ACTA MECHANICA SOLIDA SINICA model could be developed, the spatial distribution of crack initiation, coalescence, propagation, and thus a real fractural process in composite could be studied which is of great importance and significance for understanding of the mechanism of the fractural process and toughening of composite ceramics IL. NUMERICAL MODELING Solid material is always heterogeneous because of the presence of micro- weakness or fault in meso- scale. Thus the heterogeneity of material is essential to model verification. Compared to the traditional numerical method, where laminated composite ceramic is usually treated as homogeneous material by many researchers, great improvement has been made with a code named RFPA3D(3D Realistic Failure Process Analysis The ceramic matrix here is divided into cells of a hexahedron, and the heterogeneity of mechanical features is represented with a two parameters Weibull distribution. The probability distribution function can be described as follow f(a) e(ayao)m where a stands for mechanical characteristic parameters, such as Youngs modulus, strength, the P( sons ratio, etc. ao is the mean value of a, m is the shape factor of Weibull, defined as the homogeneity index of material The choice of a proper fracture criterion is crucial to cracking simulation. A Coulomb criterion[ 8 envelope with a tensile cut-off is adopted in this paper for brittle materials. The tensile failure will occur when the principal stress in an element is greater than its tensile strength, see formula(2). Meanwhile to simulate the shear failure. the second valve criterion as formula (3) is also used 01≥ 1+sin o > where o is the frictional angle, o1 and o3 are the maximum and minimum principal stresses, respectively, ot and c are the uniaxial tensile and compression strength of an element, respectively. After a displacement vector is applied to the model, stress and deformation in each element are then computed. When the fracture criterion is met in an element, the element is considered to be weak or failed. The failed element has been applied a very low elastic modulus instead of being removed from the mesh. The stress and the deformation distribution throughout the model are adjusted instantaneously after each element ruptures in an equilibrium state. In the areas with increased stress due to stress redistribution, the stress may exceed the critical value so that further ruptures will occur. The process would be repeated until no more elements exceed the fracture criterion under the same load. Then, the calculation would move to the next step by a small increment and the procedure can be repeated until the whole specimen fractures In this calculation, the stress and strain can be obtained by introducing a damage factor D E(1-D)E0 Fig. 1. Three-dimensional numerical model of laminated ceramic· 142 · ACTA MECHANICA SOLIDA SINICA 2007 model could be developed, the spatial distribution of crack initiation, coalescence, propagation, and thus a real fractural process in composite could be studied, which is of great importance and significance for understanding of the mechanism of the fractural process and toughening of composite ceramics. II. NUMERICAL MODELING Solid material is always heterogeneous because of the presence of micro-weakness or fault in meso￾scale. Thus the heterogeneity of material is essential to model verification. Compared to the traditional numerical method, where laminated composite ceramic is usually treated as homogeneous material by many researchers, great improvement has been made with a code named RFPA3D (3D Realistic Failure Process Analysis). The ceramic matrix here is divided into cells of a hexahedron, and the heterogeneity of mechanical features is represented with a two parameters Weibull distribution. The probability distribution function can be described as follows: f (α) = m α0 · α α0 m−1 · e−(α/α0)m (1) where α stands for mechanical characteristic parameters, such as Young’s modulus, strength, the Pois￾son’s ratio, etc. α0 is the mean value of α, m is the shape factor of Weibull, defined as the homogeneity index of material. The choice of a proper fracture criterion is crucial to cracking simulation. A Coulomb criterion[8] envelope with a tensile cut-off is adopted in this paper for brittle materials. The tensile failure will occur when the principal stress in an element is greater than its tensile strength, see formula (2). Meanwhile, to simulate the shear failure, the second valve criterion as formula (3) is also used. σ1 ≥ σt (2) and/or 1 + sin φ 1 − sin φσ1 − σ3 ≥ σc (3) where φ is the frictional angle, σ1 and σ3 are the maximum and minimum principal stresses, respectively, σt and σc are the uniaxial tensile and compression strength of an element, respectively. After a displacement vector is applied to the model, stress and deformation in each element are then computed. When the fracture criterion is met in an element, the element is considered to be weak or failed. The failed element has been applied a very low elastic modulus instead of being removed from the mesh. The stress and the deformation distribution throughout the model are adjusted instantaneously after each element ruptures in an equilibrium state. In the areas with increased stress due to stress redistribution, the stress may exceed the critical value so that further ruptures will occur. The process would be repeated until no more elements exceed the fracture criterion under the same load. Then, the calculation would move to the next step by a small increment and the procedure can be repeated until the whole specimen fractures. In this calculation, the stress and strain can be obtained by introducing a damage factor D. ε = σ E = σ (1 − D)E0 (4) Fig. 1. Three-dimensional numerical model of laminated ceramic.