Journal of the Mechanics and Physics of Solids 61 (2013) 1798–1821
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Journal of the Mechanics and Physics of Solids
journal homepage: www.elsevier.com/locate/jmps
The impact of sand slugs against beams and plates: Coupled
discrete particle/finite element simulations
T. Liu a, N.A. Fleck a, H.N.G. Wadley b, V.S. Deshpande a,n
a
b
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK
Department of Material Science and Engineering, School of Engineering and Applied Science, University of Virginia, Charlottesville, VA 22903, USA
a r t i c l e in f o
abstract
Article history:
Received 23 January 2012
Received in revised form
12 March 2013
Accepted 22 March 2013
Available online 6 April 2013
The impact of a slug of dry sand particles against a metallic sandwich beam or circular
sandwich plate is analysed in order to aid the design of sandwich panels for shock
mitigation. The sand particles interact via a combined linear-spring-and-dashpot law
whereas the face sheets and compressible core of the sandwich beam and plate are
treated as rate-sensitive, elastic–plastic solids. The majority of the calculations are
performed in two dimensions and entail the transverse impact of end-clamped monolithic
and sandwich beams, with plane strain conditions imposed. The sand slug is of
rectangular shape and comprises a random loose packing of identical, circular cylindrical
particles. These calculations reveal that loading due to the sand is primarily inertial in
nature with negligible fluid–structure interaction: the momentum transmitted to the
beam is approximately equal to that of the incoming sand slug. For a slug of given
incoming momentum, the dynamic deflection of the beam increases with decreasing
duration of sand-loading until the impulsive limit is attained. Sandwich beams with thick,
strong cores significantly outperform monolithic beams of equal areal mass. This
performance enhancement is traced to the “sandwich effect” whereby the sandwich
beams have a higher bending strength than that of the monolithic beams. Threedimensional (3D) calculations are also performed such that the sand slug has the shape
of a circular cylindrical column of finite height, and contains spherical sand particles. The
3D slug impacts a circular monolithic plate or sandwich plate and we show that sandwich
plates with thick strong cores again outperform monolithic plates of equal areal mass.
Finally, we demonstrate that impact by sand particles is equivalent to impact by a
crushable foam projectile. The calculations on the equivalent projectile are significantly
less intensive computationally, yet give predictions to within 5% of the full discrete
particle calculations for the monolithic and sandwich beams and plates. These foam
projectile calculations suggest that metallic foam projectiles can be used to simulate the
loading by sand particles within a laboratory setting.
& 2013 Elsevier Ltd. All rights reserved.
Keywords:
Fluid–structure interaction
Discrete/continuum coupling
Dynamic loading
1. Introduction
The air and water-blast resistance of structures has recently received increased attention with the overall aim of
designing lightweight, blast resistant structures. Several recent theoretical studies have shown that sandwich structures
n
Corresponding author.
E-mail address: [email protected] (V.S. Deshpande).
0022-5096/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.jmps.2013.03.008
T. Liu et al. / J. Mech. Phys. Solids 61 (2013) 1798–1821
1799
subjected to water-blast outperform monolithic structures of equal mass, see for example Fleck and Deshpande (2004) and
Xue and Hutchinson (2004). These predictions have been confirmed experimentally by Wadley et al. (2008) and Wei et al.
(2008). The enhanced performance is mainly due to fluid–structure interaction effects such that a smaller fraction of the
impulse is transmitted into sandwich structures compared to their monolithic counterparts. In contrast, under air blast,
sandwich structures provide only marginal benefits over monolithic structures as the fluid–structure interaction effects are
less important (Kambouchev et al., 2007). The main aim of the present study is to investigate the relative performance of
monolithic and sandwich structures subjected to loading by the impact of high velocity granular media that have been
accelerated by the explosion of shallow buried explosives (such as landmines).
The prototypical problem is sketched in Fig. 1: the under-body of a land vehicle comprises a sandwich plate, and is
impacted by granular media from an underground explosion. The phenomena in play are complex and can be separated into
three sequential phases: (i) transfer of impulse from the explosive to the surrounding soil/sand, leading to the formation of a
dispersion of high velocity particles, (ii) propagation and expansion of the soil/sand ejecta and (iii) impact of the soil/sand
ejecta against the structure, with attendant momentum transfer. The focus of this study is on the third phase: the
interaction of high velocity granular media with a structure and the subsequent response of the structure.
The experimental characterisation of buried explosive events (Bergeron et al., 1998; Braid, 2001; Weckert and Andersen,
2006; Neuberger et al., 2007) has led to the development of empirical models to quantify the impulsive loads imposed by
soil ejecta (Westine et al., 1985) as well as to structural design codes such as those proposed by Morris (1993). However, the
predictive capability of these empirical models is limited and cannot be extrapolated to new structural concepts such as the
sandwich structure as sketched in Fig. 1.
In addition to the above empirical approaches, a number of numerical codes have been developed in an attempt to
predict the response of a structure to soil ejecta. This previous work has focussed on the development of appropriate
constitutive models for the soil that can be implemented within Eulerian numerical codes. The Eulerian codes are coupled to
Lagrangian finite element (FE) calculations in order to simulate the structural response. Readers are referred to Grujicic et al.
(2006, 2008) for a detailed analysis of soil models that have been used to simulate landmine explosions. Notable among
these are the so-called three phase model of Wang and Lu (2003) and Wang et al. (2004) which is a modified form of the
Drucker-Prager (1952) model, and the porous-material/compaction model as developed by Laine and Sandvik (2001). The soil
models listed above are restricted to a regime where the packing density of the soil is sufficiently high that the particle–
particle contacts are semi-permanent. While these models are appropriate during the initial stages of a buried explosion or
during an avalanche or land slide, their applicability is questionable when widely dispersed particles impact a structure.
More recently, Deshpande et al. (2009) modified the constitutive model of Bagnold (1954) to develop a continuum soil
model applicable to soils in both the densely packed and dispersed states. However, the successful implementation of this
model within a coupled Eulerian–Lagrangian computational framework has been elusive due to computational problems
associated with the analysis of low density particle sprays; see for example Wang et al. (2004) for a discussion of these
numerical issues.
An alternative modelling strategy has recently been employed by Borvik et al. (2011) and Pingle et al. (2012), as follows.
The low density soil is treated as an aggregate of particles and the contact law between particles dictates the overall
aggregate behaviour. This approach has several advantages: (i) there is no need to make a priori assumptions about the
constitutive response of the aggregate (this becomes an outcome of the simulations), (ii) it provides a fundamental tool to
study the essential physics of the sand–structure interaction and (iii) given that the sand is represented by a discrete set of
particles we do not face the usual numerical problems associated with solving the equations associated with the equivalent
continuum descriptions. Pingle et al. (2012) have recently investigated the response of a rigid target to impact by a column
Facesheet
Support
Sandwich
panel
Support
Core
Ejecta
Ejecta
Ejecta
Facesheet
Detonation
Fig. 1. Sketch of a prototypical problem of a clamped sandwich structure loaded by a shallow mine explosion.
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T. Liu et al. / J. Mech. Phys. Solids 61 (2013) 1798–1821
of particles. This idealised but fundamental fluid–structure interaction (FSI) problem is the “sand-blast” analogue to the
classical water-blast FSI problem as studied by Taylor (1963). In a parallel study, Borvik et al. (2011) analysed the response of
monolithic plates to a spherically expanding sand shell and compared their predictions with measurements. Their study
demonstrated the capability of this methodology in modelling complex sand–structure interaction events. However, they
did not examine explicitly the fluid–structure interaction phenomenon.
1.1. Approach and scope
Our aim is to consider the impact of an idealised sand slug against monolithic and sandwich structures in the form of
end-clamped beams and circular plates. The sand slug is idealised by a spatially uniform column of discrete, identical
particles travelling at a uniform initial velocity. This idealisation of structural impact by a landmine explosion enables us to
develop a physical understanding of the main phenomena at play and to compare the dynamic responses of monolithic and
sandwich structures for a well-characterised loading scenario.
