COMPUTATIONAL MATERIALS SCIENCE ELSEVIER Computational Materials Science 16(1999)17-24 Modelling and simulation of the mechanical behavior of ceramic matrix composites as shown by the example of Sic/sic H. Ismar. F. Streicher Lehrstuhl fur Technische Mechanik, Universitat des Saarlandes, Postfach 15 11 50, D-66 041 Saarbrucken, Germany Abstract modelling of the mechanical behavior of unidirectionally fiber-reinforced ceramic matrix composites(CMC) is presented by the example of SiC/Sic. The starting point of the modelling is a substructure (elementary cell) which includes on a micromechanical scale the statistical properties of the fiber, matrix and fiber-matrix interface and their interactions. The substructure is chosen in such a way that a macrostructure representative of the whole structure can be modelled from a suitable number of substructures. The typical damage behavior of ceramic composites is modelled by king fiber and matrix cracks into account. Cracks are inserted into the substructure by reducing the elastic coefficients of the material. The fracture criterion used is a surface represented by a spheroid in the principal stress space. The crack direction is determined by the criterion of the energy release rate. Interfacial behavior is simulated by consideration of fiber-matrix debonding and frictional sliding. The numerical evaluation of the model is accomplished by means of the finite element method (FEM). The effect of important parameters such as the fiber volume fraction or the fiber Weibull- shape parameter on the nonlinear behavior of the substructure is examined. Finally, a macrostructure is modelled to show the effects of these important parameters on the mechanical behavior of the whole structure. 1999 Elsevier Science B.v. all rights reserved. Keywords: Reinforced ceramics: Micromechanical-statistical modelling: Fiber/matrix cracking: Debonding: Friction; Fiber pull-out 1. Introduction structure. One means of making ceramics more reliable is to introduce amplifications such as ce- In the past years ceramic materials have become ramic fibers. Unlike the bulk materials, ceramic increasingly important. Especially for applications fiber composites show toughness because of energy which require high-strength at elevated tempera- dissipating mechanisms. These mechanisms are tures ceramics show a superior behavior. The shown schematically in Fig. 1. A matrix crack, problem of using ceramic materials in construction either newly formed or caused by the productio is their brittle damage behavior. A single defect process, grows because of external loading. When can lead to the total brittle damage of the whole the crack reaches the fiber surface debonding be- tween the fiber and the matrix takes place. The fiber is increasingly loaded until it reaches its ul nding author. Tel: +49-681-302-2157: fax: +49 timate strength and fiber failure occurs. During 68l-302-3992. subsequent fiber pull-out energy is dissipated by E-mail address friction in the fiber-matrix interface 0927-0256/99/. see front matter e 1999 Elsevier Science B v. All rights reserved PI:S0927-0256(99)00041-5
Modelling and simulation of the mechanical behavior of ceramic matrix composites as shown by the example of SiC/SiC H. Ismar, F. Streicher * Lehrstuhl fur Technische Mechanik, Universitat des Saarlandes, Postfach 15 11 50, D-66 041 Saarbrucken, Germany Abstract Modelling of the mechanical behavior of unidirectionally ®ber-reinforced ceramic matrix composites (CMC) is presented by the example of SiC/SiC. The starting point of the modelling is a substructure (elementary cell) which includes on a micromechanical scale the statistical properties of the ®ber, matrix and ®ber±matrix interface and their interactions. The substructure is chosen in such a way that a macrostructure representative of the whole structure can be modelled from a suitable number of substructures. The typical damage behavior of ceramic composites is modelled by taking ®ber and matrix cracks into account. Cracks are inserted into the substructure by reducing the elastic coecients of the material. The fracture criterion used is a surface represented by a spheroid in the principal stress space. The crack direction is determined by the criterion of the energy release rate. Interfacial behavior is simulated by consideration of ®ber±matrix debonding and frictional sliding. The numerical evaluation of the model is accomplished by means of the ®nite element method (FEM). The eect of important parameters such as the ®ber volume fraction or the ®ber Weibullshape parameter on the nonlinear behavior of the substructure is examined. Finally, a macrostructure is modelled to show the eects of these important parameters on the mechanical behavior of the whole structure. