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Papers Published

1. D. G. Schaeffer, M. Shearer, and Thomas P. Witelski, One-dimensional solutions in an elastoplasticity model of granular materials, Mathematical Models and Methods in Applied Sciences, vol. 13 no. 11 (November, 2003), pp. 1629-1671 [html] .
(last updated on 2004/07/22)

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 ${\calE}$. For large and moderate values of ${\calE}$, the model has stable steady-state solutions with uniform shearing except for one shear band; indeed, almost all solutions tend to one of these as $t \to \infty$. However, when ${\calE}$ becomes sufficiently small, the single-shear-band solutions lose stability through a Hopf bifurcation. The value of ${\calE}$ 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, solutions containing both elastic and plastic waves, and (ii)~the bifurcation diagram representing these solutions exhibits bi-stability.