The main aim of the paper is to gain insight into the interaction of the incoming high velocity sand with structures. In
keeping with this aim, the bulk of the parametric studies carried out as part of this study are for two-dimensional (2D)
problems because full three-dimensional (3D) calculations are as yet sufficiently computationally expensive so as to make
such parametric studies in 3D not feasible. However, a selection of 3D results are presented in order to demonstrate that the
conclusions drawn from the 2D studies carry forward to the 3D case.
The outline of the paper is as follows. First, the numerical methods for the discrete particle calculations, for the
Lagrangian finite element (FE) simulations and for their coupling are described. Second, the response of clamped monolithic
and sandwich beams to impact by circular, cylindrical sand particles is discussed. Finally, we extend the analysis to threedimensions (3D) and present the response of clamped circular plates to dynamic loading by spherical sand particles.
2. Summary of the plane strain coupled discrete element/finite element methodology
Two plane strain boundary value problems are sketched in Fig. 2: a slug of low density granular medium (subsequently
referred to as sand) impacts a clamped (a) monolithic or (b) sandwich beam. The sand is modelled by two-dimensional (2D)
discrete identical circular cylindrical particles using the GRANULAR package in the multi-purpose molecular dynamics code
LAMMPS (Plimpton, 1995) while the beams are modelled using a 2D finite strain Lagrangian finite element (FE) code. We
first describe the discrete particle and FE numerical methods and then provide details of the coupling between these two
techniques.
2.1. Discrete element calculations
The granular medium was modelled by a random array of 2D cylindrical particles, each of diameter D lying in the x1 −x2
plane (thickness b in the x3 -direction). The granular package in LAMMPS is based on the soft-particle contact model as
introduced by Cundall and Strack (1979) and extended to large scale simulations by Campbell and co-workers (Campbell
and Brennen, 1985; Campbell, 2002). This soft-particle contact model idealises the deformation of two contacting particles,
each of mass mp , as depicted in Fig. 3. The contact law comprises:
(i) a linear spring K n and a linear dashpot of damping constant γ n in parallel, governing the normal motion;
(ii) a linear spring K s and a Coulomb friction element of co-efficient μ, in series, governing the tangential motion during
contact.
The contact forces in the normal and tangential directions are now specified as follows. Write r as the separation
of particle centres and δn ¼ r−D as the interpenetration. During active contact ðδn o 0Þ, the normal force is given by
F n ¼ K n δn þ mef f γ n δ_ n
ð2:1Þ
where mef f is the effective or reduced mass of the two contacting bodies. We take mef f ¼ mp =2 for impacts between
particles, and mef f ¼ mp =2 for impacts between a particle and the beam.
The tangential force F s only exists during active contact, and opposes sliding. It is limited in magnitude to jF s j oμjF n j as
follows. Define δ_ s as the tangential displacement rate between the contacting particles. Then, F s is given by an “elastic–
plastic” relation of Coulomb type, i.e.
(
K s δ_ s if jF s j o μjF n j or F s δ_ s o 0
F_ s ¼
ð2:2Þ
0
otherwise
T. Liu et al. / J. Mech. Phys. Solids 61 (2013) 1798–1821
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Fig. 2. The plane strain boundary value problem of a slug of sand comprising circular cylindrical particles impacting a clamped (a) monolithic and
(b) sandwich beam.
The value of damping constant γ n dictates the loss of energy during normal collision and is directly related to the
co-efficient of restitution e according to
" −1=2 #
8K n
−1
e ¼ exp −π 2
:
ð2:3Þ
γ n mp
The collision time for individual binary collisions t e follows from (2.1) as
te ¼ −
2 lnðeÞ
:
γn
ð2:4Þ
Thus, in the limit of plastic collisions with e-0, the contact time is t e -∞.
The calculations with the above contact model were performed using the GRANULAR package within the molecular
dynamics code LAMMPS. The translational and rotational motions of the particles were obtained by integration of the
accelerations using a Verlet time-integration scheme (i.e. Newmark–Beta with β ¼ 0:5). The time-step within LAMMPS was
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T. Liu et al. / J. Mech. Phys. Solids 61 (2013) 1798–1821
Fig. 3. Sketch illustrating the contact law between two particles in the discrete calculations.
typically taken to be t e =10 in order to ensure accurate integration of the contact relations, Eqs. (2.1) and (2.2), and this value
was also used to define the time steps for the finite element calculations, as described below.
2.2. Two-dimensional finite element calculations
The 2D plane strain calculations were performed using an updated Lagrangian finite element (FE) scheme with the
current configuration at time t serving as the reference. The co-ordinate xi denotes the position of a material point in the
current configuration with respect to a fixed Cartesian frame, and vi is the velocity of that material point. The principle
of virtual power (neglecting effects of gravity) for a volume V and surface S is written in the form
Z
Z
V
sij δ_εij dV þ
V
Z
ρvi δvi dV ¼
ST
T i δvi dS
ð2:5Þ
where sij is the Cauchy stress, ε_ ij ≡ðvi;j þ vj;i Þ=2 is the strain rate, T i the traction on the surface due to the impacts by the
particles, and ρ is the material density in the current configuration. In the usual manner, δ denotes an arbitrary variation in
the respective quantity. The finite element discretisation employed linear, plane strain three noded triangular elements (i.e.
constant strain triangles). When the finite element discretisation of the displacement field is substituted into the principle of
virtual power (2.5) and the integrations are carried out, the discretised equations of motion are obtained as
M
∂2 U
¼F
∂t 2
ð2:6Þ
where U is the vector of nodal displacements, M is the mass matrix and F is the nodal force vector. An explicit timeintegration scheme based on the Newmark β-method with β ¼ 0 is used to integrate Eq. (2.6) to obtain the nodal velocities
and the nodal displacements. A lumped mass matrix is used in (2.6) instead of a consistent mass matrix: this is preferable
for explicit time-integration procedures as it gives both improved accuracy and computational efficiency.
T. Liu et al. / J. Mech. Phys. Solids 61 (2013) 1798–1821
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2.3. Coupling of the discrete particle and finite element calculations
The coupling between the discrete particle and finite element calculations was carried out as follows. At any time t,
suppose that a proportion of the particles are in contact with the beam. Consider such a particle. The displacement δn is
defined as δn ¼ r−D=2, where r is the distance between particle centre and contact point on the beam. The rate δ_ n is the
relative approach velocity of the particle and the point of contact on the beam surface, and likewise δ_ s is the tangential
velocity. The normal and tangential contact forces are calculated using Eqs. (2.1) and (2.2) and contribute to the vector of
global nodal forces F in Eq. (2.6). The discrete and finite element equations are then integrated as described in Sections 2.1
and 2.2. These integrations give the updated positions and velocities of (i) the particles in the discrete calculations and (ii)
the nodes of the finite element calculations.
3. Sand impact of a monolithic and sandwich beam in plane strain
The two plane strain boundary value problems that were investigated in this study are sketched in Fig. 2a and b. In both
problems a slug of height H and width 2a impacts the clamped beams centrally and normally at time t ¼ 0. The slug
comprises discrete identical cylindrical particles of diameter D made from solid material of density ρs ; the cylinders are
randomly packed and give rise to a uniform macroscopic relative density ρ, and absolute density ρρs . Each particle has an
initial velocity v0 in the negative x2 direction, as shown.
