A Variational Inequality Approach to the Friction Problem of Structures with Convex Strain Energy Density and Application to the Frictional Unilateral Contact Problem

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Abstract
The analysis of structures with friction boundary conditions is considered. The stress-strain relations are generally monotone nonlinear allowing for jump, locking, and slacking effects, and are derived from a convex strain energy density by “subdifferentiation.” It is proved that for the considered boundary value problem the principles of virtual and of complementary virtual work hold in an inequality form which constitutes a variational inequality. The theorems of minimum potential and complementary energy are proved to be valid for this type of boundary conditions. These theorems are used to formulate the analysis as a nonlinear programming problem. A numerical example concerning the calculation of a structure having friction boundary conditions combined with unilateral contact boundary conditions illustrates the theory.