Math @ Duke

Publications [#304499] of Thomas P. Witelski
search www.ams.org.Papers Published
 Shearer, M; Schaeffer, DG; Witelski, TP, Stability of shear bands in an elastoplastic model for granular flow: The role of discreteness,
Mathematical Models and Methods in Applied Sciences, vol. 13 no. 11
(2003),
pp. 16291671 [doi]
(last updated on 2018/08/17)
Abstract: Continuum models for granular flow generally give rise to systems of nonlinear partial differential equations that are linearly illposed. 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 largetime behavior of solutions in this model depends on an elastic shear modulus ε. For large and moderate values of ε, the model has stable steadystate solutions with uniform shearing except for one shear band; almost all solutions tend to one of these as t → ∞. However, when ε becomes sufficiently small, the singleshearband solutions lose stability through a Hopf bifurcation. The value of ε 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, timeperiodic solutions appear, containing both elastic and plastic waves, and (ii) the bifurcation diagram representing these solutions exhibits bistability.


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