《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_A micromechanical model for thermostructural composites

COMPOSITES SCIENCE AND TECHNOLOGY ELSEVIER Composites Science and Technology 60(2000)1095-1 102 A micromechanical model for thermostructural composites C Rospars,E Le Dantec, F. Lecuyer MEDYSYS, Mecanique et Dynamique des Systemes, 29 Rue Jean Rostand, F-91893 Orsay Cedex, france Received 23 June 1999: received in revised form 4 January 2000: accepted 13 January 2000 Abstract a damage model based on a micromechanical approach was studied in order to predict damage development in ceramic-matrix composites under thermomechanical loading conditions. Complex damage phenomena like matrix microcracking, fibre or bundle debonding and fibre breakage can occur. All these mechanisms can be modelled at the microscopic scale. Thus, correlation between the different scales and reliable homogenization procedures have been developed. This micro-macro model is applied to various Sic/SiC and a C/SiC composites By the use of this model, the intrinsic mechanical properties of the classical CVi Sic matrix have been identified. Implementation of this model was done by means of the finite-element code ABAQUS. An application was carried out on a notched specimen made of CerasepN3-1 composite C 2000 Elsevier Science Ltd. All rights reserved. Keywords: Ceramic-matrix composites(CMCs): Thermomechanical properties; Micro-macro modelling: Damage mechanics 1. Introduction Finally, this model was implemented in ABAQUS and a calculation on a notched specimen was performed. Ceramic-matrix composites have reached a critical stage in their development and application, partly because of a lack of specific design procedures and life- 2. Multi-scale approach prediction methodologies. The development of reliable models for describing the thermomechanical behaviour The different scales involved in the description are: of such composite is necessary. In this study, we pro- pose a micromechanical me e Micro-scale: constituents fibre. matrix interface the development of damage in various CMCs under Meso-scale: elementary ply. complex thermomechanical loading up to failure. For Macro-scale: composite, structur ceramic-matrix composites, several damage mechanisms Information obtained from the description of damage can occur, including matrix microcracking, fibre/matrix development at the micro-structural level are integrated debonding, and fibre breakage. These mechanisms are in the construction of the model of a woven-fibre strongly anisotropic: cracks can be perpendicular to the composite (or the laminate). At the micro-scale the loading direction, or partly deviated by the reinforce- thermomechanical behaviour is obtained through an ment orientation. Furthermore, these cracks can be homogenisation calculation considering matrix, fibres, opened or closed depending on the loading and on the and interfaces to have damageable properties. The thermal residual stresses induced by cure processing. prediction of the ply behaviour is obtained by using a These aspects are introduced in the proposed model classical homogenisation model [1, 2]. It is based We focus this paper on the damage kinematics intro- closed-form analytical solutions, which are available for duced at the micro-scale. Various applications on any thermomechanical loading conditions. Finally, the woven-fibre composites are presented(SiC/SiC, C/SiC). classical thin-laminate theory is applied to calculate the Simulations are compared to experimental data taken behaviour of the laminate [3, 4]. The laminate failure from the literature occurs by an instability condition on its overall stiffness, which is related to catastrophic damage propagation, or with the failure of the reinforcements(maximum stress or strain criteria 0266-3538/00/S. see front matter C 2000 Elsevier Science Ltd. All rights reserved. PII:S0266-3538(00)00010-5

