Direct identification of the damage behaviour of composite materials using the virtual fields method

Composites: Part A 35 (2004) 841–848 www.elsevier.com/locate/compositesa

Direct identification of the damage behaviour of composite materials using the virtual fields method H. Chalala, F. Meraghnia,*, F. Pierrona, M. Gre´diacb a

Laboratoire de Me´canique et Proce´de´s de Fabrication Nationale Supe´rieure des Arts et Me´tiers Saint-Dominique, BP 508, 51006 ChaWlons-en-Champagne Cedex, France b Laboratoire d’Etudes et de Recherches en Me´canique des Structures, Universite´ Blaise Pascal Clermont II, 24, avenue des Landais, BP 206, 63174 Aubie`re Cedex, France

Abstract In the present work the virtual fields method (VFM) has been used to extract the whole set of material parameters governing a nonlinear behaviour law for composite materials. The nonlinearity considered here is due to the damage inherent to the in-plane shear response. The identification method is performed by applying the principle of virtual work knowing the whole strain field onto the surface of a tested specimen. The test chosen here is a shear bending test using a rectangular coupon loaded in a Iosipescu fixture. To illustrate the capabilities of the method, the identification is performed on data provided by finite element simulations. First, the nonlinear finite element model is described. Then, numerical aspects of the VFM are discussed, in particular the stability of the technique with respect to noise in the data. Finally, first elements of test optimisation are given by studying the effect of the length of the active area and the effect of the material anisotropy. This work contributes to the development of the VFM as a tool adapted to the processing of full-field measurement to identify parameters from general constitutive equations. q 2004 Published by Elsevier Ltd. Keywords: Optical properties/techniques; C. finite element analysis; Damage behaviour

1. Introduction The analysis of the overall damage response of composite structures requires the knowledge of parameters governing the mechanical constitutive material law. Generally, damage 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 heterogeneous strain/stress fields (by ‘heterogeneous’, it is understood that all stress/strain components are present and non uniform). Indeed, in this case, these parameters are expected to be all involved in the response of the specimen. Therefore, intrinsic material constants can potentially be extracted through an identification strategy using kinematic fields obtained from one single coupon. Among the identification procedures are those based on the updating of finite element models [2,3]. In fact, in the case of heterogeneous tests, no closed-form relations between loads * Corresponding author. E-mail address: [email protected] (F. Meraghni). 1359-835X/$ - see front matter q 2004 Published by Elsevier Ltd. doi:10.1016/j.compositesa.2004.01.011

and displacements are available. In the general case, the material parameters are thus estimated through an optimisation procedure performed iteratively until the experimental data match the simulated ones. However, these mixed numerical/experimental procedures exhibit some drawbacks which are discussed by Gre´diac et al. [4]. Recently, another approach has been proposed by Claire et al. [5] based on the local equilibrium, to identify local damage. An alternative strategy, called the Virtual Fields Method and noted hereafter (VFM), developed by Gre´diac [6], has been successfully applied to determine the in-plane [7] and through-thickness [8,9] mechanical stiffnesses 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 well-known principle of virtual work expressed with specific virtual displacement fields. In this case, one obtains a set of linear equations which are inverted to extract the material unknown parameters. Currently, several improvements of the VFM have been developed notably those concerning the automatic generation of optimal virtual displacement fields [4,10].

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Fig. 1. Bending-shear test based on the Iosipescu configuration.

In practice, it is worth noting that the strain field measurements are generally performed using optical techniques such as moire´ or speckle interferometry, digital image correlation, grid method, etc. [11]. However, to focus on the performance of the identification strategy, the strain fields handled here are simulated by finite element computations. These numerical calculations have required first the implementation of the nonlinear behaviour law into the implicit finite element code ABAQUS by means of a user supplied material subroutine (UMAT) programmed in the Fortran 90 language [12]. The Iosipescu shear test applied in Refs. [13,14] consists of a V-notched specimen mounted in a Iosipescu fixture. In this configuration, the shear strain/stress fields are essentially uniform in the area between the notches. In the present paper, the aim consists in extracting simultaneously all of the behaviour law constants. This requires to carry out an experimental test involving heterogeneous strain/stress fields. The shear/bending test on a straight orthotropic plate according to the Iosipescu fixture, shown in Fig. 1, proved to be appropriate for giving rise to heterogeneous stress/strain fields. In fact, besides the predominant shear stress, the central zone of the tested specimen is subjected to normal stresses due to specimen bending and also to a transverse compression stress because of the load introduction at the inner loading points. In the present paper, the VFM is first recalled. Then, the finite element model is presented and some numerical features of the VFM are illustrated, among which the effect of noise in the data on the stability of the identification procedure. Finally, first elements of test optimisation are presented, such as the influence of the length L characterising the central part of the specimen (see Fig. 1) on which the identification is processed, and the influence of the orthotropy ratio of the tested material.

