Loss of Hyperbolicity in Yield Vertex Plasticity Models under Nonproportional Loading

Schaeffer, David G.; Shearer, Michael

doi:10.1007/978-1-4613-9049-7_15

Cited by 5 publications

(3 citation statements)

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“…In particular, we are investigating this for one class of incrementally nonlinear constitutive laws, viz. yield vertex models such as described in Christoffersen & Hutchinson [3] and Schaeffer & Shearer [16].…”

Section: Discussion Of Related Issuesmentioning

confidence: 99%

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

Schaeffer

Shearer

1992

Eur. J. Appl. Math

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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.

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Section: Discussion Of Related Issuesmentioning

confidence: 99%

“…A less serious difficulty regarding existence occurs if 3 X U L is in region (a) of figure 16. In this case, U L (x, t) is connected to U,(x, t) by a fast £-> P front, and a wave curve of states 3 X Up connected to 9^.…”

Section: Solution Of the Si-problemmentioning

confidence: 99%

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

Schaeffer

Shearer

1992

Eur. J. Appl. Math

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“…That is, when a repeated eigenvalue of Qn has an algebraic multiplicity greater than its geometric multiplicity, so that Qn is not diagonalizable. (Schaeffer and Shearer, 1990;Brannon and Drugan, 1993;Bigoni, 1995). We shall not discuss this point any further.…”

Section: Introductionmentioning

confidence: 93%

A numeric-symbolic approach to the problem of localization of plastic flow

Boussaa¹,

Aravas²

2001

Computational Mechanics

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The genericity of flutter ill-posedness in 3-dimensional elastic-plastic models

An¹

1994

Quart. Appl. Math.

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0. Introduction. In elastic-plastic models for granular material, it is common that the governing partial differential equations become ill-posed as plastic deformations are accumulated. In dynamical formulations, ill-posedness occurs if the governing equations lose their hyperbolicity. Equivalently, ill-posedness occurs if wave speeds become complex. Ill-posedness due to wave speed becoming zero and then pure imaginary has been studied extensively [M, S], It was believed that this type of illposedness is related to the formation of shear bands. In our paper, we shall investigate the case that wave speeds become equal (the equations are not strictly hyperbolic) and then complex with nonzero real part. Following Rice [R], we call it flutter illposedness.For two-dimensional models, An and Schaeffer [A, A-S] investigated the same problem. It was found that, even in the simplest of elastic-plastic models, the condition for the onset of flutter ill-posedness-wave speeds being equal-may be reached. By a topological argument, it was shown that a generic perturbation leads to equations with flutter ill-posedness in a neighborhood of a certain hardening modulus. In these papers, a readily applicable criterion for the occurrence of flutter ill-posedness is derived. It is demonstrated that flutter ill-posedness occurs in widely accepted models. Recently, Loret [L] extended the results to three-dimensional models. It was shown that, whatever the hardening modulus, the dynamical equations of motion are never strictly hyperbolic; that is, in some direction, two wave speeds are always equal. Moreover, for some discrete values of the hardening modulus, the three wave speeds become equal. By algebraic calculation, he showed that, when the flow rule deviates from deviatoric associativity, the governing equations could exhibit flutter ill-posedness in a neighborhood of the discrete values of the hardening modulus.In our paper, we continue to discuss flutter ill-posedness in three-dimensional models. For the case of three wave speeds being equal, we employ a topological argument,

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Loss of Hyperbolicity in Yield Vertex Plasticity Models under Nonproportional Loading

Cited by 5 publications

References 10 publications

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

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

A numeric-symbolic approach to the problem of localization of plastic flow

The genericity of flutter ill-posedness in 3-dimensional elastic-plastic models

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