A monolithic beam, of span 2L and made from a material of density ρm is shown in Fig. 2a. The beam has a thickness 2h
and is fully clamped at both ends. The equivalent clamped sandwich beam of span 2L is sketched in Fig. 2b. The fully dense
face sheets, each of thickness hf , and porous core, of density ρc and thickness c, are made from the same solid as that of the
fully dense monolithic beam. The impact resistance of the monolithic and sandwich beams will be evaluated for the same
value of areal mass mb , as given by
mb ¼ 2ρm h ¼ 2ρm hf þ ρc c:
ð3:1Þ
In order to reduce the variables needed to specify the sandwich geometry we restrict attention to the situation where the
mass of the sandwich beam is divided equally between the two faces and the core, i.e.
ρm hf ¼ ρc c ¼
mb
3
ð3:2Þ
This mass distribution is near optimal for sandwich beams under impulsive loading (Xue and Hutchinson, 2004).
The monolithic beam, of thickness 2h ¼ 50 mm, is made from DH 36 steel of density ρm ¼ 8000 kg m−3 and consequently
the areal mass is mb ¼ 400 kg m−2 . The sandwich and monolithic beams have a half-span of L ¼ 0:5 m, and two sandwich
geometries are investigated below with aspect ratio c=L ¼ 0:11 and 0.33. For each for these sandwich beams, the thickness of
the face sheets and the core density follow from Eq. (3.2).
The assumed symmetry of beam and loading imply that only half of the beam span needs to be modelled: accordingly,
symmetry boundary conditions are imposed at mid-span (x1 ¼ 0), and all degrees of freedom are constrained at x1 ¼ L in
order to simulate end-clamped boundary conditions. The beams are discretised by quadrilateral finite elements comprising
four constant strain triangles (Peirce et al., 1984) of dimension h=15 and hf =8 in the monolithic and sandwich beams,
respectively.
3.1. Material properties
The monolithic beam and sandwich beam face sheets were made from DH36 steel. This steel was treated as an elastic–
plastic J2 flow theory solid of Young’s modulus E ¼ 200 GPa, Poisson's ratio ν ¼ 0:3 and density ρm ¼ 8000 kg m−3 . The rate
dependent uniaxial yield strength of the steel was specified via the relation (Vaziri et al., 2007)
"
!#
h
i
ε_ Ρef f
sY ¼ Y 1 þ 1:5ðεΡef f Þ0:4 1 þ 0:015 ln
;
ð3:3Þ
ε_ 0
where the yield strength is Y ¼ 470 MPa, the reference strain rate is taken as ε_ 0 ¼ 1 s−1 and ε_ Ρef f is the von Mises plastic
strain rate.
The compressible core in the sandwich beams was modelled as a compressible, visco-plastic orthotropic solid following
Tilbrook et al. (2006). Assume the orthotropic axes xi of the core are aligned with the axes of the beam, as sketched in Fig. 2,
i.e x1 and x2 are aligned with the longitudinal and transverse directions, respectively. Now introduce the stress and plastic
strain matrices in the usual Voigt vector notation as
s ¼ ðs1 ; s2 ; s3 ; s4 ; s5 ; s6 ÞT ≡ðs11 ; s22 ; s33 ; s23 ; s13 ; s12 ÞT ;
ð3:4Þ
εp ¼ ðεp1 ; εp2 ; εp3 ; εp4 ; εp5 ; εp6 ÞT ≡ðεp11 ; εp22 ; εp33 ; 2εp23 ; 2εp13 ; 2εp12 ÞT ;
ð3:5Þ
and
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T. Liu et al. / J. Mech. Phys. Solids 61 (2013) 1798–1821
respectively. It is assumed that each component of plastic strain rate ε_ pi is a function only of the work-conjugate stress
component si , with no sensitivity to the other stress components. We define the plastic strain rate ε_ pi via an over-stress
relation as
8
p < jsi j−Y i ðεi Þ
if jsi j 4 Y i ðεpi Þ
p
η
ε_ i ¼
ð3:6Þ
:
0
otherwise;
where the yield strength Y i ðεpi Þ is a function only of the plastic strain εpi and η is the material viscosity. The total strain rate ε_ i
is obtained by supplementing the above anisotropic plasticity model with isotropic elasticity such that
ε_ i ¼ Lij s_ j þ ε_ pi signðsi Þ
ðsummation over jÞ:
ð3:7Þ
In the case of isotropic elasticity, the compliance matrix Lij of the core material is specified in terms of Young’s modulus
Ec and Poisson's ratio νc as
0
1
1=Ec −νc =Ec −νc =Ec
0
0
0
B
C
−νc =Ec
0
0
0
1=Ec
C
B
B
C
B
C
0
0
0
1=Ec
B
C
L¼B
ð3:8Þ
C
Þ=E
0
0
2ð1
þ
ν
B
C
c
c
B
C
B
C
0
sym
2ð1 þ νc Þ=Ec
@
A
2ð1 þ νc Þ=Ec
We employ an isotropic elastic response for simplicity; this suffices as the core response is dictated by the plastic branch.
(Numerical experiments confirmed that the predictions are insensitive to the precise magnitude of elastic modulus of the
core.) As already mentioned, the core has a density ρc and the cell walls are made from the same material as that of the solid
face sheets, of density ρm . The core is ascribed Young’s modulus Ec ¼ ρc E in terms of the relative density of the core ρc ≡ρc =ρm ,
and Poisson's ratio νc ¼ 0:25. The above elastic–plastic constitutive relation is expected to be adequate to model sandwich
cores such as the square-honeycomb core or the corrugated core; see for example Xue et al. (2005).
We assume that all strengths Y i are equal and strain hardening is neglected in all directions other than the transverse
direction. This constitutive response is considered to be representative of a square-honeycomb core (Xue et al., 2005). The
transverse strength Y 2 is assumed to be independent of the plastic strain εp2 up to a nominal densification strain εD : beyond
densification, a linear hardening behaviour is assumed with a very large tangent modulus Et ¼ 0:1E. The nominal
densification strain εD is related to the core relative density via the relation (Ashby et al., 2000)
εD ¼ 1−1:5ρ;
ð3:9Þ
while the transverse strength Y 2 ≡sc is varied in the parametric studies reported below. The viscosity η of the core was
chosen such that the shock width, as specified by (Radford et al., 2005)
ηεD
l¼
;
ð3:10Þ
ρc v0
is of magnitude c/10, with v0 interpreted as the initial impact velocity of the sand particles. This prescription ensures that
the shock width is always much less than the core depth yet exceeds the mesh size. Note that large gradients in stress and
strain occur over the shock width and thus a mesh size smaller than l is required in order to resolve these gradients.
3.2. Properties of the impacting particles
The properties of the particles were chosen to mimic silica sand particles. Unless otherwise specified, the following
parameters were employed in all calculations presented here. The cylindrical particles, of diameter D ¼ 200 μm and
thickness b ¼ 1 mm (in the x3 direction), have a solid material density ρs ¼ 2700 kg m−3 . The normal stiffness between the
particles is taken to be K n ¼ 7300 kN m−1 and the co-efficient of restitution is e ¼ 0:7for impacts between particles, and also
for impacts between the particles and the beam. (To achieve the same value of e for both types of impact, the value of γ n for
inter-particle contacts is twice that for impacts between the particles and the beam). Following Silbert et al. (2001), we set
the ratio K s =K n to equal 2/7, and the reference value of the friction co-efficient is μ ¼ 0:7.
