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)00122
Inelastic 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 nonlinear 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 prediction/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 ultrasonic 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 properties; D. Ultrasonics 1. Introduction The macroscopic mechanical behaviour of ceramicmatrix composites is strongly influenced by the onset and the development of microcracks [1,2]. The behaviour 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 combination of these mechanisms leads to a highly non-linear behaviour (Fig. 2). In most CMCs, debonding is accompanied by frictional 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 presence 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 measurements 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 considered: the micro-scale at which the damage mechanisms 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
S. Baste / Composites Science and Technology 61(2001)2285-2297 2. Experimental velocity of the pulses is done by a signal processing method using Hilbert transform [9] An ultrasonic device [7 and an extensometer used To fully describe the elastic behaviour of an ortho imultaneously provide a useful method for carrying out tropic material, the nine elastic constants Cii are identi a strain partition under load [8]. Applied to CMCs, the fied by measuring the phase velocities of bulk waves contribution of the damage mechanisms to their highly transiting in two accessible principal planes [planes( non linear behaviour can be evaluated 2)and(1, 3), Fig. 3] and in a non principal plane [plane (1, 45] described by the bisectrix of axis 2 and 3)of the 2.. Ultrasonic method sample[10]. The identification in plane(1, 2)gives four elastic constants: Cll, C22, C66 and Cl and three others The ultrasonic device consists in an immersion tank are obtained in plane(1, 3): C33, Css and C13. The two associated to a tensile machine [7]. It allows to study the remaining coefficients C23 and C44 are identified by complete stifness tensor variation under load thus it is propagation in the non-principal plane(1, 45). The possible to know which coefficients are affected during confidence interval associated to each identified con- the damage process. stant is then estimated by a statistical analysis [11] Wave speed measurements are performed by using ultrasonic pulses which are refracted through the sample 2. 2. Under load strain partition immersed in water. The measurement of the phase measure the plastic strains. In ticity of metals, dislocations are the main faws lead to relative discomposition th unloading, and which found expression, from a macro scopic point of view, in permanent strains. The intro- duction of the term "inelastic"is lead by a possible part- recovery of the strains beyond the yield point. It is all the more justified for brittle matrix composites. The matrix microcracking does not modify a pre-existing elastoplastic behaviour but creates residual strains that could not exist otherwise. the inelastic strains in Cmcs are the macroscopic indication of the transverse crack matrix sliding [12]. This sliding direction of the load and thus partly reversible. A Fig. 2. Consequence of matrix cracking and fibre-matrix sliding on ultrasonic device used to investigate the stress-induced development of the stress/strain behaviour the damage of CMcs
2. Experimental An ultrasonic device [7] and an extensometer used simultaneously provide a useful method for carrying out a strain partition under load [8]. Applied to CMCs, the contribution of the damage mechanisms to their highly non linear behaviour can be evaluated. 2.1. Ultrasonic method The ultrasonic device consists in an immersion tank associated to a tensile machine [7]. It allows to study the complete stiffness tensor variation under load thus it is possible to know which coefficients are affected during the damage process. Wave speed measurements are performed by using ultrasonic pulses which are refracted through the sample immersed in water. The measurement of the phase velocity of the pulses is done by a signal processing method using Hilbert transform [9]. To fully describe the elastic behaviour of an orthotropic material, the nine elastic constants Cij are identi- fied by measuring the phase velocities of bulk waves transiting in two accessible principal planes [planes (1, 2) and (1, 3), Fig. 3] and in a non principal plane [plane (1, 45] described by the bisectrix of axis 2 and 3) of the sample[10]. The identification in plane (1, 2) gives four elastic constants; C11, C22, C66 and C12 and three others are obtained in plane (1, 3): C33, C55 and C13. The two remaining coefficients C23 and C44 are identified by propagation in the non-principal plane (1, 45). The confidence interval associated to each identified constant is then estimated by a statistical analysis [11]. 2.2. Under load strain partition Unloading–reloading cycles are usually carried out to measure the plastic strains. In the plasticity of metals, dislocations are the main flaws. They lead to relative discompositions that remain stable after complete unloading, and which found expression, from a macroscopic point of view, in permanent strains. The introduction of the term ‘‘inelastic’’ is lead by a possible partrecovery of the strains beyond the yield point. It is all the more justified for brittle matrix composites. The matrix microcracking does not modify a pre-existing elastoplastic behaviour but creates residual strains that could not exist otherwise. The inelastic strains in CMCs are the macroscopic indication of the transverse crack opening displacement (COD) due to interfacial fibrematrix sliding [12]. This sliding is subordinate to the direction of the load and thus partly reversible. A Fig. 1. Damage mechanisms in CMCs. Fig. 2. Consequence of matrix cracking and fibre-matrix sliding on the stress/strain behaviour. Fig. 3. Sample instrumented for strain partition under load in the ultrasonic device used to investigate the stress-induced development of the damage of CMCs. 2286 S. Baste / Composites Science and Technology 61 (2001) 2285–2297
