The mechanical constitutive behavior of geomaterials is quite complex, involving pressure-sensitive yielding, differences in strength in triaxial extension vs. compression, the Bauschinger effect, dependence on porosity, and other factors. Capturing these behaviors necessitates the use of fairly complicated and expensive non-linear material models. For elastically isotropic materials, such models usually involve three-invariant plasticity formulations. Spectral decomposition has been used to increase the efficiency of numerical simulation for such models for the isotropically hardening case. We modify the spectral decomposition technique to models that include kinematic hardening. Finally, we perform some numerical simulations to demonstrate quadratic convergence.
A shallow earthquake focus mechanism is modeled by a plane strain shear crack extending at a nonuniform rate under the action of general loading. An equation of motion of the crack tip is determined in terms of a specific fracture energy, i.e., a surface energy, by applying an overall energy balance fracture criterion. If the specific fracture energy is specified, the motion can be determined. On the other hand, if any particular motion of the fault boundary is specified, the requisite specific fracture energy can be calculated from the equation of motion. A process of destruction of unbroken material is included in which the relative sliding of the faces of the fracture surface leads to complete removal of stress in the fractured area. The strength of the dynamic stress singularity at the crack tip is written as a product of an equivalent static stress singularity multiplied by a universal function of crack velocity, the universal function decreasing with increasing crack tip velocity.
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