A piezoelectric contact problem with slip dependent coefficient of friction

Abstract

We consider a mathematical model which describes the static frictional contact between a piezoelectric body and an obstacle. The constitutive relation of the material is assumed to be electroelastic and involves a nonlinear elasticity operator. The contact is modelled with a version of Coulomb's law of dry friction in which the coefficient of friction depends on the slip. We derive a variational formulation for the model which is in form of a coupled system involving as unknowns the displacement field and the electric potential. Then we provide the existence of a weak solution to the model and, under a smallness assumption, we provide its uniqueness. The proof is based on a result obtained in [14] in the study of elliptic quasi‐variational inequalities.

Pjezoelektriko sąlyčio su priklausomu nuo slydimo trinties koeficiento uždavinys

Santrauka. Mes nagrinėjame matematinį modelį, kuris aprašo sąlytį˛ tarp pjezoelektriko ir kliūties. Laikoma, kad medžiaga yra elektroelastinė ir nusakoma netiesiniu elastingumo operatoriumi. Sąlytis modeliuojamas remiamtis sausos trinties Coulomb’o dėsniu, kuriame trinties koeficientas priklauso nuo slydimo. Mes gavome variacinį modelio formulavimą lygčių sistemos formoje, kurios nežinomaisiais yra perkeltasis laukas ir elektrinis potencialas. Įrodomas sprendinio silpnąja prasme egzistavimas ir su nedidelėmis prielaidomis vienatis. Įrodymas paremtas rezultatais gautais [14] darbe, kuriame tiriamos elipsinės kvazivariacinės nelygybės.

First Published Online: 14 Oct 2010

Keyword : piezoelectric material, electroelasticity, static frictional contact, Coulomb's law, slip dependent coefficient of friction, quasivariational inequality, weak solution