On a frictional contact problem with adhesion in piezoelectricity

Abstract

We consider a mathematical model describing the quasistatic frictional contact between an electro-elasto-viscoplastic body and an adhesive conductive foundation. The contact is described with a normal compliance condition with adhesion, the associated general version of Coulomb's law of dry friction in which the adhesion of contact surfaces is taken into account and a regularized electrical conductivity condition. The existence of a unique weak solution is established under smallness assumption on the surface conductance. The proof is based on arguments of time-dependent variational inequalities, differential equations and Banach fixed point theorem.

A Dynamic Frictionless Contact Problem with Adhesion in Thermo-elasto-viscoplasticity

“…Using (75) and keeping in mind that Λη * = η * , to find that the quadruplet (u, σ, β, θ) is a solution of the problem P V. This solution has the regularity expressed in (47)-(50) and which follow from the regularities of the solution of problems P V η , P V β , P V θ and P V σ . Moreover, it follows from (47), (22) and ( 24) that σ ∈ L 2 (0, T ; H). Choosing now v = ±ϕ in (40), where ϕ ∈ C ∞ 0 (Ω) d , and using ( 28), (38) to find Now assumptions ( 28), (30), the fact that .. u∈L 2 (0, T ; V ) and the above equality imply that Div σ ∈L 2 (0, T ; V ), which shows that σ satisfies (48).…”

“…Following [9,10], the bonding field satisfies the restrictions 0 ≤ β ≤ 1, when β = 1 at a point of the contact surface, the adhesion is complete and all the bonds are active, when β = 0 all the bonds are inactive, severed, and there is no adhesion, when 0 < β < 1 the adhesion is partial and only a fraction β of the bonds is active. The reader is referred to the extensive bibliography on the subject in [16,17,22,25,27] . The aim of this paper consists on the study of a dynamic process of a frictionless contact between a thermo-elasto-viscoplastic body and an adhesive foundation.…”

“…Integrating this equality with respect to time, using the initial conditions v 1 (0) = v 2 (0) = v 0 and condition (22) to find…”