2. Damage mesomodelling 2.1. Mechanical constitutive law Phenomenological damage for composite materials can be efficiently modelled by means of the mesomodel proposed

by Ladeve`ze [15]. The latter was extensively used for predicting the initiation and growth of many forms of damage in composite materials [15]. 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. In addition, the method of the local state [16] allows to link the internal damage variables to thermodynamic dual forces associated to the strain energy. In the case of an in-plane stress problem, the damage state can be described by three internal variables. In addition, with the assumption that only the in-plane shear strain/stress response is nonlinear, the damage affects solely the shear rigidity. The damaged strain energy density can thus be written as follows [15]: " # nxy sx sy s2y 1 s2x s2s ð1Þ 22 þ þ ED ¼ 2 Exx Exx Eyy Gxy ð1 2 dss Þ where Exx ; Eyy ; nxy and Gxy are, respectively, the longitudinal Young’s modulus, the transverse Young’s modulus, major Poisson’s ratio and the shear modulus of the considered orthotropic material. dss is a scalar internal variable which remains constant through the specimen thickness. It describes the stiffness degradation inherent to the in-plane shear. The damage dual variable is the thermodynamic force associated to the in-plane shear stress. Considering the effective stress concept [16], it is given by: Ys ¼

›ED s2s 1 ls ¼ ¼ Qss g2s with Qss ¼ Gxy 2 ›dss 2Qss ð1 2 dss Þ2 ð2Þ

In this work, for a material subjected to a monotonic loading, damage accumulation can be modelled as a quadratic function of the shear strain and depends on one parameter, such as: dss ¼ ag2s

ð3Þ

It must be noted that other more complex polynomial forms could have been proposed. This would increase the number of parameters to be identified but would not change the method presented here. If more sophisticated models were to be used (taking into account the effects of loading history, for instance), then the problem may become non-linear and other numerical strategies would have to be sought. However, the present paper is only a first attempt and aims mainly at checking the feasibility of the method. According to Eq. (3), the nonlinear shear stress/strain relationship is given by:

ss ¼ Qss gs 2 K g3s with K ¼ aQss

ð4Þ

This relationship is typical of the stress softening that is observed on experimental shear responses of long fibre composites [17]. The global orthotropic stress/strain

H. Chalal et al. / Composites: Part A 35 (2004) 841–848

relationship can therefore be expressed as follows: 0

sx

1

2

Qxx

B C 6 B sy C ¼ 6 Qxy @ A 4

ss

0

Qxy

0

30

Qyy

0

7B C 7B 1y C 5@ A

0

Qss ð1 2

ag2s Þ

1x

the virtual work as follows: ð ð ð 1x 1px dS þ Qyy 1y 1py dS þ Qxy ð1x 1py Qxx

1

S2

þ 1y 1px ÞdS þ Qss

gs

3. Virtual fields method 3.1. Theoretical background The VFM is based on the principle of the virtual works (PVW). Assuming in-plane stresses and quasi-static loading, the PVW can be written as, for any kinematically admissible (KA) virtual displacement field: ð V

