Post on 04-May-2023
Experimental identification of a nonlinear model for composites
using the grid technique coupled to the virtual fields method
H. Chalal, S. Avril*, F. Pierron, F. Meraghni
LMPF, ENSAM, rue Saint Dominique, BP508, 51006 Ch?Ions en Champagne, France
Received 23 November 2004; revised 8 April 2005; accepted 16 April 2005
Abstract
The present paper shows some experimental results of the identification of the full set of parameters driving a nonlinear model for the in-
plane behaviour of a unidirectional composite laminate. The strain field at the surface of a rectangular coupon submitted to a shear/bending
loading is measured using the grid technique. The strain fields are then processed by the virtual fields method to retrieve the six constitutive
parameters: the linear elastic orthotropic in-plane stiffnesses Qxx, Qyy, Qxy, Qss and softening parameters K and 30s driving the shear
nonlinearity. It is shown that the shear response is correctly identified, with coefficients of variation similar to the ones of standard tests. The
other parameters are identified with larger errors and higher coefficients of variation because their identification is affected by the lower
normal strain levels in the specimen. It is therefore necessary to design new tests providing balanced strain levels for improving the
identification procedure.
q 2005 Elsevier Ltd. All rights reserved.
Keywords: B. Optical techniques; C. Damage mechanics; D. Mechanical testing; Virtual fields method
1. Introduction
The analysis of the overall response of composite
structures requires the knowledge of the parameters
governing the mechanical constitutive material behaviour.
Generally, nonlinear models for composite materials are
formulated through tensorial approaches involving coupling
terms [1]. To reach simultaneously all of the behaviour law
parameters, experimental tests must give rise to hetero-
geneous strain/stress fields. Indeed, these parameters are
expected to be all involved when the global response is
heterogeneous. Therefore, intrinsic material constants can
be extracted through an identification strategy using
kinematic fields obtained from one single coupon. Among
the identification procedures, those based on the updating of
finite element models [2,3] are the most widespread. In fact,
in the case of heterogeneous tests, no closed-form relations
between loads and displacements are available. In the
general case, the material parameters are estimated through
1359-835X/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.compositesa.2005.04.020
* Corresponding author.
E-mail address: stephane.avril@chalons.ensam.fr (S. Avril).
an optimisation procedure performed iteratively until
the experimental data match the simulated ones. This
technique is general and very flexible, full-field measure-
ments are not necessarily required. However, it suffers from
a number of shortcomings such as the sensitivity to the
modeling of the boundary conditions and to initial
parameter values, for instance (Grediac et al., [4]).
An alternative strategy specifically developed to process
full-field data is the so-called virtual fields method (VFM).
It was first imagined by Grediac [5] and has been
successfully applied to the cases of in-plane [6] and
through-thickness [7,8] mechanical rigidities of anisotropic
composite materials. The identification technique relies on
the processing of the strain fields when expressing the global
equilibrium of a structure through the wellknown principle
of virtual work expressed with specific virtual kinematic
fields. In this case, one obtains a set of linear equations
which can be inverted to extract the material unknown
parameters. Currently, several improvements to the VFM
are now available, notably those concerning the automatic
generation of optimal virtual fields [4,9,10].
A validation of the numerical procedure has been
published recently using simulated strain fields [11], in
order to focus on the performance of the identification
Composites: Part A 37 (2006) 315–325
www.elsevier.com/locate/compositesa
H. Chalal et al. / Composites: Part A 37 (2006) 315–325316
strategy. The numerical stability of the identification
strategy was illustrated by estimating linear and nonlinear
in-plane material constants from these simulated strain
fields containing random noise. The present paper is a
follow up that examines the experimental application of this
methodology.
2. Non-linear material model
2.1. Experimental justification of the nonlinear shear
behaviour of a unidirectional composite
The in-plane shear behaviour of unidirectional polymer
matrix fibre composites is widely recognized to be highly
nonlinear [12–14], and full characterisation requires the
entire stress–strain curve. The purpose of this analysis is to
provide reference values of the parameters governing the in-
plane shear response of a unidirectional composite.
The material used for this study is a unidirectional
glass/epoxy (M10) composite. Its shear behaviour has first
been investigated with a 108 off-axis tensile test [15] to
investigate the shear nonlinearity. A typical load-unload
curve has been plotted in Fig. 1.
