| Localized shear deformation plays an important role in a number of geotechnical and geological processes. Slope failures, the formation and propagation of faults, cracking in concrete dams, and shear fractures in subsiding hydrocarbon reservoir rock are examples of important consequences of shear localization. Traditional engineering analyses of these phenomena, such as limit equilibrium techniques, make certain assumptions on the shape of the failure surface as well as other simplifications. While these methods provide reasonable, if conservative, results for the applications for which they were designed, many problems have more complex geometries and loading patterns.; The finite element method is one approach to modeling the behavior of bodies with complex geometries. Standard finite elements capture varied loading and deformation patterns on arbitrary bodies quite well, and advancements in material modeling have led to more accurate prediction of the stresses and the onset of localization. However, for numerical reasons standard finite elements have difficulty capturing the softening and anisotropic damage that accompanies localized deformation. To solve this difficulty, an enhanced element has been developed that is capable of capturing a failure surface at an arbitrary position and orientation within the element.; Previous studies with this element have been limited to simple surface constitutive models. Physical experiments of rock, however, show more complex behavior. Initially, the shear strength drops rapidly as a coherent macrocrack develops. As the cohesive strength drops, it is replaced by a frictional response. The friction may vary with slip speed, wear on a changing population of contacts, temperature, and other factors. A rate- and state-dependent friction model developed by Dieterich, Ruina, Rice and others captures this behavior and models stable and unstable slip on surfaces. To capture the response of the localizing material in an arbitrary body, we embed this coupled slip weakening-frictional response into the enhanced element. The initiation of localization is captured by tracking the onset of a material instability (bifurcation condition). The coupled model is embedded numerically in the element using a generalized trapezoidal scheme. The slip speed and state variable are solved via element-level Newton iteration. |