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: Reinforced ceramics; Micromechanical±statistical modelling; Fiber/matrix cracking; Debonding; Friction; Fiber pull-out 1. Introduction In the past years ceramic materials have become increasingly important. Especially for applications which require high-strength at elevated temperatures ceramics show a superior behavior. The problem of using ceramic materials in construction is their brittle damage behavior. A single defect can lead to the total brittle damage of the whole structure. One means of making ceramics more reliable is to introduce ampli®cations such as ceramic ®bers. Unlike the bulk materials, ceramic ®ber composites show toughness because of energy dissipating mechanisms. These mechanisms are shown schematically in Fig. 1. A matrix crack, either newly formed or caused by the production process, grows because of external loading. When the crack reaches the ®ber surface debonding between the ®ber and the matrix takes place. The ®ber is increasingly loaded until it reaches its ultimate strength and ®ber failure occurs. During subsequent ®ber pull-out energy is dissipated by friction in the ®ber±matrix interface. Computational Materials Science 16 (1999) 17±24 * Corresponding author. Tel.: +49-681-302-2157; fax: +49- 681-302-3992. E-mail address: f.streicher@rz.uni-sb.de (F. Streicher). 0927-0256/99/$ - see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 9 9 ) 0 0 041-5
H. Ismar, F. Streicher /Computational Materials Science 16(1999)17-24 Debonding of the fiber Matrix crack Pull-Out of the fiber Fig. 1. Mechanisms of energy dissipation in a ceramic com- posite structure Unidirectionally reinforced Substructure used in simulations Fig. 2. Substructure (elementary cell)used in simulations. The mechanisms which lead to the macrome. chanical nonlinear behavior of fiber reinforced ceramics have been distinguished [1] as follows Failure of the matrix crack initiation crack de 2. Underlying principles flection Failure of the fiber-matrix interface: debonding Even an uniaxial loading of fiber composite Fiber failure: crack initiation leads to a multiaxial stress state in the single Fiber pull-out: frictional effects components: fiber, matrix and fiber-matrix inter- By means of suitable selection and combination face. To take the damage under this multiaxial single components the properties of the stress state into account. a failure criterion for posite can be varied over a wide range. To obtain each component is defined. A general failure cri- the most suitable properties for the considered terion is given by Tsai and Wu [2]as application an optimization process is necessary which is often purely empirical and thus requires a1+an00=1 (1) great experimental effort The aim of the model described below is to where the contracted notation is used and study the influence of important parameters on the J=1, 2, .., 6. a, and ay are tensors of the second behavior of the fiber composite. Starting from the and fourth rank, respectively. This criterion is thermal and mechanical properties of the compo- ad oL regarded. Because of its general formula- adapted to the special requirements of the com- ents and especially their statistical distributed strength, it permits statements concerning the tion it can also be adopted in an easy way to a nonlinear behavior of the substructure presented more complex material behavior than the isotropic in Fig. 2. This substructure(elementary cell) takes behavior considered below matrix interface and their specific damage behav ior into account It is chosen in such a way that a 2.1. Matrix macrostructure characteristic for the whole struc re can be built-up by a suitable number of sub- The elastic constants of the matrix material sic. structures. To study the influence of separate Youngs modulus E, Poissons ratio v and the component parameters on the behavior of the total shear modulus G were found in our experiments to material the model has been implemented in a fi- be E=251 000 MPa, v=0.16 and G=108 000 nite element method(FEM) code. Therefore, the MPa. The coefficient of thermal expansion is substructure from Fig. 2 is transformed into a fi- a=4x 10-6K-. For the observed isotropic be- nite element mesh, involving 1920 fiber, 2720 ma- havior of the matrix material the failure criterion trix and 480 interfacial elements defined in relation (1) is reduced to
