Numerical treatment of problems involving nonmonotone boundary or stress-strain laws

Abstract

In order to describe the softening behavior of the materials, nonmonotone possible multivalued laws have been recently introduced. These laws are derived by nonconvex, generally nonsmooth energy functions called superpotentials that give rise to hemivariational inequalities. Due to the lack of convexity and the nonsmoothness of the underlying superpotentials these problems have generally nonunique solutions. On the other hand, problems involving monotone laws lead to variational inequalities that can be easily treated using modern convex minimization algorithms. The present paper proposes a new method for the solution of the nonmonotone problem by approximating it using monotone ones. The proposed method finds its justification in the approximation of a hemivariational inequality by a sequence of variational inequalities. This approach leads to effective reliable and versatile numerical alogrithms for large-scale hemivariational inequalities. The numerical method proposed is illustrated using appropriate examples.