INVERSE METHOD FOR PARAMETER DETERMINATION OF BIAXIALLY
LOADED CRUCIFORM COMPOSITE SPECIMENS
C. Ramault1, A. Makris1, A. Smits1, D. Lecompte2, D. Van Hemelrijck1, H. Sol1, W. Van Paepegem3
1
Vrije Universiteit Brussel, Mechanics of Materials & Constructions, Brussels, Belgium
2
Royal Military Academy (RMA) - Department of Materials and Construction, Brussels, Belgium
3
Ghent University - Department of Mechanical Construction and Production, Ghent, Belgium
[email protected], [email protected], [email protected], [email protected],
[email protected]
ABSTRACT
This paper presents an inverse method for the identification of the in-plane orthotropic apparent engineering constants of biaxially loaded composite materials using cruciform specimens. The full field displacements are identified by a digital image
correlation technique. From the displacement field a strain field is computed and compared with finite element strain results of
the experiment. The apparent engineering constants are unknown parameters in the finite element model. Starting from initial
values, these parameters are updated till the computed strain field matches the experimental strain field. In a first stage, global
apparent engineering constants were determined. In this stage a more local determination of the engineering constants is
studied.
1. INTRODUCTION
In general, composite laminates are developing multi-axial stress states [1]. However, there is little existing experimental
capability to evaluate the multi-axial response of composite materials, even though large demand for such information exists
[2,3]. The commonly used method to apply biaxial loads to a composite specimen is the combined torsion and
tension/compression or pressure and tension/compression of a thin-walled tubular specimen. However, real construction
components in fibre reinforced composite materials are often made in the form of flat or gently curved panels. The biaxial
behaviour of the tubes is consequently different from the real behaviour. The most appropriate method for bi-axial testing
consists of applying in-plane biaxial loads to cruciform specimens. Therefore, a plane biaxial test device has been developed
at the Free University of Brussels [4]. Due to the geometry of the specimens however, the mechanical material parameters
cannot be obtained as with standard uni-axially loaded beamlike specimens.
Therefore, an inverse method was used. This method has been developed at the Royal Military Academy (RMA), which
integrates an optimization technique, a full-field measurement technique and a finite element method [5,6]. In this paper, a
method is proposed for the identification of the in-plane engineering constants E1, E2, G12 and ν12 of an orthotropic material
based on surface measurements. Also, a manner to evaluate the functioning of the biaxial testing machine will be presented,
by parameterize the slope of the distributed load in both directions and the angle between the x axis and the direction of the
first principal material axis. The responses of the system, i.e. the surface displacements are measured with digital image
correlation. Strains are subsequently calculated, based on the measured displacement field. A finite element model of the
cruciform specimen serves as numerical counterpart for the experimental set-up. The difference between the experimental and
numerical strains (the cost function) is minimized in a least squares sense by updating the values of the engineering
constants. The optimization of the parameters is performed by a Gauss-Newton method.
Fig. 1 - Biaxial cruciform device
Fig. 2 – Forces on the cruciform specimen :
ideal situation (L) and real situation (R)
2. BIAXIAL TEST SET-UP
2.1 Testing device
Different experimental techniques and specimens have been used in the past to produce biaxial stress states. These
techniques may be mainly classified into two categories [7]: (i) tests using a single loading system and (ii) tests using two or
more independent loading systems. In the first category the biaxial stress ratio depends on the specimen geometry -which is
their main disadvantage-, whereas in the second category it is specified by the applied load magnitude. Examples of the first
category are bending tests on cantilever beams, anticlastic bending of rhomboidal shaped plates and bulge tests. Examples of
the second category are thin-walled tubes subjected to a combination of tension/compression with torsion or internal/external
pressure [8] and cruciform specimens under biaxial loading [9]. The most direct way to create biaxial stress states consists of
applying in-plane loads along two perpendicular arms of cruciform specimens.
The plane biaxial test device, developed at the Free University of Brussels (Fig.1), has four independent servo-hydraulic
actuators. The use of hydraulic actuators represents a very versatile technique for the application of the loads. When one
actuator per loading direction is used [10], the centre of the specimen will move. This causes a side bending of the specimen,
which results in undesirable non-symmetric strains. Systems -like the one used in this paper- with four actuators [11-12] with a
close-loop servo control using the measured loads as feedback system, allow the centre of the specimen standing still.
The device has a maximum capacity of 100 kN in both directions, but is restricted to tensile loads. As no cylinders with
hydrostatic bearing were used, failure or slip in one arm of the specimen will result in sudden radial forces which could
seriously damage the servo-hydraulic cylinders and load cells. To prevent this, hinges were used to connect the specimen to
the load cells and the servo-hydraulic cylinders to the test frame. Using four hinges in each loading direction results in an
unstable situation in compression and consequently only tension loads can be applied. The maximum displacement ranges
from +75mm till -75mm. The loading of both test specimen types may be static or dynamic up till 20Hz.
In an ideal situation no displacement of the centre point of the specimen is observed. Even when using four actuators, a small
displacement might occur in a real situation and an imbalance arises in the forces and for instance a component Fy is added in
the y-direction (Fig.2). Due to the displacement, forces P and P’ are no longer equal. However as four load cells are used, it is
possible to measure this small load difference and this is used as a control signal.
