Existence and multiplicity of solutions in frictional contact mechanics. Part II: Analytical and numerical case study

Abstract

In the first Part (Part I) of this series, the plane quasi-static incremental problem was formulated as a mixed complementarity-inclusion problem and as a linear complementarity problem. The current paper (Part II of the series) presents a numerical examination for the existence or the occurrence of multiple solutions to the quasi-static incremental problem of a two degree of freedom elastic system in the presence of a frictional rectilinear obstacle. The onset of multiplicity of solutions is studied and numerical examples are presented. All the solutions for two dimensional finite element examples are computed and their dependence on some parameters is discussed. The conditions for the occurrence of multiple solutions were discussed for finite element models. A particular attention was devoted to finite element models of rectangular blocks and finite element versions of the one particle model of Klarbring. For finite element blocks, it was found that the critical friction coefficient for solution uniqueness depends on the Poisson's ratio (μI decreases with ν). It was also noticed that slender structures with respect to the obstacle lead to smaller coefficients of friction. The same behavior was detected for the finite element version of Klarbring's example except that Poisson's ratio had a weak influence on the critical value of the friction coefficient. An algorithm is used for the computation of all the solutions of the incremental problem and to verify the sharpness of the frictional coefficient estimates corresponding to the several criteria. Comparison of different criteria for the uniqueness of the solution (for any loading) of the quasi-static incremental problem in finite element models is presented.