《复合材料 Composites》课程教学资源（学习资料）第五章 陶瓷基复合材料_fibrous molithic-51

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 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

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 ortho￾tropic 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 con￾stant 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 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 opening displacement (COD) due to interfacial fibre￾matrix 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 under￾estimated 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 measure￾ment 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 homo￾geneously 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  asso￾ciated with the parameters needed to define the geo￾metry 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 3D￾defined 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 determina￾tion of the tensor Q defined by: Q ¼ C CPC ð9Þ and the determination of the tensor P whose compo￾nents 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 macro￾scopically 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 crr; " 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 micro￾scopic 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 rein￾forced 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 locali￾sation 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 approxima￾tion 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 iden￾tified 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