A micromechanical model for thermostructural composites C. Rospars *, E. Le Dantec, F. Lecuyer MEDYSYS, MeÂcanique et Dynamique des SysteÁmes, 29 Rue Jean Rostand, F-91893 Orsay Cedex, France Received 23 June 1999; received in revised form 4 January 2000; accepted 13 January 2000 Abstract A damage model based on a micromechanical approach was studied in order to predict damage development in ceramic-matrix composites under thermomechanical loading conditions. Complex damage phenomena like matrix microcracking, ®bre or bundle debonding and ®bre breakage can occur. All these mechanisms can be modelled at the microscopic scale. Thus, correlation between the di€erent scales and reliable homogenization procedures have been developed. This micro-macro model is applied to various SiC/SiC and a C/SiC composites. By the use of this model, the intrinsic mechanical properties of the classical CVI SiC matrix have been identi®ed. Implementation of this model was done by means of the ®nite-element code ABAQUS. An application was carried out on a notched specimen made of Cerasep1 N3-1 composite. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Ceramic-matrix composites (CMCs); Thermomechanical properties; Micro-macro modelling; Damage mechanics 1. Introduction Ceramic-matrix composites have reached a critical stage in their development and application, partly because of a lack of speci®c design procedures and life￾prediction methodologies. The development of reliable models for describing the thermomechanical behaviour of such composite is necessary. In this study, we pro￾pose a micromechanical model which is able to simulate the development of damage in various CMCs under complex thermomechanical loading up to failure. For ceramic-matrix composites, several damage mechanisms can occur, including matrix microcracking, ®bre/matrix debonding, and ®bre breakage. These mechanisms are strongly anisotropic: cracks can be perpendicular to the loading direction, or partly deviated by the reinforce￾ment orientation. Furthermore, these cracks can be opened or closed depending on the loading and on the thermal residual stresses induced by cure processing. These aspects are introduced in the proposed model. We focus this paper on the damage kinematics intro￾duced at the micro-scale. Various applications on woven-®bre composites are presented (SiC/SiC, C/SiC). Simulations are compared to experimental data taken from the literature. Finally, this model was implemented in ABAQUS and a calculation on a notched specimen was performed. 2. Multi-scale approach The di€erent scales involved in the description are: . Micro-scale: constituents, ®bre, matrix, interface, . Meso-scale: elementary ply, . Macro-scale: composite, structure. Information obtained from the description of damage development at the micro-structural level are integrated in the construction of the model of a woven-®bre composite (or the laminate). At the micro-scale the thermomechanical behaviour is obtained through an homogenisation calculation considering matrix, ®bres, and interfaces to have damageable properties. The prediction of the ply behaviour is obtained by using a classical homogenisation model [1,2]. It is based on closed-form analytical solutions, which are available for any thermomechanical loading conditions. Finally, the classical thin-laminate theory is applied to calculate the behaviour of the laminate [3,4]. The laminate failure occurs by an instability condition on its overall sti€ness, which is related to catastrophic damage propagation, or with the failure of the reinforcements (maximum stress or strain criteria). 0266-3538/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(00)00010-5 Composites Science and Technology 60 (2000) 1095±1102 * Corresonding author

Thus, in a second step, a suitable model for describing the behaviour of the constituents, based on anisotropic eD m){2,(=) damage mechanics, is used. The tool being used is the E(1-4)班E(1-四) anisotropic-damage theory developed by Ladeveze [5,6. It has already been applied to various composites -2喟吗 uch as carbon/ epoxy laminate [7]. woven-fibre ceramic G(1-) composites [8, 9] or 3D C/C composites [10]. One of the difficulties of this approach is to outline the main phe- where ()+(respectively,()-)denotes the positive nomena from the experimental response and then to (negative) part of the considered quantity build the damage laws at the appropriate scale. Since To keep in the standard model framework, d12 is the damage phenomena occur on the microscopic scale, completely defined with d, and d2 it seems appropriate to study damage at this micro- scale.Moreover,this scale enhances the role of inter- dn=d + 2 and b= ETE22 phases between fibre and matrix and it accounts for thermal residual stresses effect Damage kinematics and internal variables are deduced together from the theo- Then two damage parameters remain. We defined: