On discrete and discretized non-linear elastic structures in unilateral contact (stability, uniqueness and variational principles)

Abstract

In the present paper problems related to discrete and discretized non-linear elastic structures in unilateral contact with a rigid support are considered in the range of large displacements. A finite dimensional vector matrix description, based on the concepts of generalized stresses and strains, is derived. It is shown that the problem of determining the displacements and the contact forces for a given constant external loading can be formulated alternatively as a variational inequality, representing the principle of virtual work, or as a set of Kuhn-Tucker relations, representing force equilibrium. Furthermore, the Kuhn-Tucker relations are related to a primal and a dual minimisation problem. The primal problem represents the principle of minimum of potential energy and the dual problem is a generalization to large displacements of the so-called reciprocal formulation of contact problems. Moreover, the problems of mechanical stability and that of uniqueness of incremental response are investigated. The incremental, or rate, formulation is derived together with an associated variational inequality, representing the incremental principle of virtual work. A sufficient condition for the uniqueness of the solution of this variational inequality is given. A sufficient condition for mechanical stability, on the other hand, can be obtained directly from a second-order sufficient condition for the optimum of non-linear programs. The fact that these two sufficient conditions do not coincide is discussed and a simple naturally discrete problem exemplifies this point. Furthermore, it is seen that the curvature of the rigid support has an influence on both the stability and the uniqueness of the structure. This fact is also illustrated by an example.