Entropy solutions in the study of antiplane shear deformations for elasticsolids

The concept of entropy solution was recently introduced in the study of Dir ichlet problems for elliptic equations and extended for parabolic equations with nonlinear boundary conditions. The aim of this paper is to use the me thod of entropy solutions in the study of a new problem which arise in the theory of elasticity. More precisely, we consider here the infinitesimal an tiplane shear deformation of a cylindrical elastic body subjected to given forces and in a frictional contact with a rigid foundation. The elastic con stitutive law is physically nonlinear and the friction is described by a st atic law. We present a variational formulation of the model and prove the e xistence and the uniqueness of a weak solution in the case when the body fo rces and the prescribed surface tractions have the regularity L-infinity. T he proof is based on classical results for elliptic variational inequalitie s and measure theory arguments. We also define the concept of entropy solut ion and we prove an existence and uniqueness result in the case when the bo dy forces and the surface tractions have the regularity L-1. The proof is b ased on properties of the trace operators for functions which are not in So bolev spaces. Finally, we present a regularity result for the entropy solut ion and we give some concrete examples and mechanical interpretation.