13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- athermal stress c = U0 / . For a traction smaller than threshold stress n 0 neither damage nor reaction-rupture takes place. If the initial traction is higher than the threshold n 0, the process will take place until a crack nucleates. For a traction smaller than the threshold stress n 0 before the crack nucleation, the reaction-rupture stops. In a finite element analysis, the related traction-opening is taken as ) ( c n n n n k , (2) where n is the normal traction increment, n is the prescribed opening rate, c n the damage opening rate, kn is a stiffness that is taken large enough to ensure n c n during the reaction rupture process. In this cohesive model formulation, we do not account for a contribution of the tangential mode in the reaction-rupture process. A simple elastic response is considered as t t t k , (3) with t is the tangential traction increment, t the prescribed shear rate along the cohesive surface and kt the tangential stiffness. A quasi-static finite analysis is considered which uses a total Lagrangian description, the incremental shape of virtual work for this problem as (Romero de la Osa et al. [3,4]) dS Scz V dS T V dV . u . . , (4) where V and V respectively represent the volume of the region in the initial configuration and its boundary, and SCZ is the cohesive surface considered. The index corresponds to the normal and tangential components in the cohesive formulation. In (Eq. 4), is the second Piola-Kirchhoff stress tensor, T the corresponding traction vector; and are the conjugate Lagrangian strain rate and velocity. The governing equations are solved in a linear incremental fashion based on the rate form of (Eq. 4). 2.1. Calibration of cohesive zone parameters for Zirconia We define a case study with an elastic bulk representing a single crystal (see Fig. 2a) under a static load. A linear elastic isotropic bulk is considered with an initial crack that is subjected to mode I and constant prescribed stress intensity KI. Cohesive zone is inserted along the crack symmetry plane, where the principal stress is maximum. Small scale damage confined around the crack tip is assumed and the boundary layer approach is used to investigate mode I plane strain conditions. The mesh is refined around the crack propagation path (see Fig. 2b) and cohesive element of 1nm long is used. We prescribe a constant load in terms of stress intensity factor KI as record the crack advance with time. We extract the velocity in the steady state regime and get one point in the V-KI curve. By repeating this procedure, we are able to adjust the prediction with available experimental data [8]. The values of the Young modulus E iso = 315 GPa and Poisson ratio iso = 0.24 are derived from the cubic elastic constants of zirconia single crystals (see (Table 1). We adopt n cr = 1nm, and based on Zhurkov [8] the energy barrier U0 is of the order of the sublimation energy, with U0 = 160kJ/mol. We still have to identify and 0 , controlling the slope of V-KI and 0 its position
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