC framed structures. With respect to the existing lumped models where plastic hinges can
develop only at the two ends of an element, the proposed model is based on the possible formation of
a maximum of three hinges, in any way positioned inside the finite element. This choice reduces the
computational costs associated with the employment of remeshing procedures when a hinge doesn’t
form at the ends of an element.
The basic concept is related to the possibility to split the kinematical behavior of a frame element in
two different parts when a new hinge is formed at a certain point of the element. In other words, an
entire element can be thought as the connection of two or more sub-elements in each one of them a
displacement field can be defined and linked to the others by beams of an Heaviside function:
w  x   w1  x  H  a  x   w2  x  H  x  a 
(1)
where w  x  , w1  x  , and w2  x  are the displacement fields of the entire element, the first subelement and of second sub-elements respectively, H    is the Heaviside function of the enclosed
quantity, a identifies the position of the plastic hinge.
The nonlinear behavior of the hinge is defined in the framework of a thermo-dynamically consistent
elastoplastic theory. An elastic-perfect plastic constitutive behavior of the hinge is considered.
State equations and flow rules are derived from a Helmholtz free potential energy [5,6]. A polyhedric
activation domain is defined on the internal normal force and bending moments reference system to
control the onset and evolution of plasticity at the hinge. For a reinforced concrete cross-section with
69
generic dimensions and position and quantity of steel bars, the polyhedric domain is obtained
considering a finite number of points, related to the most important states of stress acting on the
section [7].
The hinge is assumed to have a plastic behavior only with respect to the section curvatures, while an
elastic behavior is considered in the axial direction. A non associative procedure is followed: on the
base of the internal axial force registered at a certain section of the element, the polyhedric domain is
cut by a plane at a constant value of axial force to obtain a 2D activation domain depending on the
bending moments only.
The elastoplastic frame element has been introduced in a finite element analysis program to run
nonlinear simulations on 2D and 3D framed structures. To this end state equations and flow rules have
been rewritten in a discrete manner. A classical elastic predictor phase is followed by a plastic
corrector phase in the case of activation of the inelastic phenomenon. The corrector phase is based on
the evaluation of correct bending moments by employing the closest point method, which permits to
satisfy the following loading-unloading conditions:
 pi  0; pi  0;  pi pi  0
(2)
where  pi is the generic plane which determines part of the limit surface of the elastic domain and pi
is the lagrangian multiplier associated.
The formation of one or more hinges inside a finite element modifies the distribution of stresses inside
that element and its stiffness matrix. As a consequence, the global stiffness matrix is continuously
modified at each plastic load step.
Two Newton-Raphson iterative loops have been implemented. The first one is within a single time
step and permits to reach the convergence inside a single element when plasticity is activated and a
hinge formed. The second one permits to reach the convergence of the overall structure as in a
standard finite element procedure. The iterative procedure is stopped when the structure becomes
labile and the first collapse mechanism is formed.
Numerical examples are finally reported to validate the efficiency of the proposed model. The
effectiveness of the model is obtained by comparing the results with those available in literature.
References
[1] F.F. Taucer, E. Spacano, F.C. Filippou, A fiber beam-column element for seismic response
analysis of reinforced concrete structures, Report No UCB/EERC-9117 Earthquake Engineering
Research Center, Berkley, 1991.
[2] M. Giberson, The response of nonlinear multi-story structures subjected to earthquake excitations,
Earthquake Engineering Research Laboratory, Pasadena, 1967.
[3] T. Takayanagi, W. Schnobrich, Non linear analysis of coupled wall systems, Earthquake
Engineering and Structural Dynamics, 7 (1979) 1-22.
[4] C.A. Zeris, S.A. Mahin, Behavior of reinforced structures subjected to biaxial excitation, Journal
of Structural Engineering, ASCE, 117(ST9) (1991) 2657-2673.
[5] A. Spada, G. Giambanco, P. Rizzo, Damage and plasticity at the interfaces in composite materials
and structures, Comput. Methods Appl. Mech. Engrg., 198 (2009) 3884–3901.
[6] J. Lemaitre, J.L. Chaboche, Mechanics of Solid Materials, Cambridge University Press, 1990.
[7] R. Park, T. Paulay, Reinforced concrete structures, John Wiley & Sons Inc Print on, 1975.
70
Interface poroelastic laws to model fluid-induced damage in oil wells
Carlo Callari1,a * and Valentina Fasano2,b
1
University of Molise, DiBT, via Duca degli Abruzzi, 86039 Termoli (CB), Italy
University of Rome "Tor Vergata", DICII, Via del Politecnico, 1, 00133 Roma
a
carlo.callari@unimol.it, bvalentina.fasano@uniroma2.it
*corresponding author
2
Keywords: poroelasticity, interface damage, oil/gas well integrity, underground CO2 storage.
In the life-cycle assessment of oil wells, a main issue is the debonding at the two interfaces of cement
sheath with steel casing and formation rock, respectively. As far as we know, in spite of this and of
other relevant effects of sheath cracking, our recent work [1] was the first application of damage
mechanics in the analysis of well integrity.
The formation of an annular gap at well interfaces can be the main reason for upward leakage of
fluids. In oil/gas production and in CO2 storage, such a leakage risk is an environmental issue of major
concern. The same fluid pressure often acts as the driving action for the upward propagation of well
interface opening, which is denoted as "micro-annulus" in petroleum engineering. It is then apparent
the importance of the coupling between fluid flow and mechanical damage in the problem at hand.
Hence, as a further step, we have defined a thermodynamic potential for porous interfaces, in terms of
displacement jump and fluid pressure, employing poroelastic coefficients which depend on a damage
internal variable. Differentiation of this potential leads to coupled laws expressing traction, fluid
content and energy release rate at the damaged interface. The evolution of damage is obtained from an
energy criterion which employs the damage resistance law proposed for the non-porous case in [2].
We investigate the model ability in qualitatively reproducing the main features of the response of
fluid-pressurized fractures in concrete [3]. Furthermore, we contrast the poroelastic damage model
with other laws proposed for interfaces subjected to fluid pressure [4,5].
The model is incorporated in a formulation of the axisymmetric problem of a well with displacement
discontinuities and fluid pressures at sheath-casing and sheath-rock interfaces. The problem solution,
obtained from analytical integration in space and numerical integration in time, is employed to
simulate the effects of cement shrinkage and of injected fluids on well integrity.
Acknowledgments. MIUR project on CO2 storage (PRIN 2010-2011, code 2010BFXRHS-004)
References
[1] C. Callari, V. Fasano, Damage analysis for wells in CO2 storage sites, in: G. Meschke et al.
(Eds.), EURO:TUN 2013, Aedificatio Publishers, 2013, pp. 437-448.
[2] Y. Mi, M.A. Crisfield, G.A.O. Davies, H.B. Hellweg, Progressive delamination using interface
elements, J. Compos. Mater. 32 (1998) 1246–1272.
[3] E. Brühwiler, V.E. Saouma, Water fracture interaction in concrete. Part II: Hydrostatic pressure in
cracks, ACI Mater. J., 92 (1995), 383-390.
[4] G. Bolzon, G. Cocchetti, Direct assessment of structural resistance against pressurized fracture,
Int. J. Numer. Anal. Met. 27 (2003), 353-378.
[5] G. Alfano, S. Marfia, E. Sacco, A cohesive damage-friction interface model accounting for water
pressure on crack propagation, Comput. Methods Appl. Mech. Eng. 196 (2006), 192-209.
71
Formulation of rate-dependent cohesive-zone models
Giulio Alfano1, a * and Marco Musto1,b
1
School of Engineering and Design, Brunel University, Kingston Lane, Uxbridge, UB8 3PH, UK
a
giulio.alfano@brunel.ac.uk, bmarco.musto@brunel.ac.uk
Keywords: fracture energy, rate-dependence, interface elements, viscoelasticity, viscoplasticity.
Rate dependent crack initiation and propagation has been the subject of extensive experimental,
theoretical, analytical and numerical studies. This is because in many problems of great engineering
interest the dependence of fracture processes on the loading rate cannot be ignored and often plays a
key role.
The complexity of the problem and the presence of numerous competing factors is evident from the
fact the fracture toughness may not show a monotonic trend with respect to crack speed, even when
the latter is small enough not to consider inertial effects. Furthermore, even when such trend is
monotonic, fracture toughness can increase with crack speed for some materials and decrease for
others [1, 2]. In some cases, the rate dependence of the crack growth can lead to unstable crack growth
or stick-slip crack propagation, the latter being a sequence of transitions from slow and stable crack
growth to very fast and unstable crack propagation and vice versa.
Theoretically, the problem can be studied in the framework of Griffith theory of fracture, by observing
that in the rate-dependent case the fracture energy , intended as the total energy dissipated per unit
of new formed crack area, is a function of crack speed , i.e.
. In this framework it can be
shown that crack speed instabilities may occur if is decreasing in part of its domain, see for example
[3, 4]. Within this theoretical framework, models are of a rather phenomenological nature, whereby
is determined experimentally.
Cohesive-zone models (CZMs) represent a widely used alternative method to analyse crack growth. If
they are developed within a damage-mechanics formulation a damage variable ranging between 0
and 1 can be introduced with the usual meaning. The natural extension of the above described
phenomenological approach is to assume a rate-dependent evolution law for in such a way that the
entire power dissipated is a non-linear function of .
1,2, … , , within the
An alternative approach consists of introducing other internal variables ,
CZM to capture different dissipation mechanisms, so that the entire dissipated power is a function not
only of but also of
[5]. The advantage of this approach is that the internal variables and their
evolution laws can provide a much richer description of the actual dissipation mechanisms which
occur at a micro-mechanical scale. This can lead to a model which is based more on first principles
and less on phenomenological assumptions
In this contribution attention will be focussed on rate-dependent CMZs developed within the
framework of thermodynamics with internal variables using this latter of the above described
approaches. In particular, a rate-dependent model will be presented in which (i) a rate-independent
evolution law is assumed for the damage variable
and (ii) additional internal variables are
associated with either viscoelastic or viscoplastic dissipation.
The introduction of a viscoelastic dissipation potential was considered in Ref. [5] where with the use
of a simple linear viscoelastic model with an exponential kernel and a single relaxation time good
agreement was obtained between numerical and experimental results for a DCB specimen with
metallic arms bonded through a rubber interface tested at different loading rates in displacementcontrol. Figure 1 shows that the correlation between experiments and prediction is rather good for
displacement rates between 0.1 mm/min to 100 mm/min, which is the expected captured range for
continuum problems if a viscoelastic model with one relaxation time only is used.
72
Figure 1: comparison between experimental and numerical load-displacement curves for the DCB specimen made of steel
arms with a rubber interface tested at different loading rates [5].
It is possible to show that the use of a viscoplastic dissipation potential can make the overall specific
dissipation a decreasing function of the applied rate of displacement jump on the interface. In a
structural problem, this implies that the total dissipation is a decreasing function of the crack speed,
and this leads to the prediction of crack velocity jumps and stick-slip crack propagation.
In this contribution, the above outlined general approach to the derivation of rate dependent crack
propagation will be described. Furthermore, recent developments will be presented, including the
formulation of more sophisticated viscoelastic models capturing a wider range of crack propagation
speeds. Finally, the ability of viscoplastic models to capture stick-slip crack propagation will be
demonstrated by presenting selected numerical results in good agreement with experimental data.
References
[1] R. Frassine, M. Rink, and A. Pavan. Viscoelastic effects on the interlaminar fracture toughness of
epoxy/carbon fibre. International J. Comp. Mat., 27:921–933, 1993.
[2] R. Frassine, M. Rink, and A. Pavan. Viscoelastic effects on the interlaminar fracture behaviour of
thermoplastic matrix composites: II. Rate and temperature dependence in unidirectional peek / carbonfibre laminates. J. Comp. Sci. Techn., 56:1253–1260, 1996.
[3] D. Maugis. Subcritical crack growth, surface energy, fracture toughness, stick-slip and
embrittlmenent. J. Mater. Sci., Vol. 20, pp. 3041-3073, 1985.
[4] T.W. Webb and E.C. Aifantis, Oscillatory fracture in polymeric materials. Int. J. Sol. Struct., Vol.
32, pp. 2725-2743, 1995.
[5] M. Musto and G. Alfano, A novel rate-dependent cohesive-zone model combining damage and
visco-elasticity. Comput. Struct., Vol. 118, pp. 126-133, 2013.
73
Porous shape memory alloys: a micromechanical analysis
V. Sepe1, a * , F. Auricchio2,b, S. Marfia1, c, E. Sacco1,d
1
DiCeM, Department of Civil and Mechanical Engineering,
University of Cassino and Southern Lazio, Italy
2
DICAr, Department of Civil Engineering and Architecture, University of Pavia, Italy
a
v.sepe@unicas.it, b auricchio@unipv.it, c marfia@unicas.it, d sacco@unicas.it
Keywords: Shape Memory Alloys, Porous Shape Memory Alloys, Nonuniform Transformation Field
Analysis.
Shape Memory Alloys (SMA) are characterized by a very special behavior due to their capability to
undergo reversible changes of the crystallographic structure, depending on the temperature and on the
stress state. These changes can be interpreted as reversible martensitic transformations between a
crystallographic more-ordered parent phase, the austenite, and a crystallographic less-ordered product
phase, the martensite.
Thanks to their unique properties over the last decades SMA have been used for a large number of
applications in several engineering fields, from aerospace to medical device industries.
Recently, driven by biomedical applications, a great interest has arisen concerning a particular class of
SMA: the porous SMA. The possibility of producing SMA in porous form has opened new fields of
applications owing to their low-weight with high energy dissipation properties.
In the biomedical field, thanks to their high biocompatibility and their capacity to exhibit high
strength, NiTi foams have been tested as bone implant materials, effectively exhibiting a considerable
amount of bone ingrowth.
In the last years, applications of porous SMA in the field of Civil and Mechanical Engineering have
also been considered. The potential applications of porous SMA exploit their ability to carry
significant loads and their high energy absorption capability. In fact, the porous SMA show a higher
specific damping capacity under dynamic loading conditions with respect to the dense SMA, because
the pores facilitate an additional absorption of the impact energy.
In order to correctly reproduce the behavior of the porous SMA the development of accurate models
describing their properties is needed.
The porous SMA material can be treated as a composite with SMA as the matrix and pores as the
inclusions. Several works available in literature, for instance [1],[2], developed micromechanical
averaging techniques in order to derive the mechanical response of porous SMA.
Indeed, different micromechanical and homogenization techniques can be applied to model porous
SMA, such as the Eshelby dilute inclusion technique, the Mori-Tanaka method or the self-consistent
one. An interesting approach that has been adopted to study the behavior of porous materials is based
on the assumption of having a periodic distribution of pores. In this case, the problem can be solved
by using a computational homogenization technique based, for instance, on nonlinear finite element
analyses of a single unit cell with suitable boundary conditions.
The behavior of porous SMA under cyclic loading conditions has been studied in [3], where the
constitutive law has been enhanced to account for the development of permanent inelasticity due to
stress concentrations in the porous microstructure.
The aim of the present contribution is to propose a micromechanical study of porous SMA. In
particular, the response of porous SMA is derived by performing:

the nonlinear finite element micromechanical analysis for the typical repetitive unit cell,
considering suitable periodicity conditions;
74

the nonuniform TFA homogenization technique based on piecewise interpolation functions of
the inelastic strain field proposed by Sepe et al. [4].
According to the latter homogenization procedure, a unit cell with SMA as the matrix and pores as the
inclusions is considered and divided into subsets. In each subset a nonuniform distribution of the
inelastic strain, which accounts for all the nonlinear effects that arise in the SMA matrix, is adopted.
In particular, the inelastic strain in each subset is given as a linear combination of selected analytical
functions depending on the spatial variable. The coefficients of the linear combination are determined
solving the evolutive problem.
The constitutive model proposed in [5][6] and able to reproduce the key features of the Shape
Memory Alloys is adopted in both types of analysis in order to simulate the behavior of the porous
SMA.
The constitutive behavior and the dissipation energy of the porous SMA will be investigated for
different values of porosity and for different shapes of the pores.
Numerical applications will be developed in order to test the ability of the presented procedures to
well capture the overall behavior of the special composite, correctly reproducing the key features of
the Shape Memory Alloys: the pseudoelastic and the shape memory effects.
References
[1] M.A. Qidwai, P.B Entchev, D.C. Lagoudas, V.G. De Giorgi, Modeling of the thermomechanical
behavior of porous shape memory alloys. Int. J. of Solids Struct. 38 (48-49) (2001) 8653–8671.
[2] P.B. Entchev, D.C. Lagoudas, Modeling porous shape memory alloys using micromechanical
averaging techniques, Mechanics of Materials. 34 (1) (2002) 1–24.
[3] M. Panico, L.C. Brinson, Computational modeling of porous shape memory alloys, Int. J. Solids
Struct. 45 (2008) 5613–5626.
[4] V. Sepe., S. Marfia, E. Sacco, A nonuniform TFA homogenization technique based on piecewise
interpolation functions of the inelastic field, Int. J. Solids Struct. 50 (2013) 725–742.
[5] F. Auricchio, L. Petrini, A three-dimensional model describing stress-temperature induced solid
phase transformations: solution algorithm and boundary value problems, Int. J. for Numer. Meth. Eng.
61 (2004) 807–836.
[6] V. Evangelista, S. Marfia and E. Sacco, Phenomenological 3D and 1D consistent models for
shape memory alloy materials, Comput. Mech. 44 (2009) 405-421.
75
A corotational tetrahedral element with rotational degrees of freedom
for large-displacement analysis of inelastic structures
Paolo Bisegna*, Federica Caselli, Edoardo Artioli, and Nicola A. Nodargi
Department of Civil Engineering and Computer Science, University of Rome “Tor Vergata”,
Via del Politecnico 1, 00133 Rome, Italy
bisegna@uniroma2.it
Keywords: Tetrahedral finite element, corotational formulation, large displacements and rotations,
hybrid/mixed formulation, plasticity, shape-memory alloys.
Compared to hexahedral elements, tetrahedral elements are especially attractive in practical
engineering applications involving complex geometries, since they allow for a very straightforward
mesh generation, whereas automatic mesh generation is often not feasible for the former [1]. The aim
of the present paper is the development of a new tetrahedral finite element accounting for material and
geometric nonlinearities, in the framework of the corotational formulation.
The corotational approach is based on the idea of separating rigid body motions from purely
deformational ones [2]. It is especially attractive for problems involving large displacements and small
strains. In fact in those cases existing high-performance linear elements can be reused as core
elements in the geometrically nonlinear context, after large rigid body motions have been filtered out.
In this work a polar decomposition based corotational formulation is exploited [3,4] and original
closed-form formulas are derived for the efficient computation of the nodal residual vector and of the
consistent tangent stiffness tensor.
A four-node solid tetrahedron with three translational and three rotational degrees of freedom (DOFs)
per node is adopted as core element. The rationale is that the accuracy of those elements is
intermediate between that of the linear and quadratic elements with translations only, yet they
contribute to a much smaller bandwidth and thus solution time compared to the quadratic elements
[5]. Existing four-node tetrahedrons with six DOFs per node include TET4RX [6], HT4R [5],
RGNTet4 [1]. The latter exhibits a somewhat stiff behaviour, whereas the first is not frame invariant.
Hence, HT4R is a natural candidate as core element. The derivation of HT4R is based on a modified
Hellinger-Reissner functional, that treats the rotation and the skew symmetric stress as independent
fields to formulate a stabilization scheme. Unfortunately, approaches based on Hellinger-Reissner
functional may not be efficient in a nonlinear material framework, usually involving the direct strainstress relationship. An enhancement of HT4R is here proposed, based on a modified Hu-Washizu
functional:
Ϝ(u, ε, σ, ω, τ) = < ψ(ε) - σ·ε + σ·Du + τ·(Lu - ω) - ||τ||2/(2μγ) >.
(1)
where <·> is the integral operator over the element domain, ε is the strain tensor, ψ is the free energy
density, σ and τ are the symmetric and skew-symmetric parts of the stress tensor, respectively, u is the
displacement vector, ω is the independently assumed rotation vector, μ is a typical stiffness modulus,
γ is an arbitrary positive non-dimensional parameter. Moreover, D and L are the differential operators
for deriving, respectively, the strain and rotation from the displacement field. As in the original
formulation, the corner rotations are introduced by transformation of the mid-side translational DOFs
of a ten-node tetrahedron, and the same stress modes are adopted, which were reduced to minimum
without sacrificing the frame invariance and proper rank of the element [5]. Here two different choices
are made for the assumed strain field: either the same interpolation as the stress field is assumed,
amounting to fitting the displacement-derived strain field into the assumed strain field by means of a
least squares procedure, or a piece-wise constant discontinuous strain field is adopted, related to the
quadrature rule.
76
Besides elastic case studies, simulations involving plastic or shape-memory-alloy materials are
presented. In particular, stent structures used in biomedical engineering are usually designed to
significantly reduce their diameter during the insertion into a catheter. Thereby large rotations and
displacements, usually combined with small to moderate strains, occur. Dissipation pseudo-potential,
along with chemical and transformation-strain energies in the case of shape-memory alloys, are
accordingly accounted for in formulation (1), and a dissipation-based material state-update algorithm
is adopted.
References
[1] R. Tian, H. Matsubara, G. Yagawa, Advanced 4-node tetrahedrons, Int. J. Numer. Methods Eng.
68(2006):1209-1231.
[2] B. Nour-Omid, C. C. Rankin, Finite rotation analysis and consistent linearization using projectors,
Comput. Meth. Appl. Mech. Eng. 93(1991):353-384.
[3] G. F. Moita, M. A. Crisfield, A finite element formulation for 3-D continua using the co-rotational
technique, Int. J. Numer. Methods Eng. 39(1996):3775-3792.
[4] F. Caselli, P. Bisegna, Polar decomposition based corotational framework for triangular shell
elements with distributed loads, Int. J. Numer. Methods Eng. 95(2013):499-528.
[5] K. Y. Sze, Y. S. Pan, Hybrid stress tetrahedral elements with Allman’s rotational D.O.F.s, Int. J.
Numer. Methods Eng. 48(2000):1055-1070.
[6] T. P. Pawlak, S. M. Yunus, R. D. Cook, Solid elements with rotational degrees of freedom: Part II
- tetrahedron elements, Int. J. Numer. Methods Eng. 31(1991):593-610.
77
A consistency study of cohesive zone models for mixed-mode debonding
problems
Rossana Dimitri1,a *, Marco Trullo1,b, Laura De Lorenzis2,c and Giorgio Zavarise1,d
1
Dipartimento di Ingegneria dell'Innovazione, Università del Salento,
Via per Monteroni, 73100, Lecce
2
Institut für Angewandte Mechanik, Technische Universität Braunschweig,
Bienroder Weg 87, Campus Nord, 38106 Braunschweig, Germany
a
rossana.dimitri@unisalento.it, bmarco.trullo@unisalento.it, cl.delorenzis@tu-braunschweig.de,
d
giorgio.zavarise@unisalento.it
Keywords: Cohesive zone modeling, Debonding, Mixed-mode fracture.
Cohesive zone models (CZMs) are commonly used to describe mixed-mode failure and debonding
processes at material interfaces or within quasi-brittle materials. These models consist in non-linear
relationships between tractions and relative displacements across the crack. Although a large number
of CZMs have been proposed, and despite the extensive related literature, little attention has been
devoted to ensuring the consistency of these models for mixed-mode conditions, primarily in a
thermodynamical sense. For many of these models, traction-separation laws have been directly
postulated in an ad hoc manner, which may lead to unphysical dissipation behavior.
A consistency check was performed by van den Bosch et al. [1] for the exponential model by Xu and
Needleman [2]. Thereby, an adjusted non-potential-based exponential model was also proposed to
correct the unphysical behavioral features of the original model in the description of mixed-mode
decohesion. Although non-potential-based models have been adopted for many practical applications,
this class of models is not guaranteed to be thermodynamically consistent [3]. Beside
thermodynamical consistency, an important requirement of CZMs is to allow for different values of
the fracture energy in the normal and tangential directions, as measured experimentally.
In the first part of this contribution, two widely used mixed-mode CZMs [4,5] are examined. The
consistency of their predictions in both stress and energy terms is checked. A parametric analysis on
the effect of the coupling parameters on stress distributions and energy dissipation is performed in
order to evaluate physical inconsistencies such as local abnormalities in the coupled elastic or
softening mechanical response of the interface and incomplete dissipation of the fracture energy
during decohesion. The path-dependence of the mixed-mode debonding work of separation (W) and
of the failure domains are additionally evaluated. W is given by
W=   p N (g N ,g T )dg N +   pT (g N ,g T )dg T
(1)
where Γ is the separation path, pN and pT are the normal and tangential tractions, gN and gT are the
normal and tangential relative displacements across the crack. The first term in Eq. 1 is the work done
by the normal tractions (WN), while the second term is the work done by the tangential tractions (WT).
Analytical predictions are also compared with results from numerical finite element models, where the
interface is described with zero-thickness contact elements. A node-to-segment strategy as employed
in [6] is here adopted, which incorporates decohesion and contact within a unified framework. Three
case studies are analyzed for the numerical prediction of mixed-mode interface debonding: a simple
patch test, a bimaterial peel test under mixed-mode loading conditions, and the standard mixed-mode
bending test (MMB). Figure 1 illustrates sample analytical and numerical results in terms of W, WT
and WN, obtained for a patch test with the models in [4,5]. It is evident that the total work of
separation does not vary monotonically, which reveals an energetic inconsistency.
78
120
120
120
100
100
100
WN
WT
W
WN_num
WT_num
W_num
60
40
20
0
80
WN
WT
W
WN_num
WT_num
W_num
60
40
WN, WT, W [Jm-2]
80
WN, WT, W [Jm-2]
WN, WT, W [Jm-2]
In the second part of the paper, a new thermodynamically consistent mixed-mode CZM is proposed
based on a modification of the model in [1]. Based on a predefined Helmoltz energy, the interface
model is derived by applying the Coleman and Noll procedure, in accordance with the second law of
thermodynamics, whereby the inelastic nature of the decohesion process is accounted for by means of
damage variables. The model accounts monolithically for loading and unloading conditions, as well as
for decohesion and contact. Its performance is demonstrated through suitable examples.
0.2
0.4
gN/gN,u
0.6
0.8
1
0
WN
WT
W
WN_num
WT_num
W_num
60
40
20
20
0
80
0
0
0.2
0.4
gN/gN,u
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
gN/gN,u
(a)
(b)
(c)
Figure 1: Work of separation for bilinear models by Högberg (a), and by Camanho et al. for a P-L criterion (b), or a B-K
criterion (c). N=T=100 N/m.
Acknowledgements: The authors have received funding for this research from the European Research
Council under the European Union’s Seventh Framework Programme (FP7/2007-2013), ERC Starting
Researcher Grant “INTERFACES”, Grant agreement n° 279439.