During the first hysteresis loops, the shear behaviour
proves to be quasi damageableelastic with only small
residual strains observed upon complete unloading. Increas-
ing the applied stress, the material apparent modulus
progressively decreases as nonlinearity takes place. At this
stage, the hysteresis loops are strongly extended and the
residual strains observed are very significant (z2%).
This nonlinear response results from the onset/extension
of the local damage phenomenon. Three different domains
corresponding to three different stages in the onset/exten-
sion of the damage phenomenon may be noticed in the
evolution of damage [16]:
Fig. 1. Shear stress/strain curves for cyclic loading of a 108 off-axis tensile
specimen (glass/epoxy).
† The first domain corresponds to matrix micro-cracking.
The damage variable has very small variations.
† The second domain involves cracks growth and is
characterized by a noticeable increase of the damage
variable.
† The last domain corresponds to the matrix cracking
saturation and to the elongation of the fibres. The latter
are carrying the entire load until failure. The damage
variable has now very small variations and the global
nonlinear behaviour mainly results from the increase of
residual strains.
2.2. Modeling of the nonlinear shear behaviour
Phenomenological damage of composite materials can
be efficiently modeled by means of the meso-model
proposed by Ladeveze [16]. The latter was extensively
used for predicting the initiation and growth of many forms
of damage in composite materials [16]. The mesomodel is
developed within the framework of the thermodynamics of
irreversible processes and introduces the effect of damage
on the mechanical behaviour through material stiffness
reduction. It must be emphasized that in this work, the
modeling of the residual strains and the hysteretic behaviour
shown in the previous section will be ignored for this first
stage. It has been shown very recently that residual strains
[17] and dissipative effects [18] could be handled by the
VFM. Integrating the three features together to identify
the full response of Fig. 1 will be a future step. The rest of
the paper will only focus on the nonlinear response during
monotonic loading, assuming that the nonlinear behaviour is
caused by a nonlinear relationship between in-plane shear
stresses and in-plane shear strains. The objective will be to
identify the parameters driving this relationship.
bAs a consequence, the global in-plane constitutive
equations write for orthotropic materials, using the conven-
tional notation for contracted indices xx/x, yy/y, xy/s:
sx
sy
ss
0B@
1CA Z
Qxx Qxy 0
Qxy Qyy 0
0 0 Qssð3sÞ
264
375
3x
3y
3s
0B@
1CA (1)
where Q is the in-plane stiffness matrix, s the stress tensor
and 3 the strain tensor. It is recalled here that QssZGxy. Now
the problem is to express Qss(3s) as a function of some
mechanical parameters.
The apparent shear modulus Qss(3s), defined as the ratio
between the measured shear stress and the measured shear
strain sss/3ss, has been computed at the end of each process
during the cyclic 108 off-axis tensile test (Fig. 1). Looking at
the evolution of apparent modulus decrease 1KQssð3sÞ=Q0ss,
it seems that a linear function with a threshold [16] is a good
start for modeling the variations (Fig. 2).
Therefore, the stress–strain relationship writes:
ss Z Q0ss3s KK3sh3s K30
s iC (2)
Fig. 2. Linear fit of the apparent modulus decrease. Fig. 3. Tensile stress–strain response for a 08 glass/epoxy specimen.
H. Chalal et al. / Composites: Part A 37 (2006) 315–325 317
where K and 30s are the parameters driving the model (K is
the slope of the linear fit in Fig. 2 and 30s is the threshold
defined as the intersection between the linear fit and the
shear strain axis), G0xy ZG0
ss, and hXiC means X when XR0,
0 otherwise.
The next section is dedicated to the standard character-
isation of the glass/epoxy composite material (Table 1).
2.3. Model identification with standard tests
2.3.1. Tensile response
The tensile test on a rectangular specimen is used here to
measure Young’s moduli Exx and Eyy and Poisson’s
ratio nxy.
For 08 unidirectional composites, a specimen width of
12.7 mm is selected whereas 908 specimens are 25 mm
wide, following the recommendations of the relevant ASTM
standard [19]. The specimens thickness was 2.3 mm
nominally. For each fibre orientation, six specimens were
cut from a 500!500 mm2 glass/epoxy (M10) panel
manufactured in the laboratory by hot pressing. All
specimens were tabbed with straight composite tabs [20].