4. Impact of sand against the monolithic beam
Consider the impact of a monolithic end-clamped beam by a sand slug at time t ¼ 0, as sketched in Fig. 2a. Introduce the
non-dimensional loading parameters as
a
a≡ ;
L
m¼
ρρs H
;
mb
ρ;
and
Io ≡
mb
Io
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sY =ρm
ð4:1Þ
where I o ¼ ρρs Hvo is the areal momentum of the impacting slug and mb ¼ 2ρm h is the areal mass of the beam. Dimensional
analysis dictates that, for a given set of beam material and sand properties, the deflection w of the mid-span of the beam has
T. Liu et al. / J. Mech. Phys. Solids 61 (2013) 1798–1821
1805
the functional form
w
h ρ
w≡ ¼ f t; ; m ; a; I o ; m; ρ ;
ð4:2Þ
L
L ρs
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where t≡t=ðLÞ ρm =sY is the time normalised by the structural response time of a plastic string of length 2L and made from a
material of density ρm and yield strength sY . Recall that all calculations are presented for a monolithic beam of aspect ratio
h=L ¼ 0:05 and a parametric study is now reported for the sensitivity of the response to a, I o , m and ρ. Unless otherwise
stated we assume the following reference values: a ¼ 0:2, ρ ¼ 0:2, m ¼ 0:405 (corresponding to a sand slug of height
H ¼ 300 mm). Preliminary calculations revealed that the contact properties of sand particles have only a very minor effect
upon the response of the beam: varying K n by two orders of magnitude and changing e and μ by a factor of five has less than
a 2% effect upon the predicted beam deflections. This conclusion is consistent with that of Pingle et al. (submitted for
publication) for sand impacting a rigid target.
4.1. Effect of sand momentum upon beam deflection
Predictions of the normalised mid-span deflection w versus normalised time t response of the beam are plotted in Fig. 4a
for four selected values of the normalised areal momentum of the sand I o ; the momentum was varied by keeping all
parameters fixed at their reference values and increasing the sand impact velocity from v0 ¼ 100 ms−1 to 400 ms−1 . The
maximum deflection wmax ≡wmax =L occurs at t≈1, and increases with increasing I o , as shown explicitly in Fig. 4b. Snapshots
of the sand slug and beam at selected times are shown in Fig. 5 for the case I o ¼ 0:67. Initially, the sand slug compacts
against the beam by the propagation of a near-planar densification front through the slug, from the impacted to the distal
end of the slug. With increasing time, the slug spreads laterally and the densification front adopts a curved profile.
The densification front reaches the distal end of the slug at tvo =H≈1 and subsequently, the beam deflection continues to
increase while the sand flows laterally. Elastic vibrations of the beam continue after the beam has attained its peak
deflection.
We proceed to quantify the loading of the beam due to impact by the sand slug. For this purpose we calculate the
pressure p exerted by the sand particles on the beam as follows. At any time t, there are M sand particles in contact with one
of the finite elements on the impacted surface of the beam. In the undeformed state, the mid-point of that element on the
surface of the beam has a co-ordinate X m along the x1 direction. The pressure at location X m at time t is then defined as
pðX m ; tÞ ¼
i
∑M
i ¼ 1F2
le
ð4:3Þ
where F i2 is the contact force in the x2 direction between the ith sand particle and the beam, and le is the undeformed length
of the finite element on the surface of the beam. The normalised pressure p=ðρρs v2o Þ is plotted in Fig. 6 for the choice I o ¼ 0:67
as a function of the spatial co-ordinate X, where X denotes the x1 co-ordinate of a material point on the beam surface in the
undeformed configuration, i.e. X ¼ 0 is at the beam mid-span and X ¼ L is at the supports. Results are shown in Fig. 6 for
three selected times: at t ¼ 0:1 and 0.25 the sand slug has not fully compacted while at t ¼ 1 the beam has reached its peak
deflection and the sand is flowing laterally, recall Fig. 5. It is clear from Fig. 6 that while the sand slug is compacting, the
spatial pressure profile over the beam surface is approximately rectangular with the pressure mainly acting over the width
of undeformed slug, i.e. over a width of 2a. In the early stages of impact (up to t ¼ 0:1), the pressure approximately equals
ρρs v2o but then decreases with increasing time. After the sand slug has fully compacted (at t ¼ 0:1), the sand flows laterally,
and exerts a negligible pressure upon the beam. Since the pressure on the beam is approximately spatially uniform over a
central patch of width 2a and of zero magnitude elsewhere, it suffices to characterise the pressure distribution by its average
areal value, as defined by
Z 2L
1
pðtÞ ¼
p dX
ð4:4Þ
2a 0
^
The evolution of p=ðρρs v20 Þ with normalised time t≡tv
0 =H is given in Fig. 7a for two selected values of I o . For both values of
impulse we find:
(i) The initial average pressure is p=ðρρs v20 Þ≈1 and then decreases slowly over the time interval 0 o t^ o 1.
^
(ii) The pressure drops sharply to almost zero at t≈1.
We rationalise these observations as follows. First, as discussed by Pingle et al. (2012), the loading imparted by the sand
is mainly inertial and hence scales as ρρs Δv2 , where Δv is the velocity of sand relative to that of the beam. At time t ¼ 0, the
beam is stationary and hence Δv ¼ v0 resulting in a pressure p=ðρρs v20 Þ≈1. With increasing time, the velocity of the beam
increases and consequently Δv decreases. Hence the pressure p gradually reduces. Second, the distal end of the sand slug
reaches the beam at time t≈H=v0 and subsequent motion of the sand is mainly in the lateral direction. Thus, the pressure
^
drops dramatically at t≈1.
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Fig. 4. (a) The normalised mid-span deflection w versus normalised time t histories for three selected values of the normalised sand slug impulse Io .
(b) The corresponding predictions of the normalised maximum mid-span deflection wmax as a function of I o . All other parameters are held fixed at their
reference values. Predictions using the equivalent foam projectile are also included in (a) and (b).
Fig. 5. Snapshots from the coupled discrete/continuum calculations of the sand slug impacting the monolithic beam at six selected normalised times t. The
results are shown for a normalised impulse I o ¼ 0:67 with all other parameters fixed at their reference values.
It is instructive to determine the time history of momentum transmitted to the beam by time integration of pðtÞ as
follows. Define the specific transmitted momentum as
Z t
pðt′Þ dt′
ð4:5Þ
I t ðtÞ ¼
0
T. Liu et al. / J. Mech. Phys. Solids 61 (2013) 1798–1821
1807
Fig. 6. The distribution of the normalised pressure p=ðρρs v20 Þ over the span of the beam at three times t after impact of the sand slug against the monolithic
beam with a normalised impulse I o ¼ 0:67 (all other parameters fixed at their reference values). The co-ordinate X denotes the x1 co-ordinate a material
point on the beam surface in its undeformed configuration, i.e. X ¼ 0 is at the beam mid-span and X ¼ L is at the supports.
Fig. 7. Predictions of (a) the normalised average pressure p=ðρρs v20 Þ and (b) normalised transmitted momentum I t =Io a function of the normalised time
^
t≡tv
0 =H for the sand slug impact against the monolithic beam. Results are shown for two values of the normalised momentum I o (all other parameters
fixed at their reference values).
The evolution of normalised momentum I t =I o transmitted into the beam is plotted in Fig. 7b, for the two selected values
^
of I o given in Fig. 7a. The temporal variation of the momentum I t displays two “knees”. The first knee occurs at t≈1
and
corresponds to the point where the distal end of the sand slug reaches the beam at time t≈H=v0 , and there is a significant
drop in the pressure exerted by the sand on the beam, as discussed above. After t≈1, the sand flows laterally along the
deformed beam and exerts a small pressure upon it: the sand acquires a small velocity in the positive x2 due to the deformed
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profile of the beam (see snapshots in Fig. 5). The beam attains its peak deflection at t≈1 and thereafter the beam springs
back elastically. The transmitted momentum I t continues to rise until I t =I o ≈1 and the second knee then occurs; at this point
the sand loses contact with the beam (see Fig. 5) and I t reaches a plateau.
