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Publications [#14008] of Thomas P. Witelski
search www.ams.org.Papers Published
- 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.
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