0/908 rosettes were bonded back-to-back in the centre of the
specimen to obtain both Young’s modulus and Poisson’s
ratio while accounting for possible bending effects by
averaging the back-to-back data. Young’s moduli, Exx and
Eyy, and Poisson’s ratio nxy are defined as
† Exx: the initial slope of the stress–strain curve for the 08
tensile test
Table 1
Engineering constants for the glass/epoxy composite obtained from tensile
tests
Exx (GPa) Eyy (GPa) nxy
Average (6 specimens) 44.6 11.7 0.31
Coeff. of variation (%) 1.3 7.0 1.0
† nxy: the negative ratio of the transverse to longitudinal
strains (K3y/3x) for the 08 tensile test
† Eyy: the initial slope of the stress–strain curve for the 908
tensile test
A representative example of stress, sx, versus strains
3x and K3y, for a unidirectional 08 glass/epoxy composite
is shown in Fig. 3, with 3x and 3y the longitudinal and
transverse strains, respectively, and the stress sx defined
as load divided by test section area. The modulus Exx
was deduced using a least squares linear fit to the linear
initial portion of the curve (sx versus 3x). Poisson’s ratio
nxy was determined from the ratio of the initial slopes of
sx versus 3x and sx versus K3y. To determine Eyy, the
same procedure has been undertaken with the 908
specimens. The values of Exx, nxy and Eyy are reported
in Table 2.
2.3.2. Shear response
In this section, 08 double V-notch (Iosipescu) shear tests
[21] have been performed according to the procedure
described in Ref. [22] to identify the nonlinear in-plane
shear response developed in Section 2.2.
The Iosipescu specimens are 2.3 mm thick nominally.
They have been prepared by cutting roughs in a panel using
a diamond coated wheel and machined down (including
notches) to the required dimensions shown in Fig. 4 by side
milling with a tungsten carbide tool. 0/90 back-to-back
rosettes have been bonded at 458 to obtain the average shear
strain accounting for possible nonuniform loading through
the thickness [23].
Table 2
Results of the 08 Iosipescu shear tests
G0xy (GPa) K (GPa) 30
s
Average (6 specimens) 3.68 83.6 0.006
Coeff. of variation (%) 8.6 12.8 33
Fig. 4. Iosipescu test specimen.
H. Chalal et al. / Composites: Part A 37 (2006) 315–325318
The determination of the shear stress–strain response of
the material is based on shear stress calculated from:
ss ZP
A(3)
where A is the test section.
The shear strain 3s is simply calculated from the G458
strain gauge readings as:
3s Z 3ðK45ÞK3ð45Þ (4)
This is justified by the fact that for glass-epoxy unidirec-
tionals, the correction factor for the Iosipescu test is very
close to 1. Fig. 5 shows the six stress–strain curves together
with the fit corresponding to the quadratic nonlinear law
described in Section 2.2. The initial in-plane shear modulus
and the nonlinear parameters K and 30s have been identified
using a least square fit of the curve (ss versus 3s). The results
are summarized in Table 2.
Fig. 5. Experimental in-plane shear stress–strain curve together with cubic
fit corresponding to quadratic evolution of the apparent shear modulus.
Glass/epoxy composite.
3. Model identification using the virtual fields method
In the previous section, the nonlinear behaviour of the
composite material has been characterized using several
uniform tests. However, this procedure requires the use of
three different types of specimens whereas the methodology
proposed in the present paper is aimed at using only one. It
is composed of the following steps:
† the mechanical test: the specimen is loaded in a shear-
bending mode by using a Iosipescu fixture but with an
unnotched coupon in order to generate a heterogeneous
strain field;
† full-field measurements: the full strain field is measured
over the specimen using an optical technique called the
grid method.
† the identification: the parameters of the constitutive
equations (the orthotropic stiffnesses and the parameters
driving the nonlinear behaviour) are identified through
the application of the virtual fields method.