References
[1] M.J. van den Bosch, P.J.G. Schreurs, M.G.D. Geers. An improved description of the exponential
Xu and Needleman cohesive zone law for mixed-mode decohesion, Eng. Fract. Mech. 73 (2006)
1220-1234.
[2] X.P. Xu, A. Needleman. Void nucleation by inclusion debonding in a crystal matrix, Model.
Simul. Mater. Sc. 1 (1993) 111-132.
[3] J. Mosler, I. Scheider. A thermodynamically and variationally consistent class of damage-type
cohesive models. J. Mech. Phys. Solids. 59 (2011) 1647-1668.
[4] J.L. Högberg. Mixed mode cohesive law, Int. J. Fract. 141 (2006) 549-559.
[5] P.P. Camanho, C.G. Dàvila, M.F. De Moura. Numerical simulation of mixed-mode progressive
delamination in composite materials, J. Compos. Mater. 37(16) (2003) 1415-1438.
[6] P. Wriggers, G. Zavarise, T.I. Zohdi. A computational study of interfacial debonding damage in
fibrous composite materials, Comput. Mater. Sci. 12 (1998) 39-56.
79
A multilevel finite element approach for piezoelectric textiles made of
polymeric nanofibers
Claudio Maruccio1, a *, Laura De Lorenzis2, b
1
Department of Innovation Engineering, University of Salento, via Monteroni, Lecce, Italy
2
Institut für Angewandte Mechanik, Technische Universität Braunschweig, Germany
a
claudio.maruccio@unisalento.it, bl.delorenzis@tu-braunschweig.de
Keywords: Electromechanical Coupling, FE2, Multiphysics Modeling, Multiscale Modeling,
Piezoelectricity, Shell elements.
Piezoelectric effects are exploited in an increasing number of micro- and nano-electro-mechanical
systems. In particular, energy harvesting devices convert ambient energy (i.e. mechanical pressure)
into electrical energy and their study is a very important and challenging field of research. The
development of novel piezoelectric nano-generators [1] promises to exert a substantial impact in
several industrial fields, such as the automotive, aerospace, and medical fields.
In this paper, the attention is focused on piezoelectric textiles made of arrays of polymeric nano-fibers
[2]. Due to the importance of computational modeling to understand the influence that micro-scale
geometry and constitutive variables have on the macroscopic behavior, a computational
homogenization procedure is developed and implemented. The method is based on a multilevel finite
element approach (FE2) [3,4], whereby the macroscopic structure is discretised using the finite
element method and a discretised microscale reference volume element (RVE) is assigned to each
quadrature point at the macro-scale. The geometry of the RVE is based on the microstructural
properties of the material under consideration and consists in piezoelectric polymer fibers subjected to
electromechanical contact constraints. The contact element contributions to the virtual work equations
are included using the penalty method and introducing suitable electric, mechanical and coupling
potentials. A contact smoothing approach based on Bézier patches for the master surface [5] is
extended to the multiphysics framework providing an improved continuity of the parameterization.
For each macroscopic integration point, the macro-scale shell kinematics is used to formulate suitable
boundary conditions for the corresponding RVE. Once the boundary value problem for the RVE is
fully defined, from the finite element analysis at the microlevel, Figure 1, the resulting generalized
stress and strain vectors in the RVE are computed.
a)
b)
Figure 1 a) RVE mesh b) Minimum and maximum stress x corresponding to the vertical color scales are -1.6 * 10−2 to
4.3 * 10−2 [nN/nm2]
80
By averaging the RVE results over the unit cell volume and after thickness integration, the generalized
force vector of the shell is extracted and returned to the macroscopic integration point as a local
macroscopic stress resultant. At this point the local macroscopic consistent tangent is determined
performing a sensitivity analysis at the RVE level. When the analysis of all RVEs is completed, the
generalized force vector is available at each macroscopic integration point and the internal
macroscopic forces are calculated. If these forces are in balance with the external load according to
the convergence criterion, the next time step can start, otherwise the iterative procedure is continued.
Within this approach the average response resulting from the homogenization procedure at the microlevel is directly used as a multiphysics constitutive model for each quadrature point at the macroscale.
This contribution outlines the theory and its numerical implementation, and presents the main
numerical results.
References
[1] Y. Gao, Z.L. Wang, Electrostatic potential in a bent piezoelectric nanowire. The fundamental
theory of nanogenerator and nanopiezotronics, Nano Lett. 7 (2007) 2499-2505.
[2] L. Persano, C. Dagdeviren, Y. Su, Y. Zhang, S. Girardo, D. Pisignano, Y. Huang, J.A. Rogers,
High performance piezoelectric devices based on aligned arrays of nanofibers of PVDF, Nature
Communications. 1633 (2013) 4.
[3] J. Schroeder, M. Keip, Two-scale homogenization of electromechanically coupled boundary value
problems - Consistent linearization and applications, Computational Mechanics. 50 (2012) 229-244.
[4] E.W. Coenen, V.G. Kouznetsova, M.G.D. Geers, Computational homogenization for
heterogeneous thin sheet, International Journal for Numerical Methods in Engineering. 83 (2010)
1180-1205.
[5] J. Lengiewicz, J. Korelc, S. Stupkiewicz, Automation of finite element formulations for large
deformation contact problems, International Journal for Numerical Methods in Engineering. 85 (2011)
1252-1279.
81
Computational issues on multiscale FE analysis
Francesco Parrinelloa and Guido Borinob
1
Università di Palermo, Dipartimento di Ingegneria Civile, Ambientale, Aerospaziale, dei Materiali,
Viale delle Scienze Ed.8, 90128 Palermo, Italy
a
francesco.parrinello@unipa.it, bguido.borino@unipa.it
Keywords: Multiscale, Static condensation, Master slave, Periodic boundary conditions.
Multiscale analysis typically defines the complex constitutive behavior of heterogeneous material, by
an homogenization technique of the mechanical response of a material representative volume element,
which is discretized in a microscale finite element model and subjected to imposed macro strain.
The homogenized stress and tangent moduli, obtained at the microscale, can be employed in a point of
a macroscopic discretized structure, in a multilevel finite element framework, as proposed in [1].
The focus point of the multiscale analysis is the correct formulation of the scale transition laws, which
define, as a function of the assumed macroscopic deformation, the imposed miscroscale deformation
in terms of boundary displacements or boundary tractions. Moreover, the averaging theorems allow to
define the homogenized stress and, by an incremental variation approach, also the
homogenized tangent moduli.
In the context of deformation-driven multiscale analysis, three types of boundary conditions are
known in literature to verify the averaging theorems proposed by Hill [2]: linear displacement,
constant traction and periodic displacement and antiperiodic traction. The latter approach is generally
recognized to produce better results, with respect to the other two approach, for periodic
microstructures.
The three boundary conditions are applied by the Lagrangian multiplier method, in the small strain
hypothesis, by Miehe and Koch in [3], where algorithms and matrix representation of averaged stress
and tangent moduli are derived. The same problem is analyzed in finite deformation regime in [4].
The three boundary conditions are applied by penalty approach in [5], in finite deformation context,
and the averaged tangent moduli are derived by a perturbation procedure, instead of the condensation
one.
The present paper investigates the multiscale analysis techniques from the computational point of
view and proposes an original formulation, based on the master slave elimination method, for the
application of the periodic displacement and antiperiodic traction boundary condition.
The periodic displacement boundary condition has to be applied between pairs of nodes on the
boundary of microscale model and it is a non-homogenous multifreedom constrain. The boundary
condition is applied by master slave elimination method, which produces an exact solution and
reduces the number of unknowns. On the contrary, the penalty method produces an approximated
solution and the Lagrangian multiplier method increases the number of unknowns.
The paper develops the computational aspects related to the scale transition laws. The macro stress
tensor is obtained as average of the micro stress and the matrix of tangent moduli is defined as
symmetric, positive defined and it is evaluated exactly by the static condensation approach [6]. The
model stiffness matrix is expanded to the slave degrees of freedom, which are released, and the matrix
is statically condensed to the square matrix related to the slave degrees of freedom. The static
condensation is performed simply by stopping the triangular decomposition to the first released slave
degree of freedom.
Some numerical multiscale analysis are performed and the results are proposed.
82
References
[1] R.J.M. Smit, W.A.M. Brekelmans, H.E.H. Meijer, Prediction of the mechanical behavior of
nonlinear heterogeneous systems by multi-level finite element modeling, Comput. Methods Appl.
Mech. Engrg. 155 (1998) 181–192.
[2] R. Hill, On constitutive macro–variables for heterogeneous solids at finite strain, Proc R Soc
London A 326 (1972) 131–147
[3] C. Miehe, A. Koch, Computational micro-to-macro transitions of discretized microstructures
undergoing small strains, Arch. Appl. Mech. 72 (2002) 300–317.
[4] C. Miehe, Computational micro-to-macro transitions for discretized microstructures of
heterogeneous materials at finite strains based on the minimization of averaged incremental energy,
Comput. Methods Appl. Mech. Engrg. 192 (2003) 559–591.
[5] I. Temizer, P. Wriggers, On the computation of the macroscopic tangent for multiscale volumetric
homogenization problems, Comput. Methods Appl. Mech. Engrg. 198 (2008) 495–510.
[6] E. L. Wilson, The static condensation algorithm, Short Comunication on Int. Jou. Num. Meth.
Eng. 8 (1974) 198–203.
83
An efficient Bouc & Wen approach for seismic analysis of masonry
tower
Luca Facchini1,a , Michele Betti1,b *
1
Department of Civil and Environmental Engineering, University of Florence, Italy
a
luca.facchini@unifi.it, b mbetti@dicea.unifi.it
Keywords: Bouc & Wen model, Masonry towers, Nonlinear dynamics, Statistical linearization.
The assessment of existing masonry towers under exceptional loads, such as earthquake loads,
requires reliable, expedite and efficient methods of analysis. These approaches should take into
account both the randomness that affect the masonry properties (in some cases the distribution of
elastic parameters too) and the specific non-linear behavior (f.i. the small tensile strength).
As an alternative to classical finite element technique approaches, in recent years several expeditious
methods have been proposed to analyze the structural response of such structural systems. An
approach based on the modal reduction (MO-RE) to analyze the response of slender masonry walls
under turbulent wind was proposed in [1]: the material was assumed as no tensile resistant (NTR), but
the mechanical properties were assumed as deterministic. To introduce randomness in material
distribution an approach based on a Galerkin discretization was proposed in [2]: the material
properties were assumed as a stochastic field. Other possible expeditious approaches may be based on
perturbation methods, however the results of some preliminary analysis, seem to show that the use of
a perturbation method does not allow, when the seismic action is assumed as a time history, to keep
into account correctly the cracking and crushing phenomena that occur in masonry.
Based on this background, and considering the need of simplified but effective methods to assess the
seismic response of slender masonry tower, the paper aims to propose an efficient approach for
seismic assessment of masonry towers assuming the material properties as a stochastic field. As a
prototype of masonry towers a cantilever beam is analyzed assuming that the first modal shape
governs the structural motion. With this hypothesis a non-linear hysteretic Bouc & Wen model [3] is
employed to reproduce the system response which is eventually linearized to evaluate its bounds. The
results of the simplified approach are compared with the results of FE model to show the effectiveness
of the method.
References
[1] M. Betti, P. Biagini, L. Facchini, Comparison among different techniques for reliability
assessment of no-tensile structures under turbulent wind, Proceedings of the Fourth European-African
Conference on Wind Engineering, EACWE4 2005 (2005).
[2] L. Facchini, M. Betti, P. Biagini and A. Vignoli, RBF – Galerkin approach for the dynamics of
simple disordered masonry structures, Atti del XVII Congresso Nazionale AIMETA di Meccanica
Teorica ed Applicata, AIMETA 2005 (2005).
[3] Y. K. Wen, Method for random vibration of hysteretic systems, ASCE Journal of the Engineering
Mechanics Division, 103(2), 249-263 (1976).
84
Analysis of masonry arches: a NURBS based simple applicative
program
Andrea Chiozzi1, a *, Marcello Malagù2,b, Antonio Tralli1,c , Antonio Cazzani3,d
1
University of Ferrara – Department of Engineering, Via Saragat 1 – 44100 Ferrara Italy
Delft University of Technology, Faculty of Civil Engineering and Geosciences, Stevinweg 1,
2628CN, Delft, The Netherlands
1
University of Cagliari – Faculty of Architecture, Via Marengo 2 – 09123 Cagliari Italy
a
andrea.chiozzi@unife.it, b marcello.malagu@unife.it, c tra@unife.it, d antonio.cazzani@unica.it
* corresponding author
2
Keywords: NURBS, isogeometric analysis, limit analysis, masonry arches.
In this work a practical tool for the analysis of masonry arches, which is based on a combination of
isogeometric analysis and of a suitable implementation of the classic safe theorem proposed by
Heyman [1] is presented. The computer code, developed in the popular Matlab language, has been
designed in such a way that, by using it, professionals dealing with restoration or structural
rehabilitation of historical constructions can easily produce estimates of the carrying capacity of
curved members, especially, but not exclusively, arches, with a sound theoretical background.
Moreover, the developed code is also devised to handle the presence of FRP (fiber-reinforced
polymers) strips, thus allowing design of properly dimensioned reinforcement and its verification
according to recently developed building codes.
The proposed software interacts with CAD (Computer Aided Design) design softwares, which are
widely used by the community of professionals, in order to import NURBS representation of arch
geometries, which become the basis for subsequent structural analysis. As is common knowledge,
description and computation of geometries in commercial CAD packages are based on B-splines and
NURBS (Non Rational Uniform B-Splines) basis functions. In particular, when compared to standard
B-splines, NURBS basis function have the great advantage in representing the exact geometry of a
wide set of curves and surfaces such as circles, ellipses and parabolas [2].
Starting from a NURBS representation of an arch with arbitrary shape, it is possible to perform at first
a linear elastic analysis through isogeometric analysis. Isogeometric analysis is a recently developed
computational approach that offers the possibility of integrating finite element analysis (FEA) and
CAD geometry representation: the basic idea is to use the same NURBS representations both for the
geometry and FEM solution space [3, 4].
Besides elastic analysis, the proposed software allows to perform a limit analysis of the masonry arch.
As shown by Livesley [5], the equilibrium formulation can be applied to masonry arches leading to
lower bound solutions. This approach involve the discretization of an arch into a number of rigid
blocks and the solution of a linear programming problem under the classic Heyman hypothesis. In the
proposed software, the NURBS geometric representation of the arch is used to easily obtain a
discretization of the arch into rigid blocks. An interior-point algorithm is used to solve the
optimization problem. Livesley original approach has been extended in order to include finite strength
in compression for masonry, through an iterative numerical procedure, and FRP reinforcement
through FRP strips at the intradox and/or at the extradox of the arch so that FRP delamination is taken
into account as additional constraints to the linear programming problem.
Finally, some real-world applications, which the proposed software tool can deal with, are presented
and discussed.
85
References
[1] L. Piegl and W. Tiller, The NURBS book, second ed., Springer-Verlag, Berlin, 1997.
[2] J. Heyman, “The stone skeleton”, Int. J. Solids and Structures, 2 (1966) 249-79
[3] T.J.R. Hughes, J.A. Cottrell and Y. Bazilevs, “Isogeometric analysis: CAD, finite elements,
NURBS exact geometry and mesh refinement”, Computer Methods in Applied Mechanical and
Engineering, 194 (2005) 4135-95.
[4] C. de Falco, A. Reali and R. Vazquez, “GeoPDEs: a research tool for Isogeometric Analysis of
PDEs”, Adv. Eng. Sotw., 42(2011) 1020-1034
[5] R.K. Livesley, “Limit analysis of structures formed from rigid blocks”, Int. J. for Num. Meth.
Eng.12 (1978), 1853-1871.
86
Isogeometric collocation for large-deformation frictional contact
Laura De Lorenzis, Roland Kruse, Nhon Nguyen-Thanh
Institut für Angewandte Mechanik, Technische Universität Braunschweig, Germany
l.delorenzis@tu-braunschweig.de, r.kruse@tu-bs.de, n.nguyen-thanh@tu-braunschweig.de
Keywords: isogeometric analysis, isogeometric collocation, large-deformation contact, friction
Within the framework of NURBS-based isogeometric analysis, collocation methods have been
recently proposed and their accuracy and efficiency demonstrated for elastostatics and explicit
dynamics [1-3]. Recently, further progress has been achieved in the application of isogeometric
collocation to frictionless contact problems [4]. As the contact constraints were enforced as special
Neumann boundary conditions (BCs), enhancements were needed in the general imposition of these
BCs. Thereby, an enhanced collocation scheme was proposed, whereby the Neumann BCs are
imposed considering both boundary and bulk contributions. This was shown to significantly improve
the accuracy over the basic collocation method and to achieve an accuracy comparable to that of the
Galerkin method, especially for discretizations of order larger than 3, while completely eliminating
quadrature.
For contact problems between deformable bodies, the two-half-pass formulation proposed in [5] and
recently extended to friction [6] seems the most natural algorithm in the collocation framework. A
striking result obtained from this formulation in [4] was that, despite the pointwise evaluation of the
contact residual contributions at the surface collocation points, the formulation passes the contact
patch test to machine precision. Such a result has never been obtained in the Galerkin setting. The
proposed frictionless contact formulation in the collocation setting yields results of very good quality
for regular solutions and uniform meshes. In situations with highly non-uniform meshes, as for the
more general Neumann cases, the basic collocation method fails whereas the enhanced collocation
scheme proposed in [4] is an effective remedy which restores accuracy of the results and robustness of
the iterative procedure.
The investigation in [4] was limited to the frictionless setting. Moreover, although the contact
formulation was fully non-linear, the continuum was treated within the framework of linear elasticity,
so that only small deformation cases could be tackled. Herein, two main advancements are reported:
1) The continuum formulation is extended to the large-deformation elasticity case. A few
different constitutive laws are considered, namely St. Venant-Kirchhoff, compressible neoHooke, and a different neo-Hooke model version with volumetric-deviatoric decoupling.
Special attention is given to the consistent linearization of the non-linear problem, which
presents peculiar features in the collocation framework;
2) The contact formulation is extended to the frictional setting. A small-deformation case is first
solved illustrating the comparison to an analytical solution. Subsequently a large-deformation
problem is presented.
Isogeometric collocation is shown to represent a promising framework for the solution of non-linear
frictional contact problems.
References
[1] Auricchio, F., Beirão da Veiga, L., Hughes, T.J.R., Reali, A., and Sangalli, G. (2010).
Isogeometric Collocation Methods. Math. Mod. Meth. Appl. Sci., 20(11): 2075–2107.
[2] Auricchio, F., Beirão da Veiga, L., Hughes, T.J.R., Reali, A., and Sangalli, G. (2012).
Isogeometric collocation for elastostatics and explicit dynamics. Comp. Meth. Appl. Mech. Engrg.,
249–252: 2–14.
87
[3] Schillinger, D., Evans, J.A., Reali, A., Scott, M.A., and Hughes, T.J.R. (2013). Isogeometric
Collocation: Cost Comparison with Galerkin Methods and Extension to Adaptive Hierarchical
NURBS Discretizations, Comp. Meth. Appl. Mech. Engrg., accepted for publication, ICES Report 1303, University of Texas at Austin, February 6, 2013.
[4] De Lorenzis, L., Evans, J.A., Hughes, T.J.R., Reali, A. (submitted). Isogeometric collocation:
Neumann boundary conditions and contact. Submitted version available online as ICES Report 14-06.
[5] Sauer, R.A., De Lorenzis, L. (2013). A computational contact formulation based on surface
potentials. Comp. Meth. Appl. Mech. Engrg., 253:369-395.
[6] Sauer R., De Lorenzis L. (revision submitted). An unbiased computational contact formulation for
3D friction. Int. J. Num. Meth. Eng.
88
An adaptive multiscale approach for the failure analysis of fiberreinforced composite materials
Domenico Bruno1,a, Fabrizio Greco1,b*, Lorenzo Leonetti1,c,
Stefania Lo Feudo,d, Paolo Lonetti1,e
1Department of Civil Engineering, University of Calabria, Via P. Bucci, Cubo 39B, Rende (CS), Italy
ad.bruno@unical.it, af.greco@unical.it, clorenzo.leonetti@unical.it, dstefania.lofeudo@unical.it,
epaolo.lonetti@unical.it
Keywords: multiscale methods, composite materials, transverse cracking, micromechanics, failure
analysis.
Composite materials may be affected by different kinds of damage phenomena, which usually start
from manufacturing-induced and other pre-existing defects. Fiber-reinforced composites, especially
for laminate configurations, experience both intralaminar mechanisms, such as matrix cracking, fiber
splitting and fiber/matrix interface debonding, and interlaminar mechanisms, such as delamination
(see [1] for instance). These damage mechanisms, which take place at the microscopic level, strongly
influence the overall structural behavior of composite materials, leading to a highly nonlinear
structural response associated with a progressive loss in stiffness up to failure [2]. Therefore, a proper
analysis of damage mechanisms in composites would require a complete description of their
microstructural evolution, resulting in fully microscopic problems, whose numerical solution needs a
huge computational effort; as a consequence, simplified modeling is preferred when performing
failure analyses in composite materials.
In the framework of continuum damage representation, different macroscopic nonlinear constitutive
laws have been proposed, involving both scalar and tensorial damage variables, the latter being more
suitable when dealing with damage phenomena in heterogeneous media. Micromechanical approaches
have been extensively used as a powerful tool for developing microscopically-informed damage laws
(see, for instance, [3]). The accuracy of such approaches (mostly based on first-order homogenization
schemes) depends on the assumptions of both perfect periodicity and macroscopic uniformity, which
usually cease to hold in the damaging regions, where high local field gradients occur. Indeed,
softening behaviors cannot be properly analyzed by using classical homogenization approaches,
because of the mesh dependence at the macroscale due to the ill-posedness of the macroscopic
boundary value problem, as shown in [4]; moreover, when large deformation must be accounted for,
additional difficulties arise since the size of the representative volume element (RVE) is not known a
priori (see [5] for details).
In order to partially overcome such drawbacks, other homogenization methods have been introduced,
such as higher-order and continuous-discontinuous schemes (see for instance [6] and [7],
respectively). Both approaches, relying on the proper incorporation of a length scale into the
microscopic model, are usually adopted within the more general framework of multiscale methods.
According to [8], such methods can be grouped in three classes depending on the type of coupling
between the microscale and the macroscale: hierarchical, semiconcurrent and concurrent methods.
In hierarchical methods, a “one-way” bottom-up coupling is established between the microscopic and
macroscopic problems, i.e. during the “micro-to-macro” transition step the information is passed from
lower to higher scales. In semiconcurrent methods, also referred to as computational homogenization
approaches, a microscopic BVP is associated with each integration point of the discretized
microstructure, in order to obtain the local governing equation at the macroscale. This class of
methods allows one to compute the fine-scale response required by the coarse-scale model for a
specific input and passes the information to the coarser scale during the analysis; thus, a
phenomenological constitutive model at the macroscale is not needed. On the other hand, concurrent
multiscale methods abandon the concept of scale transition in favor of the concept of scale
embedding, according to which models at different resolutions are defined in adjacent regions of the
89
same domain. Such methods fall within the class of domain decomposition methods (DDMs), in
which a strong two-way coupling between different scales is established.
In this work an innovative multiscale method able to perform complete failure analyses of fiberreinforced composite materials is presented, taking advantage of a non-overlapping domain
decomposition method, combined with an adaptive technique able to continuously update the fineresolution subdomain around a propagating macroscopic crack. Although the proposed method can be
easily applied to more general failure modes, only transverse cracking is considered in the present
work, since such a mechanism, which includes both matrix cracking and fiber/matrix interfacial
debonding is one of most observed in continuous fiber-reinforced laminates; this allows to perform
numerical simulations in a 2D setting.
Ad-hoc fracture criteria have been incorporated into the model, based on linear elastic fracture
mechanics, in order to perform propagation analyses involving non-prescribed crack paths, as
explained in [9]. The competition between fiber/matrix interface debonding and kinking phenomena
from and towards the matrix is accounted for, as well the continuous matrix cracking. The latter
mechanism is simulated by using a novel shape optimization method based on the coupling between a
moving mesh technique and a gradient-free optimization solver; such an ingredient makes the present
approach different form existing concurrent multiscale methods, which usually adopt damage models
or cohesive zone models to simulate damage propagation (see [10] for instance).
The main advantages of the proposed model are: (i) the possibility to simulate the competition
between different damage modes, and (ii) the capability to capture unstable equilibrium branches in a
quasistatic setting by virtue of a path-following strategy based on the total crack length.
Numerical computations are performed with reference to the failure analysis of a single notched
composite beam subjected to different loading conditions involving both mode-I and mixed-mode
crack propagation. Comparisons with solutions obtained by a direct numerical simulation (DNS) are
presented, in order to assess the validity of the proposed multiscale approach.
References
[1] D. Bruno, F. Greco, P. Lonetti, Interaction between interlaminar and intralaminar damage in
fiber-reinforced composite laminates, Int. J. Comput. Method Eng. Sci. Mech. 9(6) (2008) 358–373.
[2] F. Greco, L. Leonetti, P. Nevone Blasi, Non-linear macroscopic response of fiber reinforced
composite materials due to initiation and propagation of interface cracks, Eng. Fract. Mech. 80 (2012)
92–113.
[3] A. Caporale, R. Luciano, E. Sacco, Micromechanical analysis of interfacial debonding in
unidirectional fiber-reinforced composites, Comput. Struct. 84(31–32) (2006) 2200–2211.
[4] I.M. Gitman, H. Askes, L.J. Sluys, Representative volume: existence and size determination, Eng.
Fract. Mech. 74(16) (2007) 2518–2534.
[5] D. Bruno, F. Greco, R. Luciano, P. Nevone Blasi, Nonlinear homogenized properties of defected
composite materials, Comput. Struct. 134 (2014) 102–111.
[6] V.G. Kouznetsova, M.G.D. Geers, W.A.M. Brekelmans, Multi-scale second-order computational
homogenization of multi-phase materials: a nested finite element solution strategy, Comput. Method
Appl. Mech. 193(48–51) (2004) 5525–5550.
[7] T. Belytschko, S. Loehnert, J.H. Song, Multiscale aggregating discontinuities: a method for
circumventing loss of material stability, Int. J. Numer. Meth. Eng. 73(6) (2008) 869–894.
[8] T. Belytschko T, J.H. Song, Coarse-graining of multiscale crack propagation, Int. J. Numer.
Meth. Eng. 81 (2010) 537–563.
[9] F. Greco, L. Leonetti, P. Nevone Blasi, Adaptive multiscale modeling of fiber-reinforced
composite materials subjected to transverse microcracking, Compos. Struct. 113 (2014) 249–263.