etical analysis of fibre reinforced matrix composites behaviour71 and from experimental investigations y2=Sp(√吗+b,y) with the conjugate quantities to d and d2: yi=amla.cst 3. Damage modelling Y t=0 This expression is similar to an energy release rate in Assuming experimental results and microscop fracture mechanics. The laws of damage accumulation observations, the damage mechanisms of CMCs have are chosen to fit the experimental data From a physical been well described. Under tensile loading, CMCs pre- point of view, we chosed an exponential law that takes ent a linear elastic response until the initiation and into account the load bearing capability of the matrix propagation of matrix micro-cracks and the partial re after the matrix crack saturation(os). A similar opening of thermal cracks. In a second stage, multi- approach has been previously proposed by Burr et al n of matrix micro-cracks and the associated [ 18]. A suitable expression is given by natrix debonding are propagating until matrix crack saturation [16]. Composites with weak interface exhibits a 'plateau-like behaviour[13]. The matrix dim(-ex Y-Ya)irY≤≤ crack saturation is rapidly achieved (load transfer being d poor)and the total failure occurred almost immediately after this point of saturation. For composites that pre- dim( )ifY≥Y sent high strain to rupture, after matrix crack satura tion, a progressive load transfer to the fibres which then fracture progressively occurs [17]. Another net of crack r is the starting point of matrix cracking and ys is corresponding to multiple cracking of bundles can occur representative of the saturation value of the damage [16]. These composites present a broad non-linear parameter modelling matrix cracking. Therefore, for domain and higher stresses without any 'plateau-like example, after saturation of matrix cracks For each damage mechanism one damage parameter vYm>ys must be defined. Such a description of damage at the micro-scale allows prediction of all aspects of the strain stress curve. like the effect of thermal residual stress d-vbrs(-dlim(-exy oS =cst 3. Matrix behaviour modelling with b=(VEmE2/G12) Progressive damage resulting from matrix cracking is defined with three damage variables r s is linked to r and Yo through the damage cri the direction of the reinforcements, and one to char- terium. Thus, only three parameters remain: ym, Yo, acterise the decreasing of the shear modulus. Therefore and dim which represents the damage induced by matrix the elastic energy relative to the matrix is: microcracking saturation state

Thus, in a second step, a suitable model for describing the behaviour of the constituents, based on anisotropic damage mechanics, is used. The tool being used is the anisotropic-damage theory developed by Ladeveze [5,6]). It has already been applied to various composites such as carbon/epoxy laminate [7], woven-®bre ceramic composites [8,9] or 3D C/C composites [10]. One of the diculties of this approach is to outline the main phe￾nomena from the experimental response and then to build the damage laws at the appropriate scale. Since the damage phenomena occur on the microscopic scale, it seems appropriate to study damage at this micro￾scale. Moreover, this scale enhances the role of inter￾phases between ®bre and matrix and it accounts for thermal residual stresses e€ect. Damage kinematics and internal variables are deduced together from the theo￾retical analysis of ®bre reinforced matrix composites behaviour [7,11] and from experimental investigations on progressive degradation [12±16]. 3. Damage modelling Assuming experimental results and microscopic observations, the damage mechanisms of CMCs have been well described. Under tensile loading, CMCs pre￾sent a linear elastic response until the initiation and propagation of matrix micro-cracks and the partial re￾opening of thermal cracks. In a second stage, multi￾plication of matrix micro-cracks and the associated ®bre/matrix debonding are propagating until matrix crack saturation [16]. Composites with weak interface exhibits a `plateau-like behaviour' [13]. The matrix crack saturation is rapidly achieved (load transfer being poor) and the total failure occurred almost immediately after this point of saturation. For composites that pre￾sent high strain to rupture, after matrix crack satura￾tion, a progressive load transfer to the ®bres which then fracture progressively occurs [17]. Another net of cracks corresponding to multiple cracking of bundles can occur [16]. These composites present a broad non-linear domain and higher stresses without any `plateau-like' domain. For each damage mechanism one damage parameter must be de®ned. Such a description of damage at the micro-scale allows prediction of all aspects of the strain/ stress curve, like the e€ect of thermal residual stress. 