Tiré à Part
Continuum Damage Mechanics: initial and induced anisotropy in composite materials J.-L. Chaboche Conference to the Memory of Professor Jean-Paul Boehler Grenoble (France), June 21-22, 1999 TP 1999-105
Continuum Damage Mechanics: initial and induced anisotropy in composite materials Mécanique de l'endommagement continu : anisotropie initiale et induite dans les matériaux composites par
J.-L. Chaboche
Conference to the Memory of Professor Jean-Paul Boehler Grenoble (France), June 21-22, 1999
Résumé : L'anisotropie induite par endommagement et des aspects connexes sont brièvement revus et discutés, dans le contexte de la Mécanique de l'endommagement continu et de son application à des matériaux CMC ou CMM. Quelques indications sont données sur les effets de désactivation des dommages, les stockages associés d'énergie élastique et l'état actuel des possibilités des modèles existants.
Ce Tiré à part fait référence au Document d’Accompagnement de Publication DMSE9917
Continuum Damage Mechanics : initial and induced anisotropy in composite materials J.L. Chaboche O.N.E.R.A. France
Abstract Damage induced anisotropy and related aspects are briefly reviewed and discussed in the context of CDM and its application to composite materials like CMC’s or MMC’s. Some indications are given about the damage deactivation effects, the associated elastic energy storage, and the present state of existing modeling capabilities. 1. Introduction Since the early work of Kachanov (1958) many progresses have been done in the development of a consistent continuum framework for Damage Mechanics. This Continuum Damage Mechanics (CDM), as a purely macroscopic tool, do not try to describe all the local, micromechanical and physical features but summarizes and approximates them through global constitutive and damage equations. Concerning damage induced anisotropy the preliminary works by Kachanov or Leckie and Hayhurst (1974) were limited to creep damage and proportional loading conditions. Complete anisotropic constitutive and damage modeling, applied to creep damage situations, were appearing at the end of the seventies, involving second rank damage tensors (Cordebois & Sidoroff, 1979, Murakami, 1983), or fourth rank tensors (Chaboche, 1979). Let us recall that the idea of a second rank tensor was also proposed by Kachanov and Vakulenko (1971), based on a micro-macro analysis for a microcracked elastic materials. During the 80’s damage induced anisotropy has been the subject of many researches, especially supported by civil engineering applications to elastic brittle materials, like concrete, rocks,… Let us mention the works by Krajcinovic and Fonseka (1981), Ju (1989). For composite materials we should mention Ladeveze's approach (1983, 1994) and works done by Talreja (1991). One of the difficulties associated to the damage modeling of brittle materials is the damage deactivation effects under compressive like loading conditions (see the discussions by Chaboche, 1993). 2. Continuum Damage Mechanics of composites The composite material is treated at the macroscopic level, as a continuum. In case of laminated structures, we consider each ply as a different material, with its own constitutive and damage equations, the laminate model being then obtained through the structural analysis. To describe delamination between plies, we assume damaging interface layers. The works conducted over the past few years at ONERA, led to developing a hierarchical approach to the constitutive equations for the various composite systems (Chaboche et al., 1995, Maire & Chaboche, 1997, Pottier, 1998, Chaboche et al., 1999) : - The basic configuration is woven ceramic matrix composites (CMC's), such as SiC/SiC, whose behavior is essentially elastic, with nonlinearities caused by damage. Very clear damage deactivation effects take place under compression loadings,
