On discrete and discretized nonlinear elastic structures in unilateral contact (stability, uniqueness and variational principles)
type: Item
instance of: scholarly article
title: On discrete and discretized nonlinear elastic structures in unilateral contact (stability, uniqueness and variational principles)
zbMATH Open document ID: 0636.73096
DOI: 10.1016/0020-7683(88)90002-9
author: Anders Klarbring
published in: International Journal of Solids and Structures
publication date: 1988-01-01T00:00:00Z
review text: Problems related to discrete and discretized nonlinear elastic structures in unilateral contact with a rigid support are considered in the range of large displacements. A finite dimensional vector matrix description, based on the concepts of generalized stresses and strains, is derived. It is shown that the problem of determining the displacements and the contact forces for a given constant external loading can be formulated alternatively as a variational inequality, representing the principle of virtual work, or as a set of Kuhn-Tucker relations, representing force equilibrium. Furthermore, the Kuhn-Tucker relations are related to a primal and a dual minimization problem. The primal problem represents the principle of minimum of potential energy and the dual problem is a generalization to large displacements of the so-called reciprocal formulation of contact problems. Moreover, the problems of mechanical stability and that of uniqueness of incremental response are investigated. The incremental, or rate, formulation is derived together with an associated variational inequality, representing the incremental principle of virtual work. A sufficient condition for the uniqueness of the solution of this variational inequality is given. A sufficient condition for mechanical stability, on the other hand, can be obtained directly from a second-order sufficient condition for the optimum of nonlinear programs. The fact that these two sufficient conditions do not coincide is discussed and a simple naturally discrete problem exemplifies this point. Furthermore, it is seen that the curvature of the rigid support has an influence on both the stability and the uniqueness of the structure. This fact is also illustrated by an example.
Mathematics Subject Classification ID: 74A55
Mathematics Subject Classification ID: 74M15
Mathematics Subject Classification ID: 74S30
Mathematics Subject Classification ID: 49M37
Mathematics Subject Classification ID: 74S05
Mathematics Subject Classification ID: 74B20
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 49S05
zbMATH DE Number: 4037400
zbMATH Keywords: bifurcation
zbMATH Keywords: nonlinear Signorini problem
zbMATH Keywords: frictionless contact
zbMATH Keywords: discrete
zbMATH Keywords: discretized
zbMATH Keywords: nonlinear elastic structures
zbMATH Keywords: unilateral contact
zbMATH Keywords: rigid support
zbMATH Keywords: large displacements
zbMATH Keywords: finite dimensional vector matrix description
zbMATH Keywords: generalized stresses
zbMATH Keywords: strains
zbMATH Keywords: constant external loading
zbMATH Keywords: principle of virtual work
zbMATH Keywords: set of Kuhn-Tucker relations
zbMATH Keywords: force equilibrium
zbMATH Keywords: minimization problem
zbMATH Keywords: primal problem
zbMATH Keywords: principle of minimum of potential energy
zbMATH Keywords: dual problem
zbMATH Keywords: reciprocal formulation
zbMATH Keywords: mechanical stability
zbMATH Keywords: uniqueness of incremental response
Wikidata QID: Q58878642
MaRDI profile type: Publication
full work available at URL: 0020-7683(88)90002-9
OpenAlex ID: W2002181588
description: scientific article; zbMATH DE number 4037400