4.2. Parametric study
We proceed to explore the sensitivity of the maximum deflection wmax upon the parameters m, ρ and a. We shall show
below that the response depends upon a particular combination of m and ρ in the form of the ratio
τ of ffisand-loading time
pffiffiffiffiffiffiffiffiffiffiffiffiffi
relative to the beam response time. Recall that the response time of the beam is approximately L ρm =sY while the loading
time is H=v0 . Thus, τ is defined as
1 H=v0
m2 h ρm
τ≡ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼
;
ð4:6Þ
2 L ρm =sY
ρI o L ρs
where we have included the factor of 0.5 in order to get an expression for τ with no numerical constants. It has already been
demonstrated by Qiu et al. (2003) and by Xue and Hutchinson (2003) that the dynamic response of the beam is only
dependent on I o for values of τ o0:01, i.e. the loading can be treated as impulsive and the response of the beam is
independent of the applied pressure. At larger values of τ, the maximum deflection decreases with increasing τ and the
response now depends upon both the applied pressure and the impulse. We shall show subsequently that the effect of both
m and ρ is adequately captured via the single parameter τ.
The ability of τ to capture the dependence of wmax upon the parameters m and ρ is demonstrated in Fig. 8. Calculations
were conducted for two selected values of I o : first, by varying m but ρ held fixed at the reference values of 0.2 (filled
squares), and second, by varying ρ but m held fixed at the reference values of 0.405 (open circles). It is evident that the
dependence of wmax on m and ρ is adequately captured via the single parameter τ.
The effect of the lateral spreading of the sand and the width a of the sand slug on the response of the beams are
discussed in the Appendix.
4.3. The foam analogue
It has been demonstrated above that constraining the sand against lateral flow has a negligible effect upon the response
of the beam. Moreover, the impact process in this constrained case as seen in Fig. 10a is very reminiscent of a foam
impacting a rigid stationary target. Pingle et al. (2012) developed an analogy between a constrained sand slug and a foam
projectile with all properties of the foam derived from the properties of the sand particle. Here, we investigate whether such
a foam model can also be used to predict the response of beams impacted by sand slugs.
Plane strain, finite deformation, FE calculations were conducted of foam projectiles impacting beams using the
commercial FE package ABAQUS. Again, only half the beams were modelled with symmetry imposed at x1 ¼ 0 and all
degrees of freedom constrained at x1 ¼ L to simulate fully clamped ends. Both the beam and foam projectile were discretised
using 4-noded plane strain elements with reduced integration (CPE4R in the ABAQUS notation). Typically, the elements
were square and of side length h=15 for the beam and H=100 for the foam projectile. The projectile was given an initial
Fig. 8. Predictions of the normalised maximum deflection wmax of the monolithic beams as a function of the normalised loading time τ. Two sets of
calculations are reported: (i) vary m with ρ ¼ 0:2 and (ii) vary ρ with m ¼ 0:405. In each case results are presented for two values of the normalised impulse
I o ¼ 0:67 and 1.0.
T. Liu et al. / J. Mech. Phys. Solids 61 (2013) 1798–1821
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Fig. 9. Predictions of the normalised deflection w versus normalised time t histories of the (a) c ¼ 0:11 and (b) c ¼ 0:33 sandwich beams impacted by sand
slugs with a normalised impulse I o ¼ 0:67. Results are shown for three values of the normalised core strength sc of the sandwich beams. Predictions of the
deflection of the monolithic beam of equal areal mass are also included. All parameters not listed here have their reference values.
velocity v0 and brought into contact with the beam at mid-span at time t ¼ 0; contact between projectile and beam was
modelled using the general contact option in ABAQUS. The beam material was modelled as an elastic–plastic solid using J2
flow theory, with properties as specified in Section 3.1. The foam projectile was identical to that as specified by Pingle et al.
(submitted for publication) and as detailed below.
The foam of initial density ρρs is a compressible solid, obeying the metallic foam constitutive
model
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi of Deshpande and
Fleck (2000). Write sij as the usual deviatoric stress and the von Mises effective stress as se ≡ 3sij sij =2. Then, the isotropic
yield surface for the metal foam is specified by
^
s−Y
¼ 0;
ð4:7Þ
where Y is the yield strength, and the equivalent stress s^ is a homogeneous function of se and mean stress sm ≡skk =3
according to
s^ 2 ≡
1
1 þ ðα=3Þ2
s2e þ α2 s2m
ð4:8Þ
Here, the material parameter α denotes the ratio of deviatoric strength to hydrostatic strength, and the normalisation
factor in the denominator of (4.8) is such that s^ reduces to the axial stress in uniaxial tension or compression. An over-stress
model is employed with the yield stress Y specified by
Y ¼ ηε_^ þ sc ;
p
ð4:9Þ
^ The characteristic sc ð^εp Þ is the static uniaxial stress
in terms of the viscosity η and plastic strain rate ε_^ (work conjugate to s).
versus plastic strain relation of the foam. Normality of plastic flow is assumed, and this implies that the “plastic Poisson's
p
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Fig. 10. Snapshots at six selected times t showing the deformation of the (a) c ¼ 0:11 and (b) c ¼ 0:33 sandwich beams with core strength sc ¼ 0:11
impacted by sand slugs with a normalised impulse I o ¼ 0:67. All parameters not listed here have their reference values.
ratio” νp ¼ −_εp22 =_εp11 for uniaxial compression in the 1-direction is given by
νp ¼
1=2−ðα=3Þ2
1 þ ðα=3Þ2
:
Young’s modulus of the foam is related to the contact stiffness of the sand particles via
pffiffiffi
3 3K n
Ef ¼
4b
ð4:10Þ
ð4:11Þ
while the elastic Poisson's p
ratio
ffiffiffi ν ¼ 0:25. Further, we assume that the foam has a plastic Poisson's ratio νp ¼ 0 which is
achieved by setting α ¼ 3= 2. The foam is used to model a loose aggregate of particles of vanishing static strength
up to densification; in order to give numerical stability, we assume a low value of static strength prior to densification,
according to
(
spl ε^ p ≤−lnð1−εD Þ
sc ¼
ð4:12Þ
∞
otherwise
where spl ¼ ρρs v20 =100 (i.e. significantly less than the dynamic strength). The nominal densification strain εD is given by
εD ¼ 1−
ρ
:
ρmax
ð4:13Þ
The discrete calculations suggest that ρmax ¼ 0:9 and this value is used in Eq. (4.13). It remains to specify the material
viscosity η. Radford et al. (2005) have shown that the shock width l within the foam is given by
l¼
ηεD
ρρs v0
ð4:14Þ
Using (4.14), we choose a viscosity so that l≪H in all calculations. This choice is motivated by the fact that Pingle et al.
(2012) observed the shock width to be below 10 particle diameters in their discrete calculations.
The discrete particle simulations are compared with the foam projectile simulations in Fig. 4: the foam projectile is able
to mimic sand impact in a satisfactory manner. The slight over-prediction of the deflection is due to the fact that the foam
projectile does not capture the reduction in the applied pressure associated with lateral spread of the sand. We conclude
T. Liu et al. / J. Mech. Phys. Solids 61 (2013) 1798–1821
1811
that the foam projectile represents a good compromise between accuracy and computational cost to simulate the loading of
beams by sand slugs.
5. Impact of sand slugs against sandwich beams
We proceed to analyse the response of sandwich beams subjected to the loading by a high velocity sand slug. These
sandwich beams have the same areal mass mb ¼ 400 kg m−2 and half-span L ¼ 0:5 m as the monolithic beams analysed in
Section 4. The deflection w of the back face of the sandwich beam at mid-span may be written in non-dimensional form as
w
ρ
w≡ ¼ f t; m ; c; sc ; a; I o ; m; ρ ;
ð5:1Þ
L
ρs
where c≡c=L is the aspect ratio of the beam and sc ≡sc =ðρc sY Þ is the normalised core strength: for a given topology, a
“stretching-dominated” cellular core will have a unique value of sc independent of its density (Wadley et al., 2003). Note
that the choice as stated by Eq. (3.2) implies that the core relative density is not an independent parameter, but is related to
the beam aspect ratio via the relation
ρc ≡
ρc
mb
:
¼
ρm
3ρm Lc
ð5:2Þ
The effect of the loading parameters (a; I o ; m; ρ) has already been clarified in Section 4. Our aim here is to contrast the
monolithic and sandwich beam responses and hence we focus attention on the sandwich beam parameter sc and present
results for two beam aspect ratios c ¼ 0:11 and 0.33 corresponding to the core relative densities ρc ¼ 0:3 and 0.1, respectively.
