# Nonsmooth Mechanics. Convex and Nonconvex Problems

## Abstract

Nonlinear, multivalued and possibly nonmonotone relations arise in several areas of mechanics. A multivalued or complete relation is a relation with complete vertical branches. Boundary laws of this kind connect boundary (or interface) quantities. A contact relation or a locking mechanism between boundary displacements and boundary tractions in elasticity is a representative example. Material constitutive relations with complete branches connect stress and strain tensors, or, in simplified theories, equivalent stress and strain quantities. A locking material or a perfectly plastic one is represented by such a relation. The question of nonmonotonicity is more complicated. One aspect concerns nonmonotonicity of a constitutive or a boundary law. Certainly, at a local microscopic level a nonmonotone relation corresponds to an unstable material or boundary law. Examples from damage or fracture mechanics may be presented. On a macroscopic level the complete mechanical behaviour of structural components can be described with such nonmonotone and possibly multivalued relations. A typical example of this kind is the delamination process of a composite structure, where local delaminations, crack propagation and interface or crack contact effects lead to a sawtooth overall load-displacement relation (see Panagiotopoulos and Baniotopoulos, 1984, Mistakidis and Stavroulakis, 1998, Li and Carlsson, 1999). The latter relation is adopted here as a constitutive law for the study of the structure at a macroscopic level. Another reason for nonmonotonicity is the large displacement or deformation effects. Let us consider that there exists a convex deformation energy potential which is a function of some appropriate strain quantity. In a kinematically nonlinear mechanical theory the geometric compatibility relation, which connects strains with displacements of the structure, is nonlinear. Therefore the same potential energy, considered as a composite function of a convex function with a nonlinear relation, is, in general, nonconvex in the displacement variables.

## Keywords

Variational Inequality Hemivariational Inequality Convex Case Unilateral Contact Strain Energy Density Function## Preview

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