[10] S. Ghosh, J. Bai, P. Raghavan, Concurrent multi-level model for damage evolution in
microstructurally debonding composites, Mech. Mater. 39 (2007) 241–266.
90
Consistent tangent operator for an exact Kirchhoff rod model
Leopoldo Greco1, Massimo Cuomo2,b,*
1
2
M&MOCS, Cisterna di Latina, Italy
Departmento of Civil Engineering and Architecture, University of Catania, Italy
b
mcuomo@dica.unict.it
Keywords: non linear Kirchhoff -Love rods, tangent stiffness, symmetry of geometric stiffness, LeviCivita connection.
In the paper it is considered an exact spatial Kirchhoff rod structural model. The configuration space
for this model, that has dimension 4, is obtained considering an ad-hoc split of the rotation operator
that implicitly enforces the constraints on the directors. The exact kinematics is presented, then the
equilibrium operator, both in strong and in weak form, is derived. The hypothesis of linear elastic
behaviour is introduced for the stresses, that allows to obtain simpler results that can be compared
with previous formulations (all of which adopt the same hypothesis). The relevant equations forgthe
model are obtained.
The tangent stiffness operator, essential for the non linear numerical simulations, has been studied. It
has been obtained as second covariant gradient of the internal energy functional for the considered
structural model, that preserves symmetry for any configuration, either equilibrated or not. The result
has been reached evaluating the Levi-Civita connection for the tangent space of the configuration
manifold. The results obtained extend to the case of Kirchhoff -Love rods those presented by Simo
(1986) for Timoshenko rods. Given the different structure of the tangent spaces in this case, it has
been necessary to introduce a specific metric that accounts for the rotation of the intrinsic triad due to
the change of the position of the centroid axis of the rod.
References
[1] Simo, J.C., Vu-Quoc, L., A three-dimensional finite strain rod model. Part II: computational
aspects, Comp. Meth. Appl. Mech. Engng., 58, 79-116, (1986).
[2] Greco, L., Cuomo, M., B-spline interpolation for Kirchhoff-Love space rod, Comp. Meth. Appl.
Mech. Engng., 256, 251-269, (2012).
[3] Greco, L., Cuomo, M., An implicit G1 multi patch B-Spline interpolation for Kirchhoff Love
space rod, Comp. Meth. Appl. Mech. Engng., 269, 173-197, (2014).
91
An implicit G1-continuity interpolation for Kirchhoff plate elements
Leopoldo Greco1, Massimo Cuomo2,b,*
1
2
M&MOCS, Cisterna di Latina, Italy
Departmento of Civil Engineering and Architecture, University of Catania, Italy
b
mcuomo@dica.unict.it
Keywords: Isogeometric analysis, Kirchhoff plate, G1 continuity, generalized FEM.
Finite element discretization of non polar theories of 1D and 2D structural models, like rod, plate or
shell models, require continuity for the rotations as well as for the displacements at the boundaries of
the elements in order to obtain a non singular stiffness matrix. Continuity on the boundary rotations
implies that a first order geometric, or G1, continuity be guaranteed. Geometric continuity is a weaker
continuity condition with respect to first order parametric continuity, denoted by C1, that imposes
constraints also on the membrane strain measures at the boundary, see for instance [1]. In classical
Finite Element methods an Hermitian interpolation is used for rods or a bi-Hermitian interpolation for
plates and shells, that if the element is distorted yields a G1 continuity only at the nodes.
In the paper we propose a generalization of the bi-Hermitian interpolation that can be easily
generalized to higher order b-spline isogeometric interpolations, based on the concept of Gregory's
patches, see [2, 3]. Introducing additional rotational degrees of freedom (related to the bi-variate
internal control points) it is obtained an implicit G1 interpolation, i.e. the continuity of the unit normal
vector along the edge of the plate element. Differently from Gregory's original approach, we don't use
as degree of freeom the second derivatives at the vertices of the element, rather we introduce
rotational degrees of freedom on the edges (connected to the two bi-variate control points), so that an
easier assemblage of the global stiffness matrix is obtained. In the framework of the isogeometric
analysis the Gregory's patch is a third order Beziér interpolation with bi-variate internal control points.
This concept can be easy generalized to the B-Spline interpolation, (and NURBS) for each polynomial
degree, as was done in [1] for rod elements, obtaining a new generalization of the bi-Hermitian
interpolation that permits a congruent tessellation of plate or shell elements. Figures (a) and (b) show
some examples of the G1 continuous interpolation achieved on a generally distorted mesh.
References
[1] L. Greco and M. Cuomo, An implicit G1 multi patch B-spline interpolation for Kirchhoff-Love
space rods, Computer Methods in Applied Mechanics and Engineering, Vol. 269, 173-197, 2014.
[2] J. Gregory, Smooth interpolation without twist constraints, in Computer Aided Geometric Design,
Eds. R. E. Barnhill and R. F. Riesenfeld, Acadmic Press, New York, pp. 7187, 1974.
[3] G. Farin and D. Hansford, Agnostic G1 Gregory surfaces. Graphical Models, Vol. 74, 346-350,
2012.
92
Pull-out strength of chemical anchors in natural stone
Loredana Contrafatto1, a *, Renato Cosenza2,b
Department of Civil Engineering and Architecture, University of Catania, Italy
Laboratory of Structural and Material Testing, University of Catania, Italy
a
loredana.contrafatto@dica.unict.it, b renato.cosenza@studium.unict.it
Keywords: adhesive anchor, epoxy resin, threaded rods, sandstone, basalt, limestone, fracture.
Masonry was in the past, and is, today, one of the most commonly used materials throughout the
world for the construction of low rise buildings. The stonework is largely widespread in different
countries and despite the variety of materials and techniques used, it has recurring problems regarding
both the vulnerability to seismic actions and the applicability of reinforcement techniques. There are
many techniques that can be implemented on masonry buildings. In particular, in the context of the
retrofitting of existing buildings, a great development has been achieved with the use of anchoring
systems. Chemical anchoring systems are commonly used in plain or reinforced concrete structures
but also in structures in lightweight material, such as wood and brick, and in masonry constructions to
rigidly couple different structural elements. A variety of metal elements are usually used, normally
steel elements such as stirrups, reinforcement bars, threaded rods. The adhesive component of the
system is generally resin.
Nevertheless the specific legislation on the architectural heritage does not allow the use of resin on
historical and monumental buildings, but suggest the usage of special mortars, there exists a lot of
cases in which the use of chemical anchoring is more suitable than mortar, especially in the case of
anchoring systems on rocks or high resistance supports. Such is the case of all the masonry buildings
that are not under a preservation order or listed buildings.
While a number of studies, both theoretical and experimental, concerns the behavior of concrete
anchors, in terms of pull-out strength and anchor depth determination, as reported in [1,2,3,4,5,6], the
lack of data concerning the behavior of chemical anchors in natural stone is incontrovertible.
The work is based on the results of an experimental research [7] related to chemical anchors in natural
stone. The specific goal is to achieve technical guidelines for the determination of the minimum
anchorage length in the case of brick masonry. The minimum embedment depth is determined for
chemical anchoring of post-installed threaded rods in basalt, sandstone and limestone support, by
using Hilti Re-500 epoxy resin of the company Hilti. Typical samples are shown in Figure 1.
Sandstone
Basalt stone
Limestome
Fig. 1 Influence of the stone type on the pull-out strength. Rod diameter 10 mm. Embedment depth equal to three times the
rod diameter.
The reliability of theoretical formulations in the literature valid for concrete is also evaluated.
The applicability of some numerical models for the prediction of the bearing capacity of the anchor is
then investigated, whereas the theoretical formulations are not feasible. A specific algorithm based on
the Strong Discontinuity Approach for the prediction of the cracks in brittle material is applied as an
explanatory example of the failure mechanism (Figure 2).
93
Fig. 2 SDA simulation of the failure mechanism.
References
[1] A. H. Nilson, Internal Measurement of Bond Slip, ACI Structural Journal. 69 (1972) 439-441.
[2] R. A. Cook, Behavior of chemically bonded anchors, Journal of Structural Engineering. 119
(1993) 2744-2762.
[3] R. A. Cook, R. C. Konz, Factors Influencing Bond Strength of Adhesive Anchors, ACI Structural
Journal. 98 (2001) 76-86.
[4] A. Colak, Parametric study of factors affecting the pull-out strength of steel rods bonded into
precast concrete panels, International Journal of Adhesion & Adhesives. 21 (2001) 487-493.
[5] T. S. Bickel, A. Fattah Shaikh, Shear Strength of Adhesive Anchors, PCI Journal. Sept-Oct
(2002) 92-102.
[6] R. Eligehausen, R. A. Cook, J. Appl, Behavior and Design of Adhesive Bonded Anchors, ACI
Structural Journal. 103 (2006) 822-831.
[7] L. Contrafatto, R. Cosenza, Experimental behaviour of post-installed adhesive anchors in natural
stone, submitted to Construction and Building Materials, (2014).
94
Strain gradient elasticity within the symmetric BEM formulation
S. Terravecchia1,a* , T. Panzeca2,b and C. Polizzotto3,c
1,2,3
a
DICAM, Viale delle Scienze, 90128 Palermo, Italy
silvioterravecchia@gmail.com, b teotista.panzeca@unipa.it, c castrenze.polizzotto@unipa.it
Keywords: Strain gradient elasticity, Symmetric Galerkin BEM.
The symmetric Galerkin Boundary Element Method is used to address a class of strain gradient elastic
materials featured by a free energy function of the (classical) strain and of its (first) gradient. With
respect to the classical elasticity, additional response variables intervene, such as the normal
derivative of the displacements on the boundary, and the work-coniugate moment tractions. The
fundamental solutions - featuring a fourth order partial differential equations (PDEs) system - exhibit
singularities which in 2D may be of the order 1 / r 4 .
This research communication shows a part of the results being elaborated within a more general paper
[1].
After the pioneering work of Mindlin [2], theories of strain gradient elasticity have become very
popular, particularly within the domain of nano-technologies, that is, for problems where the ratio
surface/volume tends to become very large and there is a need to introduce at least one internal length.
However the model introduced by Mindlin and then improved by Mindlin et al. [3] and Wu [4] leads
to an excessive number of material coefficients, which at the best for isotropic materials reduce to the
number of seven. In the early 1990, Aifantis [5] introduced a signified material model of strain
gradient elasticity which requires only three material coefficients, that is, the Lame' constants and one
length scale parameter. The latter model was then developed further following the so-called Form II
format given by Mindlin et al. [3], that is a theory centered on the existence of a free energy
functions of the (classical) strain and of its first gradient, which leads to the generation of symmetric
stress fields (see Askes et al. [6] for historical details about the latter formulations and its
applications).
Formulations in the boundary element method based on the strain gradient elasticity were pioneered
by Polyzos et al. [7], Karlis et al. [8] who provide a collocation BEM formulation where the simplified
constitutive equation by Aifantis [5] has been adopted. In latter papers only the fundamental solutions
used in the collocation approach to BEM are provided.
In the case of the symmetric formulation of the BEM, Somigliana Identities (SIs) for the tractions and
for the double tractions are also needed. These new SIs are necessary in order to get, through the
process of modeling and weighing, a solving equation system having symmetric operators. In
Polizzotto et al. [1] all the set of fundamental solutions is derived starting from the displacement
fundamental solution given in [7,8].
The symmetric formulation is motivated by the high efficiency achieved within classical elasticity by
the method in [9] with regard to the techniques used to eliminate the singularities of the fundamental
solutions, the evaluation of the coefficients of the solving system and the computational procedures
characterized by great implementation simplicity. This has already led to the birth of the computer
code Karnak sGbem [10] operating in the classical elasticity.
The objective of this paper is to experiment new techniques and procedures that, applied in the context
of strain gradient elastic materials, may permit one to obtain the related solving system.
The class of strain gradient elastic materials herein considered is featured by the following strain
elastic energy
1
2
W  ε : E : ε  E :: (ε)T  ε 
2
2
(1)
95
where  is the internal length, ε  (u) S is the classical strain and E is the classic isotropic elasticity
tensor. Eq. (1) provides the "primitive" stresses
σ (0)  E ε;
σ (1)   2 2σ (0)
(2)
The constitutive equation relating the total stress σ to the strain is
σ  E : (ε   2 2 ε)
(3)
where ε is the classical strain.
In [7,8] the fundamental solution of the displacements for 2D solids is obtained through which, using
the well-known procedure given in [11], it is possible by exploiting the known properties of symmetry
of the fundamental solutions to derive the entire tableau of the fundamental solutions.
Computational techniques have been pursued in order to eliminate the singularities of order 1 / r ,
1/ r 2 , in the blocks of the coefficients related to the corners of the solid and new techniques based on
the rigid motion strategy have been introduced in order to test the coefficients of the blocks of the
solving system. The displacement and internal deformation fields were obtained.
Numerical techniques in order to remove the singularity of higher order like 1/ r 3 and 1/ r 4 are in
advanced study.
References
[1] C. Polizzotto, T. Panzeca, S. Terravecchia, A symmetric Galerkin BEM formulation for a class
of gradient elastic materials of Mindlin type. Part I: Theory, (2014) in preparation.
[2] R.D.Mindlin, Second gradient of strain and surface tension in linear elasticity, Int. J. Solids
Struct. 1 (1965) 417-438.
[3] R.D. Mindlin, N.N. Eshel, On first strain-gradient theories in linear elasticity, Int. J. Solids
Struct. 28 (1968) 845-858.
[4] C.H. Wu, Cohesive elasticity and surface phenomena, Quart. Appl. Math. L(1) (1992) 73-103.
[5] E.C. Aifantis, On the role of gradients in the localization of deformation and fracture,. Int. J.
Eng. Sci. 30 (1992) 1279-1299.
[6] H. Askes, E.C. Aifantis, Gradient elasticity in statics and dynamics: An overview of
formulations, length scale identification procedures, finite element implementations and new results,
Int. J. Solids Struct. 48 (2011) 1962-1990.
[7] D. Polyzos, K.G. Tsepoura, S.V. Tsinopoulos, D.E. Beskos, A boundary element method for
solving 2-D and 3-D static gradient elastic problems. Part.I: integral formulation, Comput. Meth.
Appl. Mech. Engng. 192 (2003) 2845-2873.
[8] G. F. Karlis, A. Charalambopoulos, D. Polyzos, An advanced boundary element method for
solving 2D and 3D static problems in Mindlin's strain gradient theory of elasticity, Int. J. Numer.
Meth. Engng. 83 (2010) 1407-1427.
[9] T. Panzeca, F. Cucco, S. Terravecchia, Symmetric Boundary Element Method versus Finite
Element Method, Comp. Meth. Appl. Mech. Engrg. 191 (2002) 3347-3367.
[10] F. Cucco, T. Panzeca, S. Terravecchia, The program Karnak.sGbem Release 2.1, Palermo
University (2002).
[11] C. Polizzotto, An energy approach to the boundary element method. Part.I: elastic solids,
Comput. Meth. Appl. Mech. Engng. 69 (1988) 167-184.
96
Multidomain Symmetric Galerkin BEM for non-linear analysis of
masonries in-plane loaded
L. Zito1,a*, S. Terravecchia2,b and T. Panzeca3,c
1,2,3
a
DICAM, Viale delle Scienze, 90128 Palermo, Italy
liborio.zito@unipa.it, b silvioterravecchia@gmail.com, c teotista.panzeca@unipa.it
Keywords: Multidomain SGBEM, Displacement approach, Masonries.
The preservation of the historical and monumental buildings, but also of the considerable heritage of
old constructions made by traditional techniques, is one of the actual problems of the structural
mechanics. The level of knowledge of their structural behavior in presence of external actions is made
through calculus methods and simple procedures in order to allow a reading of the material suffering
degree and as a consequence of the related safety.
In this paper an elastic analysis of walls, also in presence of geometrical nonlinearity consisting in the
contact/detachment phenomenon among stone blocks. The wall having any shape and zone-wise
variable physical characteristics is loaded in its plane. For these structures some interventions of
structural strengthening have as aim to improve the wall behavior by reducing the stress
concentration, so to have a better safety in comparison with its initial value.
Among the more considerable aspects within the protection of the masonry buildings, and in particular
of the historical and monumental patrimony, there is the identifying of the static instability causes
found in the walls. The difficulties in studying these structural systems depend on several elements, as
the complexity of structural behavior and the uncertainties in the physical-mechanical characterization
of the materials.
At same time, the choice of the interventions in order to guarantee the safety wanted, respectful of the
rules of the restoration, is full of dangers. Often the architect or engineer uses empiric rules, also
supported by rough structural schematizations, attaining to solutions often not answering to the
necessary safety of the building.
As a consequence the use of the calculus code supported by appropriate computational methodologies
is necessary to make numerical simulations which, once known the external actions, have the aim:
- to identify the possible causes of the instabilities through the solution of an inverse problem;
- to perform interventions on masonry panels, having feature of prevention towards possible static
instabilities;
- to establish what, among the technical solutions of intervention to be activated, is this more
appropriate, by making a comparison among different solutions of reinforcement.
The aim is to increase the safety conditions of the masonry buildings through the improving the
stiffness of each wall panel in order to reach a global structural response with a more uniform field of
the stress state.
In this paper the analysis is performed in the hypotheses that the actions (boundary and body forces,
displacements of the constraint) act in the plane of each panel. It can be made through numerical
simulations using analysis methodologies able to perform the necessary checks within reduced
calculus times.
The usual calculus codes employed to analyze the masonries have as theoretical basis the Finite
Element Method (FEM). But this method shows several drawbacks meanly connected to the
discretization employed in the walls, usually foreseeing the use of elements having the same
geometry.
The present paper suggests an innovative analysis based on the use of a multidomain strategy within
the symmetric Galerkin formulation of the Boundary Element Method (SGBEM).
97
Within the SGBEM, in this paper, a strategy which uses the displacement method, proposed by
Panzeca et al. [1, 2], is utilized. The latter method shows symmetry of the algebraic operators and is
characterized as follows:
a) the subdivision of each masonry panel by substructures having any shape and dimension and
different physical properties;
b) the boundary distinction of each substructure into constrained, free and interface with other
substructures;
c) the writing of a characteristic elasticity equation for each substructure connecting weighted
tractions evaluated on the interface boundary to nodal displacements of the same interface and to the
load vector;
d) the use of the equation system through the writing of the compatibility strong form at the interface
nodes and through the related weak form involving weighted (or generalized) tractions at the interface
boundary elements,
e) the computation in closed form of all the double integrals making up the equation system
coefficients, having hypersingular, singular or regular kernels [3],
f) permit the transformation of the domain integrals into boundary ones. The reader can refer to
Panzeca et al. [4].
The discretization of the panel into substructures can be made through its subdivision into macrozones
characterized by a homogenization of the physical parameters of the stone-mortar system and single
elements constituting the panel as stone blocks and mortar layers, possibly considered separately.
The simultaneous use of the two different levels of discretization, one more sparse in macrozones and
another more dense, is characteristic of the potenziality of the method. The presence of substructures
having big or small dimensions does not involve numerical instabilities because all the coefficients of
the equation system were computed in closed form.
The analysis of the masonry is made by using the Karnak.sGbem code [5]. This program allows to
evaluate the response of the structural system subjected to all the possible static actions, whether
volumetric or surface loads, but also to volumetric and linear distortions and to displacements
imposed in the constraints.
Besides, it is possible to make a nonlinear analysis of cohesive detachment, so reproducing the
evolution of the probable disconnectedness between stones through a strategy developed by some
authors [6].
References
[1] T. Panzeca, F. Cucco, S. Terravecchia, Symmetric boundary element method versus finite
element method. Comput. Methods Appl. Mech. Engrg. 191 (2002) 3347-3367.
[2] T. Panzeca, M. Salerno, S. Terravecchia, Domain decomposition in the symmetric boundary
element method analysis. Comput. Mech. 28 (2002) 191-201.
[3] M. Bonnet, Regularized direct and indirect symmetric variational BIE formulation for threedimensional elasticity. Eng. Anal. Boundary Elem. 15 (1995) 93-102.
[4] T. Panzeca, S. Terravecchia, L. Zito, Computational aspects in 2D SBEM analysis with domain
inelastic actions. Int. J. Numer. Meth. Engng. 82 (2010) 184-204.
[5] F. Cucco, T. Panzeca, S. Terravecchia, Karnak.sGbem. Release 1.0, www.bemsoft.it, Palermo
2002.
[6] T. Panzeca, M. Salerno, S. Terravecchia, L. Zito, The symmetric Boundary Element Method for
unilateral contact problems. Comput. Methods Appl. Mech. Engrg. 197 (2008) 2667-2679.
98
SOMMARI
GMA
Gruppo Materiali
AIMETA
Congresso GIMC-GMA-2014
11-13 giugno 2014
99
100
Invited Lecture
Sorption of low molecular weight compounds in polymers:
thermodynamic issues and plasticization effects
Giuseppe Mensitieri1,a *, Giuseppe Scherillo1,b and Pellegrino Musto2,c
1
Dept. of Chemical, Materials and Production Engineering
University of Naples Federico II, P.le Tecchio 80, 80125, Italy
2
Institute for Polymers, Composite and Biomaterials
National Research Council of Italy, Viale Campi Flegrei 34, 80078 Pozzuoli (Na), Italy
a
mensitie@unina.it, bgscheril@unina.it, cpellegrino.musto@ictp.cnr.it
* corresponding author
Keywords: Sorption Thermodynamics; Low molecular weight compounds; Compressible lattice
fluid, Glassy polymer; Rubbery Polymer
Introduction
Sorption and transport of low molecular weight compounds is a key issue in assessing the durability
of polymer matrix composites. In fact, absorbed compounds (water, solvents, high pressure gases) can
adversely affect mechanical properties of the matrix and fiber-matrix interface integrity. In this
contribution, the general issue of sorption thermodynamics and mass transport of low m.w. penetrants
in glassy and rubbery polymers is addressed providing a consistent thermodynamic interpretation of
the experimental investigations, accounting also for the effects of possible self- and cross-hydrogen
bonding interactions. In particular, the analysis of water and methanol sorption in polyimides,
polyetherimide, polyetheretherketone and in polycaprolactone are presented.
Sorption thermodynamics
Interpretation of sorption thermodynamics in rubbery polymers is here approached by using a lattice
fluid theory accounting also for the effect of possible self- and cross-interactions (hydrogen bonding)
in polymer–penetrant systems, the so called ‘Non Random lattice fluid Hydrogen Bonding’ (NRHB)
model [1, 2]. Thermodynamic models accounting for both interactions and non-equilibrium nature of
glassy polymers have been proposed only recently [3]. In fact, physical properties of glassy polymers
significantly differ from those of the same polymer in the rubbery state. Consistently, also sorption
thermodynamics differ substantially and modeling should properly account for non-equilibrium state.
To this aim, a successful approach is provided by a theoretical framework developed to extend the
equilibrium mixture theories suitable for rubbery polymers to the non-equilibrium glassy polymerpenetrant mixtures (the so-called Non-Equilibrium Thermodynamics for Glassy Polymers, NETGP)
by introducing internal state variables, which act as order parameters quantifying the departure from
the equilibrium conditions at fixed pressure and temperature [4]. In the present contribution, we have
applied this procedure to extend NRHB theory to non-equilibrium glassy systems [3] to provide a
suitable model (NETGP-NRHB) for interpretation of sorption of interacting penetrants in glassy
polymers.
Experimental analysis
Gravimetric and vibrational spectroscopy experiments have been performed. The former provides
information on the total mass of sorbed penetrant (sorption isotherms) while the latter provides
qualitative and quantitative information regarding the ‘state’ of penetrant molecules within the
polymer matrix, with particular reference to the interactions which are established with
macromolecules. Results of in situ FTIR spectroscopy on rubbery and glassy polymers exposed to
penetrants vapour at different pressures have been analyzed by 2D correlation spectroscopy to
improve the resolution by spreading the data over a second frequency axis providing, at the same time,
information about the dynamics of the evolving system [5]. The results of experimental analyses are
compared with predictions of NRHB and NETGP-NRHB models, respectively for the case of rubbery
and glassy polymers. As an example, the results of spectroscopic analysis of water sorption in 6FDA101
ODA polyimide (2 F atoms per repeating unit, Tg = 308°C) are reported in figure 1. The whole of the
2D results (figure 1, left) points to the occurrence of two distinct water species. Likely structures for
the H-bonding aggregates are reported in figure 1, right. In figure 2 is, instead. shown, for the same
system, the capability of the NETGP-NRHB model to rovide good quantitative estimates of the
different kinds of water species present in the system.
Figure 1: Left:2D-FTIR correlation spectra (asynchronous) obtained from the time-resolved spectra collected during water
sorption in 6FDA-ODA polyimide at a relative pressure = 0.6. Right : Schematic representation of the H-bonding
interactions in the investigated water/PI system
Figure 2: Comparison of predictions of NETGP−NRHB model with experimental results for 6FDA-ODA: moles of water
wp
self-HBs (n11 ) and moles of hydrogen bonding between absorbed water molecules and proton acceptor groups on the
wp
polymer backbone (n12 ) per gram of dry polymer as a function of water mass fraction in the polymer/water mixture.
References
[1] C. Panayiotou, I. Tsivintzelis, I.G.Economou, Nonrandom hydrogen-bonding model of fluids and
their mixtures. 2. Multicomponent mixtures Ind. Eng. Chem. Res., 46 (2007) 2628-2636.
[2] I. Tsivintzelis, G.M. Kontogeorgis, Modeling the vapor-liquid equilibria of polymer-solvent
mixtures: Systems with complex hydrogen bonding behavior Fluid Phase Equilib., 280 (2009) 100109.
[3] G. Scherillo, L. Sanguigno, M. Galizia, M. Lavorgna, P. Musto and G. Mensitieri, Nonequilibrium compressible lattice theories accounting for hydrogen bonding interactions: Modelling
water sorption thermodynamics in fluorinated polyimides Fluid Phase Equilibria, 334 (2012) 166-188.
[4] F. Doghieri, G. C. Sarti, Nonequilibrium lattice fluids: A predictive model for the solubility in
glassy polymers Macromolecules, 29 (1996) 7885-7896.
[5] P. Musto, G. Mensitieri, M. Lavorgna, G. Scarinzi, G. Scherillo, Combining Gravimetric and
Vibrational Spectroscopy Measurements to Quantify First- and Second-Shell Hydration Layers in
Polyimides with Different Molecular Architectures J. Phys. Chem. B, 116 (2012) 1209−1220.
