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RESEARCH PRODUCT

ENTROPY SOLUTIONS IN THE STUDY OF ANTIPLANE SHEAR DEFORMATIONS FOR ELASTIC SOLIDS

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The concept of entropy solution was recently introduced in the study of Dirichlet problems for elliptic equations and extended for parabolic equations with nonlinear boundary conditions. The aim of this paper is to use the method of entropy solutions in the study of a new problem which arise in the theory of elasticity. More precisely, we consider here the infinitesimal antiplane shear deformation of a cylindrical elastic body subjected to given forces and in a frictional contact with a rigid foundation. The elastic constitutive law is physically nonlinear and the friction is described by a static law. We present a variational formulation of the model and prove the existence and the uniqueness of a weak solution in the case when the body forces and the prescribed surface tractions have the regularity L∞. The proof is based on classical results for elliptic variational inequalities and measure theory arguments. We also define the concept of entropy solution and we prove an existence and uniqueness result in the case when the body forces and the surface tractions have the regularity L1. The proof is based on properties of the trace operators for functions which are not in Sobolev spaces. Finally, we present a regularity result for the entropy solution and we give some concrete examples and mechanical interpretation.