2.2 Specimen design
Traditional biaxial tests on with cruciform specimens require at least the following conditions: (i) maximization of the region of
uniform biaxial strain and (ii) minimization of the shear strains in the biaxially loaded test zone [13]. These conditions are
required if strains are measured at the centre of the specimen with a strain gage or mechanical extensometer since one
average strain value is obtained over their length. Since in this paper a full field method is used for the strain determination,
non-uniformities and occurring shear strains can be quantified and thus eliminate the requirement of a uniform strain field . It is
even found out that for the purpose of material parameter identification, strain heterogeneities are necessary since the aim is
to create a heterogeneous deformation field and to examine its information content with respect to the four independent
orthotropic parameters. In this paper two geometry types were tested: one completely flat and one with a hole of 8 mm
diameter in the centre to increase the strain heterogeneity. The total length of the specimens is 250 mm, the width of the arms
25 mm and the radius of the corner fillet 20 mm (Fig.3).
Fig. 3 - Specimen design with hole
Fig. 4 – Material lay-up
The material used is glass fibre reinforced epoxy with a [(+45° -45° 0°)3(+45°-45°)]-lay-up (Fig.4). This material with this
specific lay-up is often used for wind turbine rotor blades as they are globally submitted to bending combined with torsion, a
complex stress state. The (+/-45°)-layers have a thickness of 0.61mm; the 0°-layers of 0.88mm resulting in a total nominal
thickness of 5.08mm. The engineering constants E1, E2, ν12 and G12 that are targeted in this paper are the homogenized
values over the thickness; the so-called apparent engineering constants.
3. DIGITAL IMAGE CORRELATION TECHNIQUE
Digital image correlation is a full field measurement technique that detects deformations using image processing. Once the
deformations of a speckle pattern applied to the specimen have been measured, the strain distribution in the material can be
calculated. This technique allows studying qualitatively as well as quantitatively the mechanical behaviour of materials under
certain loading conditions and has been used in various technological domains. [14-16]. Each picture taken with a CCD
camera corresponds to a different load step. The cameras of the current set-up use a small rectangular piece of silicon, which
has been segmented into 1392 by 1040 pixels. Each pixel stores a certain grey scale value ranging from 0 to 4095 (12 bit), in
accordance with the intensity of the light reflected. Two images of the specimen at different states of loading are compared by
using a pixel grey value in the undeformed image and searching for the pixel in the deformed image, in order to maximize a
given similarity function. A grey-value is not a unique signature of a pixel, so neighbouring pixels are used. Such a collection of
pixels is called a subset. The displacement result, expressed in the centre of the subset, is an average of the displacements of
the pixels inside the subset. The step size defines the number of pixels over which the subset is shifted in x- and y- direction to
calculate the next result. For the actual tests the subset size is 19 pixels and the step size 5 pixels. The image correlation
routine allows locating every subset of the initial image in the deformed image. Subsequently, the software determines the
displacement values of the centres of the subsets, which yields an entire displacement field. (Fig. 5) depicts the sequence of
taking a picture of an object before and after loading, storing the images onto a computer trough a frame grabber, performing
the correlation of both images and finally calculating the corresponding displacement of the centres of the subsets, which
finally yields the desired displacement field. The strain field is then calculated by numerical differentiation of the smoothened
displacement field.
Fig. 5 - Digital Image Correlation
Fig. 6 - Flow-chart of determination process
4. INVERSE METHOD
4.1 Introduction
The mixed numerical experimental technique used in this paper belongs to the category of inverse problems. In contrast to a
direct problem which is the classical problem where a given experiment is simulated in order to obtain the stresses and the
strains, inverse problems are concerned with the determination of the unknown state of a mechanical system, using
information gathered from the response to stimuli on the system. [17] Not only the boundary information is used, but relevant
information coming from full-field surface measurements is integrated. The inverse method described here can be narrowed to
parameter identification, as the only item of interest is the determination of the constitutive parameters. The values of these
parameters cannot be derived immediately from the experiment due to the specimen geometry. A numerical analysis is
necessary to simulate the experiment. However, this requires that the material parameters are known. The identification
problem can be formulated as an optimization problem where the function to be minimized is an error function that expresses
the difference between numerical simulation and experimental results. In the present case the strains are used as output data.
(Fig. 6) represents the flow chart of the inverse modeling problem.
4.2 Optimization algorithm
The optimization of the apparent engineering constants is performed by a Gauss-Newton method. The cost function that is
minimized is a simple least squares formulation. Expression (1) shows the form of the least-squares cost function that is
minimized. The residuals in the function are formed by the differences between the experimental and the numerical strains.
n
ε inum ( p ) − ε iexp
i =1
ε exp
i
∑
C( p ) = C( ε( p ), p ) =
2
(1)
The necessary condition for a cost function to attain its minimum is expressed by equation (2). The partial derivative of the
function with respect to the different material parameters has to be zero. By developing a Taylor expansion of the numerical
finite element strains around a given parameter set, an expression is obtained in which the difference between the present
parameters and their new estimates is given (3).
∂C( p )
∂p i
ε
num
i
num
exp
num
1 n  ε i ( p ) − ε i  ∂ε j
=
∑
 ∂p = 0
C( p ) j=1 
ε exp
i
i

