13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- 2.4. Crystal plasticity formulation Plastic deformation at the macroscopic scale in metals is a manifestation of dislocation motion and interaction at the microscopic scale. The details are intimately related to the basic crystallographic nature of the material as well as the current state of the microstructure. Macroscopic models of plasticity lack the ability to link these fundamental mechanisms to the bulk material response without very substantial experimental characterization. Furthermore, these models give relatively little physical connection between the actual deformation processes and the observed material behavior. Many formulations of constitutive laws for the elastic-plastic deformation of single and polycrystals have long been proposed (cf. Talor [12], Hill and Rice [13], Asaro and Rice [14], Peire et al. [15] and McGinty and McDowell [16]). The basic premise of these theories is that macroscopic plastic deformation is related to the cumulative process of slip (or twin) system shearing relative to the lattice. This methodology provides a physical link between the processes at radically different length scales. The two basic components of crystal plasticity model are the kinematic and kinetic relations. The kinematic relations provide the mathematical framework for describing the physical process of dislocation motion based on continuum deformation fields, whereas the kinetic relations incorporate the material and/or mechanism dependence of non-equilibrium dislocation motion and interaction. Fig. 4 Shell mesh region around the unsmoothed microstructure mesh. The multiplicative decomposition of the total deformation gradient is given by e p F F F , (1) where eF is the elastic deformation gradient representing the elastic stretch and rotation of lattice, and pF is the plastic deformation gradient describing the collective effects of dislocation motion along the active slip planes relative to a fixed lattice in the reference configuration. Unit vectors 0 s
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