Variational analysis for some frictional contact problems

Leila Ait Kaki,M Denche

doi:10.5269/bspm.v38i7.44258

Abstract

We consider a class of evolutionary variational problems which describes the static frictional contact between a piezoelectric body and a conductive obstacle. The formulation is in a form of coupled system involving the displacement and electric potentiel fieelds. We provide the existence of unique weak solution of the problems. The proof is based on the evolutionary variational inequalities and Banach's xed point theorem.

Highlights

Preliminaries and notationsWe consider that spaces V and H are real Hilbert spaces satisfying V ⊂ H ⊂ V ′ and W ⊂ H ⊂ W ′ with continuous and dense injections, where H = L2(Ω) and H = L2(Ω) d
Stress is applied, and in the presence of the electric potential the mechanical stress is generated
This work is a continuation in this line of research and we study an abstract weak formulation of quasistatic frictional contact problem for an electro-viscoelastic material, in the framework of the MTCM, when the foundation is deformable and conductive and the friction is described by the normal compliance and versions of Coulomb’s law

Summary

Preliminaries and notations

We consider that spaces V and H are real Hilbert spaces satisfying V ⊂ H ⊂ V ′ and W ⊂ H ⊂ W ′ with continuous and dense injections, where H = L2(Ω) and H = L2(Ω) d. (φ, ς)W = (∇φ, ∇ς)H , where ε is linear map defined from V to H.We suppose that we may apply the Sobolev trace theorem, it means that there exists two constants c0 and c0, depending only on Ω, and parts of Γ, such that. The functional h given in the equation (1.2) is defined with the Riesz representation theorem by (h(u, φ), ζ)W = ψ(uν − g)φL(φ − φ0)ζ da, ΓC (2.5). The functional j : X × X → R satisfies:. The solution satisfies u ∈ W 2,p(0, T ; V ), φ ∈ W 1,p(0, T ; W )

Proof of the main result

Application

Conclusion