3.1. Matrix behaviour modelling Progressive damage resulting from matrix micro￾cracking is de®ned with three damage variables, two in the direction of the reinforcements, and one to char￾acterise the decreasing of the shear modulus. Therefore the elastic energy relative to the matrix is: em D ˆ 1 2 " m 11 2 ‡ Em 11 1 ÿ dm 1 ÿ
‡ m 11 2 ÿ Em 11 ‡ m 22 2 ‡ Em 22 1 ÿ dm 2 ÿ
‡ m 22 2 ÿ Em 22 ÿ 2 m 12 Em 11 m 11m 22 ‡ m2 12 Gm 12 1 ÿ dm 12 ÿ
# where h i‡ (respectively, h iÿ) denotes the positive (negative) part of the considered quantity. To keep in the standard model framework, d12 is completely de®ned with d1 and d2. d12 ˆ d1 ‡ d2 b and b ˆ  Em 11Em 22 p Gm 12 Then two damage parameters remain. We de®ned: Y i ˆ Sup 4t  Ym 12 ‡ bYm i p ; Ym 0   with the conjugate quantities to d1 and d2: Ym i ˆ @e m D @d m i j :cst and Ym 0 ˆ Ym 1 ; t ˆ 0. This expression is similar to an energy release rate in fracture mechanics. The laws of damage accumulation are chosen to ®t the experimental data. From a physical point of view, we chosed an exponential law that takes into account the load bearing capability of the matrix after the matrix crack saturation (m S ). A similar approach has been previously proposed by Burr et al. [18]. A suitable expression is given by: dm i ˆ dlim…1 ÿ exp Ym i ÿ Ym 0 Ym c  † if Ym 0 4Ym i 4Ym S dlim…1 ÿ Ym S exp YmA i ÿ Ym 0 Ym c   Ym i † if Ym i 5Ym S 8 >>>>>< >>>>>: Ym c is the starting point of matrix cracking and Ym S is representative of the saturation value of the damage parameter modelling matrix cracking. Therefore, for example, after saturation of matrix cracks: 8Ym i 5Ym S m ii ˆ  2Eii b r Ym S 1ÿdlim 1ÿ exp Ym 0 ÿ Ym S Ym c       ˆ m S ˆ cst with b ˆ …  Em 11Em 22 p =Gm 12† Ym S is linked to Ym c and Ym 0 through the damage cri￾terium. Thus, only three parameters remain: Ym c ; Ym 0 ; and dlim which represents the damage induced by matrix microcracking saturation state. 1096 C. Rospars et al. / Composites Science and Technology 60 (2000) 1095±1102

C. Rospars et al./ Composites Science and Technology 60(2000)1095-1102 3. 2. Fibre behaviour modelling 4. Experimental-model comparisons ultimate tensile strength properties of fibre-rein- All the composites presented hereafter are made by CMC are usually directed by the strength of the SNECMA- Division S E P( France). Various SiC fibres. The fibre failure is characterised with one damage SIC (NLM 202 fibres and pyC interface)have been stu- parameter, d. A simple failure criteria based on a max- died; the SiC/Sic 0.2%(referring to its ultimate strain imum strain value or stress value is used. This maximum under tensile load in orthotropic direction), previously strain is easily extracted from literature (for example, studied by Aubard [8] and Ladeveze et al. [9], and the [19]. Then the elastic energy is: Sic-SiC 0.6% studied by Guillaumat and Lamon [16] The main difference between these two SiC/Sic batches is the quality of fibre/matrix interface. For the.6% fibres were treated in order to avoid silica around the ((1) fibres in the composite. Thus, the load-transfer cap- E(-4)+E1+2-2eB12+ ability is improved [13]. The Cerasep N3-1(NL 207 fibres and pyrolitic carbon interface) and a C/SiC (PAN-based carbon fibres and pyrolytic carbon inter The law of damage is defined using an exponential law. yo characterises the begining of fiber breakage and Yf 4.L. SiC CVI matrix properties identification depends on the maximum tensile stress supported by fibres. As previously mentioned, direct identification of the damage parameters is almost impossible. As the matter of fact, the constituent's properties are difficult to obtain from measurement on the composite. Fibre properties and interface properties can be found in the literature, but the mechanical properties of the matrix From literature data, we expressed these parameters inside the composite should be identified with at least (Yo. yo with their corresponding stress values(o, o) one calculation. The examples presented here are com [19]. a represents the beginning of fibre breakage, posites with CVI( Chemical Vapour Infiltration) SiC similar to a statistical parameter, and o/ is the max matrix. We used the results on SiC/SiC 0. 