13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- and 0 n denote the slip direction and the slip plane normal direction, respectively for the th slip system in the undeformed configuration. The resolved shear stress on each slip system is related to the Cauchy stress tensor σ according to , σ: s n (2) where the slip vectors have been rotated into the current configuration. Under the application of the resolved shear stress, the shearing rates on the slip systems are related to the plastic velocity gradient in the intermediate configuration according to 0 0 , p L s n (3) with ascribed to follow the rate-dependent flow rule as 0 sgn , m g (4) where m is the reverse strain rate sensitivity exponent and g and are drag stress and back stress, respectively on the th slip system. These quantities evolve according to 1 , and . kin dyn g H q A A (5) Here q is the latent hardening coefficient, H, kin A and dyn A are the isotropic hardening, kinematic hardening and dynamic recovery coefficients, respectively. These non-linear coupled differential equations are solved via the fourth order Runge-Kutta integration scheme using VUMAT. 3. Conclusion A cohesive finite element based multi-scale framework is developed to simulate crack initiation and propagation in 3D polycrystalline microstructures. Crystal plasticity is incorporated into the constitutive relations so as to capture how anisotropic deformation and texture evolution affect
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