102
Propagation of elastic waves and generation of band-gaps in diffusively
damaged structures
Giorgio Carta1, 2, a *, Michele Brun1, 2,b and Alexander B. Movchan2,c
1
Dipartimento di Ingegneria Meccanica, Chimica e dei Materiali, Università di Cagliari, Italy
2
Department of Mathematical Sciences, University of Liverpool, UK
a
giorgio_carta@unica.it, bmbrun@unica.it, cabm@liverpool.ac.uk
Keywords: elastic waves; damaged structures; cracks; dispersion curves; band-gaps.
Discontinuities in homogeneous elastic solids generate band-gaps, that are ranges of frequencies for
which waves do not propagate. Such discontinuities may be represented by cross-section reductions
due to cracks or to design choices. For instance, in multi-span simply-supported bridges the
connection between two adjacent spans is usually constituted by the upper deck; therefore, in
correspondence of the connection the bridge possesses a lower cross-section.
The dynamic behaviour of an elastic solid with equispaced cracks is discussed [1]. The solid is firstly
studied as a two-dimensional strip, and the propagation of transverse waves is examined. It is shown
that the eigenfrequencies of finite strips with different lengths fall inside the propagation bands of
infinite strips with periodically-distributed cracks. In addition, we consider different boundary
conditions at the ends of finite strips of different dimensions.
The two-dimensional damaged strip is also analysed as a beam with elastic junctions [1]. The latter
simulate the cracked cross-sections and are characterised by a rotational (bending) and a translational
(shear) spring. The stiffnesses (per unit thickness of the beam) of the two springs are derived by
means of a static asymptotic analysis [2] and are given respectively by
Kb 
π E 2
4  5  2 1  
(1)
πE
,
4 1   log  h /  
(2)
and
Ks 
2
where E is Young's modulus, ν is Poisson's ratio, while ρε and h are the depths of the cracked and
intact cross-sections. The dispersion curves of the beam reproduce well the propagation bands of the
two-dimensional strip in the low- and medium-frequency ranges, as shown in Figure 1. In the axes of
this figure, k l is the product of the wavenumber and the distance between the cracks, while  is a nondimensional parameter related to the radian frequency. It is proved that the limits of the band-gaps
coincide with the eigenfrequencies of simple beams with appropriate boundary conditions [1].
Moreover, we investigate the effect of changing the slenderness of the strip on the efficiency of the
beam model.
103
Figure 1: Comparison between the dispersion curves of an infinite beam (solid grey lines) and of an infinite strip (dots),
both with periodically-distributed cracks and characterised by the following properties: E = 200 GPa, ν = 0.3, mass density
ρ = 7800 kg/m3, h = 0.2 m, ρε = 0.04 m.
The outcomes of this work may be useful in the context of Structural Health Monitoring for the
detection of damages and defects in structures. Furthermore, they could be exploited to design
structures with appropriate discontinuities (e.g. reduced cross-sections) that can filter waves of
specified frequencies. Finally, we point out that it still remains a challenge to study the dynamic
response of a solid with randomly-spaced cracks or defects.
References
[1] G. Carta, M. Brun, A.B. Movchan, Dynamic response and localisation in strongly damaged
waveguides, Proc. R. Soc. A, accepted for publication.
[2] M. Gei, I.S. Jones, A.B. Movchan, Junction conditions for cracked elastic thin solids under
bearing and shear, Quart. J. Mech. Appl. Math. 62 (2009), 481-493.
104
On the compressive strength of glass-microballoons/thermoset-matrix
syntactic foams
Lorenzo Bardella1, a * and Andrea Panteghini2,b
1
2
DICATAM, University of Brescia, via Branze 43, 25123, Brescia, Italy
DICATAM, University of Brescia, via Branze 43, 25123, Brescia, Italy
a
lorenzo.bardella@unibs.it, bandrea.panteghini@unibs.it
* corresponding author
Keywords: Syntactic foam; Glass microballoon; Computational homogenization; Finite element
modeling; Strength; Thermoset matrix composites.
This work is concerned with particulate composites filled with hollow spherical inclusions, i.e.,
syntactic foams. We propose a micromechanical model to evaluate the uniaxial compressive strength
for the most relevant case of glass inclusions of wall thickness of few micrometers (microballoons)
filling a thermoset matrix. The studied failure modality is experimentally characterised by shear bands
inclined of about 45 degress with respect to the loading axis, and a prominent softening behaviour. We
develop a three-dimensional Finite Element (FE) modelling which extends and improves that recently
proposed by our group [1,2]. Different microstructures are described by cubic unit cells containing
fifty hollow spheres accounting for different filler polydispersions and filler volume fraction up to
60%. Each microballoon is assumed to undergo brittle failure according to a structural criterion. Here,
we account for the matrix nonlinear behaviour and, in a phenomenological way, for the detriment of
its mechanical properties, proportional to its defectiveness, which increases with the filler content and
becomes extremely relevant at filler volume fraction larger than 50%. Our findings agree with
experimental observations from the literature and reveal room for improvement in the effective
mechanical properties by acting on the manufacturing process.
References
[1] L. Bardella, A. Sfreddo, C. Ventura, M. Porfiri, N. Gupta, A critical evaluation of
micromechanical models for syntactic foams, Mech. Mater. 50 (2012) 53-69.
[2] L. Bardella, F. Malanca, P. Ponzo, A. Panteghini, M. Porfiri, A micromechanical model for quasibrittle compressive failure of glass-microballoons/thermoset-matrix syntactic foams, J. Eur. Ceram.
Soc. (2014) in print, DOI: 10.1016/j.jeurceramsoc.2013.11.045.
105
Elastically deformable scale through configurational forces
Francesco Dal Corso1,a, Davide Bigoni1,b, Federico Bosi1,c and Diego Misseroni1,d
1
University of Trento, via Mesiano 77, 38123 Trento, Italy
a
francesco.dalcorso@ing.unitn.it, bbigoni@ing.unitn.it,
c
federico.bosi@ing.unitn.it, ddiego.misseroni @ing.unitn.it
Keywords: Eshelby force, Elastic Structure, Frictionless constraint.
An Eshelbian (or configurational) force has been theoretically and experimentally shown to act on
elastic structures constrained by a frictionless sliding sleeve [1].
This force is here exploited to develop a ‘deformable arm scale’ (completely different from a
traditional rigid arm balance), Fig 1. The principle of the scale is based on nonlinear equilibrium
kinematics of rods inducing configurational forces, so that deflection of the arms becomes necessary
for the equilibrium, which would be impossible for a rigid system.
Figure 1: (Left) A prototype and (right) the scheme of the deformable arm scale. The lamina (loaded at the two free ends)
can slide into a frictionless sliding sleeve inclined at an angle α=60◦ with respect to the vertical direction. The equilibrium
in the direction of the sliding is realized by the presence of configurational force.
References
[1] D. Bigoni, F. Dal Corso, F. Bosi, D. Misseroni. Eshelby-like forces acting on elastic structures:
theoretical and experimental proof. Mechanics of Materials (2014) in press, doi:
http://dx.doi.org/10.1016/j.mechmat.2013.10.009
106
Flaw-tolerance of nonlocal discrete systems and interpretation
according to network theory
Andrea Infuso1, a *, Marco Paggi2,b
1
2
Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino, Italy
IMT Institute for Advanced Studies, P.zza San Francesco 19, 5510 Lucca, Italy
a
andrea.infuso@polito.it, bmarco.paggi@imtlucca.it
Keywords: nonlocality, discrete systems, MDFEM, network theory, numerical methods.
Nonlocal continuum theories based on gradient models, integral formulations or fractional calculus
have been widely explored in mechanics to describe long-range interactions (see, e.g., [1-4] among the
most recent contributions on this topic). At the same time, discrete systems composed of particles or
molecules have been proposed in the physics community to analyze the behavior of materials. Lattice
beam models [5], albeit suffering from mesh dependency due to the local nature of the bonds/links,
have been extensively used to simulate the meso-scale behaviour of concrete. Efforts accounting for
nonlocal effects, such as the three-dimensional Born model [6], have bee proposed to study the
distribution of broken bonds within a homogeneous discrete mechanical system. With the progress in
computer technology, wide tri-dimensional discrete systems can nowadays be modeled by molecular
dynamics (MD), accounting for nonlinear interatomic potential laws and nonlocal interactions among
the discrete molecules. Attempts to couple MD with FEM have also been explored [7]. In this study,
the focus regards the analysis of the ability of nonlocal molecular discrete systems to tolerate flaws.
To this aim, nonlinear springs whose constitutive relation is ruled by the van der Waals potential are
implemented in the finite element analysis programme FEAP. The tensile and compressive responses
of 1D and 2D discrete systems of molecules with or without flaws are numerically simulated, by
varying the range of nonlocal interactions. For each system, the statistical distribution of the force
field at different load levels is carefully analyzed in relation to the topological properties of the
underlying network, in order to understand the force redistribution mechanisms occurring in nonlocal
discrete systems in the presence of defects.
References
[1] M. Di Paola, M. Zingales, Long-range cohesive interactions of non-local continuum faced by
fractional calculus, International Journal of Solids and Structures, 45 (2008) 5642-5659.
[2] A. Carpinteri, P. Cornetti, A. Sapora, M. Di Paola, M. Zingales, Fractional calculus in solid
mechanics: Local vs Non-Local Approach, Physica Scripta, T136 (2009) 014003-014010.
[3] M. Di Paola, F. Marino, M. Zingales, Integral and fractional model of elastic foundation based on
long-range interactions, International Journal of Solids and Structures 46 (2009) 3124-3117.
[4] M. Di Paola, G. Failla, M. Zingales, Physically based approach to the mechanics of strong nonlocal linear elasticity theory, Journal of Elasticity, 97 (2009) 103-130.
[5] J.G.M. van Mier, E. Schlangen, A. Vervuurt, Lattice type fracture models for concrete, H.B.
Mühlhaus (Ed.), Continuum Models for Materials with Microstructure, John Wiley & Sons, 1995, pp.
341-377.
[6] A. Parisi, G. Caldarelli, L. Pietronero, Roughness of fracture surfaces, Europhysics Letters, vol. 52
(3), (2000), pp. 304-310.
[7] L. Nasdala, A. Kempe, R. Rolfes, The molecular dynamic finite element method (MDFEM),
Computer, Materials & Continua 19 (2010) 57-104.
107
A model to interpret the wedge-shaped spalling in pull-out tests of FRP
from concrete
Roberto Ballarini1, a, Annalisa Franco2,b* and Gianni Royer Carfagni2,c
1
Department of Civil Engineering, University of Minnesota, 500
Pillsbury Drive S.E. Minneapolis, MN 55455-0116, USA
2
Department of Industrial Engineering, University of Parma,
Parco Area delle Scienze 181/A, I 43100 Parma, Italy
a
broberto@umn.edu, bannalisa.franco@unipr.it, cgianni.royer@unipr.it
Keywords: Fiber Reinforced Polymer (FRP); stiffener; pull-out test; substrate cracking; inclined
crack propagation; elasticity; complex potential; distributed dislocations.
Fiber-Reinforced-Polymer (FRP) materials are commonly used to repair and refurbish buildings and
bridges by gluing strips or plates to the external surfaces of concrete structural elements in order to
increase their bending strength. Pull-out tests on FRP joints have provided a wealth of evidence that
the dominant failure mode is debonding, occurring a few millimeters underneath the adhesive
interface. The failure is characterized by the formation of a process zone, which nucleates at the
loaded end of the stiffener, where a relative slip between stiffener and substrate occurs at non-zero
shear stress. During the pull-out test, such zone progresses in a stable way until it reaches a critical
length, coinciding with the effective bond length of the bonded joint, which marks the start of
debonding. Pulling further, delamination propagates and the cohesive zone simply translates towards
the free end of the joint until a softening phase begins [1]. The final failure occurs when the FRP
stiffener completely separates from the substrate. Remarkably, such a stage is characterized by the
initiation of an inclined crack at the free end of the stiffener, penetrating into the substrate. The
subsurface crack produces a characteristic wedge-shaped spall (Figure 1).
Figure 1: Wedge-shaped spall of the substrate in FRP-to concrete bonds with different initial bond lengths, as per [2].
Initial bond length: a) l = 30 mm, b) l = 90 mm; c) l = 150 mm.
There is not unanimous consensus on the reasons for the transition from cracking along the bond to
cracking within the substrate. Here, a linear elastic fracture mechanics model problem is presented
that provides improved understanding of the formation of the subsurface crack. The stiffener is
assumed to transmit shear stresses to a substrate modeled as a homogeneous isotropic elastic halfplane in generalized plane stress. Therefore, the formulation of the propagation of a crack at the free
end of the stiffener relies on the superposition of two effects: i) the effect of tangential forces per unit
area on the surface of the half plane and ii) the effect of distributed edge dislocations along the crack.
The condition that the crack surfaces are traction-free furnishes an integral equation, which is solved
using the properties of Chebyshev polynomials.
In order to comprehend the transition between the interface delamination and the substrate cracking,
two competing mechanisms of degradation are considered; a) failure of the adhesive joint, which
progresses at the stiffener-substrate interface when the corresponding shear stress is greater than the
108
strength of the interface, say q0; b) inclined cracking, which can develop in the substrate when the
strain energy release associated with its propagation, G* ,n , is greater than the corresponding fracture
energy  of the material. A key hypothesis for this model is that fracture does not progress
continuously and uniformly, but in discrete steps. In other words, there is a quantized length for crack
propagation, namely a*, that is comparable to the characteristic dimensions of the material
microstructure. Therefore G* ,n must be interpreted as the quantized energy release rate corresponding
to an increase a* of crack length. From the competition between the two mechanisms, with the same
rationale proposed by [3], one can evaluate when the inclined crack starts to form and the
characteristic angle of the wedge-shaped bulb. In particular, the inclined crack becomes energetically
more favorable as soon as
G* ,n q02 /   1 .
(1)
When G q /   1 propagation along the interface (debonding) occurs first.
*
, n
2
0
For the same material parameter of
[2], Figure 2 shows the ratio
G* ,n q02 /  as a function of the
2
*
a = 1 mm
1.8
*
a = 2 mm
*
a = 5 mm
1.6
crack inclination angle ω for
different values of a*. The value
G* ,n q02 /   1 defines the limit case
a*= 7 mm
a*= 10 mm
1.4
a*= 12 mm
G*ω,n τ2 / Γ
a*= 15 mm
1.2
a*= 20 mm
that separates the two different
damage mechanisms. For any a*,
one can evaluate the limit angle ω
0.8
which marks the transition from
0.6
one damage mechanism to the
0.4
other [4].
In general, the quantum length a* is
0.2
associated with the material
0
0
10
20
30
40
50
60
70
80
90
intrinsic length scale that, for
Inclination angle, ω [°]
conglomerates like concrete, is
Figure 2: Normalized strain energy release as a function of the correlated with the average size of
inclination angle ω for different values of the crack quantum length a∗ the aggregate. For the tests of [2],
(mechanical parameters of [2], bond length l = 30 mm, q0 = τc).
one can conveniently consider a* =
10-15 mm. With this choice, the
critical angle ω varies in the
interval 24-31°. From the pictures of Figure 1, it is evident that the concrete wedges are defined by
angles comprised in the interval 18-33°, in good agreement with the prediction of this approach.
c
G*ω,n τ2c / Γ=1
1
References
[1]
A. Franco, G. Royer-Carfagni, Cohesive debonding of a stiffener from an elastic substrate.
Comp. Struct. 111 (2014) 401–414.
[2] P. Carrara, D. Ferretti, F. Freddi, G. Rosati, Shear tests of carbon fiber plates bonded to concrete
with control of snap-back. Eng. Fract. Mech. 78 (2011) 2663 – 2678.
[3] J. Willis, A comparison of the fracture criteria of Griffith and Barenblatt. J. Mech. Phys. Solids 15
(1967) 151 – 162.
[4] R. Ballarini, A. Franco, G. Royer-Carfagni, Wedge-shaped fracturing in the pull out of FRP
stiffeners from quasi-brittle substrates, submitted to Int. J. Solids Struct. (2014).
109
Morphoelastic rods
Alessandro Tiero1,a, Giuseppe Tomassetti1,b *
1
Universita` di Roma Tor Vergata
Dipartimento di Ingegneria Civile e Ingegneria Informatica
Via Politecnico 1 00133 Roma - Italy
a
tiero@ing.uniroma2.it, btomassetti@ing.uniroma2.it
* Corresponding author
Keywords: growth, remodeling, thin structures, configurational forces, material forces.
Unlike common engineering materials, living matter can adapt to environmental changes by growing
and by actively modifying its structure. When trying to accommodate these features in the
infrastructure of continuum mechanics, a key issue is to distinguish growth from strain. Beginning
with [1], this issue has been addressed through a multiplicative decomposition f FGof the
deformation gradient into a part F accounting for mechanically-induced strain, and a part G
accounting for growth.
The additional degrees of freedom brought in by the growth tensor G demand additional evolution
laws whose choice and interpretation set a formidable challenge. In the format set forth in [2] the laws
governing the evolution of the growth tensor are obtained by combining suitable constitutive
prescriptions with a balance equation for a system of remodeling couples which expend power over
temporal changes of G. An appropriate version of the dissipation principle yields that the energetic
part of the inner remodeling couple coincides with the Eshelby stress E ψI F S, a tensorial
quantity introduced by Eshelby [3] in its discussion of a force acting on a material defect.
For a thin body, such as a rod, one may ask what is the equivalent of the standard multiplicative
decomposition and what mechanical statement should take the place of the Eshelby stress. In [4] we
provide a possible answer by deriving a theory describing elastic rods which, like plant organs, can
grow and can change their intrinsic curvature and torsion. The equations ruling accretion and
remodeling are obtained by combining balance laws involving non-standard forces with constitutive
prescriptions filtered by a dissipation principle that takes into account both standard and non-standard
working. In this theory, the Eshelby stress is replaced by the scalare ψ U ⋅ S u ⋅ s, where U and
u are the standard strain descriptors taking into account curvature and shear, whereas S and s are the
corresponding stress descirptors. We illustrate the theory with two examples: in the first, growth takes
place at the expense of working performed by applied loads; in the second growth is accompanied by
relaxation of elastic energy.
References
[1] E.K. Rodriguez, A. Hoger, and A.D. McCulloch. Stress-dependent finite growth in soft elastic
tissues. J. Biomech., 27 (1994) 455-467.
[2] A. DiCarlo and S. Quiligotti. Growth and balance. Mech. Res. Commun. 29 (2002) 449-456.
[3] J. Eshelby. The Continuum Theory of Lattice Defects. Solid State Phys. 3 (1956) 79-144.
[4] A. Tiero and G. Tomassetti. Morphoelastic rods. Submitted (2014).
110
Bending of shape-memory alloys’ beams: constitutive modeling and
structural response
Silvia Di Caprera, Michele Marinoa* and Giuseppe Vairob
Department of Civil Engineering and Computer Science,
University of Rome “Tor Vergata”, Via del Politecnico 1, 00133 Rome, Italy
a
m.marino@ing.uniroma2.it, bvairo@ing.uniroma2.it
Keywords: Shape-memory alloys, beam model, pseudo-elastic effect, bending.
In order to provide some additional functionalities and smart properties in the mechanical response of
structures, non-conventional materials attracted the interest of many researchers in the last years. Due
to the increasingly demand of lighter, stronger and distance-controlled devices, the attention of many
engineers recently moved to the so-called active materials, able to modify theirs own thermo-electromechanical properties in accordance with the applied loading. Among different smart materials
available for engineering purposes, shape-memory alloys (SMAs) occupy a leading role and have
recently seen an increasing employment in a number of industrial applications [1]. Devices made up
of such materials have been initially referred for obtaining pseudo-elastic behaviors, and for the
conversion of thermal energy into mechanical one, allowing the control of material deformation by
temperature variations [2]. This concept has been applied over the years in the fields of aeronautic,
telecommunications, civil and biomechanical engineering: innovative devices playing the role of
actuators have thus been conceived, leading to impressive technological progress [1]. Moreover, the
latest trend is also to exploit the energy dissipation characterising thermoelastic transformations
proper to SMAs in order to design high-energy absorption devices, such as pre-crash systems and
shock absorbers.
Given the importance of prospective SMA applications, there is the great need of developing suitable
analytical models for the mechanical behaviour of SMA structures in order to provide novel and
effective technical procedures for their design. The challenge of modelling shape-memory materials
resides in the complex coupling between micro and macro scales: accordingly, changes in
microstructure could reflect in macroscopic movements as well as external thermomechanical
loadings are able to modify the alloy’s composition [1,2].
Many studies have been devoted to the constitutive modelling of SMA properties [3-5] and to the
development of numerical models for the structural response of SMA structures [6] . Recently,
treating SMA within the framework of standard generalized materials, a novel SMA constitutive
model based on an internal-constrained variational approach has been proposed in [7] generalizing the
model proposed in [8]. The convexity arguments on the pseudo-potential of dissipation are exploited
for satisfying a-priori, within the variational formulation, inequality prescriptions enforced by the
second law of thermodynamics.
In this paper, the modeling of a SMA cantilever beam is addressed, focusing a pure bending load. The
model is based on an incremental formulation that allows to obtain step-by-step analytical results,
suitable for developing low-consuming numerical simulations. Phase change mechanisms and the
strong thermomechanical coupling effects are described starting from the three-dimensional model
proposed in [7]. Accordingly, suitable expressions for the free-energy and the pseudopotential of
dissipation are chosen, by introducing physical restrictions through arguments based on convex
analysis. Innovative composition laws and equilibrium equations are introduced, leading to a
consistent integration of the predictive multi-dimensional constitutive relations in a one-dimensional
beam model.
The model is implemented in a homemade parametric code and validated by means of comparison
with available experimental data. Accordingly, the main role of pseuodelastic effects and
thermomechanical coupling mechanisms in the structural response of SMA beams under bending is
111
clearly highlighted. Furthermore, a wide campaign of numerical simulations is conducted and
obtained results show the effectiveness of present approach in predicting the evolution of nonhomogenous austenitic/martensitic composition during the functioning behavior of SMA beams.
Moreover, parametric studies on material properties allow to obtain guidelines for the design of
innovative beam-like structures with a smart behavior (such as actuators, dissipative and damping
devices) based on SMA rationale.
References
[1] D.C. Lagoudas (Ed.), Shape Memory Alloys. Modeling and Engineering Applications.
SpringerScience+BusinessMedia, LLC, New York, USA, 2008.
[2] C.B. Churchill, J.A. Shaw, M.A. Iadicola, Tips and tricks for characterizing shape memory alloys
wire: part 4 -Thermo-mechanical coupling. Exp. tech. 34 (2010) 63-80.
[3] F. Auricchio, R.L. Taylor, J. Lubliner, Shape-memory alloys: macromodelling and numerical
simulations of the superelastic behavior, Comp. Meth. Appl. Mech. Engrg. 146 (1997) 281-312.
[4] V. Evangelista, S. Marfia, E. Sacco. Phenomenological 3D and 1D consistent models for shapememory alloy materials. Comput. Mech. 44 (2009) 405-421.
[5] D. Christ, S. Reese, A finite element model for shape memory alloys considering thermomechanical
couplings at large strains, Int. J. Solids Struct. 46 (2009) 3694-3709.
[6] F. Auricchio, E. Sacco, Thermo-mechanical modelling of a superelastic shape-memory wire under
cyclic stretching-bending loadings, Int. J. Solids Struct. 38 (2001) 6123-6145.
[7] M. Marino, Pseudopotentials and thermomechanical response of materials and structures: a convex
analysis approach, PhD dissertation, University of Rome “Tor Vergata”, July 2013.
[8] M. Frémond, Phase Change in Mechanics. Lecture Notes of the Unione Matematica Italiana.
Springer-Verlag, Berlin-Heidelberg, 2012.
112
Pre-buckling behavior of composite beams: an innovative approach
Francesco Ascione1, a *, Geminiano Mancusi2,b and Marco Lamberti3,c
1,2,3
Department of Civil Engineering, University of Salerno, Fisciano (SA), Italy
a
fascione@unisa.it, bg.mancusi@unisa.it, cmalamberti@unisa.it
Keywords: FRP profiles, pre-buckling behavior, connection deformability.
Fiber-reinforced composite materials have been used over the past years in several different civil
structures, acquiring a leading role as structural elements [1-4]. In particular, FRP profiles are
manufactured by so-called automated process of pultrusion.
From a mechanical point of view, they can be considered as linear elastic, homogeneous and
transversely isotropic, with the plane of isotropy being normal to the longitudinal axis (i.e. the axis of
pultrusion). It is generally asserted that their mechanical behavior is highly affected by warping strains
due to their small thickness. In addition, low shear moduli, more or less the same as those of the
polymeric resin, can provoke a non-negligible increase in lateral deflections, thus affecting both the
local and global buckling loads. Consequently, FRPs members exhibit significant non classical effects
such as transverse shear, warping displacements and non-uniform torsional rigidity that make
deformability and stability requirements more relevant than the strength limits in the design process.
Recently, experimental studies by Mosallam [5] and Feo et al. [6] showed that the condition of a rigid
connection should be replaced by a more appropriate assumption due to the presence of a higher local
resin concentration in the connection region between the flange and web. Furthermore, taking into
account that pultrusion guarantees very high strength and stiffness along the longitudinal direction of
the beam, a deeper investigation of this topic is required.
In this paper, which is a continuation of previous ones [7-8], a geometrically nonlinear model for
studying the lateral global buckling problem of a generic open/closed composite beam is presented.
The model is based on a full second-order deformable beam theory and accounts for both the warping
effects and possible displacement discontinuities at the web/flange interface. Equilibrium nonlinear
equations are derived from the Principle of Virtual Displacements. A displacement-based onedimensional finite element model is also developed. Numerical results are obtained for thin-walled
composite beams with open and closed section under flexural/torsional loads. The main aim is to
investigate the lateral buckling behavior taking into account the effects of shear and web/flange
junction deformability as well as the initial geometric imperfections.
The reliability of the mechanical model is assured by comparisons with other numerical and
experimental results available in literature.
Preliminary results show that deformability and stability requirements are fundamental in the safety
analysis of such members.
References
[1] CNR DT205/2007. Guide for the Design and Construction of Structures made of FRP Pultruded
Elements. Advisory Committee on Technical Recommendations for Constructions, Italian National
Research Council..
[2] CAN/CSA S806_02-2002. Design and Construction of Building Components with Fibrereinforced Polymers. Canadian Standard Association, Rexdale, Canada.
[3] JSCE 1997. Recommendation for Design and Construction of Concrete Structures using
Continuous Fiber Reinforcing Materials. Japan Society of Civil Engineering.
[4] ACI 440.1R-06-2006. Guide for the Design and Construction of Concrete Reinforced with FRP
Bars. American concrete institute.
113
[5] Mosallam AS, Elsadek AA, Pul S. Semi-rigid behaviour of web-flange junctions of open-web
pultruded composites. Proceedings of the International Conference on FRP Composites 2009, San
Francisco, California.
