COMPOSITES SCIENCE AND TECHNOLOGY ELSEⅤIER Composites Science and Technology 61(2001)2285-2297 www.elsevier.com/locate/compscitech Inelastic behaviour of ceramic-matrix composites Stephane baste Universite Bordeaux 1, Laboratoire de mecanique Physique, CNRS UMR 5469, 3.51, Cours de la liberation, 33405 Talence, france Received 7 November 2000: received in revised form 9 November 2000: accepted 5 July 2001 Abstract A methodology for the formulation and identification of the constitutive laws of ceramic-matrix composites is summarised. It relies on an anisotropic damage evaluation that accurately separates the effects of the various damage mechanisms on the non- linear behaviour. A mixed approach takes into account the basic strain and damage mechanisms by using a homogenisation method that provides the relationship between the mechanical response and the intensity of damage in the individual modes. That leads to a non-arbitrary choice of internal variables in the macroscopic constitutive relationships. A successive process of predic- tion/ experimental-data confrontation allows the optimal determination of the evolution laws of those internal variables. This methodology is illustrated on various behaviours of various CMCs; several crack arrays, tilted cracks, tensile test, cyclic loading, fi-axis solicitation, in ID SiC-SiC, 2D C-SiC, 2D C/C-SiC ceramic-matrix composites. Predictions of the three-dimensional changes in elasticity and of the inelastic strains are shown to compare favourably with experimental data measured with an ultra- sonic method. C 2001 Elsevier Science Ltd. All rights reserved ties d. ultrasonics modelling: Non-linear behaviour; Matrix cracking: C. Anisotropy; Damage mechanics; Elastic prop- 1. Introduction frictional sliding. Limited hysteresis loops account for negligible frictional sliding while debonding, on the The macroscopic mechanical behaviour of ceramic- other hand, can be broadly present matrix composites is strongly infuenced by the onset To formulate the constitutive laws of such materials and the development of microcracks [1, 2 ]. The beha- it is important to separate the effects of initiation and viour of CMCs is the result of the combination of two growth of microcracks from the effects due to the pre main damage mechanisms [3]: matrix microcracking sence of cracks. The various damage mechanisms normal to the tensile axis, deflection of these cracks at induced by mechanical loading and their influence on the fibre-matrix interface if the interface is weak enough the tensile behaviour were determined and analysed by (Fig. 1). The matrix microcracking induces a loss of comparing experimental variations of the components stifness Mode II cracking prevents the composite from of the stiffness tensor obtained from ultrasonic mea- failing too early because the fibre-matrix debonding surements and prediction of effective stiffness properties leads to a fibre-matrix sliding with friction depending on of medium permeated by cracks. The relationships the nature of the interface [4, 5]. Other mechanisms between the effective stifness tensor and the intensity of increase failure energy absorption like fibre pull-out and damage in individual modes, provide coherent and out of matrix crack-plane fibre fracture [6]. The combi- comprehensive physical explanations for the observed nation of these mechanisms leads to a highly non-linear experimental phenomenology. Various scales are con- behaviour(Fig. 2) sidered: the micro-scale at which the damage mechan- In most CMCs, debonding is accompanied by fric- isms are described and the macro-scale where the tional sliding which turns loading/unloading cycles into volume element is large enough to consider that the hysteresis loops(Fig. 2). The extent of the inelastic discrete damage mechanisms are well represented by a trains and the area of the hysteresis loops result from mean leading to continuous variables. The major point both an intense interfacial debonding and fibre-matrix of the methodology lies in the non-arbitrariness of the choice of the internal variables with a concrete physical meaning which reflect the underlying processes on the *Tel:+33-5-5684-6225;fax:+33-5-5684-6964 microscale, as example, the cracks density or the crack dress: baste(@ Imp. ul-bordeaux fr opening displacement 0266-3538/01/S- see front matter c 2001 Elsevier Science Ltd. All rights reserved. PII:S0266-3538(01)00122Inelastic behaviour of ceramic-matrix composites Ste´phane Baste* Universite´ Bordeaux 1, Laboratoire de Me´canique Physique, CNRS UMR 5469, 351, Cours de la Libe´ration, 33405 Talence, France Received 7 November 2000; received in revised form 9November 2000; accepted 5 July 2001 Abstract A methodology for the formulation and identification of the constitutive laws of ceramic-matrix composites is summarised. It relies on an anisotropic damage evaluation that accurately separates the effects of the various damage mechanisms on the non￾linear behaviour. A mixed approach takes into account the basic strain and damage mechanisms by using a homogenisation method that provides the relationship between the mechanical response and the intensity of damage in the individual modes. That leads to a non-arbitrary choice of internal variables in the macroscopic constitutive relationships. A successive process of predic￾tion/experimental-data confrontation allows the optimal determination of the evolution laws of those internal variables. This methodology is illustrated on various behaviours of various CMCs; several crack arrays, tilted cracks, tensile test, cyclic loading, off-axis solicitation, in 1D SiC–SiC, 2D C–SiC, 2D C/C–SiC ceramic-matrix composites. Predictions of the three-dimensional changes in elasticity and of the inelastic strains are shown to compare favourably with experimental data measured with an ultra￾sonic method. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: A. Ceramic-matrix composites; B. Modelling; Non-linear behaviour; Matrix cracking; C. Anisotropy; Damage mechanics; Elastic prop￾erties; D. Ultrasonics 1. Introduction The macroscopic mechanical behaviour of ceramic￾matrix composites is strongly influenced by the onset and the development of microcracks [1,2]. The beha￾viour of CMCs is the result of the combination of two main damage mechanisms [3]: matrix microcracking normal to the tensile axis, deflection of these cracks at the fibre-matrix interface if the interface is weak enough (Fig. 1). The matrix microcracking induces a loss of stiffness. Mode II cracking prevents the composite from failing too early because the fibre-matrix debonding leads to a fibre-matrix sliding with friction depending on the nature of the interface [4,5]. Other mechanisms increase failure energy absorption like fibre pull-out and out of matrix crack-plane fibre fracture [6].The combi￾nation of these mechanisms leads to a highly non-linear behaviour (Fig. 2). In most CMCs, debonding is accompanied by fric￾tional sliding which turns loading/unloading cycles into hysteresis loops (Fig. 2). The extent of the inelastic strains and the area of the hysteresis loops result from both an intense interfacial debonding and fibre-matrix frictional sliding. Limited hysteresis loops account for negligible frictional sliding while debonding, on the other hand, can be broadly present. To formulate the constitutive laws of such materials, it is important to separate the effects of initiation and growth of microcracks from the effects due to the pre￾sence of cracks. The various damage mechanisms induced by mechanical loading and their influence on the tensile behaviour were determined and analysed by comparing experimental variations of the components of the stiffness tensor obtained from ultrasonic mea￾surements and prediction of effective stiffness properties of medium permeated by cracks. The relationships between the effective stiffness tensor and the intensity of damage in individual modes, provide coherent and comprehensive physical explanations for the observed experimental phenomenology. Various scales are con￾sidered: the micro-scale at which the damage mechan￾isms are described and the macro-scale where the volume element is large enough to consider that the discrete damage mechanisms are well represented by a mean leading to continuous variables. The major point of the methodology lies in the non-arbitrariness of the choice of the internal variables with a concrete physical meaning which reflect the underlying processes on the microscale, as example, the cracks density or the crack opening displacement. 0266-3538/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(01)00122-1 Composites Science and Technology 61 (2001) 2285–2297 www.elsevier.com/locate/compscitech * Tel.: +33-5-5684-6225; fax: +33-5-5684-6964. E-mail address: baste@lmp.u-bordeaux.fr