STABILITY OF SHEAR BANDS IN AN ELASTOPLASTIC MODEL FOR GRANULAR FLOW: THE ROLE OF DISCRETENESS

Author:

SHEARER MICHAEL1,SCHAEFFER DAVID G.2,WITELSKI THOMAS P.2

Affiliation:

1. Department of Mathematics and Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695, USA

2. Department of Mathematics and Center for Nonlinear and Complex Systems, Duke University, Durham, NC 27708-0320, USA

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.

Publisher

World Scientific Pub Co Pte Lt

Subject

Applied Mathematics,Modeling and Simulation

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