[6] Feo L, Mosallam AS, Penna R. Mechanical behavior of web-flange junctions of thin-walled
pultruded I-profiles: An experimental and numerical evaluation. Composites: Part B 2013; 48: 18-39.
[7] Ascione F, Mancusi G. The influence of the web-flange junction stiffness on the mechanical
behavior of thin-walled pultruded beams. Composites: Part B 2013; 55: 599-606.
[8] Francesco Ascione (2014). Influence of initial geometric imperfections in the lateral buckling
problem of thin walled pultruded GFRP I-profiles. Composite Structures 2014; doi:
10.1016/j.compstruct.2014.02.002
114
Effective modeling of multilayered composites with cohesive and
imperfect interfaces
Roberta Massabò*, Francesca Campi
DICCA, University of Genoa, Via Montallegro 1, 16145 Genova, Italy
email: Roberta.massabo@unige.it; francesca.campi@unige.it
Keywords: multilayered composites, cohesive interfaces, delamination, modeling
In multilayered composites with interfacial imperfections, such as imperfect bonding of the layers or
delaminations, or where the plies are separated by thin interlayers allowing relative motion, the
displacement field is highly discontinuous in the thickness, with a characteristic zig-zag pattern and
jumps at the layer interfaces. Stresses also have large variations, especially when the interlayers are
highly deformable or when the layers are fully debonded. These effects cannot be captured using
classical, first- or higher-order single-layer theories for beams, plates and shells, and are typically
addressed with discrete-layer approaches. Recently, the authors formulated a mechanical model for
multilayered plates with an arbitrary number of imperfect interfaces and delaminations loaded
dynamically [1-4]. The formulation is in the framework of the discrete-layer approach and the
interfaces are described through affine traction laws which relate interfacial tractions and relative
displacements at the layer interfaces. Homogenization and variational techniques have been used to
define novel equilibrium equations depending on only six generalized displacement functions. The
proposed model revises, corrects and extends to systems with cohesive interfaces, theories which have
been proposed for fully bonded plates in [5,6] and for plates with linearly elastic interfaces in [7-10].
Comparison with 2D elasticity solutions shows that complex discontinuous fields in thick, highlyanisotropic plates with an arbitrary number of sliding-interfaces are accurately predicted. The
approach extends the range of problems which can be solved analytically compared to discrete-layer
models where the unknowns depend on the number of layers and interfaces used to discretize the
system (e.g. [11,12]). The affine traction laws assumed in the derivations may describe arbitrary
branches of piecewise linear functions approximating nonlinear traction laws to represent different
interfacial mechanisms. At the meeting preliminary results on the application of the proposed theories
to study the evolution of systems with generally nonlinear cohesive traction laws will be presented.
Acnowledgements: work supported by U.S. Office of Naval Research, no. N00014-05-1-0098 and
Italian MIUR, Prin09 no. 2009XWLFKW.
References
[1] Massabò R. and Campi F., Modeling laminated composites with cohesive interfaces: a
homogenization approach, proceedings of the XXI Congress of the Italian Ass. of Theor. & Applied
Mech. AIMETA 2013, Torino, Sept. 2013, 1-10, ISBN 978-88-8239-183-6.
[2] Massabò, R., (2013), A homogenized model for progressive delamination of laminated structures
with cohesive interfaces loaded dynamically, proceedings of the U.S. O.N.R. meeting of the Solid
Mechanics Program, Arlington, VA, U.S.A., 87-98
[3] Massabò, R., Campi, F., Assessment and correction of theories for multilayered plates with
imperfect interfaces, submitted for consideration for publication in Int. Journal, 2014.
[4] Massabò, R., Campi, F., An efficient approach for multilayered beams and wide plates with
imperfect interfaces and delaminations, submitted for consideration for publication in Int. Journal,
2014.
115
[5] Di Sciuva, M., (1986), Bending, vibration and buckling of simply supported thick multilayered
orthotropic plates: an evaluation of a new displacement model, J. Sound and Vibrations, 105 (3), 425442.
[6] Di Sciuva, M., (1987), An improved shear-deformation theory for moderately thick multilayered
anisotropic shells and plates, J. Applied Mechanics, 54, 589-596.
[7] Cheng, Z. Q., Jemah, A. K., and Williams, F. W., (1996), Theory for multilayered anisotropic
plates with weakened interfaces, J. Applied Mechanics, 63, 1019-1026.
[8] Schmidt, R., and Librescu, L., “Geometrically nonlinear theory of laminated anisotropic composite
plates featuring interlayer slips,” Nova Journal of Mathematics, Game Theory, and Algebra, 5, 131147 (1996).
[9] Di Sciuva, M., (1997) An improved shear-deformation theory for moderately thick multilayered
anisotropic shells and plates, AIAA Journal, 35 (11), 1753-1759.
[10] Librescu, L., and Schmidt, R., (2001) A general theory of laminated composite shells featuring
interlaminar bonding imperfections, Int. J. Solids and Structures, 3355-3375.
[11] Williams, T. O., and Addessio, F. L., “A general theory for laminated plates with
delaminations,” Int. Journal of Solids and Structures, 34, 2003-2024 (1997).
[12] Andrews, M.G., Massabò, R., Cavicchi, A., B.N. Cox (2009), Dynamic interaction effects of
multiple delaminations in plates subject to cylindrical bending, Int. Journal of Solids and Structures,
46, 1815-1833.
116
Micropolar and second-gradient homogenization of chiral cellular solids
Andrea Bacigalupo1,a and Luigi Gambarotta2,b *
1
Department of Civil, Environmental and Mechanical Engineering, University of Trento,
via Mesiano, 77, 38123, Trento, Italy
2
Department of Civil, Chemical and Environmental Engineering, University of Genoa,
via Montallegro, 1, 16145, Genoa, Italy
a
andrea.bacigalupo@unige.it, bluigi.gambarotta@unige.it
Keywords: chiral microstructure, auxetic materials, non-local homogenization, cellular materials.
The homogenization of auxetic cellular solids having periodic hexachiral and tetrachiral
microstructure is dealt with two different techniques. The first approach is based on the representation
of the cellular solid as a beam-lattice to be homogenized as a micropolar continuum. The second
approach is developed to analyse periodic cells conceived as a two-dimensional domain consisting of
deformable portions such as the ring, the ligaments and possibly an embedded matrix internally to
these. This approach is based on a second displacement gradient computational homogenization
proposed by the Authors [1, 2, 3]. The elastic moduli obtained by the micropolar homogenization are
expressed in analytical form from which it appears explicitly their dependence on the parameter of
chirality, which is the angle of inclination of the ligaments with respect to the grid of lines connecting
the centers of the rings. For hexachiral cells, the solution given in [4] is found, showing the auxetic
property of the lattice together with the elastic coupling modulus between the normal and the
asymmetric strains; a property that has been confirmed here for the tetrachiral lattice. Unlike the
hexagonal lattice, the classical constitutive equations of the tetragonal lattice turns out to be
characterized by the coupling between the normal and shear strains through an elastic modulus that is
an odd function of the parameter of chirality. Moreover, this lattice is found to exhibit a remarkable
variability of the Young's modulus and of the Poisson's ratio with the direction of the applied uniaxial
stress. The properties of the equivalent micropolar continuum are qualitatively detected also in the
equivalent second-gradient continuum. Moreover, for both the hexachiral and the tetrachiral cellular
material, the second-order elastic moduli obtained through the homogenization technique are in
agreement with the invariance properties defined in [5]. This investigation, that is justified by the need
of understanding the effects of the compliance of the rings and of the filling material, has shown that it
is sufficient a very soft filling material to get significant increases in the Poisson's ratio, until to lose
the auxetic property of these cellular solids. Finally, the experimental and numerical results obtained
by Alderson et al. [6] are compared to the theoretical ones obtained by the homogenization techniques
here considered.
References
[1]
Bacigalupo A. Gambarotta L., Second-order computational homogenization of heterogeneous
materials with periodic microstructure, ZAMM Z. Angew. Math. Mech., 90, 796–811, 2010.
[2]
Bacigalupo A. Gambarotta L., Second-gradient homogenized model for wave propagation in
heterogeneous periodic media, Int J Solids Struct, 51, 1052–1065, 2014.
[3]
Bacigalupo A. Second-order homogenization of periodic materials based on asymptotic
approximation of the strain energy: formulation and validity limits, Meccanica, DOI 10.1007/s11012014-9906-0, 2014 (to appear).
[4]
Liu X.N., Huang G.L., Hu G.K., Chiral effect in plane isotropic micropolar elasticity and it’s
application to chiral lattices, J Mech Phys Solids, 60, 1907-1921, 2012.
[5]
Auffray N., Bouchet R., Bréchet Y., Derivation of anisotropic matrix for bi-dimensional strain
gradient elasticity behavior, Int J Solids Struct, 46, 440-454, 2009.
117
[6]
Alderson A., Alderson K.L., Attard D., Evans K.E., Gatt R., Grima J.N., Miller W., Ravirala
N., Smith C.W., Zied K., Elastic constant of 3-, 4- and 6-connected chiral and anti-chiral honeycombs
subject to uniaxial in-plane loading, Composite Science and Technology, 70, 1042-1048, 2010.
118
TWSME of NiTi strips in free bending conditions: experimental and
theoretical approach
A. Fortini1, a *, M. Merlin1,b , R. Rizzoni1,c and S. Marfia2,d
1
Department of Engineering (EnDiF), University of Ferrara, Italy
2
Department of Civil and Mechanical Engineering (DiCeM),
University of Cassino and Lazio Meridionale , Italy
a
annalisa.fortini@unife.it, bmattia.merlin@unife.it, craffaella.rizzoni@unife.it, dmarfia@unicas.it
Keywords: NiTi-based alloys, two-way shape memory effect, thermomechanical training, bending.
Shape memory alloys (SMAs) are a class of materials with unique thermomechanical characteristics,
which stem from a crystalline diffusionless and reversible phase transformation between austenite and
martensite phases. The ability of these alloys to recover seemingly permanent strains is strongly
dependent upon thermal and/or stress loading conditions. As an inherent property of the material,
SMAs show the one-way shape memory effect (OWSME) whereby it is possible to recover large
strains when the material is deformed in the martensitic phase and heated upon reaching the austenitic
finish temperature. In addition to the well-known OWSME, shape memory alloys can exhibit, through
specific thermomechanical cycling treatments, the two-way shape memory effect (TWSME). The
TWSME behaviour gives the macroscopic shape change upon heating and cooling without any
external applied load: as a result, the material is able to spontaneously recover the memorised hot and
cold shapes. SMAs are frequently used in the engineering field as actuating elements in functional
structures, which enable to integrate multiple functions within a single component providing improved
system performances upon demand.
Several authors have theoretically and experimentally investigated the bending behaviour of SMA
elements embedded in composite structures [1-4]. Baz et al. considered the behaviour of a composite
beam reinforced with NiTi strips which were thermally trained to memorise a bent shape, pre-strained
to a flat shape, and then embedded into the sleeves of the composite plate [1]. On heating, they tend to
recover the memorised shape forcing the plate to bend. The actuation characteristics of NiTi strips,
working in bending and fixed to the surface of a polymeric plate, were experimentally and
theoretically investigated in [4-6]. Most of these investigations are based on the OWSME and, as a
consequence, the recovery of the composite is demanded to the elasticity of the polymeric structure.
The possibility to realise improved structures in which the recovery takes advantage of the TWSME
of the SMA elements represents the purpose of the present work. As a result, a combined experimental
and theoretical approach is performed. The study deals with the TWSME induced by
thermomechanical cycling treatments, by means of the so-called shape memory cycling method, on
equiatomic NiTi strips. To this end, the strips are thermally treated to memorise a bent shape with a
uniform curvature and subsequently trained as follow: (a) loaded in the martensitic state to the cold
shape, (b) unloaded, (c) heated above the austenitic finish temperature and (d) cooled and loaded
again. After this training procedure, to evaluate the ability of the strips to recover both the hot and
cold shapes, several thermal cycling are performed. The transformation from martensite to austenite is
achieved by a hot air stream flow while the transformation from austenite to martensite is realised
through natural convection. The TWSME is calculated through digital image analysis and evaluated
from the difference between the hot and cold curvature values.
The thermo-mechanical behaviour of the SMA undergoing bending is simulated using a
phenomenological model proposed in [7,8]. The model is based on the use of strain, ε, and
temperature, T, as control variables. Two phases are considered for the SMA material, in particular
the volume fractions of single-variant martensite and austenite, which are assumed as internal
variables. Kinetic laws linking the internal and external variables are thus assumed to describe the
evolution of the volume fractions occurring during the phase transformations. Shape recovery
119
simulations are presented for NiTi strip undergoing uniform bending and compared with the
experimental observations.
References
[1] A. Baz, T. Chen, J. Ro. Shape control of Nitinol-reinforced composite beams. Composites Part B:
Engineering 31 (2000) 631-642.
[2] S. Marfia, R. Rizzoni. One-dimensional constitutive SMA model with two martensite variants:
analytical and numerical solutions. European Journal of Mechanics / A Solids 40 (2013) 166-185.
[3] M. Merlin, R. Rizzoni. Design of a polymeric prototype with variable geometry controlled by
shape-memory strips. In: Proceedings of AGS'10 - Advances in Geomaterials and Structures. Djerba,
Tunisia, May 10-12, 2010.
[4] M. Merlin, R. Rizzoni, Thermoelastic transformation behavior of NiTi thin strips in bending:
experiments and modelling. Warsaw University of technology, CMM 2011, Computer methods in
mechanics, 1-2, 2011.
[5] R. Rizzoni, M. Merlin, D. Casari. Shape recovery behaviour of NiTi strips in bending: experiments
and modelling. Continuum Mechanics and Thermodynamics 24 (2012) 1-21.
[6] M. Merlin, C. Soffritti, A. Fortini. Studio del trattamento termico di lamine a memoria di forma
NiTi per la realizzazione di strutture funzionali. La Metallurgia Italiana 11-12 (2011) 17-21.
[7] S. Marfia, E. Sacco, J.N. Reddy, Superelastic and shape memory effects in laminated shape
memory alloy beams, AIAA Journal 41 (2003) 100-109.
[8] F. Auricchio, S. Marfia, E. Sacco, Modeling of SMA materials: training and two way memory
effects, Computers and Structures, 81 (2003) 2301–2317.
120
Discrete-to-continuum approaches for complex materials as ‘Non–
Simple’ continua
Patrizia Trovalusci
Department of Structural and Geotechnical Engineering,
Sapienza – University of Rome, via A. Gramsci 53, 00197 Rome, Italy
patrizia.trovalusci@uniroma1.it
Keywords: coarse-graining, continua with microstructure, size dependent continua, composite
materials
„Old Ideas for New Models Across Materials“1
The mechanical behaviour of complex materials, characterized at finer scales by the presence of
heterogeneities of significant size and texture, strongly depends on their microstructural features. By
lacking in material internal scale parameters, the classical continuum does not always seem
appropriate to describe the macroscopic behaviour of such materials, taking into account the size, the
orientation and the disposition of the micro heterogeneities. This calls for the need of non–classical
continuum descriptions obtained through multiscale approaches aimed at deducing properties and
relations by bridging information at proper underlying micro–level via energy equivalence criteria.
Attention will be first focused on ‘mechanistic’ corpuscular-continuous models, as originated by the
molecular models developed in the 19th century to give an explanations per causas of elasticity. In
particular, we examine the ‘mechanistic-energetistic’ approach by Voigt and Poincaré who, when
dealing with the paradoxes coming from the search of the exact number of elastic constants in linear
elasticity, respectively introduced moment and multi-body interactions models. Thus, overcoming the
difficulties related to the so-called central--force scheme (Navier, Cauchy), which led to experimental
discrepancies [1]-[3].
Current researches in solid state physics, as well as in mechanics of materials, show that energyequivalent continua obtained by defining direct links with lattice systems are still among the most
promising approaches in material science [4], [5]. Aim of this study is to emphasize the suitability of
adopting discrete to continuous models, based on a generalization of the so-called Cauchy-Born (here
recognized also as Voigt and Poincaré) rule used in crystal elasticity and in the classical molecular
theory of elasticity, which can naturally lead to the identification of continua with additional degrees
of freedom (micromorphic, multifield, etc. [6], [7]). In accordance with the definition of complex
continua in [8], these models, which are essentially ‘non-local’ models with internal length and
dispersive properties, are here called ‘non-simple’ continua. In particular, it will be shown as, within
the general framework of the principle of virtual powers, microstructured continuous formulations can
be derived on the basis of a correspondence map relating the finite number of degrees of freedom of
discrete models to the continuum kinematical fields; thus providing a guidance on the choice of nonstandard continuum approximations for heterogeneous media [9]. The circumstances in which the
inadequacy of the hypothesis of classical lattice mechanics still call for the need of improved
constitutive models, circumventing the hypothesis of homogeneous deformations or the central-force
scheme via non-convex energy models or continua with additional degrees of freedom, will also be
discussed.
Some applications of such approaches will be shown with reference to microcracked composite
materials, ranging from fibre-reinforced composites, or porous metal-ceramic composites and ceramic
matrix composites up to masonry-like material. Further developments concerning the comparison with
1
Sentence by E. Aifantis.
121
homogenization methods based on boundary problems solutions developed for generalized continua
will be finally introduced.
References
[1] P. Trovalusci, D. Capecchi and G. Ruta. Genesis of the multiscale approach for materials with
microstructure. Archive of Applied Mechanics, 79:981–997, 2009.
[2] D. Capecchi, G. Ruta, and P. Trovalusci. Voigt and Poincaré’s mechanistic–energetic approaches
to linear elasticity and suggestions for multiscale modelling. Archive of Applied Mechanics,
81(11):1573–1584, 2011.
[3] P. Trovalusci. Molecular approaches for multifield continua: origins and actual developments with
applications to fibre composites and masonry-like materials. In T. Sadowski and P. Trovalusci (Eds.).
Multiscale Modelling of Complex Materials: phenomenological, theoretical and computational
aspects. CISM, Courses and Lectures. In press.
[4] M. Ortiz and R. Phillips. Nanomechanics of defect in solids. In T. Wu E. Van der Giessen, editor,
Advances in Applied Mechanics, volume A36, pages 1–73. Academic Press, San Diego, 1999.
[5] A. Finel, D. Maziére, and M. Vèron (Eds). Thermodynamics, Microstructures and Plasticity,
volume 108 of NATO Science Series: II: Mathematics, Physics and Chemistry. Kluwer, Dordrecht,
2003.
[6] G. Capriz. Continua with Microstructure. Springer-Verlag, Berlin, 1989.
[7] A. C. Eringen. Microcontinuum Field Theories. Springer–Verlag, New York, 1999.
[8] G. Capriz and P. Podio-Guidugli. Whence the boundary conditions in modern continuum
physics. In Atti dei Convegni Lincei, volume 210, pages 19–42, 2004.
[9] P. Trovalusci, A. Pau. Derivation of microstructured continua from lattice systems via principle of
virtual works. The case of masonry-like materials as micropolar, second gradient and classical
continua. Acta Mechanica, DOI: 10.1007/s00707-013-0936-9, 2013.
122
Constitutive Behavior of FRCM Materials for Structural Plating: an
experimental study
Luigi Ascione, Anna D’Aponte, Geminiano Mancusi
Department of Civil Engineering, University of Salerno, Italy
l.ascione@unisa.it, adaponte@unisa.it, g.mancusi@unisa.it
Keywords: innovative materials, FRCM, inorganic matrix, constitutive behavior.
An experimental study on the constitutive behavior of fiber reinforced cementitious materials (FRCM)
is presented.
The basic concept of the FRCM is the combination of a fibre fraction with an inorganic matrix made
of cement or lime mortar. Generally, the reinforcing mesh is composed of dry fibres of carbon, glass,
aramid, PBO or ultra high tensile strength steel.
The growing success of FRCMs is due to their performance in terms of a good resistance in high
temperature and fire exposure, or a satisfying water vapor permeability, the possibility to be applied
on a wet substrate.
Despite of such a diffusion, the mechanical behavior of FRCMs and their failure mechanisms have not
been adequately investigated. The lack of knowledge mainly concerns the de-bonding mechanism and
the stress concentration at the FRCM to substrate interface. As a consequence, there is not a general
agreement on which criteria have to be used for the qualification, which requirements are relevant
within the acceptance procedures, the design and quality control of structural strengthening
techniques.
The aim of this scientific work is to present the state of the art of a recent experimental program -still
in progress at the Laboratory for Tests of Materials and Structures of the University of Salerno, which
is aimed at detecting the main features of the constitutive behavior of these materials.
The experimental tests were made on many FRCM specimens tested under uniaxial tensile loads.
The results here discussed belong to a larger experimental program which has been cooperated by
many academic laboratories, aimed at identifying the basic features of the constitutive behavior of
FRCM materials.
The final goal is the development of a guide-line for the qualification of FRCM and the regulation of
their use within the context of structural plating, in line with a similar technical guideline recently
licensed in the USA by the ICC Evaluation Service.
References
[1] Triantafillou, T.C., Papanicolaou, C. G., Zissimopoulos, P., Laourdekis, T., “Concrete
confinement with textile-reinforced mortar jackets, ACI Structural Journal, 103 (1), 28-37 (2006).
[2] Triantafillou, T.C., Papanicolaou, C. G., “Shear strengthening of reinforced concrete members
with Textile reinforced Mortar (TRM) jackets”, Materials and Structures, 39, 93-103 (2006).
[3] Di Tommaso, A. Focacci, F., Mantegazza, G., “PBO-FRCM composites to strengthen r.c. beams:
mechanics of adhesion and efficiency”, Proc. Fourth International Conference on FRP Composites in
Civil Engineering (CICE 2008), Zurich, Switzerland, 22-24 (2008).
[4] Ombres, L., “Flexural analysis of reinforced concrete beams strengthened with a cement based
high strength composite material”, Composite Structures, 94(1), 143-145 (2011).
[5] D'Ambrisi, A., Focacci, F., Caporale, A. Strengthening of masonry-unreinforced concrete railway
bridges with PBO-FRCM materials, Composite Structures, 102, 193-204 (2013).
123
[6] Mantegazza, G., Gatti, A., Barbieri, A. Fibre reinforced cementitious matrix (FRCM)-advanced
composite material and emerging technology for retrofitting concrete and masonry buildings,
Proceedings of the 3rd International Conference on Bridge Maintenance, Safety and Management Bridge Maintenance, Safety, Management, Life-Cycle Performance and Cost, 1069-1070 (2006).
[7] Prota, A., Marcari, G., Fabbrocino, G., Manfredi, G., Aldea, C. Experimental in-plane behavior of
tuff masonry strengthened with cementitious matrix-grid composites, Journal of Composites for
Construction, 10 (3), art. no. 007603QCC, 223-233 (2006).
[8] Carozzi F. G., Milani G., Poggi C. Mechanical properties and numerical modeling of Fabric
Reinforced Cementitious Matrix (FRCM) systems for strengthening of masonry structures, Composite
Structures, vol. 107, 711–725 (2014).
[9] Babaeidarabad, S., De Caso, F., Nanni, A. URM walls strengthened with fabric-reinforced
cementitious matrix composite subjected to diagonal compression, Journal of Composites for
Construction, 18 (2), (2014).
[10] Trapko, T. Stress-strain model for FRCM confined concrete elements, Composites Part B:
Engineering, 45 (1), 1351-1359 (2013).
[11] Ascione L., Poggi C., Savoia M. On the mechanical behaviour of FRCM composites,
Proceedings of the XXI Conference of The Italian Association of Theoretical and Applied Mechanics
(AIMETA), Turin (Italy), September 2013.
[12] D'Ambrisi, A., Feo, L., Focacci, F. Experimental and analytical investigation on bond between
Carbon-FRCM materials and masonry, Composites Part B: Engineering, 46, 15-20 (2013).
[13] D'Ambrisi, A., Feo, L., Focacci, F. Experimental analysis on bond between PBO-FRCM
strengthening materials and concrete, Composites Part B: Engineering, 44 (1), 524-532 (2013).
[14] D'Ambrisi, A., Feo, L., Focacci, F. Bond-slip relations for PBO-FRCM materials externally
bonded to concrete, Composites Part B: Engineering, 43 (8), 2938-2949 (2012).
[15] Mazzotti C., Savoia M., Ferracuti B. A new single-shear set-up for stable delamination tests on
FRP-concrete joints, Construction and Building Materials, vol. 23(4), 1529-1537, ISSN: 0950-0618
(2009).
[16] Mazzotti C., Savoia M., Ferracuti B. An Experimental Study on Delamination of FRP Plates
Bonded to Concrete, Construction and Building Materials, vol. 22, 1409-1421, ISSN: 0950-0618
(2008).
[17] ICC Evaluation Service, Acceptance criteria for masonry and concrete strengthening using
Fabric-Reinforced Cementitious Matrix (FRCM) composite systems, AC434, Draft August 1, 2011.
124
A new auxetic lattice model
Luigi Cabras1,a Michele Brun*,2,3,b
1
Dipartimento di Ingegneria Civile, Ambientale e Architettura,
Università di Cagliari, Piazza d'Armi, I-09123 Cagliari, Italy
2
Dipartimento di Ingegneria Meccanica, Chimica e dei Materiali,
Università di Cagliari, Piazza d'Armi, I-09123 Cagliari, Italy
3
Department of Mathematical Sciences, University of Liverpool, UK
a
luigi.cabras@unica.it, bmichele.brun@dimcm.unica.it
Keywords: Auxetic materials ,negative Poisson's Ratio, microstructured media, elasticity
A new model of auxetic material is proposed. The lattice structure can be design in order to have a
Poisson’s ratio approaching the limit of -1.
Experimental evidence of the extreme properties of the microstructured lattice has been obtained as
shown in Figure1. Three different microstructures have been designed and analysed, two with threefold symmetry and one with a four-fold symmetry.
Figure 1: Deformation of the auxetic lattice subjected to a horizontal tensile traction. The auxetic lattice is composed by
cross-shaped elements of thermoplastic polymer ABS
The micro-structured media are within the class of unimode materials [1,2] where the only easy mode
of deformation is pure dilatation (plane dilatation in a two-dimensional system).
The determination of the effective properties for the ideal and real microstructures have been
performed analyticaly obtaining explicit expression for the effective consitutive parameters of the
quasi-static homogenised behavior, which depend on the constitutive behavior of its constituents, on
the microstrcuture and on the actual configuration of the system.
The limiting behavior of the homogenised sstructure approaching the constitutive stability limit has
been considered in detail.
References
[1] Milton, G.W., Cherkaev, A.V.,1995. Which elasticity tensors are realizable? ASME J. Eng.
Mater. Technol. 117, 483-493.
[2] Milton, G.W., 2013. Complete characterization of the macroscopic deformations of periodic
unimode metamaterials of rigid bars and pivots. J. Mech. Phys. Solids 61(7) 1543-1560.