S. Baste / Composites Science and Technology 61(2001)2285-2297 decrease of the stress can create a sliding opposite to the elastic strains are obtained from the generalised one created by an increase in stress. Therefore, the Hooke's law: strain measured when the specimen is completely unloaded only represents the permanent part of the ge=(Ci)o;=Si; a j inelastic strain. The inelastic part of the strain is under- estimated by the classic measurements using unloading- When the total strain is measured simultaneously with reloading cycles the ultrasonic evaluation, it becomes possible to know The matrix microcracking induces an important loss precisely which part of the global strain is either elastic of the elastic modulus. It is usually [13] measured using or inelastic. As the extensometer indicates the total the slopes of unloading-reloading cycles. This measure- strain(Fig 4)and since the variation of the elastic strain ment soon becomes inaccurate as the fibre-matrix sliding is given by Eq (1)with the variation of Cii, the inelastic is accompanied by a frictional effect in the crack wake. strain is then simply obtained [8] Unload-reload curves turns into hysteresis [1](Fig. 4) The hysteresis tangent is subordinate to both the elastic modulus drop and the reversible inelastic strains. The apparent strain soon includes both the elastic part and the inelastic part, as the sliding threshold is equal or very close to the stress reached before unloading(or to 3. Constitutive laws the minimal stress reached before reloading). In another way, measuring the elastic modulus from the slopes of The failure mechanisms favour the generation of unloading-reloading cycles can only be correct if the microcracks oriented normal to the tensile stress. The sliding occurs for a stress that is different enough from microcracks tend to propagate rapidly inside the brittle the one reached at the loading or unloading point. matrix across the entire width of fibre spacing. This Therefore, the choice of the origin of a hysteresis tangent width is thus quasi-immediately the length of each is a particularly hazardous task. It is all the more crack. The damage process is then the multiplication of hazardous when unloading-reloading cycles induce matrix cracks that propagate perpendicular to the tensile some closing-opening effects on the microcracks as direction(Fig. 5). These cracks appear to be homo- pointed out by the changes of slope in the hysteresis [14]. geneously distributed. The number of cracks does not Consequently, the usual identification underestimates change continuously but may increase step by step the inelastic strains and overestimates the elastic modulus However, this increment is very small and the density of drop because the reversible part of the inelastic strains the matrix microcracks can be assumed to be continuous can not be separated from the elastic strain To separate The chosen damage variable is the crack density B asso- and to identify the two damage mechanisms responsible ciated with the parameters needed to define the geo- for the non-linear behaviour of CMCs, it is necessary to metry of a cracks system with a given orientation make a strain partition under load measures the average distance between cracks The complete determination of the compliance tensor The inelastic strain along the tensile axis represents a together with its variation during a tensile test allows to large part of the total strain in CMCs. It is now commonly know the elastic part of the material behaviour. The recognised that they have their source in the sum of the -2/2b=u ( Crack Op 2aβ Apparent Modulus latrix Crack Elastic slope sfic inelastic atrIx Fig. 4. Strain partitio Fig. 5. Unit cell of a cracked body
decrease of the stress can create a sliding opposite to the one created by an increase in stress. Therefore, the strain measured when the specimen is completely unloaded only represents the permanent part of the inelastic strain. The inelastic part of the strain is underestimated by the classic measurements using unloading– reloading cycles. The matrix microcracking induces an important loss of the elastic modulus. It is usually [13] measured using the slopes of unloading-reloading cycles. This measurement soon becomes inaccurate as the fibre-matrix sliding is accompanied by a frictional effect in the crack wake. Unload-reload curves turns into hysteresis [1] (Fig. 4). The hysteresis tangent is subordinate to both the elastic modulus drop and the reversible inelastic strains. The apparent strain soon includes both the elastic part and the inelastic part, as the sliding threshold is equal or very close to the stress reached before unloading (or to the minimal stress reached before reloading). In another way, measuring the elastic modulus from the slopes of unloading–reloading cycles can only be correct if the sliding occurs for a stress that is different enough from the one reached at the loading or unloading point. Therefore, the choice of the origin of a hysteresis tangent is a particularly hazardous task. It is all the more hazardous when unloading–reloading cycles induce some closing–opening effects on the microcracks as pointed out by the changes of slope in the hysteresis [14]. Consequently, the usual identification underestimates the inelastic strains and overestimates the elastic modulus drop because the reversible part of the inelastic strains can not be separated from the elastic strain. To separate and to identify the two damage mechanisms responsible for the non-linear behaviour of CMCs, it is necessary to make a strain partition under load. The complete determination of the compliance tensor together with its variation during a tensile test allows to know the elastic part of the material behaviour. The elastic strains are obtained from the generalised Hooke’s law: "e ¼ Cij 1 j ¼ Sij j: ð1Þ When the total strain is measured simultaneously with the ultrasonic evaluation, it becomes possible to know precisely which part of the global strain is either elastic or inelastic. As the extensometer indicates the total strain (Fig. 4) and since the variation of the elastic strain is given by Eq. (1) with the variation of Cij, the inelastic strain is then simply obtained [8]: "in ¼ " "e : ð2Þ 3. Constitutive laws The failure mechanisms favour the generation of microcracks oriented normal to the tensile stress. The microcracks tend to propagate rapidly inside the brittle matrix across the entire width of fibre spacing. This width is thus quasi-immediately the length of each crack. The damage process is then the multiplication of matrix cracks that propagate perpendicular to the tensile direction (Fig. 5). These cracks appear to be homogeneously distributed. The number of cracks does not change continuously but may increase step by step. However, this increment is very small and the density of the matrix microcracks can be assumed to be continuous. The chosen damage variable is the crack density associated with the parameters needed to define the geometry of a cracks system with a given orientation. It measures the average distance between cracks. The inelastic strain along the tensile axis represents a large part of the total strain in CMCs. It is now commonly recognised that they have their source in the sum of the Fig. 4. Strain partition under load. Fig. 5. Unit cell of a cracked body. S. Baste / Composites Science and Technology 61 (2001) 2285–2297 2287
