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

Abstract

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,