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 dierent 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 lifeprediction methodologies. The development of reliable models for describing the thermomechanical behaviour of such composite is necessary. In this study, we propose 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 reinforcement 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 introduced 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 dierent 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 stiness, 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 diculties of this approach is to outline the main phenomena 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 microscale. Moreover, this scale enhances the role of interphases between ®bre and matrix and it accounts for thermal residual stresses eect. Damage kinematics and internal variables are deduced together from the theoretical 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 present a linear elastic response until the initiation and propagation of matrix micro-cracks and the partial reopening of thermal cracks. In a second stage, multiplication 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 present high strain to rupture, after matrix crack saturation, 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 eect of thermal residual stress. 3.1. Matrix behaviour modelling Progressive damage resulting from matrix microcracking is de®ned with three damage variables, two in the direction of the reinforcements, and one to characterise the decreasing of the shear modulus. Therefore the elastic energy relative to the matrix is: em D 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 d1 d2 b and b Em 11Em 22 p Gm 12 Then two damage parameters remain. We de®ned: Y i Sup 4t Ym 12 bYm i p ; Ym 0 with the conjugate quantities to d1 and d2: Ym i @e m D @d m i j :cst and Ym 0 Ym 1 ; t 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 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 2Eii b r Ym S 1ÿdlim 1ÿ exp Ym 0 ÿ Ym S Ym c m S cst with b Em 11Em 22 p =Gm 12 Ym S is linked to Ym c and Ym 0 through the damage criterium. 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-reinforced 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 maximum 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 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 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 maximum 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 stiness. Its constitutive law is expressed with a stiness (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 Sup 4t 2 rz 2 r i 2k0 rz 1 ÿ d0 i 2 ! s d0 i 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 studied; 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 dierence 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 capability 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 interface) 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 dicult 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 composites 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 stiness 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 aected by temperature during the processing. Besides, the coecient 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 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 progressively 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 2.20.7 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 composite, is 110 MPa. Finally, we calculated the interfacial parameters. The corresponding shear maximum value is found 12 20 MPa for the SiC/SiC 0.2%, which seems acceptable. In fact, Brenet et al. [24] measured using p u curve 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 missing damage parameters. These results are satisfactory because the knee point of free thermal residual stresses is well predicted, while the identi®cation was performed 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 interfacial parameters Yi 0 3:10ÿ1; Yi c 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
