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Scale-invariant initial value problems in one-dimensional dynamic elastoplasticity, with consequences for multidimensional nonassociative plasticity

Published online by Cambridge University Press:  16 July 2009

David G. Schaeffer  and
Michael Shearer
David G. Schaeffer
Affiliation:
Department of Mathematics, Duke University, NC 27706, USA
Michael Shearer
Affiliation:
Department of Mathematics, North Carolina State University, NC 27607, USA
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Abstract

This paper solves a class of one-dimensional, dynamic elastoplasticity problems for equations which describe the longitudinal motion of a rod. The initial conditions U(x, 0) are continuous and piecewise linear, the derivative ∂U/∂x(x, 0) having just one jump at x = 0. Both the equations and the initial data are invariant under the scaling Ũ(x, t) = α−1U(αx, αt), where α > 0; hence the term scale-invariant. Both in underlying motivation and in solution, this problem is highly analogous to the Riemann problem from gas dynamics. These ideas are applied to the Sandler–Rubin example of non-unique solutions in dynamic plasticity with a nonassociative flow rule. We introduce an entropy condition that re-establishes uniqueness, but we also exhibit problems regarding existence.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 3 , Issue 3 , September 1992 , pp. 225 - 254
Copyright
Copyright © Cambridge University Press 1992

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References

[1] Antman, S. & Szymczak, W. 1989 Nonlinear elastoplastic waves. Contemp. Math. 100, 27–54.CrossRefGoogle Scholar
[2] Barenblatt, G. I. 1952 On self-similar motions of a compressible fluid in a porous medium. Prikl. Math. Mech. 16, 679–698 (in Russian).Google Scholar
[3] Christoffersen, J. & Hutchinson, J. 1979 A class of phenomenological corner theories of plasticity. J. Mech. Phys. Solids, 27, 465–487.CrossRefGoogle Scholar
[4] Clifton, R. & Bodner, S. 1966 An analysis of elastic-plastic pulse propagation. J. Appl. Mech. 33, 248–255.CrossRefGoogle Scholar
[5] Courant, R. & Friedrichs, K. 1948 Supersonic Flow and Show Waves. New York: Interscience.Google Scholar
[6] Darve, F. 1990 Incrementally nonlinearconstitutive relationships. In Geomaterials: Constitutive Equations and Modelling (ed. Darve, F.), London: Elsevier.CrossRefGoogle Scholar
[7] Duvaut, G. & Lions, J. L. 1976 Inequalities in Mechanics and Physics. New York: Springer.CrossRefGoogle Scholar
[8] Gel'fand, I. M. 1963 Some problems in the theory of quasilinear equations. AMS Trans. 29 (2), 295–381.Google Scholar
[9] John, F. 1982 Partial Differential Equations. New York: Springer.CrossRefGoogle Scholar
[10] Kevorkian, J. & Cole, J. 1981 Perturbation Methods in Applied Mathematics. New York: Springer.CrossRefGoogle Scholar
[11] Lax, P. D. 1957 Hyperbolic systems of conservation laws II. Comm. Pure Appl. Math. 10, 537–566.CrossRefGoogle Scholar
[12] Lee, E. 1952 A boundary value problem in the theory of plastic wave propagation. Quart. Appl. Math. 10, 335–346.CrossRefGoogle Scholar
[13] Nouri, A. 1991 Problémes hyperboliques en élasto-plasticité dynamique. Thesis in Applied Mathematics, University of Nice.Google Scholar
[14] Rudnicki, J. & Rice, J. 1975 Conditions for the localization of deformation in pressure-sensitive dilatant materials. J. Mech. Phys. Solids 23, 371–394.CrossRefGoogle Scholar
[15] Sandler, I. & Rubin, D. 1987 The consequences of non-associated plasticity in dynamic problems. In Constitutive Laws for Engineering Materials: Theory and Applications (eds. Desai, C. S. et al. ), New York: Elsevier.Google Scholar
[16] Schaeffer, D. & Shearer, M. 1990 Loss of hyperbolicity in yield vertex plasticity models under nonproportional loading. In Nonlinear Evolution Equations that Change Type (eds. Keyfitz, B. & Shearer, M.), New York: Springer.Google Scholar
[17] Shearer, M. & Schaeffer, D. The Initial Value Problem for a System Modelling Unidirectional Longitudinal Elastic-Plastic Waves. Preprint.Google Scholar
[18] Smoller, J. 1983 Shock Waves and Reaction-Diffusion Equations. New York: Springer.CrossRefGoogle Scholar
[19] Témam, R. 1985 Mathematical Problems in Plasticity. Paris: Bordas.Google Scholar
[20] Trangenstein, J. & Pember, R. 1991 The Riemann problem for longitudinal motion in an elastic-plastic bar. SIAM J. Sci. Stat. Comp. 12, 180–207.CrossRefGoogle Scholar
[21] Xu, X.-P., Ortiz, M. & Needleman, A. 1991 Computational methods for localization problems. Proc. 15th Conference on Th. Appl. Mech. (SECTAM). Georgia Tech, USA.Google Scholar