Nonsmooth Mechanics. Convex and Nonconvex Problems

• Jaroslav Haslinger
• Markku Miettinen
• Panagiotis D. Panagiotopoulos
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 35)

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
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Authors and Affiliations

• Jaroslav Haslinger
• 1
• Markku Miettinen
• 2
• Panagiotis D. Panagiotopoulos
• 3
1. 1.Charles UniversityCzech Republic
2. 2.University of JyväskyläFinland
3. 3.Aristotle UniversityGreece