No abstract

We investigate unilateral contact problems for micropolar hemitropic elastic solids. Our study includes Tresca friction (given friction model) along some parts of the boundary of the body. We equivalently reduce these problems to boundary variational inequalities with the help of the Steklov-Poincaré type operator. Based on our boundary variational inequality approach we prove existence and uniqueness theorems for weak solutions. We prove that the solutions continuously depend on the data of the original problem and on the friction coefficient. We treat also the case, when the body is not fixed, but only submitted to force and couple stress vectors along some parts of the boundary and is in unilateral frictional contact with a rigid foundation. In this situation we present necessary and sufficient conditions of solvability.

We study the interior and exterior contact problems for hemitropic elastic solids. We treat the cases when the friction effects, described by Tresca friction (given friction model), are taken into consideration either on some part of the boundary of the body or on the whole boundary. We equivalently reduce these problems to a boundary variational inequality with the help of the Steklov-Poincare´type operator. Based on our boundary variational inequality approach we prove existence and uniqueness theorems for weak solutions. We prove that the solutions continuously depend on the data of the original problem and on the friction coefficient. For the interior problem, necessary and sufficient conditions of solvability are established when friction is taken into consideration on the whole boundary.

The frictionless contact problems for two interacting hemitropic solids with di¤erent elastic properties is investigated under the condition of natural impenetrability of one medium into the other. We consider two cases, the so-called coercive case (when elastic media are fixed along some parts of their boundaries), and the semicoercive case (the boundaries of the interacting elastic media are not fixed). Using the potential theory we reduce the problems to the boundary variational inequalities and analyse the existence and uniqueness of weak solutions. In the semicoercive case, the necessary and su‰cient conditions of solvability of the corresponding contact problems are written out explicitly.

In the present paper we investigate a three-dimensional boundary-contact problem of dynamics for a homogeneous hemitropic elastic medium with regard to friction. We prove the uniqueness theorem using the corresponding Green formulas and positive de niteness of the potential energy. To analyze the existence of solutions we reduce equivalently the problem under consideration to a spatial variational inequality. We consider a special parameter-dependent regularization of this variational inequality which is equivalent to the relevant regularized variational equation depending on a real parameter and study its solvability by the Faedo-Galerkin method. Some a priori estimates for solutions of the regularized variational equation are established and with the help of an appropriate limiting procedure the existence theorem for the original contact problem with friction is proved.