C. Rospars et al. Composites Science and Technology 60 (2000)1095-1102 By using the same set of data, with the manufacturer progressively, is observed [17]. Almaz et al. [22] pro- properties for the Nicalon NL 207 fibre, we simulated posed a value for fibre ultimate tensile stress o/=3.5 he tensile behaviour of the Cerasep N3-1, tested in GPa. Introducing this value and the Pan fibre proper ne reinforcements direction under 20 and 1200 C. The ties(Table 1), a simulation was performed. This simu- simulations are given Fig. 5. lation, corresponding to tensile/compressive loading, or C/SiC, the non-linearity starts almost from the dicts the of thermal stress The onset of loading. Then, after matrix saturation(<200 simulation is compared to the test results from Camus et MPa), a significant domain corresponding to a pro- al. [ 14](Fig. 6). In the case of this composite, the ther- gressive load transfer to the fibres, which then fracture mal residual stresses are involved in the inelastic strain +:95605 F#2:3#8 2:5 2373616 2 +:512+0 8. Damaged ply property Enl (Pa) after the first load and the Fig 10. Damaged ply property Enl(Pa)after the second load in a 90o nd in a 90 ply of a notched Cerasep N3I specimen. ply of a Cerasep N3-l specimen Fig. 9. Stress level on (MPa)after the second load, in a 90 ply (left)and in a 0o ply(right), for a CerasepN3-1 notched specimen
By using the same set of data, with the manufacturer properties for the Nicalon NL 207 ®bre, we simulated the tensile behaviour of the Cerasep1 N3-1, tested in the reinforcements direction under 20 and 1200C. The simulations are given Fig. 5. For C/SiC, the non-linearity starts almost from the onset of loading. Then, after matrix saturation (200 MPa), a signi®cant domain corresponding to a progressive load transfer to the ®bres, which then fracture progressively, is observed [17]. Dalmaz et al. [22] proposed a value for ®bre ultimate tensile stress f c 3:5 GPa. Introducing this value and the PAN ®bre properties (Table 1), a simulation was performed. This simulation, corresponding to tensile/compressive loading, predicts the point of thermal stress free origin. The simulation is compared to the test results from Camus et al. [14] (Fig. 6). In the case of this composite, the thermal residual stresses are involved in the inelastic strain Fig. 8. Damaged ply property E11 (Pa) after the ®rst load and the unload in a 90 ply of a notched Cerasep1 N3-1 specimen. Fig. 9. Stress level 11 (MPa) after the second load, in a 90 ply (left) and in a 0 ply (right), for a Cerasep1 N3-1 notched specimen. Fig. 10. Damaged ply property E11 (Pa) after the second load in a 90 ply of a Cerasep1 N3-1 specimen. 1100 C. Rospars et al. / Composites Science and Technology 60 (2000) 1095±1102
C. Rospars et al. / Composites Science and Technology 60(2000) 1095-1102 00X020.00400006.0O8001000120.00140.0160.18o.02 Strain (Ell) Fig. 11. au versus Ell for elements situated at the corner of the notch under a tensile stress. measured with the periodic unloadings. As the matter of a 90 ply after the unloading and after the second load fact, it is well predicted by the model Composite failure During the unloading, the ply damage level remain occurred with fibre failure constant to the value reached the first loading. Near the notch, the matrix damage parameter d is almost 0.7 5. Application to a Cerasep n3-l notched specimen As a verification of these results we could plot the stress component on versus the strain component Ell, To carry out structural calculations, the model pre- for each ply and then for the woven ply(Fig. 11). This sented here has been integrated into the finite-element result is given for the most stresses element which is the code ABAQUS, by way of the 'User Subroutine element situated in the tensile area at the corner of the UMAT. The mechanical properties of the constituents notch. This element is submitted, under elastic loading, are the one presented in Table 1. The interfacial damage to a stress almost 3. 5 times the applied stress. This value parameters are the one presented above. The composite is higher than the factor normally obtained for a hole in is oriented in the orthotropic direction with respect to an orthotropic composite( 3.25). After the second load the loading direction. A classical mesh describes half of the released of the stress field, leads to a factor of 2.68 a notched In the thickness one element defines a ply and two consecutive plies, respectively oriented in 0o and 90, described the woven ply. The 6. Conclusions stacking sequence is defined to reproduce the real structure of the CerasepN3-1. ABAQUS calculates A damage micro-macro model, including information for each element, at each integration point of each on the micromechanical damage mechanisms, has been iteration the damage state of the constituents and the developed. The damage accumulation laws, written at damaged properties of the ply. All these information are the micro-scale, accounts for the main damage phe stored in ABAQUS database, and can be recover and nomena observed for CMCs. This model is able to per- study easily. The load is given as a force which is form simulations on various CMCs, woven composite applied on the section of the structure opposite to the and UD ply, under complex thermomechanical loading notch and perpendicular to the direction I(Figs. 7-10) p to the final failure. Such a simulation needs knowl Simulations were performed: one with an unloading edge of the constituent's thermomechanical properties of the structure after the beginning of damage and a Thus, we identified the Sic CVI matrix properties, using reloading up to a level that ensures a stress of 200 MPa the test results on the Sic/SiC 0. 