( p) ≅ ε
num
i
m
(p ) + ∑
k
k
∂ε num
(p )
i
∂p j
j =1
(p
j
(2)
− p kj )
(3)
When substituting this last expression into expression (2) and after rearranging some terms, expression (4) yielding the
parameter updates is obtained.
( ) (
−1
t
( ))
t
∆p = S S S ε exp − ε num p
k
(4)
in which the following elements are:
∆p :
column vector of the parameter updates of E1, E2, G12 and v12
ε exp :
ε
num
column vector of the experimental strains
k
(p ) :
column vector of the finite element strains as a function of the parameters at iteration k
pk:
the four parameters at iteration step k
S:
sensitivity matrix
4.3 Sensitivity calculation
The sensitivity matrix (5) groups the sensitivity coefficients of the strain components in every element of the finite element
mesh with respect to the elastic material parameters. The index n in equation (5) stands for the total number of elements. The
components of this sensitivity matrix can be derived analytically from the constitutive relation between stress and strain, which
is given by expression (6) in the case of a plane stress problem.
 ∂ε 1x
 ∂E
 11
 ∂ε y
 ∂E
 11
 ∂γ xy
 ∂E
1
S= M
 n
 ∂ε x
 ∂E1
 ∂ε n
 y
 ∂E1
 ∂γ n
 xy
 ∂E1
∂ε 1x
∂E 2
∂ε 1y
∂ε 1x
∂G 12
∂ε 1y
∂E 2
∂γ 1xy
∂G 12
∂γ 1xy
∂E 2
M
∂ε nx
∂E 2
∂ε ny
∂G 12
M
∂ε nx
∂G 12
∂ε ny
∂E 2
∂γ nxy
∂G 12
∂γ nxy
∂E 2
∂G 12
∂ε 1x 
∂ν12 
∂ε 1y 
∂ν12 