- Irreversible strains due to damage and plasticity type effects were introduced for C/SiC and C/C composites. - For C/PMR15 woven composites used in some engine parts, damageable viscoplasticity and/or viscoelasticity models were developed to account for the anisotropies, the hysteresis effects during the loading and unloading cycles, creep, relaxation and recovery effects. - These macroscopic models are now being applied to organic matrix composites (OMC’s) used as unidirectional laminates. Different but similar versions have been developed for SiC/Ti metal matrix composites (MMC’s) using approaches with micro/macro transitions. The general approach is developed in the context of Continuum Thermodynamics with Internal Variables, using both scalar damage variables, associated with microcracks oriented by the highest strength components (fibers, yarns), and tensorial ones, to describe microcracks which orientation is related to the directions of the applied stresses. The models are able to describe possible damage caused by compression (splitting type cracks). 3. Damage deactivation The difficulty associated to the damage deactivation modeling appears in writing the elastic potential and Hooke’s Law. Most of the developed CDM theories in the eighties, with anisotropy and deactivation, suffer of unconsistencies (symmetry loss or stress-strain response discontinuities; see Chaboche, 1993, or Curnier et al. 1993). Some recent theories have incorporated more correctly the damage deactivation. Based on an anisotropy description via a second rank damage tensor, they are applicable either to an initially isotropic material (Halm & Dragon, 1995; Ladevèze et al. 1994) or to initially anisotropic ones (Maire & Chaboche, 1997). This class of models, even if mathematically sound, still presents some modeling deficiencies : - either damage growth under a purely compressive loading is not allowed, or the damage deactivation effect is not complete, provided the shear modulus (and Poisson's terms) are not affected by the deactivation ; - the true effects of microcrack closures and slidings are not incorporated in a framework consistent with micromechanical based approaches (see Andrieux et al., 1986, for exemple). 4. Damage deactivation, energy storage and friction mechanisms This is the reason why a significantly different approach of deactivation has been proposed recently (Boursin et al., 1996 ; Pottier, 1998), based directly on consequencies of a micromechanics analysis (Chaboche & Suquet, 1998). When damage deactivates, i.e. when a system of microcracks closes, no restriction is imposed to the change of the stiffness tensor (or compliance) : the damage parameters or damage components corresponding to the "closure direction" are replaced by zero values everywhere in the elastic potential (recovering the initial stiffness in the extreme case where all damage components are deactivated, under triaxial compression for instance). The corresponding strain (or stress) discontinuities in the stress-strain response are "stored" in the material, through a progressive elastic energy storage that takes place as the stress or strain changes after deactivation. This new condition has not the previously mentioned shortcomings. It was applied first to the modeling of a metal matrix composite, SiC/Ti with unidirectional long fibers (Pottier, 1998), in the framework of a macroscopic CDM model that takes into account the matrix behaviour through a micro-macro approach. It is under application and extension to CMC's, like SiC/SiC, trying to conciliate several aspects related to micro-friction : - when damage has deactivated (microcrack closed), there is a coupling effect between elastic energy storage and dissipation by friction (Coulomb's like friction of the crack lips);
- after elastic energy storage any re-opening (damage reactivation) is necessarily preceded by a sliding mechanism (see Andrieux et al., 1986, Chaboche and Suquet, 1998). Only in case of an infinite friction coefficient we could speak of a stress-strain "discontinuity" at re-opening (Boursin et al., 1996); - the friction/storage mechanism can also take place at a lower scale, associated with the interaction between a matrix microcrack and the bridging fibers, including debonding and sliding along the fiber/matrix interface. The involved debonding is practically of no effect on the macroscopic elastic response but has to be incorporated in order to describe better inelasticity ans hysteretic effects for quasi-elastic unloadings, like in micromechanics based composite models (Levasseur, 1999). 5.
