Variational Principles of Contact Elastostatics

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 withoutuse of functional analysis. The theory contains a number ofnew elements, but the elegant existence proofs of the French-Italianschool are missing. A start is made by extending the principleof virtual work to normal and frictional contact in such a mannerthat it needs no longer be known beforehand whether—inthe case of normal contact—actual contact is or is notestablished, or—in the case of frictional contact—slipdoes or does not occur. Then the principle of minimal potentialenergy is set up for a non-linear elastic body in contact witha rigid base. Uniqueness and minimality of the solution areproved under certain conditions, the Reissner principle is established,and the principle of minimal complementary energy is derived.Finally the principles are cast in what is termed surface mechanicalform, and two examples are given: the variational principlefor normal half-space contact problems, and a new principlefor time-dependent frictional half-space contact. Upon discretization,these principles provide a quadratic object function to be minimizedunder linear or quadratic inequality constraints. The positionsof the contact area and of the regions of slip and adhesionappear as by-products of the calculation.