The mechanisms which lead to the macromechanical nonlinear behavior of ®ber reinforced ceramics have been distinguished [1] as follows: · Failure of the matrix: crack initiation, crack de- ¯ection. · Failure of the ®ber±matrix interface: debonding. · Fiber failure: crack initiation. · Fiber pull-out: frictional eects. By means of suitable selection and combination of single components the properties of the composite can be varied over a wide range. To obtain the most suitable properties for the considered application an optimization process is necessary which is often purely empirical and thus requires great experimental eort. The aim of the model described below is to study the in¯uence of important parameters on the behavior of the ®ber composite. Starting from the thermal and mechanical properties of the components and especially their statistical distributed strength, it permits statements concerning the nonlinear behavior of the substructure presented in Fig. 2. This substructure (elementary cell) takes the single components matrix, ®ber, the ®ber± matrix interface and their speci®c damage behavior into account. It is chosen in such a way that a macrostructure characteristic for the whole structure can be built-up by a suitable number of substructures. To study the in¯uence of separate component parameters on the behavior of the total material the model has been implemented in a ®- nite element method (FEM) code. Therefore, the substructure from Fig. 2 is transformed into a ®- nite element mesh, involving 1920 ®ber, 2720 matrix and 480 interfacial elements. 2. Underlying principles Even an uniaxial loading of ®ber composites leads to a multiaxial stress state in the single components: ®ber, matrix and ®ber±matrix interface. To take the damage under this multiaxial stress state into account, a failure criterion for each component is de®ned. A general failure criterion is given by Tsai and Wu [2] as airi aijrirj 1; 1 where the contracted notation is used and i;j 1; 2; . . . ; 6. ai and aij are tensors of the second and fourth rank, respectively. This criterion is adapted to the special requirements of the component regarded. Because of its general formulation it can also be adopted in an easy way to a more complex material behavior than the isotropic behavior considered below. 2.1. Matrix The elastic constants of the matrix material SiC: Young's modulus E, Poisson's ratio m and the shear modulus G were found in our experiments to be E 251 000 MPa, m 0:16 and G 108 000 MPa. The coecient of thermal expansion is a 4 10ÿ6 Kÿ1 . For the observed isotropic behavior of the matrix material the failure criterion de®ned in relation (1) is reduced to Fig. 2. Substructure (elementary cell) used in simulations. Fig. 1. Mechanisms of energy dissipation in a ceramic composite structure. 18 H. Ismar, F. Streicher / Computational Materials Science 16 (1999) 17±24
H. Ismar, F. Streicher Computational Materials Science 16(1999)17-24 a(o1+a2+)+a1(+2+3+2 and compressive loading. If the stress vector in a matrix element reaches the fracture surface. a +2G3+206)+2a(0102+0203 crack is initiated. The modelling is carried out with +0301-02-G3-03)=1 (2) a crack model that does not change the topology of the finite element structure. but varies the ele The remaining constants al, au and aj can be determined from the values of strength under ment specific elastic constants [5]. The orientation of the crack is determined by using the energy re- uniaxial tension (ut), uniaxial compression (uc) lease rate [6]. The difference in the potential energy between the state before and the state after crack strength values is taken into account by statistics. initiation in relation to the crack surface has to be They can be considered to be Weibull distributed [3]. The Weibull distribution describes the proba- maximum. The energy release rate is given by bility of failure of a specimen at a given stress o as a function of its volume v. with△A=A-A· denoting the change in the crack P(a)=1-exp-vx urface, and AU=U-U the change in the po- tential energy. The parameters without an asterisk The Weibull-shape parameter m characterizes the represent the state before crack initiation; the pa- dispersion of the failure stress. The smaller the rameters with an asterisk. the situation afterwards value of m the greater dispersion. The constant After the crack orientation is determined the crack Go can be interpreted as the stress up to which 63% is modelled by reducing elastic constants in of the specimens have failed. Vo is the reference in the direction of the normal of the crack surface volume used to determine go. Ou is the stress below x, the Poissons ratios vry, nd the shear which no failure occurs. In the case of ceramic moduli Gv, Grg within the elasticity matrix are materials one cannot exclude failure even in the reduced. Afterwards the tensor of elastic constants case of very small stresses because of inhomoge- is transformed back to the original system(x, y, z) neities that are always present. Therefore, u is assumed to be zero [4]. Thus, Eq(3)can be re- By introducing this first crack the behavior of the written a matrix element changes from isotropic to trans- versalisotropic. It is possible to insert three or- (4) thogonal cracks into each element. If, in the case of load reversal. a crack is loaded under com- pression, the original stiffness is restored, because the crack closes and can transfer load again damage under compression occurs, the material G (5) collapses completely and all stiffnesses of the ele- ment are reduced Failure is checked independently for each finite element. The average volume of a matrix element 2. 