Variational Principles of Contact Elastostatics

Abstract

In this paper, the variational principles of contact elastostatics, which were proposed and proved by Fichera (1964) and Duvaut & Lions (1972) are developed in an engineering fashion without use of functional analysis. The theory contains a number of new elements, but the elegant existence proofs of the French-Italian school are missing. A start is made by extending the principle of virtual work to normal and frictional contact in such a manner that it needs no longer be known beforehand whether—in the case of normal contact—actual contact is or is not established, or—in the case of frictional contact—slip does or does not occur. Then the principle of minimal potential energy is set up for a non-linear elastic body in contact with a rigid base. Uniqueness and minimality of the solution are proved under certain conditions, the Reissner principle is established, and the principle of minimal complementary energy is derived. Finally the principles are cast hi what is termed surface mechanical form, and two examples are given: the variational principle for normal half-space contact problems, and a new principle for time-dependent frictional half-space contact. Upon discretization, these principles provide a quadratic object function to be minimized under linear or quadratic inequality constraints. The positions of the contact area and of the regions of slip and adhesion appear as by-products of the calculation.