Abstract

Methastable Shock Waves in the Problems of Nonlinear Elasticity Theory

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

e-mail: kulik@mi.ras.ru

There is a non-uniqueness in self-similar piston problems and arbitrary discontinuity disintegration problems. In many cases the solution contains the shock, which can disintegrate according to the conservation law. Such a shock can be termed the methastable shock. If methastable shocks are excluded from solutions, the solutions of the above mentioned problems are unique. To analyze the problem of self-similar solutions uniqueness the viscous stresses are taken into account in the motion equations. Many problems are computed for which asymptotics correspond to inviscous self-similar solutions. It was shown, that in cases of non-uniqueness all self-similar solution can represent the asymptotic of a viscous solution depending on problem formulation details, which disappear in self-similar problem if the viscosity vanish. The stability of methastable shocks is investigated from linear and nonlinear one-dimensional and two-dimensional points of view. It was obtained, that methastable shocks are stable if they are considered as viscoelastic waves. The obtained results lead to the conclusion that non-uniqueness is intrinsic property of the classical nonlinear elasticity theory. To select a unique solution additional terms such as viscous ones should be introduced in the equations.

Section : 1