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On the nonuniqueness of solutions to the nonlinear equations of elasticity theory. (English) Zbl 1104.74035
Summary: We consider one-dimensional self-similar problems for waves in an elastic half-space generated by a sudden change of boundary stress (the “piston” problem) and problems of disintegration of an arbitrary discontinuity. 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 viscoelastic 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 viscoelastic solutions. The type of asymptotics depends on those details of the viscoelastic problem formulation which are absent when formulating inviscid self-similar 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 \(t \rightarrow \infty)\) 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.

74J30 Nonlinear waves in solid mechanics
74J40 Shocks and related discontinuities in solid mechanics
74D05 Linear constitutive equations for materials with memory
74H40 Long-time behavior of solutions for dynamical problems in solid mechanics
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