125
Cloaking in flexural waves
D. Colquitt1,a, M. Brun2,3 b *, M. Gei4,c, A.B. Movchan3,d, N.V. Mochan3,e
and I.S. Jones5,f
1
Department of Mathematics, Imperial College London, South Kensington, London, SW7 2AZ, U.K.
2
Dipartimento di Ingegneria Meccanica, Chimica e dei Materiali, Universit´a di Cagliari, Piazza
d’Armi, I-09123 Cagliari, Italy
3
Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 3BX, U.K.
4
Department of Civil, Environmental and Mechanical Engineering, University of Trento,
I-38123 Trento, Italy
5
School of Engineering, John Moores University, Liverpool, L3 3AF, U.K.
a
D.J.Colquitt@liverpool.ac.uk, bmbrun@unica.it, cmgei@unitn.it, cabm@liv.ac.uk@unitn.it,
d
nvm@liv.ac.uk@unitn.it, I.S.Jones@ljmu.ac.uk
Keywords: flexural waves, cloaking, transformation elastodynamics, plates, invisibility.
We address the probem of transformation elastodynamics in the design of invisibility cloaks for
flexural waves in thin elastic plates. The problem was already considered in [1,2] where approximated
cloaking theories were developed. Based on the previous models experiments were recently
performed by the group led by Wegener [3], showing good cloaking effects in a limited range of
frequencies.
In contrast with the Helmholtz equation, the general form of the partial differential equation is not
invariant with respect to the cloaking transformation. However, we show here that the transformed
equations can have a physical interpretation in the framework of the linear theory of pre-stressed
plates.
References
[1] Farhat M, Guenneau S, Enoch S. 2009 Ultrabroadband elastic cloaking in thin plates. Physical review letters 103, 024301.
[2] Farhat M, Guenneau S, Enoch S, Movchan AB. 2009 Cloaking bending waves propagating
in thin elastic plates. Physical Review B 79, 033102.
[3] Stenger N, Wilhelm M, Wegener M. 2012 Experiments on elastic cloaking in thin plates. Physical
Review Letters 108, 014301.
126
A contact problem in couple-stress thermoelasticity
Thanasis Zisis1,a and Francesco Dal Corso1,b
1
University of Trento, via Mesiano 77, 38123 Trento, Italy
a
zisis@metal.ntua.gr, bfrancesco.dalcorso@ing.unitn.it
Keywords: Contact Mechanics, Thermal Effects, Micromechanics
When contact scales in contact problems reduce progressively, the micro-mechanical response
becomes important and strongly affects the macroscopic behaviour.
General solutions for a characteristic plane strain contact problem are provided within the framework
of the generalized continuum theory of couple-stress thermoelasticity [1], extending previous results
for classical thermoelastic materials [2]. Considering indentation of a deformable half-plane by a
heated flat punch where heat is conducted into the half-plane through the contact area (Figure 1), the
solution is derived by solving singular integral equations resulted from a treatment of the mixed
boundary value problems via integral transforms and generalized functions.
As the internal material length increases, significant departure from the classical solution is predicted,
showing inadequateness of classical contact mechanics to analyze indentation problems when reduced
contact scale is considered.
Figure 1: Schematic representation of the plane problem. The flat punch under the action of the force P is pressed into the
surface. A temperature difference between the two bodies under contact induces a heat flux Q and the state of thermal
stress is expected to alter the macroscopic contact characteristics, essentially modifying the contact area b. The general
response is governed by the micromechanical length introduced by the generalized continuum theory under consideration.
References
[1] W. Nowacki. Couple-Stress in the theory of thermoelasticity. Irreversible Aspects of Continuum
Mechanics and Transfer of Physical Characteristics in Moving Fluids IUTAM Symposia 1968, pp
259-278
[2] M. Comninou, J. R. Barber, J. Dundurs. Heat conduction through a flat punch. Journal of Applied
Mechanics 48 (4), 871-875
127
Flutter analysis of piezoelectric laminate beams in MEMS
Raffaele Ardito1, a * and Rocco Musci1,b
1
Department of Civil and Environmental Engineering, Politecnico di Milano
Piazza Leonardo da Vinci 32, 20133 Milan, Italy
a
raffaele.ardito@polimi.it, brocco.musci@mail.polimi.it
* Corresponding author
Keywords: Fluid-Structure Interaction; Aeroelastic effects; Micro-electro-mechanical systems;
piezoelectric behavior.
Piezoelectric materials are widely used in micro-electro-mechanical systems (MEMS), which
represent a huge class of devices characterized by the coupling between electronics and mechanics in
order to obtain microscopic sensors and actuators. Piezoelectric materials are exploited considering
both the “direct effect”, e.g. in energy harvesters, and the “indirect effect”, for the cases of resonators,
micropumps and other actuators. This paper is devoted to the study of piezoelectric laminate beams in
the presence of aeroelastic effects due to the interaction of the structure with a fluid flow. More
specifically, the analytical conditions for the onset of flutter instability are studied, with the purpose of
providing a sound basis for further studies focused on energy harvesting from fluid flows.
The “classical flutter” [1] is characterized by two degrees of freedom, torsional rotation and vertical
translation, coupled in a flow-driven, unstable oscillation. According to the characteristics of the
instability mechanisms, the motion of the structure will either decay or diverge according to whether
the energy of motion extracted from the flow is less than or exceeds the energy dissipated by the
system. In the case of standard beams, dissipation is related to structural damping; for piezoelectric
laminate beams, connected to an external circuitry, an electrical source of damping is superposed.
Consequently, the flutter analysis should be rearranged in the case of piezoelectric materials.
The multi-physics simulation of piezoelectric effect can be obtained by considering that the structural
members are represented by laminate composites with piezoelectric and silicon layers [2], the active
layer is then attached to an external load resistance which reproduces the circuitry employed for the
power management. The sectional behavior of the beam is studied through the Classical Lamination
Theory (CLT, specifically modified in order to introduce the piezoelectric coupling) and a reduced
order model is built through separation of time and space variables and the introduction of a suitable
shape function for the beam deformation. Finally, the coupled equations of motion (written in a
matrix-vector notation) include the equilibrium equation in the transverse direction, the equilibrium
equation for torsion and the balance for electric charges. The two equilibrium equations contain the
self-excited aerodynamic lift force and torque, which in turn depend on the flux velocity and on the
displacement components and their first derivatives by means of specific coefficients (“flutter
derivatives”). The latter values can be obtained by numerical simulations or by wind-tunnel
experimental data. A specific difficulty, when dealing with MEMS, is related to the low Reynolds
number, which is a strict consequence of the small dimensions and fluid velocities. Some specific data
for flutter derivatives in this case have been found in [3]. An eigenvalue analysis of the governing
equations is carried out, by considering a Quadratic Eigenvalue Problem [4]. Since the governing
matrices are not symmetric, complex eigenvalues are obtained (in conjugated pairs): the critical flutter
velocity is obtained as the condition which divides a decaying solution (positive real parts of the
eigenvalues) from divergent oscillation (negative real parts of the eigenvalues.
The reference problem (see Figure 1) is a simple cantilever beams with a two-layer cross-section. The
external circuitry can be represented by a simple resistive element or by a resistive-inductive system.
An example of the numerical results is reported in Figure 2, which is referred to the following
cantilever dimensions: length 200 μm, width 25 μm, overall thickness 8 μm, piezoelectric thickness 2
μm. The standard mechanical parameters of silicon and PZT are adopted. The critical velocity,
128
obtained in the absence of piezoelectric coupling, is Ucr = 3.25m/s. By introducing the electromechanical coupling this value increases according to the growth of damping. The eigenvalue
response is very different for the two external circuits applied, presenting a greater damping in the
case of resistive-inductive solution, with a critical velocity Ucr = 15.1 m/s . Several other comparative
considerations are carried out on the basis of the obtained results.
In view of its ability to reduce the dissipative component, FI can be exploited for the energy
harvesting purpose, joining this aeroelastic phenomenon with another type of excitation, like an
inertial forcing.
Figure 1: Schematic view of the reference problem.
a) R=30Ω
b) R=100Ω, L=0.024mH
Figure 2: Damping ratio with electro-mechanical coupling: a) RC circuit b) RLC circuit.
References
[1]
R.H. Scanlan, E. Simiu, Wind Effects on Structures, third ed., John Wiley and Sons, 1996.
[2]
R. Ardito, E. Bertarelli, A. Corigliano, G. Gafforelli, On the application of piezolaminated
composites to diaphragm micropumps. Compos. Struct. 99 (2013) 231-240.
[3]
L.Bruno, D.Fransos, Evaluation of the Reynolds number effects on the flutter derivatives of a
flat plate by means of a new computational approach, J. of Fluid and Struct. 24 (2008) 1058-1076.
[4]
F. Tisseur, K. Meerbergen. The quadratic eigenvalue problem, SIAM Rev. 43 (2001) 235-286.
129
Variational approach to damage mechanics with plasticity and
nucleation of cohesive cracks
Roberto Alessi1,a*, Achille Paolone1,b and Stefano Vidoli1,c
1Department
of Structural and Geotechnical Engineering, Sapienza, University of Rome, Italy
roberto.alessi@uniroma1.it, bachille.paolone@uniroma1.it, cstefano.vidoli@uniroma1.it
a
Keywords: cohesive fracture, plasticity, gradient damage model, variational methods, stability.
It is now well established that gradient damage models are very efficient to account for the behavior
of brittle and quasi-brittle materials. Indeed, they have been used in the variational theory of fracture
as a regularization of the revisited Griffith’s law [1,2]. In this approach, the evolution of cracks is
governed by a principle of least energy (called global stability condition in the present paper) and it
turns out that it is possible to prove that (a family of) gradient damage models converge (in the sense
of Gamma-convergence) to Griffith’s model when the internal length contained in those models goes
to zero.
However, this type of “quasi-brittle” models are not able to account for residual strains and
consequently cannot be used in ductile fracture. Moreover there is no discontinuity of the
displacement in the damage strip before the loss of rigidity at its center, i.e. before the nucleation of a
crack. In other words such models cannot account for the nucleation of cohesive cracks, i.e. the
existence of surface of discontinuity of the displacement with a non vanishing stress. The natural way
to include such effects is to introduce plastic strains into the model and to couple their evolution with
damage evolution. Of course, this idea is not new and a great number of damage models coupled with
plasticity have been developed from the eighties in the spirit of [3].
But our purpose is to construct such models in a softening framework with gradient of damage terms
and to see how these models can account for the nucleation of cracks in presence of plasticity. In our
knowledge, the previous works are not able to go so far. Here we will adopt a variational approach for
rate-independent systems. The main ingredients are the following ones: (i) one defines the total energy
of the body in terms of the state fields which includes the displacement field and the internal variable
fields, namely the damage, the plastic strain and the cumulated plastic strain fields; (ii) one postulates
that the evolution of the internal variables is governed by the three principles of irreversibility,
stability and energy balance. In particular, the stability condition is essential as well for constructing
the model in a rational and systematic way as for obtaining and proving general properties. Besides,
we have the chance that the variational approach works and has been already developed both in
plasticity and in damage mechanics, even though only separately up to now. So, it “suffices” to
introduce the coupling by choosing the form of the total energy to obtain a model of gradient damage
coupled with plasticity.
Specifically, the presented model is in a three-dimensional setting, contains three state functions,
namely E(α), d(α) and σp(α) which give the dependence of the stiffness, the local damage dissipated
energy and the plastic yield stress on the damage variable. So, our choice of coupling is minimalist in
the sense that it simply consists in introducing this dependence of the yield plastic stress σp (α) on the
damage variable (with the natural assumption that σp(α) goes to 0 when the damages goes to 1). In
turn, by virtue of the variational character of the model, the product σp'(α) p of the derivative of the
function σp'(α) by the cumulated plastic strain p enters in the damage criterion and this coupling plays
a fundamental role in the nucleation of a cohesive crack [4].
Moreover the variational approach leads to a natural and rational way to define efficient numerical
algorithms since its intrinsic discrete nature. The adopted numeric scheme is an alternate minimization
algorithm in a finite element framework. Some preliminary numeric simulations will be presented in
order to highlight the potentialities of the model, Figure 1.
130
Figure 1: Numerical results for a simple 2D traction test, [4]: (a) the reference configuration; (b) the damage field plotted
on the deformed shape when U = 2 (blue, α=0; red, α=1); (c) the accumulated plastic strain field plotted on the deformed
shape when U = 2 (blue, p=min; red, p=max); (d) the average normal stress at x=L versus the applied external
displacement U.
References
[1] Francfort, G. A., Marigo, J.-J., 1998. Revisiting brittle fracture as an energy minimization
problem. Journal of the Mechanics and Physics of Solids 46 (8), 1319–1342.
[2] Ambrosio, L., Lemenant, A., & Royer-Carfagni, G. (2010). A variational model for plastic slip and
its regularization via, (2000), 1–32.
[3] Lemaitre, J., Chaboche, J., 1985. Mécanique des matériaux solides. Bordas.
[4] Alessi, R., Marigo, J.-J., & Vidoli, S. (2014). Gradient damage models coupled with plasticity:
variational formulation and main properties. Mechanics of Materials. In press.
131
Geometric numerical integrators based on the magnus expansion in
bifurcation problems for non-linear elastic solids
Anna Castellano1, Pilade Foti2*, Aguinaldo Fraddosio3, Salvatore Marzano4, and Mario
Daniele Piccioni5
1
DICAR – Politecnico di Bari, Via Orabona 4, 70125 Bari, Italy
2
3
4
anna.castellano@poliba.it, p.foti@poliba.it, a.fraddosio@poliba.it, s.marzano@poliba.it,
5
md.piccioni@poliba.it
Keywords: nonlinear elasticity, bifurcation, geometric numerical integrators, Magnus expansion.
In the framework of three-dimensional non-linear elasticity, a number of “small on large” bifurcation
problems lead to the analysis of a system of linear second order ODE’s with varying coefficients. It is
a common practice trying to reduce the system of linear second order ODE’s to a simpler nonautonomous first order linear ODE system. Nevertheless, the resulting differential system may be
somewhat complex and only numerically tractable; thus, it is crucial to adopt an adequate strategy for
obtaining an accurate numerical solution for determining the singular value and the corresponding
normalized bifurcation deformation field. For example, approximate expressions of the matricant of
non-autonomous linear ODE systems, based either on the multiplicative integral of Volterra or on the
truncated Peano expansion, have been recently proposed for bifurcation problems analyzed by means
of the Stroh approach within the so-called sextic formalism (see [1], [2] and [3]).
Here, we discuss an alternative numerical method – based on the Magnus expansion (cf. [4]) – which
furnishes an approximate exponential representations of the matricant of first order linear ODE
systems. The Magnus expansion belongs to the so-called geometric numerical integrators, based on
Lie-Group methods (see [5], [6] and [7] for appropriate references). To our knowledge, the
applications of this method are not widespread in the continuum mechanics literature, but it has been
shown in [8] that for problems involving singularities, bifurcations and wave propagation these
methods may be useful and accurate. In particular, the Magnus expansion method may be very
efficient in a number of applications on bifurcation analysis in continuum mechanics, since it features
the determination of approximate solutions that preserve at any order of approximation the same
qualitative properties of the exact (but unknown) solution; moreover, this method exhibits an
improved accuracy with respect to other frequently used numerical schemes.
As an application of the Magnus method, here we study the possibility of toroidal twist-like
bifurcations for an isotropic incompressible elastic tube subject to a primary pure circular shear. In
this context, we recently investigated in [9] the possibility for a compressible Levinson-Burgess
hollow cylinder subject to azimuthal shear to support axially periodic toroidal twist-like modes similar
to the Taylor-Couette axially periodic cellular patterns observed when a viscous fluid confined
between two differentially rotating concentric cylinders becomes unstable. We performed a numerical
analysis based on the Magnus expansion of the underlying non-autonomous system of linear ODE,
and conclude that severe state of shear may lead in solid bodies to periodic twist-like bifurcations of
the Taylor-Couette form.
Here, we extend the bifurcation analysis developed in [9] to the case of incompressible isotropic
elastic solids. We first show that for an arbitrary incompressible isotopic elastic material there exists
an equilibrium axisymmetric circular shear deformation. Then, in order to investigate whether this
primary deformation may bifurcate into an axially periodic toroidal twist-like mode, we study the
incremental boundary-value problem by restricting our attention to a class of incremental
displacements characterized by three unknown scalar functions of the radial coordinate and having
axial periodic structure. This leads to a inhomogeneous system of three second order ODE’s, which
we conveniently transform into a non-autonomous homogeneous system of six first order linear
ODE’s with homogeneous boundary conditions. We then determine existence conditions for the
132
emergence of bifurcating periodic cell patterns and apply our general results to the case of an elastic
tube modeled by an extended version of the Gent constitutive equation. The approximate matricant of
the resulting explicit differential problem and the first singular value of the bifurcating load
corresponding to a non-trivial twist-like solution are determined by employing a simplified version of
the Magnus method, characterized by a truncation of the Magnus series after the second term.
References
[1] A. Goriely, R. Vandiver, M. Destrade. Nonlinear Euler buckling. Proc. R. Soc. A, 464: 30033019, 2008.
[2] A. L. Shuvalov. A sextic formalism for the three-dimensional elastodynamics of cylindrically
anisotropic radially inhomogeneous materials. Proc. R. Soc. Lond., 459: 1611-1639, 2003.
[3] A. L. Shuvalov, O. Poncelet, M. Deschamps. General formalism for plane guided waves in
transversely inhomogeneous anisotropic plates. Wave Motion, 40: 413-426, 2004.
[4] W. Magnus. On the exponential solution of differential equations for a linear operator. Comm.
Pure Appl. Math., VII: 649-673, 1954.
[5] S. Blanes, F. Casas, J.A. Oteo, J. Ros. The Magnus expansion and some of its applications.
Physics Reports, 470: 151-238, 2009.
[6] A. Iserles, H.Z. Munthe-Kaas, S.P. Norsett, A. Zanna. Lie-group methods. Acta Numer., 9: 215365, 2000.
[7] A. Iserles, S.P. Norsett. On the solution of linear differential equations in Lie groups. Phil. Trans.
R. Soc. A, 357: 983-1019, 1999.
[8] C. J. Budd, M.D. Piggott. The geometric integration of scale-invariant ordinary and partial
differential equations. J. of Comput. and Appl. Mathematics, 128: 399-422, 2001.
[9] R. Fosdick, P. Foti, A. Fraddosio, S. Marzano, M. D. Piccioni. Taylor-like bifurcations for a
compressible isotropic elastic tube. Mathematics and Mechanics of Solids, ISSN: 1081-2865, doi:
10.1177/1081286513496576, 2013.
133
Experimental and numerical approaches for the ultrasonic
characterization of composite materials
Anna Castellano1, Pilade Foti2, Aguinaldo Fraddosio3, Salvatore Marzano4, and Mario
Daniele Piccioni5
1
DICAR – Politecnico di Bari, Via Orabona 4, 70125 Bari, Italy
2
3
4
anna.castellano@poliba.it, p.foti@poliba.it, a.fraddosio@poliba.it, s.marzano@poliba.it,
5
md.piccioni@poliba.it
Keywords: Composite materials, Non-Destructive Testing, Ultrasonic Immersion Test, Anisotropic
materials, Wave propagation.
The study of the mechanical behavior of non-traditional materials, such as composites and fiberreinforced materials, biological and polymeric materials imposes to face challenging problems both at
the theoretical and experimental level. In particular, an effective and emerging approach for the
mechanical characterization of complex materials is offered by ultrasonic tests. Indeed, it has been
observed that suitable measures of ultrasonic velocity waves allow for a enough detailed experimental
identification of materials whose response is determined by a large number of moduli. For such
applications, numerical simulations reveal very useful for a deeper understanding of the experimental
measurements and for efficiently arranging the experimental set up.
In this paper, we propose an innovative experimental set up for immersion ultrasonic testing aimed at
characterizing anisotropic materials. We focused our attention on unidirectional carbon fiber
reinforced composites that we modelled as transversely isotropic elastic materials. A new
experimental device together with an innovative software developed for managing the tests allow us to
identify the elastic constants of the material by measuring ultrasonic velocities along according
appropriate angles of incidence of the ultrasound beam. As a major result, we developed a
multiphysics numerical model for simulating experimental phenomena related to the propagation of
the ultrasonic beam like, for example, the reflection and the refraction of inclined beams at the
interface between fluid and solid phases, which typically occur in immersion tests.
The results of the numerical analysis closely reproduce the experimental behavior in immersion
ultrasonic tests and consequently may be considered as a good benchmark for designing future
experiments involving much more complex materials characterized by higher degree of anisotropy.
References
[1] S. Siva Shashidhara Reddy, K. Balasubramaniam, C.V. Krishnamurthy, M. Shankar. Ultrasonic
goniometry immersion techniques for the measurement of elastic moduli. Composite Structures, 67:
3–17, 2005.
[2] H. Seiner, M. Landa. Sensitivity analysis of an inverse procedure for determination of elastic
coefficients for strong anisotropy. Ultrasonics, 43: 253–263, 2005.
[3] T. Kundu. Ultrasonic Nondestructive Evaluation: Engineering and Biological Material
Characterization. CRC Press, 2004.
[4] A. Castellano, P. Foti, A. Fraddosio, S. Marzano, M.D. Piccioni, D. Scardigno. Simulation of an
Ultrasonic Immersion Test for the Characterization of Anisotropic Materials. COMSOL Conference
2012, Milan, 2012.
[5] D. Royer, E. Dieulesaint. Elastic Waves in Solids I. Free and Guided Propagation. Springer, 1996.
[6] A. Castellano, P. Foti, A. Fraddosio, S. Marzano, M.D. Piccioni. Mechanical characterization of
CFRP composites by ultrasonic immersion tests: experimental and numerical approaches. Submitted
to Composites Part B, 2014.
134
[7] A. Castellano, P. Foti, A. Fraddosio, S. Marzano, M.D. Piccioni. Mechanical characterization of
Apricena marble by ultrasonic immersion tests. SMART BUILT International Conference on
Structural Monitoring of Artistic and Historical Building Testimonies, Bari, March 2014.
135
A micromechanical four-phase model to predict the compressive failure
surface of cement concrete
Andrea Caporale1,a *, Raimondo Luciano1,b
1
Department of Civil and Mechanical Engineering, University of Cassino and Southern Lazio,
G. Di Biasio 43, 03043 Cassino (FR) Italy
a
a.caporale@unicas.it, bluciano@unicas.it
* Corresponding author
via
Keywords: Cement concrete, micromechanics, compressive strength.
In this work, a micromechanical model is proposed in order to predict the failure surface of cement
concrete subject to multi-axial compression. Cement concrete is one of most used materials in civil
constructions. Different strengthening techniques have been introduced in order to increase the
mechanical performances of cement concrete [1,2] and to strengthen existing structures made of
concrete [3,4]. Beyond the structural behavior, attention must be paid to the behavior of plain
concrete.
In a recent work [5], a four-phase micromechanical model has been proposed in order to simulate the
non-linear instantaneous pre-peak response of cement concrete subjected to monotonically increasing
loads; the concrete material is modeled as a composite with the following constituents: coarse
aggregate (gravel), fine aggregate (sand) and cement paste. The cement paste contains some voids
which grow during the loading process. In fact, the non-linear behavior of the concrete is attributed to
the creation of cracks in the cement paste; the effect of the cracks is taken into account by introducing
equivalent voids (inclusions with zero stiffness) in the cement paste. The three types of inclusions
(namely gravel, sand and voids) have different scales, so that the overall behavior of the concrete is
obtained by the composition of three different homogenizations; in the sense that the concrete is
regarded as the homogenized material of the two-phase composite constituted by the gravel and the
mortar; in turn, the mortar is the homogenized material of the two-phase composite constituted by the
sand inclusions and a (porous) cement paste matrix; finally, the (porous) cement paste is the
homogenized material of the two-phase composite constituted by voids and the pure paste; the pure
paste is an ideal material that does not contain voids or other defects. The above mentioned three
homogenizations are realized with the predictive scheme of Mori-Tanaka in conjunction with the
Eshelby method, frequently used in the homogenization of composites [6,7]. The micromechanical
method described in [5] provides the stress in the concrete material subject to a prescribed strain and,
vice versa, the strain in concrete material subject to a prescribed stress. In the load case of prescribed
uni-axial compression, the uni-axial stress can be plotted against the uni-axial strain so as to obtain the
compressive stress-strain curve of concrete. Assuming that the compressive stress and strain are
positive, the stress-strain curves provided by [5] in the load case of uni-axial compression exhibit a
maximum compressive stress denoted by  p , which represents the compressive strength  c of
concrete. In [5], this micromechanical model has been used in order to capture peculiar aspects of the
stress-strain curve in the load case of uni-axial compression:
in most concrete materials, a higher compressive strength is associated with a higher initial tangent
Young’s modulus E0 ;


the formation and evolution of voids in the cement paste cause a reduction of the tangent line to
the stress-strain curve;
a higher w c ratio of water to cement involves a concrete with a lower compressive strength  c
and a lower tangent line E0 at the origin of the stress-strain curve;
136

the concrete materials having the same initial stiffness E0 also have the same  p  p ratio of peak
stress to peak strain, as predicted by phenomenological curves [8];  p is the strain corresponding
to  p in the concrete stress-strain curve.
The same model used in [5] can also be used to determine the behavior of cement concrete in load
cases of multi-axial compression. This is done in this work, where the failure surface of concrete
subject to bi-axial or tri-axial compression is determined by using the model described in [5]. Next,
the generic directions in the space of principal strains and principal stresses are denoted by the unit
T
T
vectors n  ˆ1 ˆ2 ˆ3  and n  ˆ1 ˆ 2 ˆ 3  , respectively. The proposed method determines the
vector ε of the principal strains in cement concrete subject to a prescribed stress σ   n and, vice
versa, the vector σ of the principal stresses in cement concrete subject to a prescribed strain ε   n ,
where  is a positive parameter which increases during the loading process. In the load case of
prescribed multi-axial compression, the stress   σ can be plotted against the strain   ε so as to
obtain a    stress-strain curve; this curve exhibits a maximum stress  p which represents the
strength in the load case of multi-axial compression defined by the vector σ   n . The maximum
value of the loading parameter  is  p and the points σ   p n define the failure surface of cement
concrete in multi-axial compression.
In the proposed model, the damage is smeared over the whole volume of concrete while in the postpeak behavior the damage localizes in limited zones. For this reason the proposed model, valid in the
pre-peak range, results not suitable to capture the post-peak behavior. In this work, the pre-peak
behavior provides essential information, such as the initial Young’s modulus of concrete and the
compressive strength, which is the peak of the stress-strain curve of cement concrete. Specifically, the
failure surfaces in the load cases of bi-axial and tri-axial compression are determined for different
types of cement concrete and are compared with experimental failure surfaces.