S Baste/ Composites Science and Technology 61(2001)2285-2297 crack opening displacements of the transverse crack x2 system,U(Fig. 6)[12]. The distance between two crack a++≤, is related to the crack density. So, in an extensometer length L, the number n of cracks, 2a deep and 2b thick, is: with stiffness Cr and compliance Sr, which is embedded in an infinite homogeneous solid whose stiffness and n=L (3) compliance tensors are, respectively, Cand S.The material is loaded by uniform stress, o, or subjected to uniform strain, a, at infinity. Let the stress and strain The relationship between the inelastic strain and the fields in the inclusion be o and Er, respectively, so that crack opening displacement is then [15]: σr=CrEr,Er=S10r It is well known that the elastic field in the ellipsoidal inclusion is uniform [17] and can be evaluated as Thus, the inelastic strain is simply the density of transverse matrix microcracks multiplied by their aspect σr=B10,Er=AE The extension of the cracks is limited by the waviness Ar and Br are the crack localisation tensors of the r of the bundles and experimental observation of inelastic inclusion strains implies that crack opening displacement is not negligible. Therefore, it is necessary ider 3D Br=[+Q(S-S),A1=[-P(Cr-C)-.(8) defined cracks in order to evaluate the effective stiffness tensor of the damaged material [16]. The cracks are The solution of this problem requires the determina modelling by ellipsoidal voids. Their volume concentration tion of the tensor Q defined by: is defined through a unit cell; it represents the largest volume of material containing a single crack 0=C-CPC By using a homogenisation method that provides the elationship between the effective stiffness tensor andand the determination of the tensor P whose compo- the intensity of damage in the individual modes, it is nents are given by [18] possible to relate the micro- and macro-level damage measurement. The cracked material is substituted by an Pal- 4 Jo(a-a?+c202+bag>d2 abc Dik(o (10) equivalent homogeneous medium. Effects of damage are then described by the changes of the effective properties of the equivalent medium. where $2 is the surface of the unit sphere centred at the Cracks are consider as an ellipsoidal inclusion origin of(ol, (2, a3)space. The fourth order tensor D is defined by Dijk=o)@)gik with gik=[Cmnpa@n@g Jik (11) Turning now to the basic equations for composites, we note that in order for the concept of overall moduli Z2a02ab-=u Displacements, to be meaningful, it is necessary to consider macro- scopically uniform loading [19]. In such a case, the lal to the phase average stresses and strains are related to the o= bro and E=a e 2a/阝 Let c. denote the volume concentration of the rth ∑ Fig. 6. Inelastic strains; a macroscopic consequence of the micro. it follows[20] that the overall stifness C and compliance
crack opening displacements of the transverse crack system, U (Fig. 6) [12]. The distance between two cracks is related to the crack density. So, in an extensometer length L, the number n of cracks, 2a deep and 2b thick, is: n ¼ L 2a : ð3Þ The relationship between the inelastic strain and the crack opening displacement is then [15]: "in ¼ Lin L ¼ n2U L ¼ n2b L ¼ : ð4Þ Thus, the inelastic strain is simply the density of transverse matrix microcracks multiplied by their aspect ratio =b/c. The extension of the cracks is limited by the waviness of the bundles and experimental observation of inelastic strains implies that crack opening displacement is not negligible. Therefore, it is necessary to consider 3Ddefined cracks in order to evaluate the effective stiffness tensor of the damaged material [16]. The cracks are modelling by ellipsoidal voids. Their volume concentration is defined through a unit cell; it represents the largest volume of material containing a single crack. By using a homogenisation method that provides the relationship between the effective stiffness tensor and the intensity of damage in the individual modes, it is possible to relate the micro- and macro-level damage measurement. The cracked material is substituted by an equivalent homogeneous medium. Effects of damage are then described by the changes of the effective properties of the equivalent medium. Cracks are consider as an ellipsoidal inclusion: x2 1 a2 þ x2 2 c2 þ x2 3 b2 41; ð5Þ with stiffness Cr and compliance Sr, which is embedded in an infinite homogeneous solid whose stiffness and compliance tensors are, respectively, C and S. The material is loaded by uniform stress, ; or subjected to uniform strain, " ; at infinity. Let the stress and strain fields in the inclusion be r and r, respectively, so that: r ¼ Cr"r; "r ¼ Srr: ð6Þ It is well known that the elastic field in the ellipsoidal inclusion is uniform [17] and can be evaluated as r ¼ Br ; "r ¼ Ar" ð7Þ Ar and Br are the crack localisation tensors of the r inclusion: Br ¼ ½ I þ Q Sð Þ r S 1 ; Ar ¼ ½ I P Cð Þ r C 1 : ð8Þ The solution of this problem requires the determination of the tensor Q defined by: Q ¼ C CPC ð9Þ and the determination of the tensor P whose components are given by [18]: Pijkl ¼ abc 4 ð O Dijklð Þ !n a2!2 1 þ c2!2 2 þ b2!2 3 3=2 d ð10Þ where is the surface of the unit sphere centred at the origin of (!1, !2, !3) space. The fourth order tensor D is defined by: Dijkl ¼ !l!jgik with gik ¼ Cmnpq!n!q 1 ik : ð11Þ Turning now to the basic equations for composites, we note that in order for the concept of overall moduli to be meaningful, it is necessary to consider macroscopically uniform loading [19]. In such a case, the applied stress is equal to the average stress, , and the phase average stresses and strains are related to the overall averages through r ¼ Br and " r ¼ Ar" : ð12Þ Let cr denote the volume concentration of the rth phase. Since X r cr ¼ 1; ¼ X r cr r; " ¼ X r cr" r; ð13Þ it follows [20] that the overall stiffness C and compliance S are given by: Fig. 6. Inelastic strains; a macroscopic consequence of the microscopic crack opening displacement. 2288 S. Baste / Composites Science and Technology 61 (2001) 2285–2297