2%[20]. These identi close to the notch. We focussed on the results obtained fied properties were input in the model to perform on the 90 ply, which is significantly damaged. Change simulations on other composites made with the same in the ply properties versus time increment is plotted. matrix (SiC/Sic 0.6%, Cerasep N3-1 and C-Sic) After the beginning of matrix cracking, the ply proper- Interfacial properties and fibre properties are extracted ties decrease. Figs. 8 and 9 showed the change in El of from literature (Table 1). The predictions obtained for
measured with the periodic unloadings. As the matter of fact, it is well predicted by the model. Composite failure occurred with ®bre failure. 5. Application to a Cerasep1 n3-1 notched specimen To carry out structural calculations, the model presented here has been integrated into the ®nite-element code ABAQUS, by way of the `User Subroutine' UMAT. The mechanical properties of the constituents are the one presented in Table 1. The interfacial damage parameters are the one presented above. The composite is oriented in the orthotropic direction with respect to the loading direction. A classical mesh describes half of a notched specimen. In the thickness, one element de®nes a ply and two consecutive plies, respectively oriented in 0 and 90, described the woven ply. The stacking sequence is de®ned to reproduce the real structure of the Cerasep1 N3-1. ABAQUS calculates for each element, at each integration point of each iteration the damage state of the constituents and the damaged properties of the ply. All these information are stored in ABAQUS database, and can be recover and study easily. The load is given as a force which is applied on the section of the structure opposite to the notch and perpendicular to the direction 1 (Figs. 7±10). Simulations were performed: one with an unloading of the structure after the beginning of damage and a reloading up to a level that ensures a stress of 200 MPa close to the notch. We focussed on the results obtained on the 90 ply, which is signi®cantly damaged. Change in the ply properties versus time increment is plotted. After the beginning of matrix cracking, the ply properties decrease. Figs. 8 and 9 showed the change in E11 of a 90 ply after the unloading and after the second load. During the unloading, the ply damage level remain constant to the value reached during the ®rst loading. Near the notch, the matrix damage parameter d1 is almost 0.7. As a veri®cation of these results we could plot the stress component 11 versus the strain component "11, for each ply and then for the woven ply (Fig. 11). This result is given for the most stresses element which is the element situated in the tensile area at the corner of the notch. This element is submitted, under elastic loading, to a stress almost 3.5 times the applied stress. This value is higher than the factor normally obtained for a hole in an orthotropic composite (3.25). After the second load the released of the stress ®eld, leads to a factor of 2.68. 6. Conclusions A damage micro-macro model, including information on the micromechanical damage mechanisms, has been developed. The damage accumulation laws, written at the micro-scale, accounts for the main damage phenomena observed for CMCs. This model is able to perform simulations on various CMCs, woven composite and UD ply, under complex thermomechanical loading up to the ®nal failure. Such a simulation needs knowledge of the constituent's thermomechanical properties. Thus, we identi®ed the SiC CVI matrix properties, using the test results on the SiC/SiC 0.2% [20]. These identi- ®ed properties were input in the model to perform simulations on other composites made with the same matrix (SiC/SiC 0.6%, Cerasep1 N3-1 and C-SiC). Interfacial properties and ®bre properties are extracted from literature (Table 1). The predictions obtained for Fig. 11. 11 versus "11 for elements situated at the corner of the notch under a tensile stress. C. Rospars et al. / Composites Science and Technology 60 (2000) 1095±1102 1101