∂γ 1xy 
∂ν12 

M 
n
∂ε x 
∂ν12 
∂ε ny 

∂ν12 
∂γ nxy 

∂ν12 
(5)
 1

 ε x   E1
   v12
 ε y  = −
γ   E1
 xy 
 0

v12
E1
1
E2
−
0

0 
σx 
 
0 σy 

 
1  τ xy 
G12 
(6)
The stresses that are used in the calculation of the derivatives are taken from the converged simulation in the actual iteration
step. The values of the parameters are taken from the previous iteration step.
5. EXPERIMENTAL RESULTS
5.1 Tests on rectangular specimen
An extended database of experimental static and fatigue results on beamlike glass fibre reinforced epoxy specimens with a
[(+45° -45° 0°)3(+45°-45°)]-lay-up has been set-up within the framework of the Optimat Blades project [18]. For the glass fibre
reinforced composite laminate with the mentioned lay-up the average and standard deviation material parameter results of
about four hundred traditional beamlike tests are given in (Tab. 1). No information about the shear modulus is available for this
lay-up. Only for a single unidirectional ply, information exists from off-axis tests and tests on a (+45°/-45°) lay-up given in
(Tab.2). Based on the ply data, the theoretically expected properties of the laminate can be calculated using classical laminate
theory (Tab. 3).
E1
E2
G12
ν12
GPa
GPa
GPa
-
average
27.03
14.21
-
0.455
standard deviation
1.19
0.85
-
0.042
Tab. 1 - Material properties of the laminate obtained on beamlike specimens
E1
E2
G12
ν12
GPa
GPa
GPa
-
average
39.10
14.44
5.39
0.294
standard deviation
2.10
0.98
1.77
0.027
Tab. 2 - Material properties of the ply used in the laminate obtained on beamlike specimens
average
E1
E2
G12
ν 12
GPa
GPa
GPa
-
28.48
16.27
8.33
0.407
Tab. 3 - Calculated material properties of the laminate using classical laminate theory
(the apparent engineering constants of the laminate)
5.2 Tests on cruciform specimen (engineering constants)
For the identification of the four independent elastic orthotropic parameters, a perforated and a non-perforated specimen are
used. The reason of testing a specimen with a hole is the aim to influence the overall deformation field and to make the
measured strain fields more sensitive to the different material parameters. Because we are dealing with an experimentally
obtained strain field, this can be important. The specimens are subjected to three different ratios of biaxial tensile loads :
2.56/1, 3.85/1 and 5.77/1. Five successive load steps are imposed per ratio, so this means that fifteen independently
measured strain field triplets are available per specimen for the identification process. The same loads are used in the finite
element simulation. A plane stress model is used with a uniformly distributed load as boundary condition. The convergence
criterion used in the optimization phase ends the iteration process when the relative value of the parameter updates is inferior
to 0.01%. In all of the optimization runs, the convergence criterion is reached in less than 13 iterations.
The results of the identification process are shown in (Tab. 4) and (Tab. 5) for both perforated and non-perforated specimen, in
terms of the mean parameter value and its corresponding standard deviation. They are obtained based on the fifteen imposed
load steps considered per specimen. The starting values for each of the parameters are mentioned as well.
It can be observed that the difference between the results for both specimen types is reasonably small. The stability of the
results obtained with the non-perforated specimen is slightly larger. This is probably due to the fact that the strain field is less
complex and therefore easier to measure with the digital image correlation technique than in the case of the perforated
specimen.
E1
E2
G12
ν12
GPa
GPa
GPa
-
15
10
10
0.3
25.11
12.17
7.05
0.483
5.4
6.8
8.9
7.7
starting values
average
standard deviation (%)
Tab. 4 – Material properties of perforated cruciform specimen
E1
E2
G12
ν12
GPa
GPa
GPa
-
15
10
10
0.3
25.11
13.31
7.69
0.467
2.8
6
6.8
6.6
starting values
average
standard deviation (%)
Tab. 5 – Material properties of non-perforated cruciform specimen
5.3 Tests on cruciform specimen (fibre orientation and load asymmetry)
In the FEM-simulation, two plane stress models can be considered (Fig. 7). In the model used for the definition of the material
parameters in 5.2, a uniformly distributed load is applied and the principal material axes are fixed and aligned with the load
directions. However, this is not always the case in reality. Therefore, a second model is developed, where the load exhibits a
linear variation along the width of the loaded arms and the fiber orientation is not fixed. This model allows considering the