S. Baste / Composites Science and Technology 61(2001)2285-2297 cm=∑c4 and Sem=∑SB (14) depends on the both the geometry of the crack and the elastic properties of material surrounded the crack here. the initial non cracked material The effective medium is considered as a two-phase medium [21]. Let phase I be the uncracked fibre rein- forced composite material. Phase 2 is a set of ellipsoids 4. Several crack arrays consisting of voids. The overall stifness tensor for the two-phase medium follows from Eq (14) 4..D Sic-Sic Cet=C-Cr C(C-Cr)Ar, Sem =S+crS- Sr)B The 3D representation of the cracks was applied to multiple arrays microcracking in a unidirectional Sic. (15) Micrographs(Fig 8)and experimental stiffness tensor where f index is for ellipsoidal voids. The crack locali- changes [23](Fig. 12)allow us to identified three arrays sation tensors are of microcracks: a transverse microcracks. which is topped and deviated in longitudinal cracks at the fibre Ar=(l-PC) and Br=-o (16) matrix interface. Those longitudinal arrays are growing cracks along the interface(Fig 9) and the effective elasticity tensor is given by A succ cessive process of prediction experimental data confrontation allows the optimal determination [24] of Ceff=C-cr C(I-PC) and Sefr=S+cr2-(17) the fixed sizes of the multiplication of the transverse cracks and of evolution laws of the cracks density, of with ce the volume concentration of the cracks the cracks thickness aspect ratio(Fig. 10) and of the semi-axes of the growing longitudinal cracks(Fig. 11 Fig. 12 plots the changes of the nine stiffnesses iden- 32x12x2x3 tified from the phase velocities as a function of tensile stress for the ID SiC-SiC. There is a good agreement where 2x are the average distances between two cracks in the i direction. They give the unit cell of the cracked material(Fig. 7) P and Q tensors appearing in Eq(9)depend upon the shape of the considered inclusion and the stiffness of the effective medium C. That leads to the self-consistent scheme. Here, we replace C with Co, the stiffness tensor of the uncracked material, in Eqs. (9). (16) and (17) This method, similar to the mori-Tanaka method [22] requires less calculations and gives a good approxima ion of the effective stiffness tensor for reasonable ig. 8. Micrograph of the transverse cracks in the ID Sic-Sic volume concentration of cracks [16] Eq.(17)is the equation used to evaluate the effe stiffness tensor for an anisotropic medium permeated by ellipsoidal cracks. It requires the determination of the tensor Q and P by a numerical evaluation of Eq. (10) [16]. P, the symmetrized derivative of the Green's tensor, fibre transverse crack Fig. 7. Unit cell of a cracked body Fig 9. Unit cell of the cracked ID SiC-SiC
Ceff ¼ X r crCrAr and Seff ¼ X r crSrBr: ð14Þ The effective medium is considered as a two-phase medium [21]. Let phase 1 be the uncracked fibre reinforced composite material. Phase 2 is a set of ellipsoids consisting of voids. The overall stiffness tensor for the two-phase medium follows from Eq. (14): Ceff ¼ C cfC Cð Þ Cf Af ; Seff ¼ S þ cf ð Þ S Sf Bf ð15Þ where f index is for ellipsoidal voids. The crack localisation tensors are Af ¼ ð Þ I PC 1 and Bf ¼ Q1 ; ð16Þ and the effective elasticity tensor is given by Ceff ¼ C cfCðI PCÞ 1 and Seff ¼ S þ cfQ1 ð17Þ with cf the volume concentration of the cracks: cf ¼ 4 3 abc 2x12x22x3 ð18Þ where 2xi are the average distances between two cracks in the i direction. They give the unit cell of the cracked material (Fig. 7). P and Q tensors appearing in Eq. (9) depend upon the shape of the considered inclusion and the stiffness of the effective medium C. That leads to the self-consistent scheme. Here, we replace C with C0, the stiffness tensor of the uncracked material, in Eqs. (9), (16) and (17). This method, similar to the Mori–Tanaka method [22], requires less calculations and gives a good approximation of the effective stiffness tensor for reasonable volume concentration of cracks [16]. Eq. (17) is the equation used to evaluate the effective stiffness tensor for an anisotropic medium permeated by ellipsoidal cracks. It requires the determination of the tensor Q and P by a numerical evaluation of Eq. (10) [16]. P, the symmetrized derivative of the Green’s tensor, depends on the both the geometry of the crack and the elastic properties of material surrounded the crack: here, the initial non cracked material. 4. Several crack arrays 4.1. 1D SiC–SiC The 3D representation of the cracks was applied to a multiple arrays microcracking in a unidirectional SiC– SiC. Micrographs (Fig. 8) and experimental stiffness tensor changes [23] (Fig. 12) allow us to identified three arrays of microcracks: a transverse microcracks, which is stopped and deviated in longitudinal cracks at the fibre matrix interface. Those longitudinal arrays are growing cracks along the interface (Fig. 9). A successive process of prediction experimental data confrontation allows the optimal determination [24] of the fixed sizes of the multiplication of the transverse cracks and of evolution laws of the cracks density, of the cracks thickness aspect ratio (Fig. 10) and of the semi-axes of the growing longitudinal cracks (Fig. 11). Fig. 12 plots the changes of the nine stiffnesses identified from the phase velocities as a function of tensile stress for the 1D SiC–SiC. There is a good agreement Fig. 7. Unit cell of a cracked body. Fig. 9. Unit cell of the cracked 1D SiC–SiC. Fig. 8. Micrograph of the transverse cracks in the 1D SiC–SiC. S. Baste / Composites Science and Technology 61 (2001) 2285–2297 2289