3.1. Mechanical set-up
The experimental set-up is a Iosipescu fixture. A constant
thickness rectangular 75!20!2.3 mm3 specimen is loaded
in a shear-bending mode by using this fixture (Fig. 6). The
strain and the stress fields arising from this mechanical test
are heterogeneous. An FEM based numerical study [11] has
shown that the mechanical response involved all the
material parameters to be identified. An optimized length
LZ30 mm between the grips has been determined for
maximizing the average identifiability of the unknown
parameters [11].
3.2. Full-field measurements with the grid method
The displacement fields required for the model identifi-
cation have been measured using an optical method called
the grid method [24–26]. A grid pattern is deposited onto the
surface of the specimen [27]. The grid period is 100 mm.
Images of the undeformed and deformed gratings are
digitized through a 1280!1024 CCD camera connected to a
PC (Fig. 7). The lens of the camera is set such that a period
of the grid is sampled by approximately four pixels. The
30!20 mm region of interest covers all the area of the
specimen located between the two grips of the Iosipescu
fixture (S2 in Fig. 6).
The basic rules leading to the measurement of the
displacement fields with the grid method are recalled in this
section.
3.2.1. Printing of the grid
The spatial carrier consists in the superposition of
horizontal and vertical black lines printed over a white
surface, with a natural spatial frequency f0Z10 lines per
Fig. 6. Schematic view of the bending/shear test based on the Iosipescu fixture.
H. Chalal et al. / Composites: Part A 37 (2006) 315–325 319
mm. This is achieved by printing the grid on a
photosensitive film consisting of a polymer backing and a
thin layer of photosensitive coating. Then, the coating is
transferred onto the specimen by adhesive bonding using a
white epoxy glue ensuring final contrast. The backing film is
finally removed after the curing of the adhesive so that no
additional stiffness is added to the specimen. The detailed
procedure can be found in Ref. [27].
3.2.2. Characterization of the displacements by a phase
modulation
At any state, the digitized light at a given pixel M0 is the
light reflected by a material point M determined by its
position vector ðRðX;YÞ in the reference cartesian frame
ðO; ði; ðjÞ. Its intensity writes:
Ið ðRÞ Z I0f1 Cg frng½2p ðF$ ðR�g (5)
where:
† I0 is the local intensity bias,
† g is the contrast,
Fig. 7. Photography of the experimental set-up.
† frng is a 2p-periodic continuous function,
† denotes the dot product,
† 2p ðF : ðR is called the phase of function frgn,
† ðF is the spatial frequency vector. It is orthogonal to the
grid lines and its amplitude is the spatial frequency of the
grid. If the grid lines are vertical, they are parallel to ðj,meaning that the spatial frequency vector writes ðFðf0; 0Þ.
If the grid lines are horizontal, they are parallel to ði,meaning that the spatial frequency vector writes ðFð0; f0Þ.
When a load is applied, there is a deformation of the
structure and the grid is also deformed. The phase of
the function frng at pixel M0 varies ofK2p ðF :ðuð ðRÞ from the
undeformed to the deformed state, where ðuð ðRÞ is the
displacement vector of M. The first component ux(x, y) of
the displacement is calculated from the phase of the function
frng for the vertical lines, the second component uy(x, y) for
the horizontal lines. The phase fields are computed by using
the spatial phase shifting method implemented in a Matlab
routine [28,29].
3.2.3. Measurement of the displacement fields
The ux(x, y) and uy(x, y) displacement components relative
to the unloaded reference condition are calculated from the
respective phase differences DfX (for vertical lines) and DfY
(for horizontal lines) introduced by the deformation:
uxðx; yÞ ZKp
2pDfXðx; yÞ (6)
uyðx; yÞ ZKp
2pDfY ðx; yÞ (7)
where pZ0.1 mm. The resolution of the method, i.e. the
smallest displacement which can be measured in absolute
value, depends on the measurement noise. The noise is a
random variable which adds to the phase during the
calculation [28]. Its mean is 0 and its standard deviation is
denoted sf. In our experiments, sf lies between 2p/100 and
2p/150, meaning that the resolution of the method for
H. Chalal et al. / Composites: Part A 37 (2006) 315–325320
measuring displacements lies between 0.6 and 1 mm. This was
evaluated by taking two images of the undeformed grid and
calculating the standard deviation of the detected phase. The
spatial resolution, i.e. the smallest distance which separates
two independent displacement values, equals 2 periodsZ0.