Unless otherwise specified, the following reference parameters are employed: a ¼ 0:2, ρ ¼ 0:2, m ¼ 0:405 and I o ¼ 0:67,
corresponding to an impact velocity vo ¼ 400 ms−1 .
The transient deflection w of sandwich beams with aspect ratios c ¼ 0:11 and 0.33 is shown in Fig. 9a and b, respectively,
for selected values of core strength sc . The response of the equivalent monolithic beam is included in Fig. 9. While the peak
deflections wmax of the c ¼ 0:11 sandwich beams with weak cores (i.e. low sc values) are comparable to their monolithic
counterparts, the c ¼ 0:33 sandwich beams have a significantly lower peak deflections compared to the equal mass
monolithic beams over the entire range of sc considered here.
Snapshots of the state of sand slug and sandwich beams (sc ¼ 0:11) are shown in Fig. 10a and b for c ¼ 0:11 and 0.33,
respectively. The overall deformation of the sand slug and beam is similar to that given in Fig. 5 for the monolithic beam: a
densification front travels through the slug, the sand spreads laterally, the beam attains its peak deflection and then
rebounds elastically, and finally contact with the sand is lost after the sand is reflected backwards. However, the details of
the deformation of the sandwich beam are markedly different to that of the monolithic beam. In the initial phase, the
deformation comprises mainly of core compression with negligible deflection of the back face of the sandwich beam. This is
followed by the bending of the entire beam. For the cases shown in Fig. 10a and b, the core compression phase typically ends
before the peak back face deflection has been attained.
Define ΔcðtÞ as the transient reduction in core thickness at mid-span due to sand impact. The associated core
compression strain εc ≡Δc=c is plotted in Fig. 11a for c ¼ 0:11 and in Fig. 11b for c ¼ 0:33. Curves are shown for selected
values of sc , and for the usual reference values a ¼ 0:2, ρ ¼ 0:2 and I o ¼ 0:67 (corresponding to the w predictions in Fig. 9).
Note that εc increases with decreasing sc but the time t c required for the maximum core compression εmax
to be attained is
c
reasonably independent of sc .
The effect of core strength sc upon the normalised maximum deflection wmax ≡wmax =L and maximum core compression
εmax
is shown explicitly in Fig. 12a and b, respectively. Recall that the maximum deflection of the equivalent monolithic
c
beam subjected to the same loading is wmax ≈0:14. Thus, over the entire range of sc values considered, the c ¼ 0:33 sandwich
beam outperforms their monolithic counterparts in terms of having of lower wmax while the c ¼ 0:11 sandwich beams have
a superior performance only for sc values greater than 0.2. The core compression εmax
and the corresponding core
c
compression time t c were higher for the c ¼ 0:33 sandwich beam compared to the c ¼ 0:11 beam for all values of sc
considered here. This observation is rationalised as follows. The c ¼ 0:33 core has a relative density ρc ¼ 0:1 while ρc ¼ 0:3 for
the c ¼ 0:11 beam. Thus, the absolute value of core strength sc for the c ¼ 0:33 beam is less than that for the c ¼ 0:11 beam,
for any given normalised strength sc . This lower value of core strength sc leads to a larger core compression and to a longer
core compression time t c .
5.1. Analysis of the superior performance of sandwich beams
In general, the superior response of sandwich beams over monolithic design to impact by sand blast has two possible
sources, as follows:
(i) A fluid–structure interaction effect can occur whereby the pressure exerted by the sand on the sandwich beam is less
than that exerted on the monolithic beam (and consequently the momentum transmitted to the sandwich panel is less
than that transmitted into the monolithic beam). This was the dominant phenomenon responsible for the superior
performance of sandwich panels subjected to underwater blast loading (Fleck and Deshpande, 2004).
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Fig. 11. Predictions of the variation of the core compression εc with normalised time t for the (a) c ¼ 0:11 and (b) c ¼ 0:33 sandwich beams impacted by
sand slugs with a normalised impulse I o ¼ 0:67. Results are shown for the three selected values of core strength sc corresponding to Fig. 11 with all
parameters fixed at their reference values.
In order to understand the relative contributions from these two effects we first plot in Fig. 13 the temporal variation of
the normalised pressure p=ðρρs v2o Þ for the c ¼ 0:33 sandwich beam and for two extreme values of sc . The spatially
averaged pressure p is calculated using (4.3) and (4.4) in the same manner as that described for the monolithic beams in
Section 4. The corresponding prediction for the monolithic beam is included in Fig. 13. It is clear that the loading due to
the sand is nearly identical for the sandwich and monolithic beams and hence the explanation for the superior
performance of sandwich beams lies elsewhere.
(ii) The structural performance of sandwich beams is superior to that of monolithic beams. It is generally recognised that the
static bending strength of sandwich beams exceeds that of monolithic beams of equal mass due to the larger plastic
moment of area. Consequently, the sandwich beam will deflect less than the monolithic beam for a given applied load,
provided the static response of a beam is dominated by the bending strength rather than the stretch resistance; this
occurs when the deflection of the sandwich beams is less than the beam thickness.
In contrast, if the beam deflection exceeds the beam thickness, the response becomes dominated by beam stretching and
sandwich beams then lose their advantage over monolithic beams. It is inferred from Fig. 12a that wmax =c≡wmax =c is less
than unity for the c ¼ 0:33 beams over the entire range of sc considered, i.e. the sandwich beams are in the regime where
they outperform monolithic beams. In contrast, the c ¼ 0:11 beams have a smaller core thickness and, for the loadings
considered here, wmax typically exceeds c. Consequently, these beams deflect by axial stretch and have a performance
comparable to their monolithic counterparts. We further note from Fig. 12a that wmax ≡wmax =L decreases with increasing sc .
This is traced to the fact that an increase in sc leads to a reduction in εmax
(Fig. 12b), and to a preservation of the sandwich
c
effect upon sand impact.
T. Liu et al. / J. Mech. Phys. Solids 61 (2013) 1798–1821
1813
Fig. 12. Predictions of (a) normalised maximum deflection wmax and (b) maximum core compression εmax
as a function of the normalised core strength sc
c
for the c ¼ 0:11 and c ¼ 0:33 sandwich beams impacted by sand slugs with a normalised impulse Io ¼ 0:67. All parameters not listed here have their
reference values. The corresponding deflection of the monolithic beam of equal areal mass is included in (a). Predictions using the equivalent foam
projectile are included in (a) and (b).
5.2. Foam analogue
It has already been demonstrated that the foam projectile gives a similar loading pulse to that of a sand slug for
monolithic beams. Here, we demonstrate that the foam projectile analogy carries forward to sandwich beams.
The properties of the foam projectile, and associated FE representation have already been described in Section 4.4.
Predictions of wmax and εmax
as a function of sc are included in Fig. 12 for the c ¼ 0:11 and 0.33 sandwich beams. Similarly,
c
predictions of wmax versus I o are compared in Fig. 14 for the foam analogue and sand slug; results are plotted for the
monolithic beam, and for sandwich beams of dimension c ¼ 0:33 and two selected values of sc . In both Figs. 12 and 14
excellent agreement is obtained between the predictions of the sand slug impact and the foam projectile. We conclude that
a foam projectile is a useful laboratory simulator for sand slug loading of both monolithic and sandwich beams.