2%[8, 20] in imum stress before fibre complete failure identification procedures The Sic CVI matrix properties, including porosity, 3.3. Interfacial behaviour modelling are identified to predict the initial stiffness measured with a tensile test in the fibre direction. The constituents a first homogenisation calculation estimates an properties used in the simulations, including the matrix homogenized fibre, including the interfacial properties. identified properties, are given in Table 1. The given len fibre/interface zone is described as a surfacial moduli are not affected by temperature during the pro- entity with its own stiffness. Its constitutive law is cessing. Besides, the coefficient of thermal expansion is expressed with a stiffness (k), function of the loads, and taken on average between 25 and 1000C to include the that takes values from zero(representative of complete thermal residual stresses induced by processing. It cannot debonding) to infinity (perfect interface). Then the be simply estimated from room temperature values [23] damage accumulation law is: 4.2. Damage parameters identifications (27 g()) A set of damage parameters is identified by using the (r) same test conducted on Sic/SiC 0. 2%[20). From experimental investigation on this SiC/SiC 0. 2%under tensile load in the reinforcement direction [8, 16], we an)+≤d know that(Fig The matrix cracking initiates around 85 MPa(end and di is constant around the fibre The matrix crack saturation is rapidly achieved because of a weak bonding between fibre and Three damage parameters should be estimated matrix [16]. Then the pre-existing cracks begin to Yo and o open but failure occurs immediately after

3.2. Fibre behaviour modelling The ultimate tensile strength properties of ®bre-rein￾forced CMC are usually directed by the strength of the ®bres. The ®bre failure is characterised with one damage parameter, dl. A simple failure criteria based on a max￾imum strain value or stress value is used. This maximum strain is easily extracted from literature (for example, [19]. Then the elastic energy is: ef D ˆ 1 2  f 11 D E2 ‡ Ef 11 1 ÿ df 1   ‡  f 11 D E2 ÿ Ef 11 ‡  f 2 22 Ef 22 ÿ 2 f 12 Ef 11  f 11 f 22 ‡ f 2 12 Gf 12 2 6 4 3 7 5 The law of damage is de®ned using an exponential law. Yf 0 characterises the begining of ®ber breakage and Yf c depends on the maximum tensile stress supported by ®bres: df 1 ˆ 1 ÿ exp Yf 1 ÿ Yf 0 Yf c ! From literature data, we expressed these parameters …Yf 0; Yf c† with their corresponding stress values … f c;  f 0† [19].  f 0 represents the beginning of ®bre breakage, similar to a statistical parameter, and  f c is the max￾imum stress before ®bre complete failure. 3.3. Interfacial behaviour modelling A ®rst homogenisation calculation estimates an homogenized ®bre, including the interfacial properties. Then ®bre/interface zone is described as a surfacial entity with its own sti€ness. Its constitutive law is expressed with a sti€ness (k), function of the loads, and that takes values from zero (representative of complete debonding) to in®nity (perfect interface). Then the damage accumulation law is: Yi…t† ˆ Sup 4t  …2 rz…† ‡ 2 r…††i 2k0 rz…1 ÿ d0 i …††2 ! s d0 i ˆ Min Yi ÿ Yi 0  ‡ Yi c ; 1 ! h i rr ‡4t i 1 h i rr ‡> t i 8 >< >: and di is constant around the fibre: Three damage parameters should be estimated, Ym c ; Yi 0 and t i . 4. ExperimentalÐmodel comparisons All the composites presented hereafter are made by SNECMA Ð Division S.E.P. (France). Various SiC/ SiC (NLM 202 ®bres and pyC interface) have been stu￾died; the SiC/SiC `0.2%' (referring to its ultimate strain under tensile load in orthotropic direction), previously studied by Aubard [8] and Ladeveze et al. [9], and the SiC-SiC `0.6%' studied by Guillaumat and Lamon [16]. The main di€erence between these two SiC/SiC batches is the quality of ®bre/matrix interface. For the `0.6%', ®bres were treated in order to avoid silica around the ®bres in the composite. Thus, the load-transfer cap￾ability is improved [13]. The Cerasep1 N3-1 (NL 207 ®bres and pyrolitic carbon interface) and a C/SiC (PAN-based carbon ®bres and pyrolytic carbon inter￾face) are presented. 