On the Nonuniqueness of Solutions to the Nonlinear Equations of Elasticity Theory

Abstract

One-dimensional selfsimilar problems for waves in an elastic half-space generated by a sudden change of the boundary stress (the "piston" problem) and problems of disintegration of an arbitrary discontinuity are considered. For the case when small-amplitude waves are generated in a medium with small anisotropy a qualitative analysis shows that these problems have nonunique solutions when it is assumed that the solutions involve Riemann waves and evolutionary discontinuities. The above-mentioned problems are considered as limits of properly formulated problems for visco-elastic media when the viscosity tends to zero or (what is the same) that time tends to infinity. It is numerically found that all above-mentioned inviscid solutions can represent the asymptotics of visco-elastic solutions. The type of asymptotics depends on those details of the visco-elastic problem formulation which are absent when formulating inviscid selfsimilar problems. Similar considerations are made for elastic media with dispersion along with dissipation which are manifested in small-scale processes. In such media the number of available asymptotics (as t → ∞) for the above-mentioned solutions depends on a relation between dispersion and dissipation and can be large. Thus, two possible causes for the nonuniqueness of solutions to the equations of elasticity theory are investigated.

Classical and nonclassical discontinuities and their structures in nonlinear elastic media with dispersion and dissipation

“…Results on the problems of weakly nonlinear waves in anisotropic elastic media were surveyed in the above-mentioned books [8,19] and in papers [26][27][28] …”

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“…Проблемам слабонелинейных волн в анизотропных упругих средах, помимо упоминавшихся обзоров [8,19], посвящены обзо-ры [26]- [28]. § 1.…”