2 mmZ8 pixels.
3.2.4. Strain field computation
To compute the strain field components, the displace-
ment fields must be differentiated according to:
3ij Z1
2
vui
vxj
Cvuj
vxi
�; i; j2½1 K3� (8)
Several methods are available in the literature for nu-
merically differentiating the displacement fields [10,30–33].
The solution chosen here is to fit a plane to P!P pixels
subimages of the displacement fields. Then, the gradient of
the plane along the relevant axis provides an estimate of the
strain component at the central pixel of the subimage.
Here, measured displacement fields are too noisy to
compute accurate enough derivatives; they must be filtered.
A square convolution kernel is applied twice consecutively,
with a width of 33!33 pixels for the square window.
Afterwards, planes are fitted to subimages of 33!33 pixels
for computing the gradients. The resulting spatial resolution
can be approximated as 33ffiffiffi3
ppixelsx1:4 mm [34].
The strain resolution, i.e. the smallest strain value which
can be detected, has been obtained by applying a rigid body
motion to the specimen. The actual resulting strain field is
obviously null but the measured strain field, computed with
the same procedure as the one described before, is not null
because of noise in the measurements. The strain resolution
can be assessed as the standard deviation of these strain
fields. It is about 120 mstrain here.
It is also important to note that with this technique, there is
a coupling between in-plane and out-of-plane displacements.
Because of the magnification of the imaging lens, an out-of-
plane displacement will be interpreted by the camera as a
change in the period of the grating, resulting in apparent in-
plane displacements. It results in parasitic apparent strains 3x
and 3y which equal DZ/Z, where DZ is the out-of-plane
displacement and Z the distance between the CCD sensor and
the surface of the specimen. Here, Z is 250 mm, so there is a
parasitic strain of 4 mstrain which adds to 3x and 3y for each
micrometer of out-of-plane displacement; 3s is insensitive to
this effect. One has to ensure that the experimental set-up
does not cause too much out-of plane displacements during
the tests to measure reliable values of 3x and 3y.
3.3. Stiffness identification with the virtual fields method
3.3.1. Principle
Kinematical full-field data are then processed through an
inverse procedure called the Virtual Fields method (VFM).
The VFM has been introduced by Grediac [5] and is based
on the Principle of Virtual Work. According to this
principle, the static global equilibrium of a coupon having
a constant thickness e writes, for plane stress conditions and
in absence of body forces:
e
ðS
s : 3dS Z
ðSf
ðT ðMÞ:u�!ðMÞdS (9)
where:
†�s�
is the stress field;
† S is the total surface of the studied coupon;
† Sf is the surface where the load is applied;
† ðT ðMÞ are the forces per unit area (distribution of
tractions) over the boundary surface Sf;
† u�! is the virtual displacement field and�3
�
the virtual
strain field derived from the gradient of u�!.
Introducing the elastic orthotropic constitutive equations
(Eq. (1)) before the apparition of damage (linear behaviour),
Eq. (9) becomes:
Qxx
ðS
3x3x dS CQyy
ðS
3y3y dS CQxy
ðS
ð3x3y C3y3x ÞdS
CQ0ss
ðS
3s3s dS Z
1
e
ðSf
ðT ðMÞ:u�!ðMÞdS ð10Þ
Four independent virtual displacement fields fu�a�!; 3ag,
fu�b�!
; 3bg, f u�c�!; 3cg and fu�d
�!; 3d g are chosen to identify
Qxx, Qyy, Qxy and Q0ss independently. They should respect
the following rules:
† be kinematically admissible, i.e. continuous, differenti-
able and consistent with the actual boundary conditions,
† the external virtual work must only involve the applied
load P and not its distribution which is unknown in
practice,
† full-field measurements are only obtained in the middle
part of the specimen (surface S2 in Fig. 6), so the virtual
strains should be zero in S1 and S3.