2. Fiber conducts V= 4x10-8 m3. The values used re ferring to this volume are out= 500 MPa The fiber also shows isotropic behavior, so the 00= 1490 MPa, 00=675 MPa and mm=9.7. criterion defined in Eq. (2)can be used to describe For these strength values the surface of fracture is fiber failure. The elastic and thermal constants represented by a spheroid in the principle stress E= 200 000 MPa, v=0.25,G=80 000 MPa and space. The axis of the spheroid coincides with the a=3x 10-K- are taken from the literature pace diagonal of the principle stress state. Its [7, 8]. If a fiber of constant diameter is regarded center is moved towards the seventh octant be the fracture probability for a fiber with a length L cause of the differing strength values for tension is defined [9] by Eq (4)with
a1 r1 r2 r3 a11 r2 1 ÿ r2 2 r2 3 2r2 4 2r2 5 2r2 6 2a12 r1r2 r2r3 r3r1 ÿ r2 4 ÿ r2 5 ÿ r2 6 1: 2 The remaining constants a1, a11 and a12 can be determined from the values of strength under uniaxial tension (ut), uniaxial compression (uc) and pure shear (s). The strong scattering of the strength values is taken into account by statistics. They can be considered to be Weibull distributed [3]. The Weibull distribution describes the probability of failure of a specimen at a given stress r as a function of its volume V: Pf r 1 ÿ exp ÿ V V0 r ÿ ru r0 m : 3 The Weibull-shape parameter m characterizes the dispersion of the failure stress. The smaller the value of m the greater the dispersion. The constant r0 can be interpreted as the stress up to which 63% of the specimens have failed. V0 is the reference volume used to determine r0. ru is the stress below which no failure occurs. In the case of ceramic materials one cannot exclude failure even in the case of very small stresses because of inhomogeneities that are always present. Therefore, ru is assumed to be zero [4]. Thus, Eq. (3) can be rewritten as Pf r 1 ÿ exp ÿ r r 0 m 4 with r 0 V0 V 1=m r0: 5 Failure is checked independently for each ®nite element. The average volume of a matrix element conducts V 4 10ÿ18 m3: The values used referring to this volume are r 0;ut 500 MPa, r 0;uc 1490 MPa, r 0;s 675 MPa and mm 9:7. For these strength values the surface of fracture is represented by a spheroid in the principle stress space. The axis of the spheroid coincides with the space diagonal of the principle stress state. Its center is moved towards the seventh octant because of the diering strength values for tension and compressive loading. If the stress vector in a matrix element reaches the fracture surface, a crack is initiated. The modelling is carried out with a crack model that does not change the topology of the ®nite element structure, but varies the element speci®c elastic constants [5]. The orientation of the crack is determined by using the energy release rate [6]. The dierence in the potential energy between the state before and the state after crack initiation in relation to the crack surface has to be maximum. The energy release rate is given by G DU DA 6 with DA A ÿ A denoting the change in the crack surface, and DU U ÿ U the change in the potential energy. The parameters without an asterisk represent the state before crack initiation; the parameters with an asterisk, the situation afterwards. After the crack orientation is determined the crack is modelled by reducing elastic constants in the crack system (xr ; yr ;zr ). Young's modulus Exr in the direction of the normal of the crack surface xr , the Poisson's ratios mxryr , mxrzr and the shear moduli Gxryr , Gxrzr within the elasticity matrix are reduced. Afterwards the tensor of elastic constants is transformed back to the original system (x; y;z). By introducing this ®rst crack the behavior of the matrix element changes from isotropic to transversalisotropic. It is possible to insert three orthogonal cracks into each element. If, in the case of load reversal, a crack is loaded under compression, the original stiness is restored, because the crack closes and can transfer load again. If damage under compression occurs, the material collapses completely and all stinesses of the element are reduced. 2.2. Fiber The ®ber also shows isotropic behavior, so the criterion de®ned in Eq. (2) can be used to describe ®ber failure. The elastic and thermal constants E 200 000 MPa, m 0:25, G 80 000 MPa and a 3 10ÿ6 Kÿ1 are taken from the literature [7,8]. If a ®ber of constant diameter is regarded, the fracture probability for a ®ber with a length L is de®ned [9] by Eq. (4) with H. Ismar, F. Streicher / Computational Materials Science 16 (1999) 17±24 19