sij 1pij dV þ

ð ›V

Ti upi dS ¼ 0

S2

ð5Þ

¼2

2

843

ð6Þ

where ðVÞ is the volume of the solid, ›V the boundary surface of the solid, sij is the stress field, 1pij is the virtual strain field, T the external surface load density applied over ›V and up is the virtual displacement field associated to 1p : Assuming that the actual strain field is known over the active surface of the tested specimen, the objective here is to extract the whole set of parameters governing the linear and nonlinear material behaviour. It is obvious here that the form of the material constitutive model has to be assumed a priori, as it has to be for any identification strategy. It might be possible to address the relevance of the form of the constitutive relation by performing the identification procedure using sets of different constitutive models and find numerical indicators of the quality of the models. This is an important issue that will have to be addressed in the future. The idea of the VFM is to write Eq. (6) with as many virtual fields as unknown parameters [6]. Let us consider the in-plane orthotropic behaviour of a composite given by Eq. (5). The purpose here is to extract the parameter governing the non-linear shear response as well as the four elastic stiffness constants. The particular test configuration used here is based on the Iosipescu fixture (Fig. 1). Such configuration is justified by the fact that the stress/strain fields in the central part S2 of the tested specimen are heterogeneous; it should be emphasized that in this case no closed-form solution of the mechanical problem is available. If the specimen dimensions are ‘well-chosen’ (this issue will be examined later in the paper), the response of the coupon may involve simultaneously all the unknown parameters of the constitutive equations. For any KA virtual field and assuming that the material is macroscopically homogeneous (i.e. Qij are constant), the global equilibrium can be written with the principle of

ð S2

gs gps dS 2 K

S2

ð S2

g3s gps dS

P~upy ðLÞ e

ð7Þ

u~ py ðLÞ is the constant vertical virtual displacement of surface S3 (see Fig. 1) and P is the resulting global load applied on the right-hand side of the specimen. It has been shown [8,18] that an empirical choice of the virtual fields does not always lead to accurate and stable results. Substituting some virtual fields in the above equation provides a new equation in which the parameters governing the behaviour law can be considered as unknown. To improve the VFM in terms of accuracy and stability, Gre´diac et al., [4] have proposed a procedure which leads to a direct determination of the unknown parameters using socalled ‘special’ and optimised virtual fields. 3.2. Construction of the special virtual fields Feeding five different virtual fields in Eq. (7) leads to the following linear system: ½M·{Q} ¼ {R}

ð8Þ

where M is a square matrix and Q a vector which components are the five unknown parameters. The inversion of the system allows to determine simultaneously the constitutive parameters. The choice of the virtual fields constitutes a key issue in the identification procedure. The form of these virtual fields controls the degree of independence of the equations in linear system (8). Gre´diac et al. [4] have suggested that the virtual fields could be generated automatically using basis functions, for instance polynomials (other basis functions could be used, this issue is not addressed here). In the present case, the virtual fields are expressed over surface S2 as a weighted sum of these basis polynomial functions:  i  j m X n X x y p u~ x ¼ Aij ; L H i¼0 j¼0 ð9Þ  i  j p X q X x y p u~ y ¼ Bij L H i¼0 j¼0 where L and H characterise the dimensions of the coupon according to the Iosipescu fixture given in Fig. 1. X and Y are the coordinates of the points at the surface of the specimen. Over S1 and S3 ; the virtual fields have to be solidrigid like (respectively, u~ py ¼ 0 and u~ py ¼ u0y ) because it is assumed that full-field strain measurements are only available over surface S2 : It must be noted that the virtual

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displacement field must be continuous over the whole specimen, in particular along the boundaries between S1 and S2 ; and S2 and S3 : The knowledge of the virtual fields is reduced to the determination of the Aij and Bij coefficients in Eq. (9). In order to do so, it has been proposed [4] that some displacement fields, called ‘special’ hereafter, could lead directly to the identification of each parameter. For instance, a strategy to identify Qxx is to find the special virtual field u~ pð1Þ satisfying kinematic admissibility (two continuity equations) and the five following relationships:

This feature illustrates the filtering capabilities of the VFM where unknown information can be discarded, if the virtual fields are chosen adequately. 4. Finite element computation 4.1. Finite element model and constitutive equation implementation In practice, the strain field measurements can be obtained experimentally by some full-field optical technique, such as

8 m n  i21  j  i  j21 p X q XX X > i ð x y j ð x y > > e Aij 1x dS ¼ 1; Bij 1y dS ¼ 0 > > L S2 L H H S2 L H > i¼0 j¼1 i¼1 j¼0 > > > >  i21  j  i  j21 p X q > m X n X X > i ð x y j ð x y > > A 1 dS þ B 1x dS ¼ 0 > ij y ij > L S2 L H H S2 L H < i¼1 j¼0

i¼0 j¼1

 i  j21  i21  j p X q m X n > X X > j ð x y i ð x y > > Aij gs dS þ Bij gs dS ¼ 0 > > H L H L L H S2 S2 > i¼0 j¼1 i¼1 j¼0 > > > >  i  j21  i21  j p X q > m X n X >X j ð x y i ð x y > 3 > Aij gs dS þ Bij g3s dS ¼ 0 > : H L H L L H S2 S2 i¼1 j¼0 i¼0 j¼1