S. Baste/ Composites Science and Technology 61(2001)2285-2297 between experimental values and the prediction, Eq (17), plotted in straight lines [24]. 4. 2. 2DC-Sic Let us consider a woven 2D C-SiC. Micrographs [25] Fig. 13)and ultrasonic evaluation of the stifness tensor hanges(Fig. 18)allow us to identified three arrays of microcracks. The damage starts with a transverse cracking localised in the matrix mantle that surround the longitudinal bundle(Fig. 14). The waviness of the bundles stops the matrix cracking, which is deviated in mode Il by the transverse bundles. Then, longitudinal cracking arrays normal to direction 2 or I are created The successive process of prediction/experimental data confrontation allows the optimal determination of the evolution laws of the cracks density, of the cracks fad trim 2u 3-0 T 0, *30情“ Stress [PAl thickness aspect ratio(Fig. 16)and of the volume con centration of the three cracks arrays(Fig. 17)[24] Fig. 12. Stiffness changes (in GPa) during a tensile test in fibres Fig. 18 plots the experimental changes of the nine direction 3. ID SiC-S stiffnesses identified from the phase velocities as a func tion of tensile stress and the prediction, Eq.(17), plotted in straight lines Confrontation of predictions and experimental data has also been done for the strains in this composite Experimentally, an extensometer gives us the total strain(Fig. 19). The elastic strain is computed from the elasticity tensor(Fig. 18). The difference between the elastic strain and the total strain is the inelastic strain Fig. 13. Micrographs of the three cracks arrays in a 2D C-Sic ( Fig. 19). Prediction of the effective elastic properties gives us the predicted elastic strain, Eq (1). The inelastic 8=bc3 0.00075 ) 0.000 Longitudinal crack Fig.10. Variation of the crack density parameter and of the thickness Fig. 14. Schematic description of fthe three cracks arrays in a long. pect ratio for the transverse matrix microcracking as a func- itudinal bundle of a 2DC-SiC applied stress. 2x3=4.2c3 transverse crack 2x1=120u c,(um) 200300 normal to x? normal to xI Stress(MPa) L. Variation of the semi-axes of the growing longitudinal cracks as a function of applied stress. Fig. 15. Unit cell of the cracked 2D C-Sic
between experimental values and the prediction, Eq. (17), plotted in straight lines [24]. 4.2. 2D C–SiC Let us consider a woven 2D C–SiC. Micrographs [25] (Fig. 13) and ultrasonic evaluation of the stiffness tensor changes (Fig. 18) allow us to identified three arrays of microcracks. The damage starts with a transverse cracking localised in the matrix mantle that surrounds the longitudinal bundle (Fig. 14). The waviness of the bundles stops the matrix cracking, which is deviated in mode II by the transverse bundles. Then, longitudinal cracking arrays normal to direction 2 or 1 are created (Fig. 15). The successive process of prediction/experimentaldata confrontation allows the optimal determination of the evolution laws of the cracks density, of the cracks thickness aspect ratio (Fig. 16) and of the volume concentration of the three cracks arrays (Fig. 17) [24]. Fig. 18 plots the experimental changes of the nine stiffnesses identified from the phase velocities as a function of tensile stress and the prediction, Eq. (17), plotted in straight lines. Confrontation of predictions and experimental data has also been done for the strains in this composite. Experimentally, an extensometer gives us the total strain (Fig. 19). The elastic strain is computed from the elasticity tensor (Fig. 18). The difference between the elastic strain and the total strain is the inelastic strain (Fig. 19). Prediction of the effective elastic properties gives us the predicted elastic strain, Eq. (1). The inelastic Fig. 10. Variation of the crack density parameter and of the thickness crack aspect ratio for the transverse matrix microcracking as a function of applied stress. Fig. 11. Variation of the semi-axes of the growing longitudinal cracks as a function of applied stress. Fig. 12. Stiffness changes (in GPa) during a tensile test in fibres direction 3, 1D SiC–SiC. Fig. 13. Micrographs of the three cracks arrays in a 2D C–SiC. Fig. 14. Schematic description of fthe three cracks arrays in a longitudinal bundle of a 2D C–SiC. Fig. 15. Unit cell of the cracked 2D C–SiC. 2290 S. Baste / Composites Science and Technology 61 (2001) 2285–2297
S. Baste / Composites Science and Technology 61(2001)2285-2297 strain is simply the effective density of transverse matrix B=FB microcracks multiplied by their aspect ratio, Eq.(4). Then, the predicted total strain is the sum of the pre- where B is the density of cracks that have been accumu- dicted elastic and inelastic parts of the strain. The com- lated during monotonic loading. The opening-closure parison between the experimental strains and their variable F is the proportion of cracks that remains open predictions(Fig 19), allows us to validate the choice of during unloading and thus represents the damage deac the evolution laws of the two internal variables, B and 8, tivation and, so, of the description of this micro level damage mechanism [24] 5. Cyclic loading When cyo clic loading is performed, as the stress decreases during unloading, the transverse cracks that have been created during the monotonic loading are prone to close(Fig. 20). They are still present anyway but their closure simulates an increase of stiffness as if the material could recover its mechanical properties The cracks that remain open represent the active part of the microcracking. An apparent state of damage seems to decrease while the damage accumulated is greater. The closing effect induces damage deactivation leading to unilateral behaviour [26-28]. The internal variable is then representative of a damage that can be qualified as apparent damage [14]. It is necessary to make a dif- ference between the apparent state of cracking and the number of cracks that has been created effectively. Just as the active cracks were defined we can define an active Fig. 18. Stiffness changes (in GPa) during a tensile test in fibres or apparent crack density direction 3. 2D C-SiC inelastic elastic [B-4.2,2x,/ 300 080160240320 b如01624x Stress(MPa) 00.10.20.30.40.50.60.70.8 Fig. 16. Variation of the crack density parameter and of the thickness Strain(%) crack aspect ratio for the transverse matrix microcracking as a func- Fig. 19. Variation of the total strain, of the elastic strain and of the tion of applied stress inelastic strain as a function of the applied stress. 4104 Debonding crack Transverse Matrix Crau 210+ t 2u (Crack Ope 200300400 Stress(MPa) Fig. 17. Variation of the volume concentration of the three cracks laI nnen Fig. 20. States of the transverse cracks