Concluding remarks
To some extent composite materials, as oriented ones, are easier to model in terms of a CDM approach : microcracks being often guided by the constituent orientations, the damage theory can consider only scalar variables. However, in many cases one can also observe microcracks driven by the stress direction, like in CMC’s. This is the reason why the present capabilities of many models for composites are incorporating both : - a scalar representation of damage for those parallel to the constituents (not changing the initial material symmetries), - tensorial damage variables (2d or 4th ranks) for representing microcracks that more or less develop perpendicular to the maximum principal stress. Moreover the correct modeling of damage deactivation and related effects is a difficult task. Presently several models incorporate in a mathematical and thermodynamical sound way the essential features, essentially in France the models developed at LMT-Cachan, at ENSMA and at ONERA. More recent extensions for elastic energy storage and the corresponding friction mechanisms, even if already present in previous works, are still under development in the framework of a general macroscopic CDM model, with application to several classes of composites (Levasseur, 1999). 6. References Andrieux S., Marigo J.J. and Bamberger Y. (1986) Un modèle de matériau microfissuré pour les bétons et les roches, J. de Mécanique Théorique et Appliquée, 5 (3), 471-513. Boursin A., Chaboche J.L. and Roudolff F. (1996) Mécanique de l'endommagement avec conditions unilatérales et stockage d'énergie élastique, CRAS Paris, t.323, Série IIb, 369-376. Chaboche, J.L. (1979) Le concept de contrainte effective appliqué à l'élasticité et à la viscoplasticité en présence d'un endommagement anisotrope. Col. Euromech 115, Grenoble, Eds du CNRS, 1982. Chaboche J.L. (1993) Development of CDM for elastic solids sustaining anisotropic and unilateral damage, Int. J. of Damage mechanics, 2, pp. 311-329. Chaboche J.L., Lesne P.M. and Maire J.F. (1995) Continuum Damage Mechanics, anisotropy and damage deactivation for brittle materials like concrete and ceramic composites. Int. J. Damage Mechanics, 4, 5-22. Chaboche J.L. and Suquet P. (1998) Endommagement, interfaces. Ecole d'Ete "Méthodes d'Homogénéisation en Mécanique des Matériaux", La Londe-les-Maures.
Chaboche J.L., Kruch S., Maire J.F. and Pottier T. (1999) Micromechanics based modelling of inelasticity and damage in MMC’s, 7th Int. Symp. on Plasticity and its Current Applications, PLASTICITY’99, Cancun, pp.705-708. Cordebois J.P. and Sidoroff F. (1979) Anisotropie élastique induite par endommagement, Col. Euromech 115, Grenoble, Eds. du CNRS, 1982. Curnier A., He Q. and Zysset P. (1995) Conewise linear elastic materials. J. Elasticity, 37, 138. Halm D. and Dragon A. (1996) A model of anisotropic damage by mesocrack growth unilateral effects. Int. J. Damage Mechanics, 5(4), 384-402. Ju J.W. (1989) On energy-based coupled elastoplastic damage theories : constitutive modeling and computational aspects. Int. J. Solids Structures, 25(7), 803-833. Kachanov L.M. (1958) Time of the rupture process under creep conditions. Isv. Akad. Nauk. SSR. Otd Tekh. Nauk., 8, 26-31. Krajcinovic D. and Fonseka G.U. (1981) The continuous damage theory of brittle materials, Parts I and II, J. of Applied Mechanics, 48, 809-824. Ladevèze P. (1983) Sur une théorie de l'endommagement anisotrope, Rapport Interne 34, LMT-Cachan. Ladevèze P., Gasser A. and Allix O. (1994) Damage mechanics modelling for ceramic composites, Journal of Engineering Materials and Technology, 116. Leckie F.A., Hayhurst D.R. (1974) Creep rupture of structures. Proc. Royal Soc. London, 340, 323-347. Levasseur P. (1999) Mécanique multiéchelle du frottement dans les composites à fibres longues, thèse ENSMP. Maire J.F. and Chaboche J.L. (1997) A new formulation of Continuum Damage Mechanics for composite materials, Aerospace Science and Technology, 2, pp. 247-257. Murakami S. (1983) Notion of Continuum Damage Mechanics and its application to anisotropic creep damage theory. J. of Engng. Mat. Technol., 105, 99. Pottier T. (1998) Modélisation multiéchelle du comportement et de l'endommagement de composites à matrice métallique, thèse ENPC. Talreja R. (1991) Continuum modelling of damage in ceramic matrix composites, Mechanics of Materials, 12, pp. 165-180. Vakulenko A.A. and Kachanov M.L. (1971) Continuum theory of medium with cracks. Mech. of Solids, engl. Transl. of Mekhanika Tverdogo Tela (in Russian), 6(4), 145-151.
