Signorini Problem - Formal Statement of The Problem

Formal Statement of The Problem

The content of this section and the following subsections follows closely the treatment of Gaetano Fichera in Fichera 1963, Fichera 1964b and also Fichera 1995: his derivation of the problem is different from Signorini's one in that he does not consider only incompressible bodies and a plane rest surface, as Signorini does. The problem consist in finding the displacement vector from the natural configuration of an anisotropic non-homogeneous elastic body that lies in a subset of the three-dimensional euclidean space whose boundary is and whose interior normal is the vector , resting on a rigid frictionless surface whose contact surface (or more generally contact set) is and subject only to its body forces, and surface forces applied on the free (i.e. not in contact with the rest surface) surface : the set and the contact surface characterize the natural configuration of the body and are known a priori. Therefore the body has to satisfy the general equilibrium equations

(1)

written using the Einstein notation as all in the following development, the ordinary boundary conditions on

(2)

and the following two sets of boundary conditions on, where is the Cauchy stress tensor. Obviously, the body forces and surface forces cannot be given in arbitrary way but they must satisfy a condition in order for the body to reach an equilibrium configuration: this condition will be deduced and analized in the following development.