these composites proved the intrinsic character of the [7 Ladeveze P, Le Dantec E Damage modelling of the elementary matrix properties previously identified. Various Sic ply for laminated composite Comp Sci Tech 1992: 43: 257-67. CVI based composites were successfully simulated [8 Aubard x. Modelisation et identification du comportement Moreover, the simulations performed are satisfactory, ue des materiaux composites 2D SiC/SiC. Thesis of Universite Paris vI. Paris. 1992 and validated the model. This model has been imple 9 Ladeveze P, Gasser A. Allix O. Damage mechanisms modelling mented in a structural analysis code to predict the for ceramic matrix composites. J Eng Mat Tech 1994: 116: 331-6 damage state of a structure under complex loading [10] Aubard x, Cluzel C, Ladeveze P, Perie JN. Damage meso-mod- before a macrocrack appears elling of a multidirectional SEPCARB. In: Hui D, editor. Pro- Obviously, this model can be used for other CMCs ceeding of the Fifth International Conference on Composittes Engineering, ICCE, 5-1l July 1998, Las Vegas. p 721-2. made of brittle matrix and brittle fibres In order to extend capabilities of this model, we are introducing [1 Talreja R. Continuum modelling of damage in ceramic matrix mposites. Mechanics of Material 1992: 12: 165-80 creep and fatigue damage laws. By this way, we will be [12] Sorensen BF, Talreja R, Sorensen OT Damage development in able to simulate high temperature components life cramic matrix composite under mechanical loading. In: Pro- eedings of ECCM5, Bordeaux, 1992. p. 613-8 [13 Naslain R. Fiber-matrix interphases and interfaces in ceramic natrix composites processed by CVI. J CompInterfaces Acknowledgements 1993:1:253-86. [14 Camus G, Guillaumat L, Baste S. Development of damage in a This study is supported by the European Commission 2D woven C/Sic composite under mechanical loading: I project no. BRPR-CT97-0399 Development of Design [15] Se anical characterization. Comp Sci and Tech 1996: 56: 1363-72. of European Communities under the BRITE EURAM n M. Effect of residual stresses on the mechanical response of ontinuous fibre reinforced ceramic matrix composites. In: Had- Methodologies for Ceramic Matrix Composites for Industrial Gas Turbine. We thank our partners ALS- mposites. Kluwer Academic, 1998. p 297-309 TOM ETC, ANSALDO and IAM laboratory for help [6] Guillaumat L, Lamon J Microcraking of a woven SiC/Sic com- and discussion. We thank Alain Gasser for his help in In: Proceedings of JNC 10, 29-31 October 1996, Paris, p. 35 exploitation of the SiC/SiC tests [7 El Bouazzaoui R, Baste S, Camus G. Development of damage in 2D woven C/SiC composite under mechanical loading: Il ultrasonic characterisation. Comp Sci and Tech 1996: 56: 1373-82. [18] Burr A, Hild F, Leckie FA. On the mechanical behaviour under cyclic loading of ceramic matrix composites. Mat Hashin Z. Analysis of t anIcs1983;50:481-505 [19) Bunsell AR, Simon G, Abe Y, Akiyama M. Ceramic fibres. In 2 Pideri C Materiaux composites elastiques: comparaison de quel- posite M ques methodes homogenisation. Thesis Universite Paris VI rials Series 2. Elsevier, 1988. p 427-78 20 Gasser A. Sur la modelisation et Identification du comporte B Engrand D, Le Dantec E, Lecuyer F. A multiscale approach for ent mecanique des composites ceramique-ceramique a tem- modelling thermo-mechanical behaviour of laminated ceramic perature ambiant Thesis ENS Cachan, 1994 D, edi- [21 Lamon J, Bodet JL. Thermal residual stresses in ceramic matrix tors. Proceedings of the HTCMCl, 6th European Conference on mposites--I. Axisymmetrical model and finite element analy Composite Materials, 20-24 September 1993, Bordeaux. Wood sis. Acta Metall 1995: 43(6): 2241-53 head Pub.. 1993 22 Almaz A, Reynaud P, Rouby D, Fantozzi G. J Mat Sci ( Rospars C, Le Dantec E, Lecuyer F CMC Damage prediction by 1996:31:4213. micro-macro modelling. In: Proceeding of the 12th International [23] Lebrun GA Comportement thermomecanique et duree de vie de Conference on Composites Materials, Paris, 5-9 July Is composites a matrice ceramique: theorie et experience, Thesis 5 Ladeveze P. Sur une theorie de Iendommagement a Universite Bordeaux l. 1996 Rapport interne no 34. Laboratoire de mecanique et [24 Brenet P, Fantozzi G, Reynaud P, Rouby D, Measurement and gie. ENS Cachan, 1983 of the p(u) curve giving the bridging tractions vs crack [6 Ladeveze P. A damage computational approach for composites: displacement in fibre reinforced ceramic matrix compo basic aspects and micromechanical relations. Computational Naslain R. Evans AG. editors. Proceedings of the Mechanics 1995: 17: 142-50 21-24 August 1995, Santa Barbara, 1995. p 259-64
