On the existence of indeterminate solutions to the equations of motion under non-associated flow

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Under certain conditions, an indeterminate solution exists to the equations of motion for dynamic elastic-plastic deformation of materials using constitutive laws based on non-associated flow that suggests that an initially unbounded dynamic perturbation in the stress can develop from a quiescent state on the yield surface. The existence of this indeterminate solution has been alleged to discourage use of non-associated flow rules for both dynamic and quasi-static analysis theoretically. It is shown in this paper that the indeterminate solution that may solve the equations of motion is intrinsically dynamic, and it determinately goes to zero in the quasi-static limit regardless of other indeterminate parameters. Consequently, the existence of this unstable dynamic solution has no impact on stability and use of non-associated flow rules for analysis of the quasi-static problem. More importantly, for dynamic applications, it is also shown that the indeterminate solution solves the equations of motion only if critical restrictions are applied to the constitutive equations such that the effective modulus during loading is constant and the direction of the perturbation is unidirectional over a finite time interval. It is shown that common components of the constitutive laws used in metal forming and deformation analysis are inconsistent with these restrictions. So, these common models can be generalized to include non-associated flow for analysis of the dynamic problem without concern that the solution will become indeterminate. (c) 2007 Published by Elsevier Ltd.
Publisher
PERGAMON-ELSEVIER SCIENCE LTD
Issue Date
2008
Language
English
Article Type
Article
Keywords

PLASTIC DEFORMATION-BEHAVIOR; PRESSURE; RULE; STRESS; STABILITY; ALUMINUM; SOLIDS

Citation

INTERNATIONAL JOURNAL OF PLASTICITY, v.24, no.4, pp.583 - 613

ISSN
0749-6419
DOI
10.1016/j.ijplas.2007.07.002
URI
http://hdl.handle.net/10203/203798
Appears in Collection
ME-Journal Papers(저널논문)
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