Boundary Integral Formulations for the Monotone Multivalued Contact Boundary Conditions

Abstract

Let Ω be an open bounded⁵ subset of the three-dimensional Euclidean space ℝ³ with a boundary Γ (a Lipschitz boundary is sufficient). Ω is occupied by a linear elastic body in its undeformed state and we refer it to a Cartesian orthogonal coordinate system Ox ₁ x ₂ x ₃. Γ is decomposed into three mutually disjoint parts Γ₁, Γ ₂ and Γ₃ that are open subsets of T and mes Γ₁ ≠ 0, mes Γ₃ ≠ 0. It is assumed that on Γ₁ (resp. Γ₂) the displacements (resp. the tractions) are given and that on Γ₃ boundary conditions which are monotone and multivalued and which cause the inequality formulation of the problem hold. We call these conditions inequality boundary conditions. We denote as usual by n = {n _i } the outward unit normal vector to Γ, by S = {S _i } = {σ _ij n _j } the traction vector on the boundary, where σ = {σ _ij } is the stress tensor and, by u = {u _i } the corresponding displacement vector. Let ɛ = {ɛ _ij } be the strain tensor (assumption of small strains) and C = {C _ijhk }, i, j, h, k = 1, 2, 3, Hooke’s tensor of elasticity obeying the well-known symmetry and ellipticity conditions (1.151a, b). On Γ₁ we assume that

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

and on Γ₂

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

The boundary conditions on Γ₃ are written in the general subdifferential form (cf. Sect. 1.3.1)

$$ - S \in \partial j(u)$$

where j is a convex, lower semicontinuous, proper superpotential (see Sect. 1.2.1 and 1.3.1) and ∂ denotes the subdifferential.