6. Circular plates loaded by cylindrical sand slugs
We proceed to show that the above findings also hold for circular plates impacted by sand slugs comprising spherical
sand particles. Here, we analyse the response of circular monolithic and sandwich plates (Fig. 15) impacted centrally and
normally by a cylindrical sand slug of radius a and height H, comprising spherical sand particles of diameter D and with a
spatially uniform relative density ρ. Before impact, the sand particles have a velocity v0 in the negative z direction, as shown
in Fig. 15. The sand slug impacts clamped circular plates of radius R and areal mass mb at time t ¼ 0.
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2.0
p
ρρsv02
1.5
Monolithic
beam
1.0
σc = 0.64
0.5
σc = 0.11
0.0
0.0
0.1
0.2
0.3
Time after impact
0.4
0.5
0.6
0.7
t
L ρm / σ y
Fig. 13. Predicted normalised average pressure p=ðρρs v20 Þ versus normalised time t history for the c ¼ 0:33 sandwich beams impacted by the sand slug with
impulse I o ¼ 0:67. Results are shown for two extreme values of the normalised core strength sc with all other parameters equal to their reference values.
The corresponding predictions for the equal areal mass monolithic beam are also included.
Fig. 14. Predictions of the normalised maximum mid-span deflection of the back face of the c ¼ 0:33 sandwich beams as a function of the sand slug
normalised impulse I o . Results are shown for two values of the normalised core strength sc with all other parameters equal to their reference values.
Predictions using the equivalent foam projectile are also included.
6.1. Pseudo three-dimensional calculations
The boundary value problems as sketched in Fig. 15 are inherently three-dimensional (3D) as we need to model spherical
sand particles. However, in order to reduce the computational cost, we analysed these problems using a pseudo 3D
technique wherein the sand slug is modelled in a 3D setting but the plate is assumed to undergo axi-symmetric
deformation. We briefly explain the methodology.
A slice of the cylindrical sand slug subtending a polar angle ϕ about the central axis is shown in Fig. 16, and is modelled
using the GRANULAR package of the discrete code LAMMPS. The sand particles are constrained in the circumferential
direction by radial rigid stationary walls along θ ¼ 0 and θ ¼ ϕ, as shown in Fig. 16. This ensures that deformation of the
cylindrical sand slug is approximately axi-symmetric. Contact between the spherical sand particles of diameter D is the
same as that described in Section 2.1 for cylindrical particles.
The discrete particle calculations are coupled to the axi-symmetric FE calculation as follows. At any time instant t, the FE
calculation provides the radial and axial coordinates ðr p ; zp ) and radial and axial velocities ðvr ; vz Þ of every material point on
the plate surface. The FE calculation is axi-symmetric; consequently, the co-ordinate zp is independent of θ and of the
tangential velocity vθ ¼ 0. The contact forces between the particles and the plate surface (F r and F z in the radial and axial
directions, respectively) are calculated as described in Section 2.3 and a vector of nodal forces is determined. We note
in passing that the force F θ due to any motion of the sand particles in the θ direction is neglected as it cannot be included
in the axi-symmetric FE calculation. These contact forces are scaled by the factor 2π=ϕ and inserted into the vector of
T. Liu et al. / J. Mech. Phys. Solids 61 (2013) 1798–1821
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Fig. 15. Sketches illustrating the geometries of the clamped circular (a) monolithic and (b) sandwich plates impacted by a cylindrical sand slug comprising
identical spherical sand particles.
global nodal forces F in Eq. (2.5). Subsequently, the discrete and finite element equations are integrated as described
in Sections 2.1 and 2.2, and the new positions and velocities of the particles in the discrete calculations and material points
in the finite element calculations are determined at time t þ Δt.
6.2. Geometry and material properties
Clamped plates of radius R ¼ 0:5 m and areal mass mb ¼ 160 kg m−2 are analysed. Both the monolithic and sandwich
plates are made from DH36 steel, as detailed in Sections 4 and 5. The monolithic plates are of thickness 2h ¼ 20 mm and the
sandwich plates were designed in a similar manner to the sandwich beams, such that the areal mass equals that of the
monolithic plate. The mass of the sandwich plate is partitioned equally between the two face sheets, each of thickness
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Fig. 16. Sketch illustrating the slice of the circular cylindrical sand slug analysed using discrete calculations. The applied boundary conditions on the sides
of the slice are also illustrated.
hf ¼ 6:67 mm, and the core of thickness c; the value of c is related to relative density ρc via Eq. (5.2). Similar to the sandwich
beams, we shall present results for two core relative densities ρc ¼ 0:1 and ρc ¼ 0:3 corresponding to c≡c=R ¼ 0:044 and
0.133, respectively. The material properties of the monolithic plate material, the sandwich plate face sheets and the
compressible core are identical to those described in Section 3.1. The only free material parameter is the core strength sc :
results are presented for a range of values of sc in order to parameterise the effect of core strength.
In all calculations, the sand slug has a radius a ¼ 100 mm so that a=R ¼ 0:2, the initial height is H ¼ 150 mm, and the
sand particles are of diameter D ¼ 200 μm and are initially spaced to possess a relative density of ρ ¼ 0:2. The sand particles
are made from silica with density and contact properties as listed in Section 3.2.
For a given set of sand properties, the mid-span deflection of the monolithic and sandwich plates may be written in nondimensional form as
w
h ρ
w≡ ¼ f t; ; m ; a; I o ; m; ρ
ð6:1Þ
R
R ρs
and
w≡
w
ρ
¼ g t; m ; c; sc ; a; I o ; m; ρ ;
R
ρs
respectively. The various terms have already been defined in Sections 4 and 5, but with L replaced by R, such that
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
t≡t=ðR ρm =sY Þ; c≡c=R and a ¼ a=R:
ð6:2Þ
ð6:3Þ
The following non-dimensional groups are held fixed in the calculations: m ¼ 0:405, h=R ¼ 0:04, a ¼ 0:2, ρ ¼ 0:2 and
ρm =ρs ¼ 2:96. Results are presented for the effect of the sand momentum I o , sandwich beam aspect ratio c and core strength
sc upon the plate deflection. All calculations assumed a cylindrical slice of sand slug such that ϕ ¼ 10o : spot calculations
demonstrated that increasing the value of ϕ has no appreciable effect upon the predictions.
T. Liu et al. / J. Mech. Phys. Solids 61 (2013) 1798–1821
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6.3. Response of monolithic and sandwich plates
The normalised maximum back face deflections wmax ≡wmax =R at mid-span, versus normalised sand slug impulse I o , is plotted
in Fig. 17a for both the monolithic and the c ¼ 0:133 sandwich plates. Results for the sandwich plate are given at two values of core
strength sc . Over the entire range of impulses considered, the sandwich plates have a lower deflection than the monolithic plates.
As for the sandwich beams (Fig. 12a), the deflection of the sandwich plate decreases with increasing core strength. Snapshots
showing the deformation of ϕ ¼ 10o slices of the monolithic and sandwich plates (c ¼ 0:133, sc ¼ 0:21) are included in Fig. 18a and
b, respectively. The deformation modes are qualitatively very similar to those of the monolithic beam (Fig. 5) and sandwich beams
(Fig. 10). Thus, we anticipate that the findings for the beams will extrapolate to the plates in a qualitative manner.