As a consequence, the virtual fields must be rigid body
like outside the region of interest of the specimen (S1 and S3
in Fig. 6). Thus, the external virtual work writes:ðSf
ðT ðMÞ:u�!ðMÞdS ZKPuay ðLÞ; a Z a; b; c or d: (11)
Using four different virtual fields, a system of four linear
equations is obtained where Qxx, Qyy, Qxy and Q0ss are the
four unknowns. One of the critical points of the method is
the choice of the four virtual fields for ensuring four
independent equations. It has been shown in Ref. [4,9] that
some virtual fields called special virtual fields render the
matrix of the system equal to unity. Eq. (10) leads therefore
to the direct identification of the unknown parameters (the
dimension of 1 s mK2 for ensuring dimensional
H. Chalal et al. / Composites: Part A 37 (2006) 315–325 321
homogeneity). Finally, the solution is:
Qxx ZK1!Pua
y ðLÞ
eQyy ZK1!
Puby ðLÞ
e
Qxy ZK1!Puc
y ðLÞ
eQ0
ss ZK1!Pud
y ðLÞ
e
(12)
3.3.2. Choice of virtual fields
At the beginning, the virtual fields were expanded as
polynomial functions defined over the whole region of
interest [4,9,11]. In the present approach, the virtual fields
are defined by subdomains, as piecewise defined functions
[35]. Shape functions such as that in the finite element
method are used to expand the virtual fields. The first asset
of this approach is that the degree of the shape functions is
lower than the degree of previously used polynomials,
which brings up more stability and robustness. The second
asset is that only the virtual displacement fields have to be
continuous. It is not necessary that their gradient is also
continuous at the edges of the subdomains, which relaxes
the conditions of virtual fields construction. However, still
an infinity of piecewise defined virtual fields can be chosen
for deriving the unknown stiffnesses from Eq. (12). A
method for selecting the virtual fields has recently been
developed [10]. It relies on the minimization of noise
effects. Only the main features of this method will be
recalled here.
The model assumed here for the noise is a Gaussian
white noise. Although the present authors are well aware
that this assumption is too simplistic, it is still thought
that it could be a helpful criterion to select good virtual
fields among the many available special fields. So, this
noise, respectively, gNx, gNy and gNs, adds to 3x, 3y and
3s. g is a strictly positive real number that represents the
random variability of the strain measurements (it is
interpreted as the uncertainty of the strain measure-
ments). The noise is assumed to be uncorrelated from
one measurement location to another in the current
model. The components are also assumed to be
uncorrelated between each other. Again, since exper-
imentally, the strains will be derived from the two in-
plane displacement components bearing the noise, this is
not strictly true but this assumption is needed to keep the
process analytical.
Since 3x, 3y and 3s contain random noise, the value of
each unknown parameter identified from 3x, 3y and 3s with
Eq. (12) is a random variable. It writes, for example for Qxx:
Qxx ZKPua
y ðLÞ
eCg Qxx
ðS2
Nx3ax dS CQyy
ðS2
Ny3ay dS
24
CQ0ss
ðS2
Ns3as dS CQxy
ðS2
½Nx3ay dS CNy3a
x dS�
35 ð13Þ
Qyy, Qxy and Q0ss are also random variables. According to
Eq. (13), it can be shown that the variances of Qxx, Qyy, Qxy
and Q0ss write [10]:
VðQxxÞ Z ðhaÞ2g2
VðQyyÞ Z ðhbÞ2g2
VðQxyÞ Z ðhcÞ2g2
VðQ0ssÞ Z ðhdÞ2g2
8>>>>><>>>>>:
(14)
where:
ðhaÞ2 ZbL
ðS2
½Q2xx CQ2
xy�ð3ax Þ2dS
8<:
C
ðS2
½Q2yy CQ2
xy�ð3ay Þ2dSC
ðS2
½Q0ss�
2ð3as Þ2dS
C2
ðS2
½QxyðQxx CQyyÞ�ð3ax 3a
y ÞdS
9=; for a Za;b;c; and d:
(15)
Therefore, computing ha quantifies the sensitivity to
noise of the method. For a given virtual field, the lowest
ha, the most accurate the identification. It has been
shown in Ref. [10] that, for a given strain field 3x, 3y et
3s, there is a unique set of special virtual fields
fu�a�!;3ag, fu�b
�!;3bg, f u�c�!
;3cgetfu�d�!