References
[1] F. Bencardino, L. Rizzuti, G. Spadea, R.N. Swamy, Experimental evaluation of fiber reinforced
concrete fracture properties, Composites Part B: Engineering 41 (2010) 17-24.
[2] R.S. Olivito, F.A. Zuccarello, An experimental study on the tensile strength of steel fiber
reinforced concrete, Composites Part B: Engineering 41 (2010) 246-55.
[3] A. Aprile, A. Benedetti, Coupled flexural-shear design of R/C beams strengthened with FRP,
Composites Part B: Engineering 35 (2004) 1-25.
[4] A. D’Ambrisi, L. Feo, F. Focacci, Experimental analysis on bond between PBO-FRCM
strengthening materials and concrete, Composites Part B: Engineering 44 (2013) 524-32.
[5] A. Caporale, L. Feo, R. Luciano, Damage mechanics of cement concrete modeled as a four-phase
composite, Composites: Part B (2014), In Press, http://dx.doi.org/10.1016/j.compositesb.2014.02.006
[6] C.C. Yang, R. Huang, A two-phase model for predicting the compressive strength of concrete,
Cement and Concrete Research 26 (1996) 1567-77.
[7] C.C. Yang, R. Huang, Approximate strength of lightweight aggregate using micromechanics
method, Advanced Cement Based Materials 7 (1998) 133-8.
[8] P. Desayi, S. Krishnan, Equation for the stress–strain curve of concrete, ACI J. 61 (1964) 345–50.
137
Multiscale analyses of a three layers osteochondral scaffold
G. Parisi2, S. Bignozzi3 ,E. Kon4, P. Vena2
1
2
Istituti Ortopedici Rizzoli, Via di Barbiano, 1/10 40136, paola.ginestra@ior.it
Politecnico di Milano, Piazza Leonardo da Vinci, 32 20133 Milano, pasquale.vena@polimi.it
3
Istituti Ortopedici Rizzoli, Via di Barbiano, 1/10 40136, s.bignozzi@biomec.ior.it
4
Istituti Ortopedici Rizzoli, Via di Barbiano, 1/10 40136, e.kon@biomec.ior.it
Keywords: Multiscale analyses, Analytical methods, Osteochondral tissue substitute.
Tissue engineering has great potential in providing the appropriate replacement of diseased articular
cartilage with a compatible substitute able to grant a stable fixation into the joint defect and reliable
integration with the subchondral bone1.
The replaced engineered tissue needs to be fully biocompatible with the individual subject and
requires specific mechanical and structural properties for adequate functioning and integration within
the articular joint. One of the most widely adopted strategies relies on the use of an artificial structure,
also regarded as scaffolds, having the function of supporting stress under loading conditions and
promoting the bio-mineralization process and the formation of new tissue.
The resulting tissue constructs generally exhibits an overall composition that resembles that of the
original tissue, but the tissue structure at nano and micro scales may be considerably different from
that of the native tissue. This difference may compromise proper functionality and integration of the
implants2.
In this work an inhomogeneous monolithic scaffold is analyzed with 3 distinct phases, a chondral
phase, an intermediate phase and a subchondral (bony) phase3. Each of the three layers is
characterized by different constituents and architectural features in terms of average porosity. In
particular, the chondral phase is the superficial layer composed by 100% deantigenated type I equine
collagen. The intermediate layer (tide mark) and the bony layer instead, are formed by a randomly
oriented network of collagen fibers with magnesium-enriched HA inclusions of different shapes (see
table 1).
MATERIAL LAYER
HA volumetric fraction ƒHA
Collagen volumetric fraction ƒcol
Chondral
Intermediate
Subchondral
0
0.4
0.7
100
0.6
0.3
Table 1: material compositions for the three layer of the HA/Collagen osteochondral tissue substitute.
The mechanical properties of the multilayer scaffold is estimated by means of a multiscale
hierarchical approach4 which spans from the collagen molecules level to the tissue level, including the
effect of the bound water at the small length scale and the large porosity at the tissue scale5. In
particular, a double porosity feature is accounted for, which allows for a large porous matrix with
pores of the order of hundreds of microns in a solid phase which exhibits micron-size pores.
Furthermore, experimental mechanical characterization has been achieved by means of
nanoindentation and micro-compression tests. Nanoindentation tests have been carried out using a flat
punch with circular cross section having a 250 m radius; whereas, microcompression tests have been
carried out using circular punches having up to 1 mm radius. Micro-compression tests have performed
both in dry and liquid environment. For the tests run in liquid medium, a water/glycol solution has
been used with the purpose to simulate the viscosity of blood (the typical working environment for the
138
scaffold). An good agreement has been obtained between the mechanical properties estimated in the
experimental tests and the one predicted using the homogenization models.
The achieved results allowed to draw the relevant conclusion that the multiscale homogenization
model has shown to be a reliable tool to estimate the mechanical properties of a multi-phase material
like the osteochondral scaffold uder consideration.
References
[1] Swieszkowski W., Ho Saey Tuan B., Kurzydlowski K. J., Hutmacher D. W., Repair and
regeneration of osteochondral defects in the articular joints, Biomolecular Engineering, Vol. 24 pp.
489-495, 2007.
[2] Kelly DJ, Prendergast PJ. Mechano-regulation of stem cell differentiation and tissue regeneration
in osteochondral defects, J Biomech., Vol. 7, pp. 1413-22, 2005.
[3] Tampieri et al., Design of graded biomimetic osteochondral composite scaffolds, Biomaterials,
Vol. 29, pp. 3539-46, 2008.C.
[4] Fritsch and C. Hellmich, ‘Universal’ microstructural patterns in cortical and trabecular,
extracellular and extravascular bone materials: Micromechanics-base prediction of anisotropic
elasticity. Journal of Theoretical Biology, Vol. 244, pp. 597-620, 2007.
[5] Hellimch, J-F Barthelemy, L. Dormieux, Mineal-collagen interactions in elasticity of bone
ultrastructure- a continuum micromechanics approach. European Journal of Mechanics A/Solids, Vol.
23, pp. 783-810, 2004.
139
Damage propagation modeling of masonry structures subjected to
dynamic loading
Jessica Toti1, a *, Vincenzo Gattulli1,b and Elio Sacco3,c
1
Department of Civil, Construction-Architectural and Environmental Engineering, University of
L'Aquila, Italy
2
Department of Civil and Mechanical Engineering, University of Cassino and Southern Lazio, Italy
a
jessica.toti@unicas.it, bv.gattulli@univaq.it, csacco@unicas.it
Keywords: dynamic analysis, masonry, nonlocal damage, finite element method.
Cohesive structures, such as masonry or concrete structures constituting a large portion of existing
building around the word. In the last forty years, an enormous growth in the development of numerical
tools for structural analysis has been achieved. The development of dynamic inelastic analyses
became an important tool for the evaluation of the safety level of the cohesive structures under
earthquakes. The computational analysis of cohesive structures, subjected to dynamic or cyclic
loadings, requires accurate stress-strain material models able to reproduce the real behavior of the
structure. Research efforts on the cyclic response of cohesive material aim at providing an efficient
model capable of predicting all the hysteretic characteristics of the material. Common models able to
reproduce the cyclic are based on damage mechanics, plasticity theory and coupling of both.
Mathematical and numerical models of cohesive material failure must correctly reflect the energy
dissipated in the fracture process zone. This is not the case if the stress-strain laws with softening is
used within the standard continuum theory. Numerical results obtained with such models suffer from
pathological sensitivity to the spatial discretization, e.g. to the size of finite elements. Upon mesh
refinement, the energy dissipated by the numerical model decreases and tends to extremely low
values, sometimes even to zero. As remedy, regularized models, based on nonlocal continuum
approaches [6], can be adopted.
Indeed, the presence of damage or of other inelastic phenomena modifies the overall structural
dynamic response; moreover, the damage propagation potentially interacts dynamically with the
element vibrations. In particular, changes in its behavior are associated to the decay of the mechanical
properties of the system [9]. Based on these considerations, many studies have been devoted to use the
variations of the dynamic response for the detection of the structural damage. Particular attention has
been focused on the use of frequencies only, on account of simplicity of measuring them and,
therefore, their experimental reliability. In this framework, dynamic analyses of damaged structures
have been performed in [5] with the aim to detect the damage state of the structure.
In the present work, a nonlocal damage-plastic model for dynamic finite element analyses of large
scale masonry structures is proposed. The developed cohesive model is able to reproduce the main
features characterizing the behavior of quasi-brittle materials under static or dynamic loadings, still
remaining quite simple, i.e. governed by few parameters which can be determined by standard
laboratory tests. In particular, the developed constitutive formulation is able to consider: the damaging
evolution in tension and in compression, the plasticity in compression and the cyclic macroscopic
behavior which accounts for the stiffness recovery due to the unilateral effect of the crack closure. The
latter effect represents an important characteristic above all in the case of dynamic loadings. The
proposed formulation is implemented as constitutive model for two-dimensional plane stress four
node quadrilateral elements. The second order equations of motion are solved adopting implicit
Newmark time integration scheme. The validation and the dynamic performance of the proposed
model are demonstrated by numerical examples concerning the analysis of existing masonry structures
(e.g. pillar, arch, vault).
References
140
[6] G. Pijaudier-Cabot, Z. Bažant, Nonlocal Damage Theory, J. Eng. Mech., 113 (1987), 1512-1533.
[7] G. Borino, B. Failla, F. Parrinello, A symmetric nonlocal damage theory, Int. J. Solids Struct, 40
(2003), pp. 3621-3645.
[8] J. Toti, S. Marfia, E. Sacco, Coupled body interface nonlocal damage model for FRP detachment,
Comput. Method Appl. M., 260 (2013), pp. 1-23.
[9] J. Toti, V. Gattulli, E. Sacco, Nonlocal damage propagation in the dynamics of masonry
structures, submitted (2014).
[10] U. Andreaus, P. Casini, F. Vestroni. Nonlinear dynamics of a cracked cantilever beam under
harmonic excitation, Int. J. Nonlin. Mech., 42 (2007), pp. 566-575.
141
A micromechanical approach for the micropolar modeling of
heterogeneous periodic media
Maria Laura De Bellis1, a *, Daniela Addessi1,b and Elio Sacco2,c
1
Dipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma “Sapienza",
Via Eudossiana 18, 00184 Roma, Italy
2
Dipartimento di Ingegneria Civile e Meccanica, Università di Cassino e del Lazio Meridionale
Via G. Di Biasio 43, 03043 Cassino, Italy
a
marialaura.debellis@uniroma1.it, bdaniela.addessi@uniroma1.it, csacco@unicas.it
Keywords: Heterogeneous materials, Homogenization, Cosserat continuum, Periodicity, Constitutive
identication.
Composite materials, both natural or manufactured, are widely used in many fields of engineering and
for different types of structures. Although they may have very distinct features, they are all
characterized by a heterogeneous micro-structure. The study of the constitutive response of such
materials is essential both to reproduce the behavior of existing structures and to design innovative
ones with optimized properties.
Various approaches, characterized by different formulations, have been proposed in literature to deal
with this issue; among others the homogenization techniques have been widely used. In particular,
these techniques analyze the actual heterogeneous material at two different scales: the macro-scale,
where an effective homogenized medium is considered, characterized by overall mechanical
properties, which are estimated from detailed information available at a lower scale, the micro-scale,
where the texture, the geometry and the constitutive laws of the constituents are accurately described.
This work is a contribution towards the study of the homogenized response of periodic composite
materials, considering a Cosserat continuum at the macro-level and a Cauchy continuum at the microlevel. The computational homogenization technique, here adopted, has proved to be very effective to
predict the macroscopic behavior of composite materials[1]. In particular, thanks to the adoption of a
generalized continuum at the macro-level it is possible to naturally account for length-scale
parameters that play a fundamental role when high strain and stress gradients at the macro-level occur,
or when the microscopic length of the constituents is comparable to the wavelength of variation of the
strain and stress mean fields at the macro-level[2].
In this paper, because of the assumed regular texture of the analyzed composite material, a Unit Cell
(UC) is selected at the micro-level and, consistently with the strain-driven approach, the two levels are
linked through a kinematic map based on a third order polynomial expansion [3].
Firstly, the problem of determining the displacement perturbation fields is investigated. To this end, a
new micromechanical approach, based on the decomposition of the perturbation fields in terms of
functions which depend on the macroscopic strain components [4,5], is proposed. A consistently
defined Boundary Value Problem is solved and the results obtained by analyzing a single UC are in a
very good agreement with the reference solution evaluated on a large RVE made from the same
material.
Then, the identification of the homogenized linear elastic 2D Cosserat constitutive parameters is
performed, by using the Hill-Mandel technique, based on the generalized macrohomogeneity
condition. The influence of the selection of the UC is analyzed and some critical issues are outlined.
By analyzing two different UCs, selected for representing the composite texture, it emerges that the
constitutive response of the homogenized medium depends on the choice of the cell. In fact, while the
elastic Cauchy coefficients are independent on the specific choice of the UC, for the bending and
skew-symmetric shear Cosserat coefficients this does not occur, at least in the framework of
computational homogenization. This fact is also confirmed by the results obtained from the structural
142
application. It is also highlighted that considering a RVE made from an assemblage of UCs, the elastic
coefficients converge to the same value, apart from the considered UC. This value corresponds to that
evaluated by simply considering at the micro-level a homogenized Cauchy medium.
References
[1] De Bellis, M., Addessi, D., A Cosserat based multi-scale model for masonry structures. Int J
Multiscale Com 9 (5), (2011)543-563.
[2] Forest, S., Trinh, D., Generalized continua and non-homogeneous boundary conditions in
homogenisation methods. ZAMM-Z Angew Math Me 91 (2) (2011) 90-109.
[3] Addessi, D., De Bellis, M. L., Sacco, E., Micromechanical analysis of heterogeneous materials
subjected to overall Cosserat strains. Mech Res Commun 54 (2013) 27-34.
[4] Yuan, X., Tomita, Y., Andou, T., A micromechanical approach of nonlocal modeling for media
with periodic microstructures. Mech Res Commun 35 (1-2), (2008) 126 -133.
[5] Bacigalupo, A., Gambarotta, L., Second-order computational homogenization of heterogeneous
materials with periodic microstructure. ZAMM-Z Angew Math Me 90 (11)( 2011) 796-811.
143
An experimental investigation on the axial and rotational behavior of
web-flange junctions of open-web pultruded glass fibre-reinforced
profiles
Luciano Feo1,a*, Ayman S. Mosallam2,b and Rosa Penna1,c
1
2
Department of Civil Engineering, University of Salerno, Italy
Department of Civil & Env. Engineering, University of California Irvine, USA
a
l.feo@unisa.it, bmosallam@uci.edu, crpenna@unisa.it
* corresponding author
Keywords: Pultruded Composites, Web-Flange Junction, Mechanical Testing
Fibre-reinforced polymer (FRP) composites represent a class of advanced materials whose use has
spread from the aeronautical, mechanical and naval industry to civil infrastructure due to their high
strength-to-weight ratio, low maintenance cost and high corrosion resistance.
The majority of the commercially-produced pultruded fibre-reinforced polymers (PFRP) have been
designed and developed by the pultrusion industry and are intended for low-stress applications.
Recently, composites have been introduced as primary structural members to replace or complement
other conventional materials, such as steel, concrete and wood, in critical applications such as bridge
decks, pedestrian bridges, and recently in highway bridges and other infrastructural systems.
As the interest in using PFRP profiles in construction applications continues to increase, it became
critical and essential to understand their short- and long-term mechanical behavior. Several recent and
relevant studies [1-10] have been conducted and focused on the performance of PFRP frame
structures. The results of these studies have highlighted the major problems associated with the
structural deficiency of unidirectional PFRP profiles, especially at the web-flange junctions (WFJ)
that lack fibre continuity. This lack of fibre continuity may lead to progressive degradation in both
axial and rotational stiffnesses and strength of these junctions, affecting both the buckling, postbuckling and the overall short- and long-term structural integrity of the PFRP profiles.
Moreover, these studies have highlighted the influence of the architecture of the web-flange junctions
on the collapse of the profiles. In fact, the mechanical properties of the WFJ are not the same as those
of the flats parts of the web and the flanges due to them also depending on the specific processing
method used by the manufacturer to produce PFRP profiles.
This work has been developed within the research activities of a multi-phase comprehensive joint
research program between University of Salerno, Italy, and the University of California, Irvine, USA,
on investigating one of the major structural issues that defines the strength limit-state of pultruded
fibre-reinforced polymer profiles. In particular, both the axial and rotational stiffness of WFJ of I-, Hand L- profiles have been investigated through in-depth experimental program in order to develop P-δ
and M-θ relations that are necessary for accurate analytical predictions of both the local and global
responses of PFRP frame structures. In fact, their failure mechanism has yet to be fully understood
and they often involve failure of the web-flange junctions. Moreover, P-δ and M-θ relations are also
essential for establishing optimum and reliable design limit-states of such structures.
Specifically, the research program has been divided into the following two phases:
- the first phase consisted of an experimental investigation carried out at the Materials and Structural
Testing Laboratory (LMS) of the Department of Civil Engineering (DICIV) of the University of
Salerno in order to evaluate the axial strength and stiffness of WFJ of PFRP I-profiles;
- the second phase was an experimental study conducted at the Structural Engineering Testing Hall
(SETH) of the University of California Irvine (UCI) to evaluate both the axial and the rotational
strength and stiffnesses of WFJ of PFRP H- and L-profiles.
144
The results gathered from this multi-phase research program provided important information on one
of the major structural deficiencies and limitations related to the inherent weakness of the web/flange
junctions of the majority of commercially produced, off-the-shelf unidirectional pultruded composites.
It is hoped that the results obtained from this research will fill exisiting gaps and provide structural
engineers with essentail engineering data to assist to secure optimum designs and obtain the maximum
benefit of PFRP materials.
References
[1]
Mosallam A.S., Elsadek A.A., Pul S. Semi-rigid behavior of web-flange junctions of openweb pultruded composites in Proceeding of the International Conference on FRP Composites, San
Francisco, California (2009).
[2]
Mosallam A.S., Bank L.C. Short-term behavior of pultruded fiber reinforced plastic frame.
Journal of Structural Engineering (ASCE), 118(7), pp. 1037–1954 (1992).
[3]
Mosallam A.S., Abdelhamid M.K. Dynamic behavior of PFRP structural Sections, in Proc. of
ASME (Energy Sources Tech. Conf. and Expo, Composite Material Tech.), 53, pp. 37–44 (1993).
[4]
Davalos J.F., Salim H.A., Qiao P., Lopez-Andio R. Analysis and design of pultruded FRP
shapes under bending. Composites Part B: Engineering, 27B(3,4), pp. 295–305 (1996).
[5]
Mosallam A.S., Abdelhamid M.K., Conway J.H. Performance of pultruded FRP connection
under static and dynamic loads. Journal of Reinforced Plastic and Composites, 13, pp. 1052–1067
(1996).
[6]
Liu X., Mosallam A.S., Kreiner J. A numerical investigation on static behavior of pultruded
composite (PFRP) portal frame structures in Proceeding of the 43rd International SAMPE Symposium
and Exhibition, Anaheim, California (1998).
[7]
Mosallam A.S. Durability of pultruded fiber reinforced polymer (PFRP) composites in mining
environments in Durability of fiber reinforced polymer (FRP) composites for construction, Edited by
B. Benmokrane and H. Rahman, pp. 649-659 (1998).
[8]
Turvey G.J., Zhang Y. Characterization of the rotational stiffness and strength of web-flange
junctions of pultruded GRP WF-sections via web bending tests. Composites Part A: applied science
and manufacturing, 37, pp. 152–164 (2006).
[9]
Turvey G.J, Zhang Y. Shear failure strength of web-flange junctions in pultruded GRP WF
profiles. Construction and Building Materials, 20, pp. 81–89 (2006).
[10] Feo L., Mosallam A. S., Penna R., Mechanical behavior of web-flange junctions of thin walled
pultruded I-profiles: An experimental and numerical evaluation. Composites Part B: Engineering, 48,
pp.18-39 (2013).
145
Development of biodegradable magnesium alloy stents with coating: the
peeling problem
Lorenza Petrini1,a *,Wei Wu2,b, Lina Altomare2,c, Barbara Previtali3,d, Maurizio Vedani3,e
and Francesco Migliavacca2,f
1
Civil and Environmental Engineering Department, Politecnico di Milano, Italy
Laboratory of Biological Structure Mechanics, Chemistry, Materials and Chemical Engineering
‘Giulio Natta’ Department, Politecnico di Milano, Italy
3
Mechanical Engineering Department, Politecnico di Milano, Italy
a
lorenza.petrini@polimi.it, bwei.wu@polimi.it, clina.altomare@polimi.it, dbarbara.previtali@polimi.it,
e
maurizio.vedani@polimi.it, ffrancesco.migliavacca@polimi.it
2
Keywords: coated stents, corrosion, biodegradable materials, coating adhesion.
Biodegradable stents are attracting the attention of many researchers in biomedical and materials
research fields since they can absolve their specific function for the expected period of time and then
gradually disappear [1]. This feature allows avoiding the risk of long-term complications such as
restenosis or mechanical instability of the device when the vessel grows in size in peadiatric patients.
Up to now biodegradable stents made of polymers or magnesium alloys have been proposed.
However, both the solutions have limitations. The polymers have low mechanical properties, which
lead to devices that cannot withstand the natural contraction of the blood vessel: the restenosis appears
just after the implant, and can be ascribed to the compliance of the stent. The magnesium alloys have a
level of toxicity similar to the one of the polymers and much higher mechanical properties.
Unfortunately, they dissolve too fast in the human body: the duration in a blood vessel is about 2-3
months and not 8-10, as should be to withstand the vessel remodeling.
In this work we present some results of an ongoing study aiming to the development of biodegradable
stents made of a magnesium alloy that is coated with a polymer having a high corrosion resistance.
The mechanical action on the blood vessel is given by the magnesium stent for the desired period,
being the stent protected against fast corrosion by the coating. The coating will dissolve in a longer
term, thus delaying the exposition of the magnesium stent to the corrosive environment. Our study
required the following steps: i) selection of a Mg alloy suitable for stent production, having sufficient
strength and elongation capability; ii) optimization of the alloy microstructure through equal channel
angular pressing; iii) optimization of the stent geometry to minimize stress and strain after stent
deployment and improve scaffolding ability; iii) selection of a coating able to assure enough corrosion
resistance and to avoid detachment from substrate during stent expansion; iv) set up of the procedure
to produce magnesium stent, in terms of laser cut and surface finishing.
In the following the study performed on the adhesion ability of the selected coating (polycaprolactone, PCL, Mn = 80 000 g/mol, Sigma-Aldrich, product number 440744-250G)) is described.
Previous studies found that peeling phenomenon (or coating delamination) sometimes happens for
coated stainless stents during stent expansion [2]. If this happens to coated magnesium stents (MAS),
it will lead to a worse result than the uncoated MAS: indeed, the corrosion will concentrate at the
unprotected location and the attack up will accelerate the rupture of the structure. Furthermore, the
peeling of coated MAS during expansion should be strictly avoided. Finite element analysis (FEA) is
a good method to study the peeling problem of MAS. Until now, only one work has used FEA to
study coating delamination of stainless steel by means of the cohesive zone method (CZM)[3]. Herein,
CZM implemented in the commercial code ABAQUS (Dassault Systèmes, Simulia Corp., USA) is
applied to study our problem. Experimental tests were performed to find the cohesive element
parameters. PCL was dissolved in chloroform at a concentration of 5% w/v and was dropped onto
AZ31 foils (70mm x 4mm x0.8mm). Samples were left under a hood for 24 hours to allow the
complete solvent evaporation and a uniform PCL coating with a thickness of 0.1 mm was obtained. A
146
90-degree peeling test (Figure 1 left) was carried out on a MTS Synergie 200H testing machine (MTS
Systems Corporation, Minneapolis, MN,USA). Three samples were tested. The experimental tests
were reproduced numerically by FEA (Figure1 right).. In particular the 2D stent strut model with a
polymer coating is extracted from the 3D stent model (Figure2 left). Only one side of the coating was
studied because of the symmetry of the location. A symmetrical boundary condition in Y-direction
was applied to one strut end and a displacement in Y-direction was applied to the other end to
simulate thestent expansion to a final diameter of 3 mm. After expansion, the cohesive elements were
stretched and the coating had different separation ranges according to the location; however, none of
them has reached the maximum traction stress thus the damage did not affect the cohesive elements
and the scalar damage variable D was consequently 0 (Figure 2 right). The simulation suggests that
under the studied conditions the peeling should not occur to the coating of the investigated stent
design. Experimental tests are under investigation to support the model findings.
Figure 1: Experimental (left) and numerical (right) peeling test on AZ31 foil coated by PCL.
Figure 2: FEA 2D model of a coated stent strut (left) and final configuration after stent expansion with results in terms of
damage parameter in the coating (right).
References
[1] Y. Onuma, J. Ormiston and P.W. Serruys, Bioresorbable Scaffold Technologies, Circ. J.75 (2011)
509-20.
[2] W M.W.Z. Basalus, K. Tandjung, T. van Westen, H. Sen, P.K.N. van der Jagt, D.W. Grijpma,
A.A. van Apeldoorn, and C. von Birgelen, Scanning Electron Microscopic Assessment of Coating
Irregularities and their Precursors in Unexpanded Durable Polymer-Based Drug-Eluting Stents.
Catheter. Cardio. Inter. 79 (2012) 644-53.
[3] C.G. Hopkins, P.E. McHugh and J.P.McGarry, Computational Investigation of the Delamination
of Polymer Coatings During Stent Deployment, Ann Biomed Eng;38 (2010) 2263-73.
147
Interface constitutive relation derived from a representative adhesive
layer
Guido Borino1, a * and Francesco Parrinello1,b
1
Università di Palermo, Dipartimento di Ingegneria Civile, Ambientale, Aerospaziale, dei Materiali,
Viale delle Scienze Ed.8, 90128 Palermo, Italy
a
guido.borino@unipa.it, bfrancesco.parrinello@unipa.it
Keywords: Interface model, Cohesive-frictional, Multiscale analysis.
The mechanical description of joined structural elements made of different materials is typically
carried out by inserting a mechanical interface in between the two clamped bulk components. The
mathematical joining surface is then the locus demanded to represent all the main mechanical
processes which develops along the joining zone.
