constraint. For this reason, the interaction of the tracked front with the underlying hyperbolic grid is nontrivial and
In this paper we describe a numerical algorithm for the study of shear band, formation and growth in two-dimensional antiplane we develop techniques to allow for the growth of the shear shear. The constitutive model uses a non-associative flow rule. The band by more than one cell per time step. numerical scheme is based on a Godunov method for updating the
In particular, the model used in this paper includes a velocity, while the stress is updated via integration along particle non-associative flow rule. Non-associative flow rules are paths. The scheme is coupled with a front tracking algorithm for recognized as an important tool in the modeling of deforcareful evolution of the shear bands. The main challenges are the non-hyperbolicity of the shear band formation and growth (front mations in granular materials (see Vardoulakis and Graf tracking avoids the catastrophic effects of the loss of hyperbolicity [21]). This present model has been studied by Schaeffer in the Godunov-type numerical scheme), the existence of endpoints [13] and offers a criterion for the formation of shear bands for the shear band (the tracked front does not separate the computaand a description of the evolution of the material inside tional domain into disconnected regions), and the non-hyperbolic the band. rate of growth of the shear band. We give examples of the success of the algorithm and show convergence tests. แฎ 1997 Academic Press
The non-associativity of the flow rule implies ill-posedness, in the sense of Hadamard, at certain levels of stress. Schaeffer [13] formulated a relation between ill-posedness