where, for instance, m ¼ 1; n ¼ 3; p ¼ 2 and q ¼ 3 [4]. These values are chosen in order to have ‘enough’ freedom to express the special virtual fields, ie to have more Aij ’s and Bij ’s than the number of parameters to identify. Other choices could be possible, this has not been thoroughly investigated yet. In this case, 7 £ 7 matrices are extracted from the rectangular main matrix and inverted when the determinant is greater than a fixed threshold, which confirms that the resulting linear system can be inverted. Thus, one obtains a set of Aij and Bij ’s that define the special virtual field allowing to identify directly the unknown parameter Qxx : It has been shown empirically on this configuration that the special virtual fields exist and are not unique [4]. Therefore, additional criteria have to be set up to select among the great numbers of special fields obtained. The criterion adopted here is based on the sensitivity of the identified parameters to noise in the data. All the details related to the selection of the optimal special virtual fields are given in references [4,10]. To extract the remaining parameters, the procedure is repeated with other special virtual fields u~ pðiÞ ; i ¼ 2; …; 5 such that the location of the term ‘1’ in Eq. (10) is moved for each case. It is interesting to note that since the vertical virtual displacement of the right hand side of the specimen is constant, it is not necessary to know the external load distribution, which depends on the specimen to fixture contact. Only the resultant load measured by the load cell appears in Eq. (7). Moreover, the contribution of any longitudinal load due to the friction between the specimen and the fixture will be cancelled out thanks to the zero horizontal virtual displacement of this area.

ð10Þ

moire´ or speckle interferometry, digital image correlation, grid method, etc. [11]. The objective of the present paper being the validation of the identification procedure, experimental data are simulated by FE modelling. These numerical calculations require the implementation of the constitutive model into the implicit code ABAQUS by means of the User MATterial subroutine (UMAT) programmed in the Fortran 90 language [12], which purpose is to provide at each integration point the consistent material ›Dsij Jacobian matrix, Jijkl ¼ ; where ›Dsij and ›D1kl are, ›D1kl respectively, the stress and strain increments. In the case of linear elasticity, the Jacobian tangent matrix is equal to the stiffness matrix of the orthotropic material. Otherwise, the user must change the Jacobian matrix definition when implementing his own constitutive model. In the present case, only one component of the matrix ðJs Þ is affected by the stiffness reduction due to the in-plane shear damage. The non-linear shear component becomes: Js ¼

›Dss ¼ Qss 2 3K g2s ›Dgs

ð11Þ

4.2. Application to the unnotched Iosipescu configuration In this part, the application of the procedure is carried out on an orthotropic glass/epoxy composite material.

H. Chalal et al. / Composites: Part A 35 (2004) 841–848 Table 1 Identified stiffnesses for the elastic behaviour of the studied glass/epoxy composite u0y ¼ 1:3 mm Reference (GPa) Identified (GPa) Relative difference (%)

Qxx

Qyy

Qxy

Qss

K

25.93

10.37

3.112

4.000

0.

25.94

10.36

3.113

3.99

46.80

20.050

0.060

20.030

0.120

–

845

5. Results and discussion 5.1. Linear elastic case

dss max 0 0.008 –

The stiffness constants are: Qxx ¼ 25:93 GPa, Qyy ¼ 10:37 GPa, Qxy ¼ 3:112 GPa, Qss ¼ 4:000 GPa. The material constant governing the nonlinearity, noted K; is K ¼ 4420 GPa. These values represent a generic material. The specimen represented in Fig. 1 was modelled. Its thickness and width are, respectively, 2 and 20 mm. The calculations were performed using the inplane stress assumption and four-noded quadrilateral elements (CPS4). The imposed vertical displacement on the right-hand side (respectively, left-hand side) of the specimen is constant and equal to u0y ¼ 1:3 mm (respectively, 0 mm). It should be pointed out that the spatial convergence of the FE computation has been checked. As result, the central area ðS2 Þ of the specimen tested has been meshed with 9600 elements (with a total of 24,320 elements for the whole specimen) for the linear elastic case, and with 15,000 elements (with a total of 19,142 elements for the whole specimen, using an adapted mesh) for the non-linear calculations. The five material parameters listed above are the input values introduced in the finite element model to simulate the actual strain fields. These computed strain fields are then used as inputs to the VFM procedure described previously and the objective is to check that the VFM retrieves the reference parameters input in the FE code.

