On the Nonuniqueness of Solutions to the Nonlinear Equations of Elasticity Theory |
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Authors: | A G Kulikovskii A P Chugainova E I Sveshnikova |
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Affiliation: | (1) Steklov Mathematical Institute of the Russian Academy of Sciences, 119991, Gubkina Street 8, Moscow, Russia;(2) Department of Mechanics and Mathematics, Moscow M.V. Lomonosov State University, 119992, Vorobyovy Gory, Moscow, Russia |
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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. |
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Keywords: | dispersion elasticity theory shock wave structure viscosity |
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