strain is simply the effective density of transverse matrix microcracks multiplied by their aspect ratio, Eq. (4). Then, the predicted total strain is the sum of the predicted elastic and inelastic parts of the strain. The comparison between the experimental strains and their predictions (Fig. 19), allows us to validate the choice of the evolution laws of the two internal variables, and , and, so, of the description of this micro level damage mechanism [24]. 5. Cyclic loading When cyclic loading is performed, as the stress decreases during unloading, the transverse cracks that have been created during the monotonic loading are prone to close (Fig. 20). They are still present anyway but their closure simulates an increase of stiffness as if the material could recover its mechanical properties. The cracks that remain open represent the active part of the microcracking. An apparent state of damage seems to decrease while the damage accumulated is greater. The closing effect induces damage deactivation leading to unilateral behaviour [26–28]. The internal variable b is then representative of a damage that can be qualified as apparent damage [14]. It is necessary to make a difference between the apparent state of cracking and the number of cracks that has been created effectively. Just as the active cracks were defined, we can define an active or apparent crack density: ¼ F ð19Þ where is the density of cracks that have been accumulated during monotonic loading. The opening-closure variable F is the proportion of cracks that remains open during unloading and thus represents the damage deactivation. Fig. 16. Variation of the crack density parameter and of the thickness crack aspect ratio for the transverse matrix microcracking as a function of applied stress. Fig. 17. Variation of the volume concentration of the three cracks arrays. Fig. 18. Stiffness changes (in GPa) during a tensile test in fibres direction 3, 2D C–SiC. Fig. 19. Variation of the total strain, of the elastic strain and of the inelastic strain as a function of the applied stress. Fig. 20. States of the transverse cracks. S. Baste / Composites Science and Technology 61 (2001) 2285–2297 2291
S. Baste/Composites Science and Technology 61(2001)2285-2297 Under cyclic tensile loading, a 2D carbon SiC com- active again. The compliances reach the value they had posite exhibits quite immediately a non-linear behaviour before unloading [15] ( Fig. 21). The cycles do not show any hysteresis but Fig. 23 shows the relative part of the elastic and nevertheless the residual strains are far from negligible inelastic strains on the total strain. As an example, the The lack of hysteresis let think that the sliding occurs cycle at 280 MPa has been chosen. The linear variation with little or no friction at the fibre matrix interface [15]. of the total strain is the sum of both the non-linear S33, S44 and Sss modified by the presence of a trans- variations of the elastic and inelastic strains. The elastic verse cracks system are the only compliances to exhibit strain naturally comes back to zero with the stress a variation during the cycles(Fig. 22). Their progressive whereas the inelastic strain is still present as a function decrease during the unloading cycles is relevant of of the transverse cracks which is still active [15] progressive closure of the cracks. As some cracks close, The successive process of prediction experimental they are active no more and they lose the effect they had data confrontation allows us the optimal determination on the compliances. When the sample is reloaded, all of the evolution laws of the cracks density and the le cracks that have been created re open and become opening closure variable (Fig. 24). Fig. 22 plots the experimental changes of the three most influenced com pliances under cyclic loading and the prediction plotted in straight lines. According to Eqs.(17)and(19), the behaviour during the cycles is fully described by the variables: B,8 and f that predict the compliances var- iations(Fig. 22). During the loading/unloading cycles, only the transverse crack system has an influence on the behaviour due to the opening-closure of the cracks [15] During unloading. it is obvious that the effect of closure of the cracks must be taken into account. The proportion Total Strain (% of transverse cracks which is still active under cyclic 040.5060.708 loading is described with the opening-closure function Fig. 21. Stress-strain curve of a 2D C-SiC under cyclic loading F. Eq(4)becomes ein=8.FB Furthermore, some cracks will probably not com- pletely close and have a residual opening. This opening can be due to the roughness of the cracks edges coming from both wear debris at the sliding interface [29] and inelastic Fig. 23. Under load strain partition during cyclic loading at 280 MPa of a 2D C-Sic 删 Fig 22. The most influenced compliances of a 2D C-SiC under cyclic Fig. 24. Variation of the crack density parameter and of the opening- loadin closure function of a 2D C-sic
Under cyclic tensile loading, a 2D carbon SiC composite exhibits quite immediately a non-linear behaviour (Fig. 21). The cycles do not show any hysteresis but nevertheless the residual strains are far from negligible. The lack of hysteresis let think that the sliding occurs with little or no friction at the fibre matrix interface [15]. S33, S44 and S55 modified by the presence of a transverse cracks system are the only compliances to exhibit a variation during the cycles (Fig. 22). Their progressive decrease during the unloading cycles is relevant of a progressive closure of the cracks. As some cracks close, they are active no more and they lose the effect they had on the compliances. When the sample is reloaded, all the cracks that have been created re open and become active again. The compliances reach the value they had before unloading [15]. Fig. 23 shows the relative part of the elastic and inelastic strains on the total strain. As an example, the cycle at 280 MPa has been chosen. The linear variation of the total strain is the sum of both the non-linear variations of the elastic and inelastic strains. The elastic strain naturally comes back to zero with the stress whereas the inelastic strain is still present as a function of the transverse cracks which is still active [15]. The successive process of prediction experimental data confrontation allows us the optimal determination of the evolution laws of the cracks density and the opening closure variable (Fig. 24). Fig. 22 plots the experimental changes of the three most influenced compliances under cyclic loading and the prediction plotted in straight lines. According to Eqs. (17) and (19), the behaviour during the cycles is fully described by the variables: , and F that predict the compliances variations (Fig. 22). During the loading/unloading cycles, only the transverse crack system has an influence on the behaviour due to the opening-closure of the cracks [15]. During unloading, it is obvious that the effect of closure of the cracks must be taken into account. The proportion of transverse cracks which is still active under cyclic loading is described with the opening-closure function F. Eq. (4) becomes: "in ¼ F ð20Þ Furthermore, some cracks will probably not completely close and have a residual opening. This opening can be due to the roughness of the cracks edges coming from both wear debris at the sliding interface [29] and Fig. 22. The most influenced compliances of a 2D C–SiC under cyclic loading. Fig. 23. Under load strain partition during cyclic loading at 280 MPa of a 2D C–SiC. Fig. 24. Variation of the crack density parameter and of the openingclosure function of a 2D C–SiC. Fig. 21. Stress-strain curve of a 2D C–SiC under cyclic loading. 2292 S. Baste / Composites Science and Technology 61 (2001) 2285–2297