The identification procedure was first applied in the case where K ¼ 0 (linear elastic behaviour). Table 1 reports the identified parameters obtained by the VFM compared to the reference values introduced in the direct computation (FEM). The identified parameters reported in Table 1 are very close to the reference values. Such results have already been obtained in Ref. [10]. Besides, the nonlinearity parameter K is very small. Its value corresponds to a negligible damage within the material, which is consistent with the assumption of elastic behaviour. 5.2. Nonlinear shear response The above identification is now extended to the nonlinear case. The shear strain field is represented in Fig. 2. The identification results given by the VFM are reported in Table 2. It can be seen that the additional identification of the non linear parameter does not affect the identification of the four elastic stiffnesses for which the relative differences are almost similar to those obtained in the linear case. Concerning the parameter governing the nonlinearity, the identified value is very close to the reference one. This result confirms the feasibility of the VFM when applied to such a nonlinear constitutive model. 5.3. Sensitivity to a random noise In practice, the identification procedure uses actual strain fields obtained by experimental techniques. Experimental data are always subjected to noise. Hence, the investigation of the sensitivity of the identification process to noisy data constitutes an important point in such a type of inverse problem. To simulate experimental errors, a white gaussian noise was added to strain values provided by the finite element computations. Experimental errors inherent to

Fig. 2. In-plane shear strain field simulated for the damaged glass/epoxy specimen.

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H. Chalal et al. / Composites: Part A 35 (2004) 841–848

Table 2 Identified stiffness for the nonlinear behaviour of the studied glass/epoxy composite u0y ¼ 1:3 mm

Qxx

Qyy

Qxy

Qss

K

Reference (GPa) Identified (GPa) Relative difference (%)

25.93 25.94 20.03

10.37 10.35 0.19

3.112 3.112 0.0

4.000 3.990 0.12

4420 4347 1.65

the resolution of the optical techniques are not considered here and will be taken into account in a future work devoted to experimental optical displacement and strain measurements. Other type of errors can also be numerically simulated, such as a shift of the coordinate system that represents an imperfect positioning of the CCD camera [10]. This type of errors is not discussed here. The average of the simulated random noise is equal to zero (white noise). In this work, two amplitudes A of the added noise were simulated (with A ¼ 2s with s the standard deviation). These correspond, respectively, to: A ¼ b Maxðmeanl1x l; meanl1y l; meanlgs lÞ with b ¼ 5% and 10%

ð12Þ

The noise amplitudes correspond, respectively, to strain values of 5 £ 1024 and 1023 : For each set of noisy strains, an identification is performed. Repeating the process 30 times with renewed noise, a distribution of identified parameters is obtained. The average and coefficients of variation of these distributions of identified parameters are reported in Table 3. The coefficient of variation represents the scatter of the results extracted from noisy data. It can be seen that Qxx and Qss are the most stable. The sensitivity of the other rigidity constants is more important. As expected, the results show that the relative difference and the scatter increase with the noise amplitude. 5.4. Influence of length L To examine the influence of length L characterising the distance between the grips, a parametric study has been

performed. The material tested here is the same as that studied in the above sections. The variation of length L of surface ðS2 Þ lies within the interval [10,50] mm with a step of 5 mm for each simulation. Fig. 3 shows the evolution of the parameters governing the behaviour law, identified by the VFM, as a function of length L: This length has a considerable importance in the Iosipescu fixture. It characterises the degree of heterogeneity of the strain/stress fields in the center part of the specimen. The results are presented in Fig. 3. Considering Qxx ; the effect of noise becomes negligible when L . 20 mm. This is consistent with the fact that bending effects increase with increasing L: On the contrary, reducing L promotes transverse compression and shear. As seen on Fig. 3, Qyy is more stable for lower L: Nevertheless, the shear modulus becomes less stable for very low values of L: This is probably caused by higher shear gradients resulting in increased sensitivity to noise. The same effect can be observed on K: As for Qxy ; as already mentioned in previous studies [7,8,10], it is more sensitive to noise than the other components, but with 20 # L # 35 mm, the results are better. This clearly shows that for this problem, there is an ‘optimal’ value for L; around 25– 30 mm. Nevertheless, the present study underlines the difficulty to define an objective criterion to choose the ‘best’ test configuration for optimal identification of the material parameters. Relative strain values between the different components are an issue, but gradients also play a role since high gradients zone will be prone to higher effects of noise. More work is necessary in this area to design optimal test configurations for identification with the VFM. A few studies exist on this topic for finite element model updating [19,20] but such approaches may not be appropriate for the VFM. 5.5. Influence of the orthotropy ratio In this section, another orthotropic material has been studied in order to investigate the influence of the anisotropy degree. The first material considered is a unidirectional glass/epoxy composite which behaviour characteristics are those given in Section 4.2. The second material is a unidirectional T300/914 carbon/epoxy composite.