The normalised maximum back face deflections, wmax , are summarised in Fig. 17b for c ¼ 0:044 and 0.133 sandwich
plates as a function of the core strength sc . The corresponding deflection of the monolithic plate of equal area mass is
included. Consistent with the findings for the sandwich beams, the sandwich plate of thicker core (c ¼ 0:133) outperforms
both the monolithic plate and the c ¼ 0:044 sandwich plate over the entire range of sc values investigated here. In contrast,
the c ¼ 0:044 sandwich plates have a smaller deflection than the monolithic plates only for sc 4 0:2, i.e. strong cores. Again,
Fig. 17. (a) Predictions of the normalised maximum deflection wmax of the monolithic and the c ¼ 0:133 sandwich plate as a function of the normalised
sand slug impulse Io . Results for the sandwich plate are shown for two values of the normalised core strength sc . (b) The deflections wmax of the c ¼ 0:044
and c ¼ 0:133 sandwich plates as a function of sc for an impulse I o ¼ 0:67. The corresponding deflection of the monolithic plate of equal areal mass is
included in (b). Predictions using the equivalent foam projectile are also included in (a) and (b).
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Fig. 18. Snapshots at six selected times t showing the deformation of the (a) monolithic and (b) c ¼ 0:133 sandwich plate with core strength sc ¼ 0:21
impacted by sand slugs with a normalised impulse I o ¼ 0:67. All parameters not listed here have their reference values. The figures show the ϕ ¼ 10o slice
analysed in the discrete calculations.
the superior performance of the sandwich plates is due to the “sandwich effect” as discussed in Section 5.1: the sandwich
effect is greater for thicker cores and hence the c ¼ 0:133 plates outperform the thinner c ¼ 0:044 plates.
6.4. Foam analogue
We have already demonstrated in Sections 4.4 and 5.2 that the loading imposed by sand slugs on monolithic and
sandwich beams loaded is adequately represented by that of a foam projectile. We now extend this analogy to the plates
analysed in this section.
We proceed to replace the cylindrical sand slug by an equivalent cylindrical foam projectile of the same overall
dimensions and density as that of the undeformed sand slug and travelling with equal initial velocity. The material
properties of the foam are identical to those detailed in Section 4.4 with two exceptions. The sand slug now comprises
spherical rather than cylindrical sand particles. These particles can pack to a maximum relative density ρmax ¼ 0:74 (rather
than the ρmax ¼ 0:9 in the 2D case). Assuming that ρmax corresponds to cubic close packing, the modulus Ef of the foam
under uniaxial straining is then determined by using the Octet truss properties as given by Deshpande et al. (2001). This
analysis gives Ef in terms of the sand properties as
pffiffiffi
4 2K n
:
ð6:4Þ
Ef ¼
3D
Thus, the only modifications to the prescription presented in Section 4.4 are: (i) (4.11) is replaced by (6.4) and (ii)
ρmax ¼ 0:74 is used in (4.13) rather than ρmax ¼ 0:9.
Axi-symmetric FE calculations of the foam projectiles impacting the monolithic and sandwich plates have been
performed using ABAQUS. Predictions of wmax are included in Fig. 17a and b and clearly show the excellent agreement
between the full discrete calculations and the calculations using the equivalent foam projectiles.
7. Concluding remarks
We have presented a coupled discrete/continuum computational framework to analyse the loading of monolithic and sandwich
structures by sand columns (slugs) travelling at high velocities. The sand is modelled as either discrete identical circular cylindrical
or spherical particles while the structures (beams or circular plates) are modelled using a Lagrangian finite element (FE) framework.
T. Liu et al. / J. Mech. Phys. Solids 61 (2013) 1798–1821
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Fig. A1. (a) Predictions of the normalised maximum deflection wmax of the monolithic beam as a function of the normalised width a of the sand slug for
two selected values of the sand slug impulse I o . (b) Snapshots showing the deformation of the monolithic beam and sand slug at six selected times t for the
a ¼ 0:8 slug with an impulse Io ¼ 0:67. All parameters not listed here have their reference values.
The clamped beams and plates are loaded by sand slugs impacting the structures normally and centrally: this is a model problem
devised to develop a fundamental understanding of the sand–structure interaction that occurs in more complex loading events
such as the loading of structures by landmine explosions.
The calculations of the monolithic beams have demonstrated that the loading due to the sand is primarily inertial in
nature. The loading and the structural response is adequately characterised by two parameters: (i) the initial momentum of
the sand slug and (ii) the loading time as given by the ratio of the height of the slug and the initial velocity of the sand
particles. The effect of the areal mass ratio of the sand slug and structure, relative density of the sand particles in the sand
slug and beam aspect ratio are all captured within a single non-dimensional parameter, viz, the ratio of the loading time to
the structural response time.
The sandwich beams with a thick strong core have a high bending strength and significantly outperforms monolithic
beams of equal areal mass. The main sand–structure interactions mechanisms as elucidated by the beam calculations also
occur in a three-dimensional setting where slugs comprising spherical sand particles impact clamped circular monolithic
and sandwich plates. Sandwich plates with thick, strong cores outperform monolithic plates of equal areal mass.
We have demonstrated that replacement of the sand slug by an equivalent foam projectile leads to computational
savings while capturing the response of the monolithic and sandwich beams and plates to remarkable accuracy. Thus, a
foam projectile can replace the sand slug in both numerical calculations and in laboratory-based testing.
Acknowledgements
This research was supported by the Office of Naval Research (ONR NICOP grant number N00014-09-1-0573) as part of a
Multidisciplinary University Research Initiative on “Cellular materials concepts for force protection”, Prime Award no.
N00014-07-1-0764 is gratefully acknowledged. The programme manager was Dr. David Shifler.
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T. Liu et al. / J. Mech. Phys. Solids 61 (2013) 1798–1821
Appendix A. The effect width of the sand slug and the lateral spread of the sand
The sensitivity of the maximum deflection wmax to the normalised width a of the sand slug is summarised in Fig. A1a for
two selected values of I o , with all other parameters held fixed at their reference values. Similar to the impulsive loading
results of Qiu et al. (2005), the deflections increase with increasing a for a given value of I o . Snapshots showing the
deformation of the beam and sand slug are included in Fig. A1b for the case with I o ¼ 0:67 and a ¼ 0:8. The deformation
mode is very similar to that seen in Fig. 5 for the a ¼ 0:2 case. Thus, the increase in the deflection is not due to a change in
the deformation mode but rather only because the total momentum of the sand slug increases with increasing a with all
other parameters held fixed.
In order to investigate the effect of lateral spread of the sand upon beam deflection, we have performed an additional set of
calculations such that the sand particles are constrained to remain within the strip jx1 j o a, i.e. the width of the sand slug remains
constant during the entire deformation history. Time snapshots showing the deformation of the beam and sand slug from a
simulation employing this constraint are included in Fig. A2a for I o ¼ 0:67 and the other parameters held fixed at their reference
values. The deformation of the sand slug in Fig. A2a is reminiscent of the impact of a foam on a target with a planar densification
front travelling from the impacted end of the slug towards the distal end (Radford et al., 2005). After the densification front reaches
the distal end it reflects as a tensile wave and the sand spalls and travels in the negative x2 direction.
The significance of lateral constraint upon the wmax versus I o is given in Fig. A2b for two values ρ and all other parameters
held fixed at their reference values. The figure includes two types of simulations: in the unconstrained simulations lateral
spreading of the sand is permitted while in the constrained simulations no spreading of the sand into the region jx1 j 4a is
permitted. The predictions of wmax for the ρ ¼ 0:2 case are nearly identical in the constrained and unconstrained cases while
for ρ ¼ 0:45, wmax is about 5% higher for the constrained case. The small increase in deflection for the constrained ρ ¼ 0:45
Fig. A2. (a) Snapshots showing the deformation of the monolithic beam and constrained sand slug with an impulse Io ¼ 0:67 at six selected times t.
(b) Predictions of the normalised maximum deflections wmax as a function of the sand slug impulse I o . Results are included for two relative densities ρ of
both the constrained and unconstrained sand slugs. All parameters not listed here have their reference values.
T. Liu et al. / J. Mech. Phys. Solids 61 (2013) 1798–1821
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case is associated with the greater bounce-back of the sand (and increase in transmitted impulse) at higher values of ρ, as
discussed by Pingle et al. (2012).
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