;3d g which
minimize, respectively, the sensitivity to noise ha, for
aZa, b, c, or d. These four virtual fields are deduced
from the resolution of a constrained minimization
problem. The unknowns of this minimization problem
are the components of the virtual fields expanded in a
suitable basis of continuous functions. The Lagrangian of
the problem is written. Its unique saddle point is found
[10], yielding the virtual field the least sensitive to noise
for each unknown constitutive parameter. Then, each
unknown constitutive parameter can be identified by
applying the VFM with the deduced virtual fields.
However, the problem is not explicit because the
unknowns Qxx, Qyy, Qxy and Q0ss are involved in the
expression of ha. The problem is solved by an iterative
algorithm where the unknown parameters are replaced by
their identified values. A first set of initial values is chosen.
Tests show that this algorithm converges in less than four
iterations whatever the choice of the initial values [10].
3.4. Nonlineary identification with the virtual fields method
When the specimen is loaded in the nonlinear part of its
behaviour (here, PO800 N), Qss is not a constant anymore
Fig. 9. Plot results used to identify nonlinear parameters K and 30s .
H. Chalal et al. / Composites: Part A 37 (2006) 315–325322
and Eq. (9) becomes:
Qxx
ðS2
3x3x dSCQyy
ðS2
3y3y dSCQxy
ðS2
ð3x3y C3y3x ÞdS
CQ0ss
ðS2
3s3s dSKK
ðS2
3sh3s K30s i
C3s dS Z
KPuy ðLÞ
eð16Þ
As Qxx, Qyy, Qxy and Q0ss have already been identified in the
linear part of the material response, only K and 30s remain
unknown. In this case, only one virtual field is required to
identify them. The field that can be considered here is the
following:
† on the left hand side of the region of interest (over S1 in
Fig. 6), ux Zu
y Z0
† in the region of interest (over S2 in Fig. 6), ux Z0 and
uy ZKx
† on the right hand side of the region of interest (over S3 in
Fig. 6), ux Z0 and u
y ZKL where L is the length of the
region of interest.
The effect of this field on the specimen is represented in
Fig. 8. It can be seen that this field has a pure virtual
shearing effect and the equation resulting from Eq. (16) will
therefore involve only Q0ss, 30
s and K.
Accordingly, using this field, K and 30s can be identified
by plotting PLe
CQoss
ÐS2 3sdS versus
ÐS2 3sh3sK30
s iCdS for
the different loads where a shear strain field is available
(here for every 50 N from 800 N up to 2000 N). 30s is chosen
so as to minimize the deviation of the plot from a linear fit.
Different values varying of 0.001 from 0 up to 0.01 are
tested and the value which leads to the straightest plot is
validated. Finally, K is obtained as the slope of the linear fit
(Fig. 9)
4. Experimental results and discussion
Four specimens have been tested. They have been loaded
up to 2000 N. The displacements have been measured using
the grid method all along each test. As expected, all the
components of the strain fields derived from those
Fig. 8. Schematic view of the virtual shear field.
displacement fields are non zero and heterogeneous
(Fig. 10). Thus, all the unknown parameters are involved
in the measured data. For each specimen, one strain field is
processed to retrieve the four elastic parameters. The results
are reported in Table 3.
The first thing to note is that the coefficients of variation
are very different from one component to the other. It is also
interesting to note that the higher the coefficients of
variation, the higher the difference with the reference
value. This seems to indicate that the main problem here is
caused by the noise level. Indeed, the reference values of
Qxx, Qyy and Q0ss lie within the bounds of the distributions of
Qxx, Qyy and Q0ss from the VFM, as a consequence of the
very high coefficients of variation (Table 4). Only Qxy does
not follow this trend, indicating a possible bias. Unfortu-
nately, the number of VFM results is too low to enable a
more accurate statistical comparison (comparing
distributions).
The above results are easily explained by looking more
carefully at Fig. 9. Indeed, the highest strain values are
clearly the shear strains. Therefore, it is to be expected that
the effect of noise will be minimal on the identification of
Q0ss, which is confirmed by the low coefficient of variation,
and therefore, by the fact that both the VFM and the
reference values agree very well. After the shear strain,
the component with the highest values is 3x although the
difference with the levels of 3y is not very important.
However, looking at both strain maps more carefully, it can
be seen that the high values of 3x concern a larger area of the
specimen whereas for 3y, the high values are concentrated
on its vertical edges.
