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 toThe mechanisms which lead to the macrome￾chanical 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 e€ects. By means of suitable selection and combination of single components the properties of the com￾posite 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 e€ort. 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 compo￾nents 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 behav￾ior into account. It is chosen in such a way that a macrostructure characteristic for the whole struc￾ture can be built-up by a suitable number of sub￾structures. 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 ma￾trix 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 inter￾face. To take the damage under this multiaxial stress state into account, a failure criterion for each component is de®ned. A general failure cri￾terion is given by Tsai and Wu [2] as airi ‡ aijrirj ˆ 1; …1† where the contracted notation is used and i;j ˆ 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 com￾ponent regarded. Because of its general formula￾tion 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 ˆ 251 000 MPa, m ˆ 0:16 and G ˆ 108 000 MPa. The coecient of thermal expansion is a ˆ 4 10ÿ6 Kÿ1 . For the observed isotropic be￾havior 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 com￾posite structure. 18 H. Ismar, F. Streicher / Computational Materials Science 16 (1999) 17±24