S. Baste/ Composites Science and Technology 61(2001)2285-2297 2293 wedging of the cracks by grain bridging [301. To main damage mode and induces a fully anisotropic describe this later fact, another aspect ratio 8 that takes degradation. In particular, the coupling stiffnesses C34, the residual opening into account is introduced [15]. C24, C14 and C56, which are naturally about zero at 0 Finally, the variations of inelastic strains during the MPa, become non-negligible at 40 MPa. After 60 MPa whole test can be described by [15] they decrease and recover their initial value of about zero. It coincides with the increase of inelastic strain e=B(F(8-8)+8) (21) The fibrous reinforcement stops and deviates the The inelastic strains are thus a function of the transverse matrix microcracking in mode Il Sliding occurs in the crack density and of the aspect ratio of the cracks whether fibre-matrix interphase and leads to other cracking modes they are completely open or not(Fig. 25) consisting in slit cracks whose orientations coincide with The predictions of the three dimensional changes in elasticity and of the inelastic strains under cyclic loading are shown to compare favourably with experimental data. While the crack density describes the inelastic strains and the drop in elastic modulus, the opening closure variable modulates these effects when cyclic ading is applied [ 15]- 6. Effective elastic stiffnesses of an anisotropic medium permeated by tilted cracks In composite materials, failure mechanisms favour the Fig. 26. The 300 off-axis solicitation sample, cut out according to generation of microcracks oriented normally to the tensile 300 angle from fibre axes, and loading in this direction. stress [1]. An off-axis tensile loading creates microcracks whose orientation does not coincide with the fibre axes [31, 32] and induces a fully anisotropic elastic degrad tion. To emphasise the induced anisotropy and the loss of elastic symmetry caused by ofi-principal solicitations, the measurement of the changes of all the stiffness tensor components has been done [33] for a 2D C-C-SiC composite material, subjected to a tensile solicitation at 30 from one of the fibre directions(Fig. 26). The load induced changes of the thirteen stiffnesses. associated 8x, with a monoclinic symmetry, have been recovered from ultrasonic velocity data [33] Damage-induced changes of the thirteen stiffnesses (Cii identified in the geometric coordinate system of the sample are plotted in Fig. 27 as a function of the applied b行 tensile stress [33]. The loss of stiffness along the tensile axis x3 is very important and occurs from the first stress levels. The variation of the experimental stiffnesses C33 Ca and Css leads us to consider that the matrix micro- cracking oriented normally to the tensile loading is the 灿出江 Applied Stress (MPay experinental vahcs with thcir 90 s confidence interval prediction af the stiffness changes Inelastic Strain (%) Fig. 25. Variation of the inelastic strains during cyclic loading of a 2D Fig. 27. Variation of the stiffness tensor as a function of a tensile C-SiC stress applied within direction
wedging of the cracks by grain bridging [30]. To describe this later fact, another aspect ratio d’ that takes the residual opening into account is introduced [15]. Finally, the variations of inelastic strains during the whole test can be described by [15]: "in ¼ ðFð 0 Þ þ 0 Þ ð21Þ The inelastic strains are thus a function of the transverse crack density and of the aspect ratio of the cracks whether they are completely open or not (Fig. 25). The predictions of the three dimensional changes in elasticity and of the inelastic strains under cyclic loading are shown to compare favourably with experimental data. While the crack density describes the inelastic strains and the drop in elastic modulus, the opening closure variable modulates these effects when cyclic loading is applied [15]. 6. Effective elastic stiffnesses of an anisotropic medium permeated by tilted cracks In composite materials, failure mechanisms favour the generation of microcracks oriented normally to the tensile stress [1]. An off-axis tensile loading creates microcracks whose orientation does not coincide with the fibre axes [31,32] and induces a fully anisotropic elastic degradation. To emphasise the induced anisotropy and the loss of elastic symmetry caused by off-principal solicitations, the measurement of the changes of all the stiffness tensor components has been done [33] for a 2D C–C–SiC composite material, subjected to a tensile solicitation at 30 from one of the fibre directions (Fig. 26). The loadinduced changes of the thirteen stiffnesses, associated with a monoclinic symmetry, have been recovered from ultrasonic velocity data [33]. Damage-induced changes of the thirteen stiffnesses (Cij) identified in the geometric coordinate system of the sample are plotted in Fig. 27 as a function of the applied tensile stress [33]. The loss of stiffness along the tensile axis x3 is very important and occurs from the first stress levels. The variation of the experimental stiffnesses C33, C44 and C55 leads us to consider that the matrix microcracking oriented normally to the tensile loading is the main damage mode and induces a fully anisotropic degradation. In particular, the coupling stiffnesses C34, C24, C14 and C56, which are naturally about zero at 0 MPa, become non-negligible at 40 MPa. After 60 MPa, they decrease and recover their initial value of about zero. It coincides with the increase of inelastic strain (Fig. 29). The fibrous reinforcement stops and deviates the matrix microcracking in mode II. Sliding occurs in the fibre-matrix interphase and leads to other cracking modes consisting in slit cracks whose orientations coincide with Fig. 26. The 30 off-axis solicitation sample, cut out according to a 30 angle from fibre axes, and loading in this direction. Fig. 25. Variation of the inelastic strains during cyclic loading of a 2D C–SiC. Fig. 27. Variation of the stiffness tensor as a function of a tensile stress applied within direction 3. S. Baste / Composites Science and Technology 61 (2001) 2285–2297 2293