Table 3 Identified stiffness and damage parameters from noisy strain fields. Glass/epoxy(M10) composite u0y ¼ 1:3 mm

Qxx

Qyy

Reference (GPa) Noise amplitude: 5 £ 1024 Identified (GPa) Relative difference (%) Coefficient of variation (%)

25.93

10.37

25.94 20.03 1.02

9.400 9.35 35.9

Noise amplitude: 1023 Identified (GPa) Relative difference (%) Coefficient of variation (%)

26.02 20.34 1.080

11.48 210.7 43.03

Qxy

Qss

K

3.112

4.000

4420

3.430 29.72 23.9

3.990 0.25 0.70

4306 2.58 5.28

4.070 30.78 37.18

4.090 22.25 7.460

5077 214.8 40.98

H. Chalal et al. / Composites: Part A 35 (2004) 841–848

847

Fig. 3. Influence of the active zone length L on the identified stiffness and damage parameters for the studied glass/epoxy composite.(errors bars correspond to ^2s; s: standard deviation).

Its rigidity constants constituting the reference used for FE computations, are: Qxx ¼ 125:9 GPa, Qyy ¼ 9:120 GPa, Qxy ¼ 3:080 GPa, Qss ¼ 4:960 GPa [21]. Furthermore, the constant K that governs the nonlinearity is K ¼ 2349 GPa. For the carbon/epoxy specimen, numerical simulations are performed for an identification length L ¼ 30 mm. Furthermore, the composite plate is subjected to

a constrained vertical displacement u0y ¼ 1:98 mm on its right-hand side as shown in Fig. 1. The identified parameters governing the behaviour law for both composite materials are reported in Table 4. The identified elastic stiffnesses and the damage parameter agree well with the reference values. However, when increasing the orthotropy ratio, one notices a slight increase of

Table 4 Identified parameters for both orthotropic composite materials Qxx =Qyy ¼ 2:5

Qxx

Reference (GPa) Identified (GPa) Relative difference (%)

25.93 25.94 20.03

Qxx =Qyy ¼ 13:7 Reference (GPa) Identified (GPa) Relative difference (%)

125.9 126.0 20.08

With noisy data which amplitude ¼ 5 £ 1024 Identified (GPa) 96.11 Relative difference (%) 23.68 Coefficient of variation (%) 56.13

Qyy

Qxy

10.37 10.35 0.19 9.120 8.970 1.57 10.05 210.26 160.0

Qss

K

3.112 3.112 0.0

4.000 3.990 0.12

4420 4347 1.65

3.080 3.080 0.0

4.960 4.950 0.14

2349 2303 1.93

1.950 36.56 194.0

4.640 6.37 96.32

256.8 89.06 4155.

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the imposed displacement. Finally, the experimental implementation of the method is currently underway.

References

Fig. 4. Relative difference for both studied composite materials.

the relative error. For the purpose of comparison, the relative difference is plotted for both materials (Fig. 4). This illustrates that the identification procedure can be affected by the orthotropy ratio of the tested material when length L is not optimised. In processing noisy data with an amplitude of 5%, it has been noticed that the relative difference increases dramatically for the carbon/epoxy specimen compared to the case of the glass/epoxy material. This may be caused by improper choice of length L: Additional work is needed to investigate this point.

