Continuum Mathematical Modeling of Slip Weakening in Geological Systems

Slip weakening is a conceptual process used to describe strength degradation during the initial stage of slip instability. For an intact or relatively undamaged rock mass the strength may consist of frictional and cohesive components along potential faults, whereas for a previously faulted rock mass the strength may be predominantly frictional. In the former case a fault may initiate and propagate, whereas in the latter case an older fault may be reactivated. We describe a framework for mathematical modeling of slip weakening in an initially intact rock mass due to shear strain localization along any arbitrary slip plane. The modeling technique considered is based on continuum mechanics and may be cast directly into a standard nonlinear finite element algorithm for the analysis of pre- and post-failure responses of geological systems in a boundary-value problem. The pre-failure behavior is represented by a continuum constitutive model; the post-failure behavior is characterized by frictional yielding on a slip surface with state- and velocity-dependent coefficient of friction. In the context of finite element analysis, slip planes are represented by an embedded strong discontinuity introduced into an initially intact finite element to signal the beginning of post-failure behavior. These discontinuity surfaces may be inserted at an arbitrary orientation and location in the element, depending on the conditions at failure. Here we focus on the narrow time interval of slip weakening, from the moment the strong discontinuity has been embedded into a finite element until the slip has grown to a large enough value for the coefficient of friction to reach steady state. To this end, we formulate a linear slip weakening constitutive law in which the weakening component decays to zero at the same time that the frictional component increases to its value at residual state. Corresponding author. E-mail: borja@stanford.edu (R.I. Borja). Supported by U.S. Department of Energy, Grant No. DE-FG02-03ER15454, and U.S. National Science Foundation, Grant No. CMG-0417521.