Actually, along the joining zone, which is a relative thin layer, several complex nonlinear mechanical
process may develop, such as, damage localization, decohesion phenomena, fracture initiation and
propagation etc. The main difficulties are promoted by the fact that, beside the different mechanical
properties of the two materials joined, the interface possesses specific properties in itself. The usual
adopted approach introduces a phenomenological zero thickness interface with very simple
characterizing properties, namely, assigning only the fracture energy constant is often enough in order
to drive a fracture propagation analysis [1, 2]. The increasing demand of more accurate description of
the complex interactions which emerges along these joining zones has produced more sophisticated
phenomenological interface models, which, in turn can deals with aspects such as cyclic loadingunloading, cohesive-frictional transition, damage and plasticity phenomena, dilatancy and so on [3,5].
On the other hand, in the last years a new mechanical approach is emerging as alternative at a pure
phenomenological description. In this new approaches, the actual constitutive relations are indirectly
derived analyzing the material as an heterogeneous substructure and evaluating its mechanical
response at its own micro scale. This general structural analysis is named multi-scale mechanical
approach and has given a significant improvement in understanding the material nonlinear constitutive
behavior starting from micro mechanical response and the relevant nonlinear homogenization
procedure. More than this, multiscale approaches have been proved to be valuable tools for material
design and for optimization of mechanical response performances.
In the present contribution a two scale approach is presented in which the concept of Representative
Adhesive Layer is introduced. The Representative Adhesive Layer is a thin layer of finite thickness in
which the adhesion surface is a rough surface with its own morphology, and also specific
microstructure adhesive features can be introduced at this scale. The idea of performing a structural
analysis with interface, adopting a multiscale approach for interface is already present in the recent
literature [6,7].
In the present formulation the analysis is performed over a microscale layer element of small length,
which is considered to reproduce a perfect periodic ligament. The investigation is then confined to the
mechanical response of this Representative Adhesive Layer employing proper periodic boundary
conditions at the two ends of the element, and deriving the relation between macro tractions and
macro displacement jumps by means of specific homogenization techniques and then employed for
the analysis at structural level.
References
[1] A. Needleman, An analysis of tensile decohesion along an interface, J. Mech. Phy. Sol. 38 (1990)
289–324.
148
[2] O. Allix, P. Ladevèze, A. Corigliano, Damage analysis of interlaminar fracture specimens, Comp.
Struct., 31 (1995), 61–74.
[3] G. Alfano, E. Sacco, Combining interface damage and friction in a cohesive-zone model Int. J.
Num. Meth. Eng., 68 (2006) 542-582
[4] F. Parrinello, B. Failla, G. Borino, Cohesive–frictional interface constitutive model, Int. J. Sol.
Struct. 46 (2009) 2680–2692.
[5] A. Spada, G. Giambanco, P. Rizzo, Damage and plasticity at the interfaces in composite materials
and structures, Comp. Meth. Appl. Mech. Engng. 198 (2009) 3884 – 3901.
[6] K. Matouš, M. G. Kulkarni, P. H. Geubelle, Multiscale cohesive failure modeling of
heterogeneous adhesives, J. Mech. Phy. Sol. 56 (2008) 1511–1533
[7] C. B. Hirschberger, S. Ricker, P. Steinmann, N. Sukumar, Computational multiscale modelling of
heterogeneous material layers, Eng. Fract. Mech. 76 (2009) 793–812
149
A cohesive-zone model simulating damage, friction and interlocking
Roberto Serpieri1,a*, Elio Sacco2,b and Giulio Alfano3,c
1
Università degli Studi del Sannio, Dipartimento di Ingegneria,
Piazza Roma n. 21 - 82100, Benevento, Italy
2
Università di Cassino e del Lazio Meridionale, Dipartimento di Ingegneria Civile e Meccanica,
Via di Biasio n. 43 - 03043 Cassino (FR), Italy
3
Brunel University, School of Engineering and Design, Uxbridge, UB8 3PH, UK
a
serpieri@unisannio.it, bsacco@unicas.it, cgiulio.alfano@brunel.ac.uk
Keywords: Interface Friction, Thermodynamics with Internal Variables, Interlocking, Interface
Elements, Fracture Energy.
Cohesive-zone models (CZMs) are widely used to simulate initiation and propagation of cracks along
structural interfaces. They represent an effective alternative approach to fracture-mechanics-based
methods for a wide variety of problems at very different scales, such as crack growth in dams, mortarjoint failure in brick masonry, bond-slip response of reinforcing bars in concrete, debonding of
adhesive joints, delamination or fibre-matrix debonding in composites, among many others. Many of
such problems entail combination of de-cohesion and frictional sliding, which is often accompanied
by dilatancy, in turn associated with the interlocking effect created by the asperities of the fracture
surface. Therefore, in order to formulate models that properly account for the underlying physics of
the problem it is essential to capture the distinct types of dissipation due to fracture and friction, the
influence of the geometry of the asperities on the interlocking effect.
Several interface models accounting for damage-friction coupling have been proposed in literature,
see e.g. Del Piero and Raous [1] and references therein. Some of them are based on nonassociative
softening plasticity, as for the multi-dissipative interface model proposed by Cocchetti et al. [2] and
the contributions given by Bolzon and Cocchetti [3] and by Červenka et al. [4] in the field of concrete
dams analysis, and by Giambanco et al. [5].
A different strategy was followed by Alfano and Sacco [6], Alfano et al. [7] and, more recently, Sacco
and Toti [8], where interface damage and friction have been combined in a cohesive zone model based
on a simplified micromechanical formulation. The main idea was to consider a representative area at a
micromechanical scale, which is assumed to be additively decomposed into an undamaged and a fully
damaged part; moreover, it is supposed that friction occurs only on the latter. The evolution of damage
is assumed to depend on the elastic energy in the undamaged part while the frictional behaviour is
governed by a Coulomb law. To simulate dilatancy and interlocking this approach was adopted by
Serpieri and Alfano [9], within a multi-scale framework in which, at a small scale, the asperities of the
interface are represented in the form of a periodic arrangement of distinct inclined planes,
denominated Representative Interface Element (RIE). On each of these planes the interaction is
governed by the formulation proposed in Alfano and Sacco (2006).
The above formulation by Serpieri and Alfano was recently revisited by Serpieri et al. [10], where it
was shown that use of a single damage variable, combined with the choice of having a threshold
damage function only depending on the damage variable itself and an equivalent displacement norm,
requires coincidence of fracture energies in modes I and II to preserve thermodynamic consistency.
Furthermore, it was shown that the enhancement of the model to account for friction and interlocking,
based on the formulation proposed by Serpieri and Alfano [9], results in retrieving the experimental
evidence that that the measured fracture energy in mode II is typically quite higher than in mode I, and
more generally, that for mixed-mode cases with positive (opening) mode I, the measured fracture
energy increases with the mode II-to-mode I ratio. Both facts are well supported by good agreement
between experimental and numerical results.
150
Open issues concerning the approach presented in Refs. [9, 10] are related to the extension of the
kinematics of the RIE-based multi-scale framework to account for relative displacements that are
large compared to the characteristic size of the interface asperities.
Further relevant issues concern the extensibility of the multi-plane description to account for the nonlinear behaviour of the asperities at the micro- or meso-scale under cyclic loading. In particular, the
asperity-related non-linear behavior can involve plasticity, damage, crushing and wear effects which
play, to different extents, a significant role in decohesion of rock interfaces [11] and rebar-to-concrete
bond-slip [12]. More generally the interaction between damage evolution and crack growth on the
interface with other dissipative processes in the bulk material adjacent to the interface is an area worth
of further investigation. Methods to account for fibre bridging could also be a possible enrichment of
the model for the simulation of delamination in fibre-reinforced composites [13].
In this contribution the above formulations [6-10] will be reviewed and more recent developments
made to extend the RIE approach in order to address the above highlighted issues will be presented.
References
[1] G. Del Piero and M. Raous, A unified model for adhesive interfaces with damage, viscosity and
friction. Eur J. Mech. A/Solids. Vol. B, 29(4) (2010) 496-507.
[2] G. Cocchetti, G. Maier and X.P. Shen, Piecewise linear models for interfaces and mixed mode
cohesive cracks. Comp Model. Eng. Sci. 3 (2002) 279–298.
[3] G. Bolzon, G. Cocchetti, Direct assessment of structural resistance against pressurized fracture.
Int. J. Num. Anal. Meth. Geomech. 27(2003) 353–378.
[4] J. Červenka, J.M. Chandra Kishen, V.E. Saouma, Mixed mode fracture of cementitious bimaterial
interfaces; part II: numerical simulation. Eng. Frac. Mech. 60 (1998) 95–107.
[5] G. Giambanco, S. Rizzo, R.. Spallino, Numerical analysis of masonry structures via interface
models. Comp. Meth. Appl. Mech. Eng., 190 (2001) 6493–6511.
[6] G. Alfano and E. Sacco, Combining interface damage and friction in a cohesive-zone model. Int.
J. Num. Meth. Eng. 68 (2006) 542-582.
[7] G. Alfano, S. Marfia, E. Sacco, A cohesive damage-friction interface model accounting for water
pressure on crack propagation. Comp. Meth. Appl. Mech. Eng. 196 (2006) 192-209.
[8] E. Sacco, and J. Toti, Interface elements for the analysis of masonry structures. Int. J. Comp.
Meth. Eng. Sci. Mech. 11 (2010) 354-373.
[9] R. Serpieri, G. Alfano, Bond-slip analysis via a thermodynamically consistent interface model
combining interlocking, damage and friction, Int. J. Num. Meth. Eng. 85 (2011) 164-186.
[10] R. Serpieri, E. Sacco, G. Alfano, A thermodinamically consistent derivation of a frictionaldamage cohesive-zone model with different mode I and mode II fracture energies. submitted to Eur J.
Mech. A/Solids. (2014).
[11] H.S. Lee, Y.J. Park, T.F. Cho, K.H. You, Influence of asperity degradation on the mechanical
behavior of rough rock joints under cyclic shear loading. Int. J. Rock Mech. Min. Sci., 38(7) (2001)
967-980.
[12] H. Shima, L.L. Chou, H. Okamura, Bond Characteristics in post-yield range of deformed bars.
Concrete Library of JSCE 10 (1987) 113-124.
[13] B.F. Sørensen, T. K. Jacobsen. Characterizing delamination of fibre composites by mixed mode
cohesive laws. Comp. Sci. Techn. 69(3) (2009) 445-456.
151
Crack detection in beam-like structures by nonlinear harmonic
identification
Paolo Casini1, a *, Oliviero Giannini2,b and Fabrizio Vestroni1,c
1
Dipartimento di Ingegneria Strutturale e Geotecnica, Università degli Studi di Roma “la Sapienza”
Via Eudossiana, 18 00184, Roma, Italy
2
Università degli Studi Niccolò Cusano Via Don Carlo Gnocchi, 3 00166, Roma, Italy
a
p.casini@uniroma1.it, boliviero.giannini@unicusano.it, cvestroni@uniroma1.it
Keywords: Fatigue crack; damage identification; nonlinear dynamics; cracked beam model
The occurrence of cracks in civil or mechanical systems leads to dangerous effects for the structural
integrity and causes anomalous behaviors. In order not to compromise the safety, the detection of a
crack in the early stage is of great interest. The presence of a crack not only causes a local variation in
the mechanical characteristics of the structure at its location, but it also has a global effect involving
the entire structure. For this reason, the dynamic characterization of cracked structures can be used for
damage detection in non-destructive tests and, among the various techniques, vibration-based methods
provide an effective means of detecting fatigue cracks in structures [1].
There are two main categories of crack models used in vibration-based detection methods: open crack
and breathing crack models. In the first case it is assumed that the crack in a structural member
remains open during vibration. This assumption is usually satisfied in notched beams and when the
damage is rather large; this model avoids the complexity resulting from nonlinear behavior when a
breathing crack is presented. On the contrary, breathing behavior is generally reported in the case of
fatigue cracks, also when the damage affects only a small portion of the cross section of the structural
element; it requires a nonlinear model to take into account its effect on the system dynamics; in fact
the breathing crack model considers that, during the vibration cycle of a structure, the edges of the
crack come into and out of contact, leading to sudden changes in the dynamic response of the
structure. Depending on the crack model, vibration based methods are also classified into two
categories: the linear and the nonlinear approaches. The first group of methods can identify only the
open cracks at an advanced stage, once changes in modal parameters become significant [2]. For this
reason, refined studies focus on the nonlinear response characteristics that can be investigated to
identify the presence of the crack in an early stage. In fact, the structures with breathing cracks behave
similarly to bilinear systems and hence exhibit nonlinear phenomena in the dynamic response even for
low damage. Therefore in the second group of methods, the identification is obtained by assuming as
damage indicators some peculiar characteristics of the nonlinear dynamic response such as the
presence of sub and super harmonics, the changes in the phase diagrams, the rise of superabundant
nonlinear normal modes and bifurcations [3-5].
The dynamic behavior of beam-like structures with fatigue cracks forced by harmonic excitation is
characterized by the appearance of sub and super-harmonics in the response even in presence of
cracks with small depth. Since the amplitude of these harmonics depends on the position and the depth
of the crack, an identification technique based on such a dependency has been developed by the
authors in [5]. The main advantage of this method relies on the use of different modes of the structure,
each sensitive to the damage position in its peculiar way.
In this study the identification method proposed in [5] is extended and detailed through numerical
examples tested on structures of increasing complexity to evaluate the applicability of the method to
engineering applications. The amount of data to obtain a unique solution and the optimal choice of the
observed quantities are discussed. Finally, a robustness analysis is carried out for each test case to
assess the influence of measuring noise on the damage identification; the robustness of the
identification, evaluated through a Monte Carlo simulation, is shown to be quite strong to both
152
measuring and modeling errors envisioning the possibility for in-field applications of this method
even in the case of very small cracks.
References
[1] A. Morassi and F. Vestroni, Dynamic methods for damage detection in structures, SpringerVerlag, ISBN: 3211787763, 2008.
[2] A. Morassi, Crack-induced changes in eigenparameters of beam structures, Journal of Engineering
Mechanics, 119(9) (1993) 1798-1803
[3] U. Andreaus, P. Casini, F. Vestroni, Nonlinear dynamics of a cracked cantilever beam under
harmonic excitation, International Journal of Non-linear Mechanics 42 (3) (2007) 566-575.
[4] P. Casini, O. Giannini, F. Vestroni, Persistent and ghost Nonlinear Normal Modes on the forced
response of non-smooth systems, Physica D 241 (2012) 2058–2067.
[5] O. Giannini, P. Casini and F. Vestroni, Nonlinear harmonic identification of breathing cracks in
beams, Computer and Structures, 129 (2013) 166-177.
153
A data fusion based approach for damage detection in linear systems
Ernesto Grande1, a and Maura Imbimbo2,b*
1
Dept. of Civil and Mechanical Engineering, University of Cassino and Southern Lazio,
Via G. Di Biasio 43, 03043 Cassino, Italy
a
e.grande@unicas.it, bmimbimbo@unicas.it
Keywords: damage identification; modal strain energy; data-fusion.
One of the main goal of dynamic identification is to derive information about the health of systems by
monitoring changes affecting their dynamic properties due to the presence of damage. Indeed,
considering damage indicators just based on modal parameters of systems before and after the
damage, three possible levels of information can be derived: presence of damage (level 1), position of
damage (level 2), severity of damage (level 3). In this context modal strain energy (MSE) is often
used as basis for defining a damage detection index for systems based on the change of its dynamic
properties and elemental stiffness of systems. The studies available in literature, among which [1-4],
show the ability of this indicator to cover all the levels of information concerning the status of
damaging of systems. Nevertheless, the same studies have also underlined some drawbacks of this
approach, which generally arise when multiple damages occur or significant levels of noise/error
affect the identified dynamic properties of systems. In these cases the major difficulties depend on
identifying only a limited number of modes and not always those more sensible to the damage
scenarios.
It is, thus, important to improve the performance of the traditional MSE based damage indicator in the
case of multi damage locations, noise-polluted data and reduced number of modes.
Recently, in order to improve the ability of classical indicators to detect damage of systems, some
literature studies [5-9] propose to extent the classical data information fusion techniques [10] to
structural damage identification with the intent of combining information from different sources and
improving the final result.
In this context, this paper presents an approach for damage identification of systems, which combines
the use of damage indicators derived through the MSE with a multi stage data-fusion procedure.
Specifically, considering different sets of the identified modes of vibration as information sources,
modal strain energy change ratios are evaluated and converted in local decisions. The single decisions
provided by each source constitute the data sent to the fusion center, where they are combined on the
basis of a fusion approach that provides the global decision. More specifically, the methodology
followed in the present paper is based on the classical Dempster-Shafer (DS) theory of evidence.
The approach is applied to some numerical examples with different damage scenarios, set of identified
modes of vibrations and, also, noise levels. The obtained results clearly show that the proposed
approach can improve the performances of the classical MSE based damage indicator in the case of
single damage scenarios and, mainly, in the case of multiple damage scenarios where generally the
classical indicators fail. The results show also a significant robustness of the approach in presence of
noises. In all these cases the proposed approach provides efficient information in terms of location and
also extent of damage.
References
[1] Doebling S., Hemez F., Peterson L., Farath C., (1997). Improved damage location accuracy using
strain energy based on mode selection criteria. AIAA Journal; 35(4), 693-699.
[2] Shi Z., Law S., Zhang L., (1998). Structural damage localization from modal strain energy change.
154
Journal of Sound and Vibration; 218(5), 825-844.
[3] Shi Z., Law S., Zhang L., (2002). Structural damage detection from elemental modal strain energy
change. Journal of Engineering Mechanics; 128(5), 521-529.
[4] Ren W., Roeck G., (2002). Discussion of "Structural damage detection from elemental modal strain
energy change, by Shi, Z., Law, S.S., Zhang, L.M.". Journal of Engineering Mechanics; 128(3), 376-377.
[5] Fei Q., Li A., Han X., (2009). Simulation study on damage localization of a beam using evidence
theory. Procedia Engineering; 1, 147-150.
[6] Guo H., Li Z., (2009). A two-stage method to identify structural damage sites and extents by using
evidence theory and micro-search genetic algorithm. Mechanical Systems and Signal Processing; 23, 769782.
[7] Bao, Y. Q., Li, H., An, Y. and Ou, J. P., (2011), Dempster-Shafer evidence theory approach to
structural damage detection, Structural Health Monitoring, 10 (3), 235-246.
[8] Bao, Y. Q., Xia, Y., Li, H., Xu, Y. L. and Zhang, P., (2012), Data fusion-based structural damage
detection under varying temperature conditions, International Journal of Structural Stability and
Dynamics, 12 (6), No. 1250052.
[9] Grande E. and Imbimbo M., (2014), A multi-stage data-fusion procedure for damage detection of
linear systems based on modal strain energy, Journal of Civil Structural Health Monitoring, 4(2), 107118.
[10] Shafer G., (1976). A mathematical theory of evidence. Princeton, NJ: Princeton University Press.
155
Superelastic and Shape Memory effects in shape memory alloy beams
Sara Malagisi1,a *, Sonia Marfia1,b and Elio Sacco1,c
1
Departement of Civil and Mechanical Engineering, University of Cassino and
Southern Lazio,
Via G. Di Biasio, 03043 Cassino (FR), Italy.
a
s.malagisi@unicas.it, bmarfia@unicas.it, csacco@unicas.it
Keywords: Shape memory alloy, SMA, beam, finite element.
Shape memory alloys (SMA) are active materials able to undergo reversible large deformations when
subjected to stress-temperature loading histories. Because of their unique properties and mechanical
response, SMA are actually used in many advanced applications in different field of engineering.
Many devices made of SMA are geometrically characterized by one dimension significantly greater
than the other two. For this reason, several one-dimensional SMA constitutive laws have been
proposed in literature which are implemented in suitable beam models. Many efforts have been
devoted to model and to predict the mechanical response of SMA beam elements. For example
Auricchio and Sacco developed finite formulations and suitable computational procedures to simulate
SMA beams behavior [1]; Marfia Reddy and Sacco proposed a laminated shape-memory-alloy beam
[2]; Zbiciak presented a formulation of initial-boundary-value problem for the Bernoulli–Euler beam
made of pseudoelastic shape memory alloy (SMA) [3].
Although beam appears a very appropriate structural model to predict the response of many SMA
devices, lastly simulations are often performed adopting full three-dimensional (3D) models, within
the finite element (FE) formulation. Of course, because of the specific geometry, the stress analysis of
these structural element developed adopting 3D FE leads to computationally expensive simulations,
requiring very fine discretizations able to capture their flexural behavior. The choice of using 3D
models is mainly due to the fact that the structural members constituting the SMA devices are often
not straight and are subjected to significant shear deformation.
The present study deals with development of a SMA model which is appropriate for beam structures
subjected to bending, shear and torsion. The SMA constitutive law is derived from the well-known 1D
model proposed by Lagoudas [4], which is properly modified. In fact, even presenting the simplicity
of the 1D approach, the proposed SMA is able to account for:
 the reorientation of the detwinned martensite,
 the development of shear inelastic strains.
The proposed SMA model is implemented in a beam finite element which considers shear
deformations according to the Timoshenko theory and to torsional effect.
A robust numerical algorithms is developed to perform simulations of SMA beams under complex
thermomechanical loading histories.
Numerical applications are presented in order to assess the ability of the constitutive model and beam
FE in simulating the response of element of SMA devices.
References
[1] F. Auricchio and E. Sacco, "A Temperature-Dependent Beam for Shape-Memory Alloys:
Constitutive Modelling, Finite-Element Implementation and Numerical Simulations", Comput.
Method. Appl. M., vol. 174, no. 1-2, pp. 171-190, 1999.
[2] S. Marfia, J. N. Reddy and E. Sacco, "Superelastic and Shape Memory Effects in Laminated
Shape-Memory-Alloy Beams", AiAA J., vol. 41(1), pp. 100-109, 2003.
156
[3] A. Zbiciak, “Dynamic analysis of pseudoelastic SMA beam” Int. J. Mech. Sci, vol. 52, pp. 56-64,
2010.
[4] D. C. Lagoudas, Shape Memory Alloys - Modelling and Engineering Applications, Springer,
2008.
157
Anisotropic Swelling in fibrous materials
Paola Nardinocchi1,a *, Matteo Pezzulla1,b and Luciano Teresi2,c
1
Dip. Ingegneria Strutturale e Geotecnica, Sapienza Università di Roma, Italy
2
Dip. Matematica e Fisica, Università Roma Tre, Italy
3
Full address of second author, including country
a
paola.nardinocchi@uniroma1.it, bmatteo.pezzulla@uniroma1.it, cteresi@uniroma3.it
Keywords: Swelling-induced deformations, fibrous materials.
Anisotropic swelling concerns swelling in anisotropic systems, such as micro--structured polymers
and natural fibrous materials. Most of the models for swelling are based on the Flory-Rehner (FR) free
energy, which couples the elasticity of the polymer network with the migration of solvent inside the
system [1,2]. Such energy has an intrinsic isotropic structure that prevents changes in shape during
free swelling processes, i.e., in the absence of constraints and applied forces. Hence, anisotropic
swelling is still an open issue within the coupled theory, even if attempts to extend the FR-based
theory of swelling date back to 1961 [3]. Moreover, a lot of recent applications where anisotropic
swelling plays a major role ask for a in--depth analysis of the problem [4,5].
Firstly, we discuss anisotropic swelling in passively fibered materials (PFM), that is, fibrous materials
whose fibers work as passive material reinforcement; this is the case of many natural fibrous systems
with cellulose myofibrils as fibers, which hamper the deformation of the system in special directions
determined by their orientation. The assembly of materials of this type can be managed with the aim
to get, under free--swelling conditions, three--dimensional curved shape starting from originally
straight configurations.
A prototype is a bilayer gel beam whose bottom and top beam components have the same length and
cross-section at the dry and straight configuration. The gel within the beam can be though as a
homogeneous matrix; in the top beam, longitudinal fibers are added, with a stiffness depending on a
scalar parameter (to be appropriately tuned). Once immersed in a solvent bath, the beam goes towards
an equilibrium swelling ratio, which can't be uniform, due to the presence, in the top layer of the
fibers, which hamper the swelling in the longitudinal direction. Hence, the final deformation at
equilibrium realizes a bending of the gel beam. The problem was studied and implemented within a
revised version of the stress--diffusion model presented and discussed in [2]. In particular, the entropic
component of the standard Flory--Rehner free energy was augmented by an anisotropic term which
assigns a higher energetic expenditure to longitudinal deformation. In general when bilayer gel beams
are considered, with different compositions of the layers i.e., both fibered with different fibers's
orientation), our aim is to identify the key parameters of the global deformation pattern induced, and
the related stress states and energetic expenditures.
Secondly, we deal with anisotropic swelling in actively fibered materials (AFM), that is, fibrous
materials whose fibers work as an active material with swelling properties different from the matrix's
ones. In particular, we refer to the bio-inspired device presented in [6], where a fibered twodimensional plate-like system was built, assembling two different gels structure to get a striped
system. The ability of neighbored strips (fibers) to swells differently allows to identify many different
three--dimensional shapes, starting from the same straight configuration and changing the orientation
of the strips. Our goal is to describe the swelling--induced deformations of this kind of systems within
our fully nonlinear stress--diffusion theory [2], based on an appropriate and modified free--energy
representation.
In the end, we focus on the swelling--induced deformations of homogeneous systems with a non-homogeneous and fibered coating. A simple prototype is a paper sheet printed on with strip patterns,
different for stripes's orientation and width; if wetted, it undergoes a de-swelling which induces, due
to the surface strip pattern, a class of different curved shapes.
158
References
[1] M. Doi, Introduction to Polymer Physics, Clarendon Press, Oxford, 1996.
[2] A. Lucantonio, P. Nardinocchi, L. Teresi. Transient Analysis of swelling-induced large
deformations in polymer gels. J. Mech. Phys. Solids 30 (2013) 159-183.
[3] S.D. Bruck. Extension of the Flory-Rehner theory of swelling to an anisotropic polymer system.
J. Res. Nbs. A Phys. Ch. 65(6) (1961) 485-487.
[4] K. Urayama, Y. O. Arai, T. Takigawa. Anisotropic Swelling and Phase Behavior of Monodomain
Nematic Networks in Nematogenic Solvents. Macromolecules 38 (2005) 5721-5728.
[5] R. M. Erb, J. S. Sander, R. Grisch, A. R. Studart. Self-shaping composites with programmable
bioinspired microstructure. Nature Communications 4 (2013).
[6] Z. L. Wu, M. Moshe, J. Greener, H. Therien-Aubin, Z. Nie, E. Sharon, E. Kumacheva. Threedimensional shape transformations of hydrogel sheets induced by small-scale modulation of internal
stresses. Nature Communications 4 (2013).
159
Compendio dei Sommari del Convegno GIMC-GMA 2014:
- XX Convegno Nazionale di Meccanica Computazionale
- VII Riunione del Gruppo Materiali AIMETA
Cassino 11 – 13 giugno 2014
Università degli Studi di Cassino e del Lazio Meridionale
Dipartimento di Ingegneria Civile e Meccanica
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