Wave propagation, localization and dispersion in a gradient-dependent medium

A continuum model that incorporates a dependence upon the Laplacian of the inelastic strain is used to regularize the initial value problem that results from the introduction of strain softening or non-associated flow. It is shown that the introduction of this gradient dependence preserves well-posedness of the initial value problem and that wave propagation in the enhanced continuum is dispersive. An analysis of the dispersive wave propagation reveals the existence of an internal length scale. Numerical analyses of one-dimensional and two-dimensional problems confirm that this internal length scale sets the localization zone and show that the results are insensitive to the fineness of the discretization and to the direction of the grid lines. This holds true with respect to the strain profiles, the energy dissipation and the extent of wave reflection.