Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Wave speeds for an elastoplastic model for two-dimensional deformations with a nonassociative flow rule

Authors: Michael Gordon and F. Xabier Garaizar
Journal: Quart. Appl. Math. 57 (1999), 245-259
MSC: Primary 74C05; Secondary 35Q72, 74E20, 74J99
DOI: https://doi.org/10.1090/qam/1686188
MathSciNet review: MR1686188
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Abstract: A system of partial differential equations describing elastoplastic deformations in two space dimensions is studied. The constitutive relations for plastic deformation include a nonassociative flow rule and shear strain hardening. After a change of variables, the characteristic speeds of plane wave solutions of the system are computed. For both plastic and elastic deformations, there are two nonzero wave speeds, referred to as fast and slow waves. It is shown that there are regions in stress space for which the speed of fast plastic waves exceeds the speed of fast elastic waves, which translates into a lack of uniqueness for certain initial value problems and introduces nontrivial difficulties for numerical methods. Finally, these regions are computed for an example using representative constitutive data.

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  • [1] Lian Jun An, Loss of hyperbolicity in elastic-plastic material at finite strains, SIAM J. Appl. Math. 53 (1993), no. 3, 621–654. MR 1218377, https://doi.org/10.1137/0153032
  • [2] Lian Jun An and David G. Schaeffer, The flutter instability in granular flow, J. Mech. Phys. Solids 40 (1992), no. 3, 683–698. MR 1157067, https://doi.org/10.1016/0022-5096(92)80009-F
  • [3] D. C. Drucker, R. E. Gibson, and D. J. Henkel, Soil mechanics and work-hardening theories of plasticity, Trans. American Soc. Civil Engrg. 122, 338-346 (1957)
  • [4] P. V. Lade and J. M. Duncan, Elastoplastic stress-strain theory for cohesionless soil, J. Geotech. Engrg. 101, 1037-1053 (1975)
  • [5] Gianni Dal Maso, Philippe G. Lefloch, and François Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. (9) 74 (1995), no. 6, 483–548. MR 1365258
  • [6] Philippe LeFloch and Tai-Ping Liu, Existence theory for nonlinear hyperbolic systems in nonconservative form, Forum Math. 5 (1993), no. 3, 261–280. MR 1216035, https://doi.org/10.1515/form.1993.5.261
  • [7] J. Mandel, Conditions de stabilité et postulate de Drucker, Rheology and Soil Mechanics, IUTAM Symposium at Grenoble (G. Kravtchenko and P. Sirieys, eds.), 58-68 (1964)
  • [8] H. B. Poorooshasb, I. Holubec, and A. N. Sherbourne, Yielding and flow of sand in triaxial compression: Parts II and III, Canadian Geotech. J. 4, 376-397 (1976)
  • [9] J. Rice, The localization of plastic deformation, Proc. IUTAM Congress at Delft (W. Koiter, ed.), 207-220 (1976)
  • [10] I. Sandler and D. Rubin, The consequences of non-associated plasticity in dynamic problems, Constitutive Laws for Engineering Materials: Theory and Applications (C. S. Desai et al., ed.), Elsevier, New York, 1987, pp. 345-352
  • [11] David G. Schaeffer, Instability and ill-posedness in the deformation of granular materials, Internat. J. Numer. Anal. Methods Geomech. 14 (1990), no. 4, 253–278. MR 1055259, https://doi.org/10.1002/nag.1610140403
  • [12] David G. Schaeffer and Michael Shearer, Scale-invariant initial value problems in one-dimensional dynamic elastoplasticity, with consequences for multidimensional nonassociative plasticity, European J. Appl. Math. 3 (1992), no. 3, 225–254. MR 1182214, https://doi.org/10.1017/S0956792500000814
  • [13] V. V. Sokolovskiĭ, D. H. Jones, and A. N. Schofield, Statics of soil media, Translated by D. H. Jones and A. N. Schofield, Butterworths Scientific Publications, London, 1960. MR 0134055
  • [14] I. Vardoulakis and B. Graf, Calibration of constitutive models for granular materials using data from experiments, Géotechnique 35, 299-317 (1985)

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DOI: https://doi.org/10.1090/qam/1686188
Article copyright: © Copyright 1999 American Mathematical Society

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