Abstract
Continuum models for granular flow generally give rise to systems of nonlinear partial differential equations that are linearly ill-posed. In this paper we introduce discreteness into an elastoplasticity model for granular flow by approximating spatial derivatives with finite differences. The resulting ordinary differential equations have bounded solutions for all time, a consequence of both discreteness and nonlinearity. We study how the large-time behavior of solutions in this model depends on an elastic shear modulus ℰ. For large and moderate values of ℰ, the model has stable steady-state solutions with uniform shearing except for one shear band; almost all solutions tend to one of these as t→∞. However, when ℰ becomes sufficiently small, the single-shear-band solutions lose stability through a Hopf bifurcation. The value of ℰ at the bifurcation point is proportional to the ratio of the mesh size to the macroscopic length scale. These conclusions are established analytically through a careful estimation of the eigenvalues. In numerical simulations we find that: (i) after stability is lost, time-periodic solutions appear, containing both elastic and plastic waves, and (ii) the bifurcation diagram representing these solutions exhibits bi-stability.
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