H Ismar, F Streicher/ Computational Materials Science 16(1999)17-24 o=/4o)1 where f is the surface area of the interface element (7) and f is a reference area. Examinations on in- terfacial properties can be found in the literature Fiber cracking usually begins at defects or flaws at [14-17]. The strength values referring to an inter the fiber surface and leads to brittle fiber damage face area F=5x 10-12 m2 are o.=300. Thereby the crack surface can be assumed to be douc 2400 and oos=300 MPa. They are results vertical to the fiber axis. To model this behavior of our parameter studies on interfacial behavior of the fiber is subdivided vertical to the fiber axis in unidirectional composites with a fiber volume pieces of the length L. If the stress in a fiber piece fraction uf= 45%. The Weibull parameter m and eaches the failure surface a crack is introduced the friction coefficient u are 8 and 0.6, respectively vertical to the fiber axis. The fiber strength values used [6, 10-12] for a regarded length of 5 x 10-m are oout= 3054, 0ouc=6980 and oos =2704 MPa3Substructure n The actual statistical fiber distribution in uni- 2.3. Fiber-matrix interface directionally reinforced ceramics is well described by a regular hexagonal fiber disposition. For this reason an elementary cell with a hexagonal fiber The fiber-matrix interfacial zone plays a key arrangement is used. The numerical simulations role in the mechanical behavior of composite ma- are carried out on substructures of different fiber terials. Its properties can be influenced by different volume fractions tr. Thereby the length of the fiber coatings such as C. bn or SiC. The most substructure . =100 10-6 m and the fiber di important task of the interface is the crack de ameter r=15x 10-6 m are the same for all fiber flection at the fiber surface after initiation of ma trix cracking. Thereby debonding length, matrix volume fractions. a different fiber content of the crack saturation and frictional sliding in the fiber substructure is realized by changing the edge lengths I, and Iy. In the case of uf= 45%, for ex- matrix interface after debonding are controlled by ample, I, and L, are 18.45 x 10-6 and 10.65x10-6 [13] to occur under the combined action of normal ress o and interfacial shear stress t In this case The calculations were performed with different q(2)can be rewritten as statistical strength distributions. At the beginning of the simulations each finite element randomly a1Gn+a1a2+2(a1-a2)2=1 obtains its strength values according to the Wei- bull distribution and the element specific fracture In the finite element mesh the interface is urface is defined. The values are distributed to the modelled as a region of negligible thickness, which elements in such a way that only the size of the forces a bonding of fiber and matrix. If the d fracture surface is varied according to statistics, bonding criterion( 8) is fulfilled for an interface but not the shape itself. element, debonding is introduced by reducing all as a result of the mismatch in coefficients of elastic constants of this element to zero. After thermal expansion between the constituer nts resid- debonding the interfacial stress caused by friction ual stresses are developed by the cooling down between the fiber and the matrix is calculated b ess after fabrication of the o take Ising Coulomb's law. Analogous to fiber and these stresses into account the substructure is matrix the strength of the interface is considered to cooled down from the stress free temperature 1025 be Weibull distributed. The constant oo is defined to 25 K [7. The side and front surfaces of the el- ementary cell are kept plane during thermal and mechanical loading. As a result of the cooling =(F (9) process there are compressive stresses in the 2-di- rection in the fibers while the matrix is loaded by