As a consequence, the scatter on Qxx is higher than for
Q0ss but much smaller than that of Qyy, with a better average
value of Qxx than Qyy which is far off the reference. The
only results that cannot be explained is Qxy with a rather low
coefficient of variation so that the reference value does fall
Fig. 10. Measured strain fields for P2Z700 N.
Table 4
Reference values and bounds on the identified stiffnesses
�QC2s (GPa) �QK2s (GPa) Q reference
(GPa)
Qxx 69.4 35.8 44.9
Qyy 15.3 K1.76 12.2
Qxy 2.45 0.43 3.86
Q0ss 4.52 3.14 3.68
H. Chalal et al. / Composites: Part A 37 (2006) 315–325 323
within the range of the VFM measurements (Table 4). The
small number of VFM measurements may be the cause of
this effect (biased evaluation of the standard deviation), or
there may be a bias in the procedure induced by out-of-plane
displacements of the specimen. This will need further
investigation.
Using the identified value of Q0ss, the values of the nonlinear
parameters K and 30s identified from the strain fields measured
in the nonlinear range are reported in Table 5.
As for the initial shear modulus, the results of the shear
nonlinear parameters are in good agreement with the
reference data. A constant value of 0.004 has been identified
for 30s . It is lower than the value obtained from standard tests
Table 3
Identified stiffnesses
Load Qxx
(GPa)
Qyy
(GPa)
Qxy
(GPa)Q0
ss
(GPa)
Test 1 700 N 44.6 10.41 1.87 4
Test 2 1010 N 46 0.8 0.7 4.2
Test 3 700 N 58.7 6.6 1.5 3.7
Test 4 900 N 61 9.3 1.7 3.4
Average 52.6 6.77 1.44 3.83
Coeff. of variation 16% 63% 35% 9%
Results with
standards tests
Average 44.9 12.2 3.86 3.68
Coeff. of variation 0.7% 2.8% 2.4% 8.6%
and more stable at the same time. This can result from the
greater amount of data available over the whole field which
can help to detect more accurately the onset of the
nonlinearity.
Table 5
Identified nonlinear parameters K and 30s
K(GPa) 30s
Test 1 100 0.004
Test 2 83 0.004
Test 3 88 0.004
Test 4 78 0.004
Average 87.2 0.004
Coeff. of variation 10.8% 0%
Results with standards tests
Average 83.6 0.006
Coeff. of variation 12.8% 33%
Fig. 11. Stress–strain curves obtained with strain gauges compared with the
model identified from full-field measurements.
Fig. 12. Distribution of the apparent shear modulus over the region of
interest for PZ1560 N.
H. Chalal et al. / Composites: Part A 37 (2006) 315–325324
Finally, the shear response from the average values
identified with the VFM fits very well the data from the
standard tests (Fig. 11). This is a very interesting result since
it proves that the VFM has filtered out the effects induced by
the shear nonlinearity from the effects induced by the other
strain/stress components which are non zero in this
heterogeneous test.
Another asset of the approach presented here is that it is
possible to map all over the field area the decrease of the
apparent modulus (Fig. 12) from the strain fields and the
identified parameters.
5. Conclusion
The present study is the first experimental application of
the virtual fields method (VFM) to the case of a nonlinear
shear behaviour of a glass-epoxy composite. The test
configuration derives from an ASTM standard (double
V-notched shear test) and provides a reasonably hetero-
geneous strain field that is captured with an optical full-field
measurement technique (grid method) and processed
through the VFM.
As a main conclusion, it can be said that the shear
response of the specimen has been very well identified
thanks to the predominance of the shear strain in the
specimen. The identification of the other parameters has
been less successful mainly because the strain levels are
insufficient to ensure a limited influence of the noise levels.
However, the quality of the identification is in full
accordance with what was expected from the VFM
sensitivity factors, which is a very nice feature of the
technique.
The main step now to improve the results is to design
mechanical tests that will provide balanced strain levels
over significant parts of the gauge section. A preliminary
study [36] has shown that rotating the fibres at 458
significantly improves the situation. In the near future, a
complete optimization scheme will be set up in order to
address this very important issue and start providing novel
test configurations fully adapted to the virtual fields method.
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