4.1. SiC CVI matrix properties identi®cation As previously mentioned, direct identi®cation of the damage parameters is almost impossible. As the matter of fact, the constituent's properties are dicult to obtain from measurement on the composite. Fibre properties and interface properties can be found in the literature, but the mechanical properties of the matrix inside the composite should be identi®ed with at least one calculation. The examples presented here are com￾posites with CVI (Chemical Vapour In®ltration) SiC matrix. We used the results on SiC/SiC 0.2% [8,20] in identi®cation procedures. The SiC CVI matrix properties, including porosity, are identi®ed to predict the initial sti€ness measured with a tensile test in the ®bre direction. The constituents properties used in the simulations, including the matrix identi®ed properties, are given in Table 1. The given moduli are not a€ected by temperature during the pro￾cessing. Besides, the coecient of thermal expansion is taken on average between 25 and 1000C to include the thermal residual stresses induced by processing. It cannot be simply estimated from room temperature values [23]. 4.2. Damage parameters identi®cations A set of damage parameters is identi®ed by using the same test conducted on SiC/SiC 0.2% [20]. From experimental investigation on this SiC/SiC 0.2% under tensile load in the reinforcement direction [8,16], we know that (Fig. 1): . The matrix cracking initiates around 85 MPa (end of linearity, m 0 ˆ 85 MPa), . The matrix crack saturation is rapidly achieved because of a weak bonding between ®bre and matrix [16]. Then the pre-existing cracks begin to open but failure occurs immediately after, C. Rospars et al. / Composites Science and Technology 60 (2000) 1095±1102 1097

C. Rospars et al. / Composites Science and Technology 60(2000) 1095-1102 Table I Undamaged characteristics of constituents(fibres, matrix and interphase) Ell(GPa) E22(GPa) G1(GPa) aL(10-6K-) ar(10-6K-) Nicalon NLM 202[ 21] 4.8×10-6 3×10 Nicalon NL 207 3×1 ex-Pan fiber 24 1.1×106a 7×10 Ref.[23] b Estimation On the contrary, for SiC/SiC 0.6%, fibres pro- frictions are responsible for the measured inelasticity gressively support the load from the saturation and those phenomena have not been considered in this up to the failure Fibre damage parameters are chosen to ensure the 4.3. Experimental- model comparisons tarting of fibre breakage around 185 MPa(point of matrix saturation). According to Bunsell et al. [19], the An application of these identification was performed maximum stress value supported by Nicalon fibres is with other SiC/SiC and C/SiC made of the classical CVI around 2.2+0.7 GPa. Interfacial damage parameters matrix. Simulation was performed, with the Sic/SiC nd matrix damage parameters are chosen to fit the 0.6% considering the better load transfer between fibre experimental tensile curve (SiC/SiC 0.2%, 0 tension; and matrix. The result is given in Figs. 3 and 4. No Fig. 1). o, which corresponds to the maximum stress other modification on constituents properties or on value supported by the matrix itself, outside the com- damage parameters, except the modification of inter posite, is 110 MPa. Finally, we calculated the interfacial facial parameters (Y0=3.10-,r(=5 x 10 parameters. The corresponding shear maximum value is [12 a 200 MPa) were done. This simulation proved the found T12= 20 MPa for the SiC/Sic 0. 2%, which seems intrinsic character of the mechanical properties of the acceptable. In fact, Brenet et al. [24] measured using identified SiC CVi matrix P(u) curve t= 17.5 MPa. The damage parameters used for the modelling are written in the following Table 2. Table 2 Figs. I and 2 show the results obtained with the Identification of the damage parameters based on the SiC/SiC 0.2% identification procedure of matrix properties and miss- [20] ing damage parameters. These results are satisfactory because the knee point of free thermal residual stresses oo is well predicted, while the identification w 85MPa110MPa0.86900MPa16GPa3×10-24×10-2 formed to stick to the global curve without considering SiC CVI Weak pyrocarbon the unloading/reloading loops. In the case of SiC/SiC, matrIx NLM 202 interface Stress (MPa) 200 SiC-SiC(0/90 Stress (M Pa) SiC·SiC(+45°7459) Gasser, 1994 Gasser. 1994 Strain (% Strain (%) Fig. 1. Identification of intrinsic SiC CVI matrix properties and Fig. 2. Identification of interfacial damage parameters by using SiC/ damage parameters by using SiC/SiC 0. 0 tensile test. SiC0.2%±45°

