Nonconvex Unilateral Contact Problems

Abstract

We consider a three-dimensional linear elastic body subjected to nonmonotone multivalued boundary conditions which are obtained from nonconvex superpotentials (cf. Sect. 1.3.3). The procedure which we follow remains valid also for shells, plates, beams etc. Let Ω be an open bounded subset of the three-dimensional Euclidean space ℝ³ with a Lipschitz boundary Γ. Ω is occupied by a linear elastic body in its undeformed state. We refer to a Cartesian orthogonal coordinate system Ox ₁ x ₂ x ₃. Γ is decomposed into nonoverlapping parts Γ₁, Γ₂ and Γ₃ open in Γ, such that mes Γ₁ ≠ 0, and mes Γ₃ ≠ 0. It is assumed that on Γ₁ (resp. Γ₂) the displacements (resp. the tractions) are given and that on Γ₃ the boundary conditions causing the inequality formulation of the problem hold. With the same notation as in the two previous Chapters we assume that on Γ₁

$$ {u_i} = {{\bar U}_i},{{\bar U}_i} = {{\bar U}_i}(x) $$

(12.1)

and on Γ₂

$$ {S_i} = {{\bar T}_i},{{\bar T}_i} = {{\bar T}_i}(x). $$

(12.2)