these composites proved the intrinsic character of the matrix properties previously identi®ed. Various SiC CVI based composites were successfully simulated. Moreover, the simulations performed are satisfactory, and validated the model. This model has been implemented in a structural analysis code to predict the damage state of a structure under complex loading before a macrocrack appears. Obviously, this model can be used for other CMCs made of brittle matrix and brittle ®bres. In order to extend capabilities of this model, we are introducing creep and fatigue damage laws. By this way, we will be able to simulate high temperature components life. Acknowledgements This study is supported by the European Commission of European Communities under the BRITE/EURAM project no. BRPR-CT97-0399 `Development of Design Methodologies for Ceramic Matrix Composites for Industrial Gas Turbine'. We thank our partners ALSTOM ETC, ANSALDO and IAM laboratory for help and discussion. We thank Alain Gasser for his help in exploitation of the SiC/SiC tests. References [1] Hashin Z. Analysis of composite materials, a survey. J of Applied Mechanics 1983;50:481±505. [2] Pideri C. MateÂriaux composites eÂlastiques: comparaison de quelques meÂthodes d'homogeÂnisation. Thesis Universite Paris VI, 1987. [3] Engrand D, Le Dantec E, Lecuyer F. A multiscale approach for modelling thermo-mechanical behaviour of laminated ceramic matrix composites. In: Naslain R, Lamon J, Doumeingts D, editors. Proceedings of the HTCMC1, 6th European Conference on Composite Materials, 20±24 September 1993, Bordeaux. Woodehead Pub., 1993. [4] Rospars C, Le Dantec E, Lecuyer F. CMC Damage prediction by micro±macro modelling. In: Proceeding of the 12th International Conference on Composites Materials, Paris, 5±9 July 1999. [5] Ladeveze P. Sur une theÂorie de l'endommagement anisotrope, Rapport interne no. 34, Laboratoire de MeÂcanique et Technologie, ENS Cachan, 1983. [6] Ladeveze P. A damage computational approach for composites : basic aspects and micromechanical relations. Computational Mechanics 1995;17:142±50. [7] Ladeveze P, Le Dantec E. Damage modelling of the elementary ply for laminated composite. Comp Sci Tech 1992;43:257±67. [8] Aubard X. ModeÂlisation et identi®cation du comportement meÂcanique des mateÂriaux composites 2D SiC/SiC. Thesis of Universite Paris VI, Paris, 1992. [9] Ladeveze P, Gasser A, Allix O. Damage mechanisms modelling for ceramic matrix composites. J Eng Mat Tech 1994;116:331±6. [10] Aubard X, Cluzel C, LadeveÁze P, PeÂrie JN. Damage meso-modelling of a multidirectional SEPCARB1. In: Hui D, editor., Proceeding of the Fifth International Conference on Composittes Engineering, ICCE, 5±11 July 1998, Las Vegas. p 721±2. [11] Talreja R. Continuum modelling of damage in ceramic matrix composites. Mechanics of Material 1992;12:165±80. [12] Sorensen BF, Talreja R, Sorensen OT. Damage development in ceramic matrix composite under mechanical loading. In: Proceedings of ECCM5, Bordeaux, 1992. p. 613±8. [13] Naslain R. Fiber-matrix interphases and interfaces in ceramic matrix composites processed by CVI. J CompInterfaces 1993;1:253±86. [14] Camus G, Guillaumat L, Baste S. Development of damage in a 2D woven C/SiC composite under mechanical loading: I. Mechanical characterization. Comp Sci and Tech 1996;56:1363±72. [15] Steen M. Eect of residual stresses on the mechanical response of continuous ®bre reinforced ceramic matrix composites. In: Haddad YM, editor. Advanced multilayered and ®bre-reinforced composites. Kluwer Academic, 1998. p. 297±309. [16] Guillaumat L, Lamon J. Microcraking of a woven SiC/SiC composite: relation between microstructure and mechanical behaviour. In: Proceedings of JNC 10, 29±31 October. 1996, Paris, p. 35. [17] El Bouazzaoui R, Baste S, Camus G. Development of damage in a 2D woven C/SiC composite under mechanical loading: II. ultrasonic characterisation. Comp Sci and Tech 1996;56:1373±82. [18] Burr A, Hild F, Leckie FA. On the mechanical behaviour under cyclic loading of ceramic matrix composites. Mat Sci Eng 1998;A250:256±63. [19] Bunsell AR, Simon G, Abe Y, Akiyama M. Ceramic ®bres. In: Fiber reinforcements for composite materials. Composite Materials Series 2. Elsevier, 1988. p 427±78. [20] Gasser A. Sur la modeÂlisation et l'identi®cation du comportement meÂcanique des composites ceÂramique-ceÂramique aÁ tempeÂrature ambiante. Thesis ENS Cachan, 1994. [21] Lamon J, Bodet JL. Thermal residual stresses in ceramic matrix composites Ð I. Axisymmetrical model and ®nite element analysis. Acta Metall 1995;43(6):2241±53. [22] Dalmaz A, Reynaud P, Rouby D, Fantozzi G. J Mat Sci 1996;31:4213. [23] Lebrun GA. Comportement thermomeÂcanique et dureÂe de vie de composites aÁ matrice ceÂramique: theÂorie et expeÂrience, Thesis Universite Bordeaux I, 1996. [24] Brenet P, Fantozzi G, Reynaud P, Rouby D, Measurement and analysis of the p u curve giving the bridging tractions vs crack opening displacement in ®bre reinforced ceramic matrix composites. In: Naslain R, Evans AG, editors. Proceedings of the HTCMC1, 21±24 August 1995, Santa Barbara, 1995. p 259±64. 1102 C. Rospars et al. / Composites Science and Technology 60 (2000) 1095±1102