S. Baste/Composites Science and Technology 61(2001)2285-2297 the directions of the fibres(Fig. 28)34 The experimental no longer orthotropic but monoclinic when tilted cracks changes of the stiffnesses Cll, C22, C66 and Css point are created. Therefore, the prediction of the effective out the generation of these cracking modes [33] stifness tensor requires the evaluation of 13 Pik in The under load strain partitio [8] in the direction x3 order to study the damage process[34] with respect to the applied stress is plotted in Fig. 29 In the 2D C/C-Sic, the damage starts with the crea- The part of inelastic strain increases with the stress. As tion of cracks oriented normally to the stress direction it is the macroscopic result of the crack opening dis-(Fig. 30). As the governing equations lie to the effective placement [12], the inelastic strain is associated with an stiffness tensor in the axes of the ellipsoidal crack, we increase in the thickness of the matrix microcracks have to consider the components of C in the principal To describe the damage on the 2D C/C-Sic by coordinate system R=(x1, x2, x3) of the ellipsoid establishing the relationship between the effective stiff The 2D C/C-Sic is orthotropic in the material prin ness tensor and the intensity of damage the matrix cipal coordinate system R=(, x2, x3) microcracking oriented normally to the tensile stress requires particular attention to fully understand the CC12C13000 experimental changes observed during the load [34). The material microstructure and the increase of inelastic strain lead us to consider 3D-defined tilted cracks for Ci(R)= C33000 C4400 (22) the evaluation of effective stiffness changes due to this cracking mode The tensile solicitation along a non-principal direction creates a matrix microcracking normal to the loading The stiffness tensor becomes in the coordinate system direction. The extension of the cracks is limited by the R=(x1, x2, x3), by using the fourth rank tensor rotation waviness of the bundles and experimental observation law of inelastic strains implies that crack opening displace- ment is not negligible. Therefore, it is necessary to con- (C2)=(MF→8)(CM→8) sider 3D-defined cracks and to calculate the p-tensor for 3D-defined tilted cracks in order to evaluate the effec where M is the transformation matrix tive stifness tensor of the damaged material. As the classical formula of the P-tensor components is given in C C13C1400 the cracks frame, Eq. (10), we have to consider a more C2C23C2400 general symmetry for the solid. Indeed, the material is (C) C33C3400 Sym C4400 As the cracks principal axes coincide with the coordi nate system R, the material remains monoclinic during () Matenal axes tracks axes Rotation Ro→R Fig. 28. Damage modes taking place during the 30 off-axis solicit- tion:(a)matrix cracking normal to the loading direction, (b) deviation at the fibre-matrix interphase, (c)"zigzag"shape of the resulting microcracking Pu Fig. 29. Variation of the total strain of the elastic strain and of the inelastic strain as a function of applied stress. Fig. 30. Principal coordinate system of the matrix microcracks
the directions of the fibres (Fig. 28).34 The experimental changes of the stiffnesses C11, C22, C66 and C55 point out the generation of these cracking modes [33]. The under load strain partitio [8] in the direction x3 with respect to the applied stress is plotted in Fig. 29. The part of inelastic strain increases with the stress. As it is the macroscopic result of the crack opening displacement [12], the inelastic strain is associated with an increase in the thickness of the matrix microcracks. To describe the damage on the 2D C/C–SiC by establishing the relationship between the effective stiff- ness tensor and the intensity of damage, the matrix microcracking oriented normally to the tensile stress requires particular attention to fully understand the experimental changes observed during the load [34]. The material microstructure and the increase of inelastic strain lead us to consider 3D-defined tilted cracks for the evaluation of effective stiffness changes due to this cracking mode. The tensile solicitation along a non-principal direction creates a matrix microcracking normal to the loading direction. The extension of the cracks is limited by the waviness of the bundles and experimental observation of inelastic strains implies that crack opening displacement is not negligible. Therefore, it is necessary to consider 3D-defined cracks and to calculate the P-tensor for 3D-defined tilted cracks in order to evaluate the effective stiffness tensor of the damaged material. As the classical formula of the P-tensor components is given in the crack’s frame, Eq. (10), we have to consider a more general symmetry for the solid. Indeed, the material is no longer orthotropic but monoclinic when tilted cracks are created. Therefore, the prediction of the effective stiffness tensor requires the evaluation of 13 Pijkl in order to study the damage process [34]. In the 2D C/C–SiC, the damage starts with the creation of cracks oriented normally to the stress direction (Fig. 30). As the governing equations lie to the effective stiffness tensor in the axes of the ellipsoidal crack, we have to consider the components of C in the principal coordinate system R=(x1, x2, x3) of the ellipsoid. The 2D C/C–SiC is orthotropic in the material principal coordinate system Rf =(x1, x2 f , x3 f ): CijðRf Þ ¼ C11 C12 C13 000 C22 C23 000 C33 000 Sym C44 0 0 C55 0 C66 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 : ð22Þ The stiffness tensor becomes in the coordinate system R=(x1, x2, x3), by using the fourth rank tensor rotation law, ðCR ij Þ¼ðMRf ! R ij ÞðCRf ij ÞðMRf ! R ij Þ t ; ð23Þ where M is the transformation matrix: ðCR ij Þ ¼ C11 C12 C13 C14 0 0 C22 C23 C24 0 0 C33 C34 0 0 Sym C44 0 0 C55 C56 C66 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 : ð24Þ As the cracks principal axes coincide with the coordinate system R, the material remains monoclinic during Fig. 28. Damage modes taking place during the 30 off-axis solicitation: (a) matrix microcracking normal to the loading direction, (b) deviation at the fibre-matrix interphase, (c) ‘‘zigzag’’ shape of the resulting microcracking. Fig. 29. Variation of the total strain, of the elastic strain and of the inelastic strain as a function of applied stress. Fig. 30. Principal coordinate system of the matrix microcracks. 2294 S. Baste / Composites Science and Technology 61 (2001) 2285–2297