[8] HKS Inc, ABAQUS Theory and Users Manuals V. 6.2.1, 2001. [9] Meraghni F., Chalal H., F. Pierron., Grédiac M., Identification directe du comportement ...
ON THE USE OF THE VIRTUAL FIELDS METHOD FOR THE IDENTIFICATION OF A PHENOMENOLOGICAL DAMAGE MODEL FOR COMPOSITE MATERIALS H. CHALAL, F. MERAGHNI, F. PIERRON and M. GRÉDIAC* Laboratoire de Mécanique et Procédés de Fabrication, Ecole Nationale Supérieure d'Arts et Métiers (ENSAM). Rue Saint-Dominique - BP 508 - 51006 Châlons-en-Champagne Cedex, France. * Laboratoire d’Études et de Recherches en Mécanique des Structures, Université Blaise Pascal Clermont II, 24, Avenue des Landais - BP 206 – 63174 Aubière Cedex, France The present work deals with the direct and simultaneous identification of parameters governing a non-linear behaviour law of composite materials. The identification procedure consists in a processing of the strain fields using the Virtual Fields Method (VFM) based on the well-known principle of virtual works. The VFM was developed by Grédiac [1] and has been successfully applied by Grédiac, Pierron et al. to determine the in-plane [2] and the through thickness [3-4] mechanical stiffness of orthotropic composite materials. The identification method relies upon the global equilibrium of a structure expressed with the principle of virtual works using specific virtual displacement fields. Currently, several improvements of the VFM have been developed concerning the automatic generation of the optimal virtual displacement fields [5]. In this work the non-linearity considered is due to damage onset and growth. The phenomenological damage model used here has been developed by Ladevèze [6] within the framework of the thermodynamics of irreversible processes using strain energy expressed through three internal damage variables. This model is extensively applied for composite materials [7]. In this study, only the damage corresponding to in-plane shear is considered. Hence, the non-linearity is mainly governed by the in-plane shear response. The damage evolution is first modelled as a quadratic function of the shear strain. The non-linear behaviour law obtained from the damage model is implemented into the finite element implicit code ABAQUS using a user material supplied subroutine UMAT programmed in Fortran 90 language [8]. To obtain heterogeneous stress and strain fields an unnotched composite specimen was numerically simulated using the Iosipescu configuration (figure 1). The simulated strain fields (figures 2) are processed by the VFM using special and optimised virtual fields. This allows the extraction of the whole set of parameters (table 1) governing the linear and the non-linear material behaviour (figure 3) with a highest accuracy [9]. The aim of the present paper is to investigate numerical aspects inherent to the method implementation and the identification procedure sensitivity in terms of accuracy and stability. On one hand, a parametric study has been performed to evaluate the influence of the length (L) characterizing the identification central area on the identified elastic stiffness and the non-linear shear parameter. On the other hand, the direct identification has been conducted upon two composite materials: glass (E) /epoxy resin (M10) and carbon/epoxy (T300/914) to emphasize the effect of the orthotropic ratio on the accuracy of the identified properties. This work showed that the identification procedure is reliable and numerically robust even when processing noisy data (table 1).
y
~uy S
S
b
S
x
L thickness : e
K1
~u
K2
y
Figure 1. Schematic view of Iosipescu fixture
Figures 2. In-plane shear strain field simulated for the damaged composite specimen Table 1: Comparison between the reference and the behaviour parameters identified from accurate and noisy strain fields (10% of each strain component) for a glass (E)/epoxy composite.
50
0 ,3 5
In-pla ne Shea r Stre ss vs Stra in
45
(GPa)
Qxx
Qyy
Qxy
Qss
K
4,0
4420
30
Identified
26,08 11,17
4,09
4092,8
25
RelativeDiff (%)
-0,57 -7,71 -6,04
-0,5
7,4
20
Noisy Identified Data
25,94 10,35 3,12
3,99
4347,05
15
-0,03
0,12
1,65
10
Relative Diff (%)
0,19
0,0
0 ,2 5
35
Reference 25,93 10,37 3,11 3,3
0 ,3
D ama ge Evolution
40
σs = Qss γ s – K γ
3 s
0 ,2
0 ,15
Dss = K .γ 2s Qss
0 ,1
0 ,0 5 5 0 0
0 ,0 0 5
0 ,0 1
0 ,0 15
0 0 ,0 2
I n - p l a n e S h e a r S tr a i n
Figure 3. Simulated non-linear in-plane shear response and damage evolution of glass (E) /epoxy composite. References [1] Grédiac M., Principe des travaux virtuels et identification. Comptes Rendus de l’Académie des Sciences. 1989, II (309) 1-5. [2] Grédiac M., Pierron F. A T-shaped specimen for the direct characterization of orthotropic materials. Int. J. Num. Meth in Engng. 1998;41:293-309. [3] Pierron F. et al. Identification of the through thickness properties of thick laminates using the virtual fields method. Int. J. Solids and Struct. 2000;37(32):4437-53. [4] Pierron F., Grédiac M. Identification of the through thickness moduli of thick composites from whole-field measurements using the Iosipescu fixture: theory and simulations. Compo. Part A 2000;31(4):309-18. [5] Grédiac M., Toussaint E. et Pierron F., Special virtual fields for the direct determination of material parameters with the virtual fields method. 1- Principle and definition. Int. J. Solids and Struct 39, (2002): 2691-2705. [6] Ladevèze P., Sur la mécanique de l’endommagement des composites. Comptes Rendus des JNC5. Paris : Pluralis Publ. 1986, pp 667-683. [7] Ladevèze P., Le Dantec E. Damage modeling of the elementary ply for laminated composites. Comp. Sci & Tech 1992;43(3):257-67. [8] HKS Inc, ABAQUS Theory and Users Manuals V. 6.2.1, 2001 [9] Meraghni F., Chalal H., F. Pierron., Grédiac M., Identification directe du comportement élastique endommageable de matériaux composites par la méthode des champs virtuels. Compte rendu 1er Colloque IME 2000. Besançon. Juillet 2002.