slope of the applied load and the fiber orientation as additional unknowns -apart from the material properties- which can be
identified.
Fig. 7 – Loading conditions : (a) only engineering constants are identified
(b) fibre orientation and load asymmetry are additional unknowns
Fiber Angle
Slopes of loads
Perforated
Non-Perforated
Perforated
Non-Perforated
Mean value
3.4°
0.1°
Between 0-2
Between 0-2
Standard deviation
1.4°
2.2°
-
-
Tab. 6 – Identification of additional parameters
Based on uni-axial tensile tests, it is impossible to retrieve these additional unknown parameters from the measured strain
fields. However, the identification of the parameters becomes possible when strain fields measured at the surface of a biaxially loaded specimen are considered. Results of these tests are showed in (Tab. 6). The values of the engineering
constants remained unchanged compared to the previous tests in 5.2.
5.4 More sophisticated tests
In the previous tests, the whole full field information field is used to determine one set of four material parameters. The
purpose is to determine the material parameters also more locally by dividing the whole field into different zones where
material properties are assumed to be equal. This will be necessary when loading the specimen beyond the linear elastic zone
since the heterogeneous strain response will cause different zones to behave and degrade differently. The arms of the
specimen are loaded uni-axially while the centre of the specimen is loaded bi-axially (Fig. 8) Attention should be paid to the
information content in each separate zone to be able to determine the four material parameters.
Another aim is to use the inverse method to determine properties of different specimens. Previous tests were performed on
rectangular or simple cruciform specimens with normal fillet corner radius at the intersection of two perpendicular arms. For
biaxial testing an advanced cruciform specimen is used, since the previously investigated specimen type will not be suitable
for obtaining bi-axial failure data. The arms will fail first due to the enlargement of the specimen area in the bi-axially loaded
zone. Smiths et al. [4] designed a new specimen to use for biaxial testing. The cruciform specimen has an adapted fillet corner
radius and material is milled away in the thickness direction to obtain biaxial failure data. (Fig. 9) Applying the current method
to this specimen design is one of the plans.
Fig. 8 – Different zones of cruciform specimen
Fig. 9 – Design of an advanced cruciform specimen
6. CONCLUSIONS
An inverse method has been proposed to determine the elastic parameters (E1, E2, G12 and v12) of a glass fibre reinforced
epoxy with a [(+45° -45° 0°)3(+45°-45°)]-lay-up. Two specimen geometries are used: a regular cruciform specimen and a
cruciform specimen into whom a central hole is drilled. The latter is made in order to enhance the already heterogeneous
deformation field. The method is based on a finite element calculated strain field of a cruciform specimen loaded in both
orthogonal axes and the measured strain field obtained by digital image correlation. The obtained material parameters agree
reasonably well with the values obtained by traditional uni-axial tensile tests. However, the results based on the regular
cruciform specimen without hole, show less variance than the results obtained with the perforated specimen. This is possibly
due to the fact that the digital image correlation technique has some difficulties measuring steep deformation gradients, hence
inducing errors in the measurement of the displacement and strain maps. Further investigation is needed to clarify this
inconvenience. The objective of the experiment is to enforce a material behaviour that exposes the different elastic material
parameters. If this is achieved by a non-perforated specimen, there is no need for a more complex geometry which will
possibly lead to more measurement errors.
Tests indicated that it is possible to identify some additional parameters like fibre orientation and load slopes. As a result, not
only the material can be characterized, but the correct functioning of the biaxial test equipment can be evaluated as well.
7. ACKNOWLEDGEMENTS
This project is supported by the Belgian Science Policy through the IAP P05/08 project and the European Commission in the
framework of the specific research and technology development program Energy, Environment and Sustainable Development
with contract number ENK6-CT-2001-00552. The authors also express their gratitude to Hans Tommerup Knudsen from LMGlassfiber in Denmark for his effort in producing the cruciform specimens.
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