r 0 L0 L 1=m r0: 7 Fiber cracking usually begins at defects or ¯aws at the ®ber surface and leads to brittle ®ber damage. Thereby the crack surface can be assumed to be vertical to the ®ber axis. To model this behavior the ®ber is subdivided vertical to the ®ber axis in pieces of the length L. If the stress in a ®ber piece reaches the failure surface a crack is introduced vertical to the ®ber axis. The ®ber strength values used [6,10±12] for a regarded length of 5 10ÿ6 m are r 0;ut 3054, r 0;uc 6980 and r 0;s 2704 MPa and mf 8. 2.3. Fiber±matrix interface The ®ber±matrix interfacial zone plays a key role in the mechanical behavior of composite materials. Its properties can be in¯uenced by dierent ®ber coatings such as C, BN or SiC. The most important task of the interface is the crack de- ¯ection at the ®ber surface after initiation of matrix cracking. Thereby debonding length, matrix crack saturation and frictional sliding in the ®ber± matrix interface after debonding are controlled by the interfacial properties. Debonding is postulated [13] to occur under the combined action of normal stress rn and interfacial shear stress s. In this case Eq. (2) can be rewritten as a1rn a11r2 n 2 a11 ÿ a12s2 1: 8 In the ®nite element mesh the interface is modelled as a region of negligible thickness, which forces a bonding of ®ber and matrix. If the debonding criterion (8) is ful®lled for an interface element, debonding is introduced by reducing all elastic constants of this element to zero. After debonding the interfacial stress caused by friction between the ®ber and the matrix is calculated by using Coulomb's law. Analogous to ®ber and matrix the strength of the interface is considered to be Weibull distributed. The constant r 0 is de®ned as r 0 F0 F 1=m r0; 9 where F is the surface area of the interface element and F0 is a reference area. Examinations on interfacial properties can be found in the literature [14±17]. The strength values referring to an interface area F 5 10ÿ12 m2 are r 0;ut 300, r 0;uc 2400 and r 0;s 300 MPa. They are results of our parameter studies on interfacial behavior of unidirectional composites with a ®ber volume fraction vf 45%. The Weibull parameter mi and the friction coecient l are 8 and 0.6, respectively. 3. Substructure The actual statistical ®ber distribution in unidirectionally reinforced ceramics is well described by a regular hexagonal ®ber disposition. For this reason an elementary cell with a hexagonal ®ber arrangement is used. The numerical simulations are carried out on substructures of dierent ®ber volume fractions vf. Thereby the length of the substructure lz 100 10ÿ6 m and the ®ber diameter rf 15 10ÿ6 m are the same for all ®ber volume fractions. A dierent ®ber content of the substructure is realized by changing the edge lengths lx and ly . In the case of vf 45%, for example, lx and ly are 18:45 10ÿ6 and 10:65 10ÿ6 m, respectively. The calculations were performed with dierent statistical strength distributions. At the beginning of the simulations each ®nite element randomly obtains its strength values according to the Weibull distribution and the element speci®c fracture surface is de®ned. The values are distributed to the elements in such a way that only the size of the fracture surface is varied according to statistics, but not the shape itself. As a result of the mismatch in coecients of thermal expansion between the constituents residual stresses are developed by the cooling down process after fabrication of the composite. To take these stresses into account the substructure is cooled down from the stress free temperature 1025 to 25 K [7]. The side and front surfaces of the elementary cell are kept plane during thermal and mechanical loading. As a result of the cooling process there are compressive stresses in the z-direction in the ®bers while the matrix is loaded by 20 H. Ismar, F. Streicher / Computational Materials Science 16 (1999) 17±24
H. Ismar, F Streicher Computational Materials Science 16(1999)17-24 tension in that direction the fiber-matrix inter- face is loaded by compressive stress normal to the fiber surface m=8 After the cooling down process the substructure loaded by tension in the fiber direction. The resultant nonlinear behavior for a fiber volume fraction uf= 45% is shown in Fig. 3. The main damage mechanisms described above can be identified in the resulting stress-strain curve. Up to a the structure shows an elastic behavior Between 0002040.6081.012141.61.8 A and B multiple matrix cracking appears. At B matrix crack saturation is reached and it follows the fiber dominant part up to C. At C the first of Fig. 4. Calculated stress-strain diagrams of the substructure the two fibers present in the substructure breaks for 20 simulations and causes a reduction of stress from d to e there is a superposition of pull-out of the broken rameter leads to an enlarged scattering of fiber fiber and the further loading of the other fiber. The strength. Comparing the calculations for the sub second fiber also breaks at E, resulting in the pull- structure with mf=4, 8 and 16 an increased in ut of both fibers until each fiber is completely terval of strain between the fracture of the first and pulled out of the matrix. the last fiber can be observed for smaller values of Fig 4 shows the dependence of the substructure behavior on the statistical distribution of the Fig. 6 shows the variation of the fiber volume strength values. There the results of 20 simulations