. On the contrary, for SiC/SiC 0.6%, ®bres pro￾gressively support the load from the saturation point, up to the failure. Fibre damage parameters are chosen to ensure the starting of ®bre breakage around 185 MPa (point of matrix saturation). According to Bunsell et al. [19], the maximum stress value supported by Nicalon ®bres is around 2.2‹0.7 GPa. Interfacial damage parameters and matrix damage parameters are chosen to ®t the experimental tensile curve (SiC/SiC 0.2%, 0 tension; Fig. 1). m c , which corresponds to the maximum stress value supported by the matrix itself, outside the com￾posite, is 110 MPa. Finally, we calculated the interfacial parameters. The corresponding shear maximum value is found 12 ˆ 20 MPa for the SiC/SiC 0.2%, which seems acceptable. In fact, Brenet et al. [24] measured using p u… † curve  ˆ 17:5 MPa. The damage parameters used for the modelling are written in the following Table 2. Figs. 1 and 2 show the results obtained with the identi®cation procedure of matrix properties and miss￾ing damage parameters. These results are satisfactory because the knee point of free thermal residual stresses is well predicted, while the identi®cation was per￾formed to stick to the global curve without considering the unloading/reloading loops. In the case of SiC/SiC, frictions are responsible for the measured inelasticity, and those phenomena have not been considered in this model. 4.3. Experimental Ð model comparisons An application of these identi®cation was performed with other SiC/SiC and C/SiC made of the classical CVI matrix. Simulation was performed, with the SiC/SiC 0.6% considering the better load transfer between ®bre and matrix. The result is given in Figs. 3 and 4. No other modi®cation on constituents properties or on damage parameters, except the modi®cation of inter￾facial parameters …Yi 0 ˆ 3:10ÿ1; Yi c ˆ 5 10ÿ1 ) 12  200 MPa† were done. This simulation proved the intrinsic character of the mechanical properties of the identi®ed SiC CVI matrix. Fig. 1. Identi®cation of intrinsic SiC CVI matrix properties and damage parameters by using SiC/SiC 0.2% 0 tensile test. Table 1 Undamaged characteristics of constituents (®bres, matrix and interphase) E11 (GPa) E22 (GPa) 12 G12 (GPa) L (10ÿ6 Kÿ1 ) T (10ÿ6 Kÿ1 ) Nicalon NLM 202 [21] 200 200 0.12 80 4.810ÿ6a 310ÿ6a Nicalon NL 207 220 220 0.12 80b 4.810ÿ6b 310ÿ6b Pyrocarbon [22] 30 11 0.12 2 3 28 SiC CVI matrix 350 310 0.2 146 4.5 4.5 C ex-Pan ®ber [22] 230 220 0.24 4.8 1.1106a 7106a a Ref. [23]. b Estimation. Table 2 Identi®cation of the damage parameters based on the SiC/SiC 0.2% [20] m 0 m c dlim  f 0  f c Yi 0 Yi c 85 MPa 110 MPa 0.86 900 MPa 1.6 GPa 310ÿ2 410ÿ2 SiC CVI matrix Nicalon NLM 202 Weak pyrocarbone interface Fig. 2. Identi®cation of interfacial damage parameters by using SiC/ SiC 0.2% ‹45. 1098 C. Rospars et al. / Composites Science and Technology 60 (2000) 1095±1102

C. Rospars et al. Composites Science and Technology 60(2000)1095-1102 Stress (M Pa) SiC·siC0.6%(0") CERASEP NE 15 Simulation sic- sic 0.6% Test ETC Alstom RT SiC-SiC 0.2%(Gasser, 199 Test etc alstom1200°c Strain(5) Strain (% Experimental results and simulation of a SiC/SiC(0.6%)0o Fig. 5. Experimental results and simulation of Cerasep N31 test performed at room temperature and under 1200C ↑sues(MPa Stress(M Pa) SiC-SiC0.6%(+/45°) 15 C/SiC Camus aL. 96 Model SiC-SiC 0.6%(Aubart, 1992) Strain(%) Simulation SiC-SiC 0.6 0.4 iC SiC 0. 2%(Gasser, 1994) Fig 4. Experimental results and simulation of a SiC/SiC(0.6%)+45 Fig. 6. Experi results and simulation of a C/SiC, 0 tensi tension test compared to the sic/sic02%+ 45 tensile test. Fig. 7. Stress level onI(MPa) after the first load in the direction I (left)and after the unload (right) in a 90 ply

Fig. 5. Experimental results and simulation of Cerasep1 N3-1 test performed at room temperature and under 1200C. Fig. 3. Experimental results and simulation of a SiC/SiC (0.6%) 0 tensile test. Fig. 4. Experimental results and simulation of a SiC/SiC (0.6%) ‹45 tension test compared to the SiC/SiC 0.2% ‹45 tensile test. Fig. 6. Experimental results and simulation of a C/SiC, 0 tension/ compression test. Fig. 7. Stress level 11 (MPa) after the ®rst load in the direction 1 (left) and after the unload (right) in a 90 ply of notched specimen made of Cerasep1 N3-1. C. Rospars et al. / Composites Science and Technology 60 (2000) 1095±1102 1099