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Boundary-Value Problems for Hyperbolic Equations Related to Steady Granular Flow

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

Michael Shearer

Department of Mathematics and Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695-8205, Duke University, Durham, NC 27708-0320, USA

Thomas P. Witelski

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

Boundary value problems for steady-state flow in elastoplasticity are a topic of mathematical and physical interest. In particular, the underlying PDE may be hyperbolic, and uncertainties surround the choice of physically appropriate stress and velocity boundary conditions. The analysis and numerical simulations of this paper address this issue for a model problem, a system of equations describing antiplane shearing of an elastoplastic material. This system retains the relevant mathematical structure of elastoplastic planar flow. Even if the flow rule is associative, two significant phenomena appear: (i) For boundary conditions suggestive of granular flow in a hopper, in which it seems physically natural to specify the velocity everywhere along a portion of the boundary, no such solutions of the equations exist; rather, we construct a solution with a shear band (velocity jump) along part of the boundary, and an appropriate relaxed boundary condition is satisfied there. (ii) Rigid zones appear inside deforming regions of the flow, and the stress field in such a zone is not uniquely determined. For a nonassociative flow rule, an extreme form of nonuniqueness—both velocity and stress—is encountered.

Key Words: granular materials • nonlinear hyperbolic partial differential equations

References

• Schaeffer, D.G. Instability and ill-posedness in the deformation of granular materials. International Journal of Numerical and Analytic Methods in Geomechanics, 14, 253—278 ( 1990).[CrossRef]
• Nedderman, R. M. Statics and Kinematics of Granular Materials, Cambridge University Press, Cambridge, 1992.
• Bouchaud, J.-P., Claudin, P., Levine, D. and Otto, M. Force chain splitting in granular materials: A mechanism for large-scale pseudo-elastic behaviour. European Physics Journal E, 4, 451—457 ( 2001).[CrossRef]
• Goldenberg, C. and Goldhirsch, I. Force chains, microelasticity, and macroelasticity. Physical Review Letters, 89, 0804302-1—4, (2002).
• Jenkins, J. and Richman, M. Plane simple shear of smooth inelastic circular disks. Journal of Fluid Mechanics, 192, 313—328 (1988).[CrossRef][Web of Science]
• Zhang, D.Z. and Rauenzahn, R.M. Stress relaxation in dense and slow granular flows. Journal of Rheology, 44, 1019—1041, (2000).[CrossRef]
• Schaeffer, D.G. A mathematical model for localization in granular flow. Proceedings of the Royal Society London Series A, 436, 217—250 (1992).[Abstract/Free Full Text]
• Ekeland, I. and Temam, R. Convex Analysis and Variational Problems, Elsevier, New York, 1976.
• Temam, R. Mathematical Problems in Plasticity, Gauthier-Villars, Paris, 1985.
• Matthews, J.V. and Schaeffer, D.G. A well-posed free boundary problem for a hyperbolic equation with Dirichlet boundary conditions. SIAM Journal of Mathematical Analysis, 36, 256—271 (2004).
• McOwen, R.C. Partial Differential Equations: Methods and Applications, Second edition, Prentice Hall, Englewood Cliffs, NJ, 2003.
• Hill, R. The Mathematical Theory of Plasticity, Oxford University Press, Oxford, 1950.
• Schaeffer, D.G., Shearer, M. and Witelski, T.P. One-dimensional solutions of an elastoplasticity model of granular material, Mathematical Models and Methods in Applied Sciences, 13, 1629—1671 ( 2003).[CrossRef]
• Witelski, T.P., Schaeffer, D.G. and Shearer, M. A discrete model for an ill-posed nonlinear parabolic PDE, Physica D, 160, 189—221 (2001).[CrossRef]
• Collins, I.F. Boundary value problems in plane strain plasticity. In Mechanics of Solids, H. G. Hopkins and M. J. Sewell (Eds.), Pergamon Press, 1982.

This version was published on December 1, 2007

Mathematics and Mechanics of Solids, Vol. 12, No. 6, 665-699 (2007)
DOI: 10.1177/1081286506067325


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This Article Abstract Free Full Text (Free PDF) Free Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Similar articles in Web of Science Alert me to new issues of the journal Add to Saved Citations Download to citation manager Request Permissions Request Reprints Add to My Marked Citations Citing Articles Citing Articles via Google Scholar Citing Articles via Scopus Google Scholar Articles by Schaeffer, D. G. Articles by Witelski, T. P. Search for Related Content Social Bookmarking                         What's this?