fraction. The Youngs moduli of the undamaged with different strength distribution can be seen. material were found to be 236 000, 228 000 and Within these 20 simulations phenomena such as 220 000 MPa for a fiber volume fraction of 30%, fiber and matrix cracking are clearly smeared over 45% and 60%, respectively. After crack saturation a wide range. The appearance of different pull-out in the matrix the stresses are mainly transferred by length can be identified by different friction stresses the fibers. Therefore, in this fiber dominant part of during the fiber pull-out the stress-strain curve after matrix crack satura- To indicate the influence of important param- tion a greater Youngs modulus for an increased eters on the behavior of the elementary cell pa- fiber volume fraction can be observed. It can also rameter studies were performed. In Fig. 5 the be seen that the stress of matrix crack initiation variation of the fiber Weibull-shape parameter mf and the subsequent stress reduction depend on Uf resented. a decreased fiber Weibull-shape pa- For a smaller fiber volume fraction matrix crack. ing is moved to higher stresses mainly because of lower residual stresses in the matrix after the a100v=45% cooling down process E 4. Macrostructure The unidirectionally reinforced Sic/SiC com- posite consists of many fibers. However, beyond a definite number of fibers. the statistical effects change only insignificantly. This is the reason why 002040.6081.0121.4 the main macroscopi enomena can be ined with a macrostructure composed of a suffi- Fig. 3. Calculated stress-strain diagram of the substructure cient number of substructures n. It was found in
tension in that direction. The ®ber±matrix interface is loaded by compressive stress normal to the ®ber surface. After the cooling down process the substructure is loaded by tension in the ®ber direction. The resultant nonlinear behavior for a ®ber volume fraction vf 45% is shown in Fig. 3. The main damage mechanisms described above can be identi®ed in the resulting stress±strain curve. Up to A the structure shows an elastic behavior. Between A and B multiple matrix cracking appears. At B matrix crack saturation is reached and it follows the ®ber dominant part up to C. At C the ®rst of the two ®bers present in the substructure breaks and causes a reduction of stress. From D to E there is a superposition of pull-out of the broken ®ber and the further loading of the other ®ber. The second ®ber also breaks at E, resulting in the pullout of both ®bers until each ®ber is completely pulled out of the matrix. Fig. 4 shows the dependence of the substructure behavior on the statistical distribution of the strength values. There the results of 20 simulations with dierent strength distribution can be seen. Within these 20 simulations phenomena such as ®ber and matrix cracking are clearly smeared over a wide range. The appearance of dierent pull-out length can be identi®ed by dierent friction stresses during the ®ber pull-out. To indicate the in¯uence of important parameters on the behavior of the elementary cell parameter studies were performed. In Fig. 5 the variation of the ®ber Weibull-shape parameter mf is presented. A decreased ®ber Weibull-shape parameter leads to an enlarged scattering of ®ber strength. Comparing the calculations for the substructure with mf 4; 8 and 16 an increased interval of strain between the fracture of the ®rst and the last ®ber can be observed for smaller values of mf. Fig. 6 shows the variation of the ®ber volume fraction. The Young's moduli of the undamaged material were found to be 236 000, 228 000 and 220 000 MPa for a ®ber volume fraction of 30%, 45% and 60%, respectively. After crack saturation in the matrix the stresses are mainly transferred by the ®bers. Therefore, in this ®ber dominant part of the stress±strain curve after matrix crack saturation a greater Young's modulus for an increased ®ber volume fraction can be observed. It can also be seen that the stress of matrix crack initiation and the subsequent stress reduction depend on vf. For a smaller ®ber volume fraction matrix cracking is moved to higher stresses mainly because of lower residual stresses in the matrix after the cooling down process. 4. Macrostructure The unidirectionally reinforced SiC/SiC composite consists of many ®bers. However, beyond a de®nite number of ®bers, the statistical eects change only insigni®cantly. This is the reason why the main macroscopic phenomena can be examined with a macrostructure composed of a su- cient number of substructures n Fig. 3. Calculated stress±strain diagram of the substructure. . It was found in Fig. 4. Calculated stress±strain diagrams of the substructure for 20 simulations. H. Ismar, F. Streicher / Computational Materials Science 16 (1999) 17±24 21