6. Conclusion The present work is one of the first attempts to use the VFM to identify material parameters driving non-linear constitutive equations. The damage model considered here is very simple but more complex descriptions could be used, for instance by increasing the number of terms in the polynomial stress/strain relationship. Nevertheless, constitutive models leading to non-explicit stress/strain relationships would be more difficult to address because the resulting system would become non-linear. Studies concerning this topic are underway in the authors research groups. The results obtained in the present study are generally satisfactory. However, the essential issue of the design of mechanical tests adapted to the VFM, though slightly addressed in the present paper, remains open. Finally, some issues remain to be addressed in the near future. Firstly, extending the current developments of the direct identification to the case of an anisotropic damage behaviour law. This latter could be described through two variables representing the coupling between the in-plane shear damage and that generated by the transverse tension. Secondly, one must improve the identification accuracy by using different load levels. This may contribute to better identified results due to information redundancy. The strain component values will be provided at intermediate values of

[1] Ladeve`ze P, Allix O, Deu J-F, Le´veˆque D. A mesomodel for localisation and damage computation in laminates. Comput Meth Appl Mech Engng 2000;183:105–22. [2] Okada H, Fukui Y, Kumazawa N. An inverse analysis determining the elastic-plastic stress–strain relationship using nonlinear sensitivities. Comput Model Simulat Engng 1999;4(3):176–85. [3] Wang WT, Kam TY. Elastic constants identification of shear defomable laminated composite plates. J Engng Mech 2001; 127(11):1117 –23. [4] Gre´diac M, Toussaint E, Pierron F. Special virtual fields for the direct determination of material parameters with the virtual fields method. Part 1—principle and definition. Int J Solids Struct 2002;39: 2691–705. [5] Claire D, Hild F, Roux S. Identification of damage fields using kinematic measurements. Comptes rendus Me´canique 2002;330: 729 –34. [6] Gre´diac M. Principe des travaux virtuels et identification. Comptes Rendus de l’Acade´mie des Sciences (in French with abridged English version 1989;II(309):1–5. [7] Gre´diac M, Pierron F. A T-shaped specimen for the direct characterization of orthotropic materials. Int J Num Meth Engng 1998;41:293 –309. [8] Pierron F, Gre´diac M. Identification of the through thickness moduli of thick. composites from whole-field measurements using the Iosipescu fixture: theory and simulations. Compos Part A 2000;31(4):309–18. [9] Pierron F, Zhavaronok S. Gre´diac. M., Identification of the through thickness properties of thick laminates using the virtual fields method. Int. J. Solids and Struct 2000;37(32):4437–53. [10] Gre´diac M, Toussaint E, Pierron F. Special virtual fields for the direct determination of material parameters with the virtual fields method. Part 2—application to in-plane properties. Int J Solids Struct 2002;39: 2707–30. [11] Rastogi P, editor. Topics in applied physics. Berlin: Springer; 1999. [12] HKS Inc, ABAQUS theory and users manuals V. 6.2.1. ; 2002. [13] Pierron F, Vautrin A. Measurement of the in-plane shear strengths of unidirectional composites with the Iosipescu test. Comp Sci Technol 1997;57:1653–60. [14] Odegard G, Kumosa M. Determination of shear strength of unidirectional composite materials with Iosipescu and 108 off-axis shear tests. Comp Sci Technol 2000;60:2917–43. [15] Ladeve`ze P, Le Dantec E. Damage modeling of the elementary ply for laminated composites. Comp Sci Technol 1992;43(3):257–67. [16] Lemaıtre J, Chaboche J-L. Mechanics of solid materials. Cambridge, UK: Cambridge University Press; 1990. [17] Broughton WR, Hodgkinson JM. Mechanical testing of advanced fibre composites. Chapter 6—shear, Woodhead Publishing Limited; 2000. p. 100– 23. [18] Gre´diac M, Auslender F, Pierron F. Applying the virtual fields method to determine the through-thickness moduli of thick composites with a nonlinear shear response. Compos Part A 2001;32:1713–25. [19] Le Magorou L, Bos F, Rouger F. Identification of constitutive laws for wood-based panels by means of an inverse method. Comp Sci Technol 2001;62(4):591–6. [20] Arafeh MH. Identification de la loi de comportement e´lastique de mate´riaux orthotropes. Doctoral Dissertation (in French). Universite´ de Technologie de Compie`gne, Compie`gne; 1995. [21] Pierron F, Vautrin A. Accurate comparative determination of the inplane shear modulus of T300/914 using the Iosipescu and 458 off-axis tests. Comp Sci Technol 1994;52(1):61–72.