Abstract

Non-Uniqueness of Solutions to Nonlinear Elasticity Equations and Its Reason

E.I.Sveshnikova (Moscow State University, Moscow, Russia); A.G.Kulikovskii (Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, Russia)

e-mail: kulik@mi.ras.ru

The self-similar arbitrary discontinuity disintegration problem and piston-problem are considered in framework of non-linear elasticity theory. The nonuniqueness of solutions to these problems is proved. The non-uniqueness manifests in the behavior of quasy-transversal waves and depends on the non-linearity of the medium properties and anisotropy. The anisotropy can be a media property or can depend on previous deformations in the discontinuity plane. In arbitrary non-linear elastic media under arbitrary small initial deformations (and stresses) there are self-similar problems with non-unique solutions. The results show that the classical model of nonlinear elasticity theory is incomplete. A sufficient criterion of the non-uniqueness of solution is found for an arbitrary hyperbolic system of the conservation laws. The criterion is formulated as a property of shocks with given state ahead a shock.

Section : 1