Evolutionary variational inequalities arising in quasistatic frictional contact problems for elastic materials. (English) Zbl 0974.58019

Motreanu, Dumitru; Sofonea, Mircea

Abstr. Appl. Anal. 4, No. 4, 255-279 (1999).

Summary: We consider a class of evolutionary variational inequalities arising in quasistatic frictional contact problems for linear elastic materials. We indicate sufficient conditions in order to have the existence, the uniqueness and the Lipschitz continuous dependence of the solution with respect to the data, respectively. The existence of the solution is obtained using a time-discretization method, compactness and lower semicontinuity arguments. In the study of the discrete problems we use a recent result obtained by the authors [Adv. Math. Sci. Appl. 10, 103-118 (2000; Zbl 0977.47057)]. Further, we apply the abstract results in the study of a number of mechanical problems modeling the frictional contact between a deformable body and a foundation. The material is assumed to have linear elastic behavior and the processes are quasistatic. The first problem concerns a model with normal compliance and a version of Coulomb’s law of dry friction, for which we prove the existence of a weak solution. We then consider a problem of bilateral contact with Tresca’s friction law and a problem involving a simplified version of Coulomb’s friction law. For these two problems we prove the existence, the uniqueness and the Lipschitz continuous dependence of the weak solution with respect to the data.

MSC:

58E35	Variational inequalities (global problems) in infinite-dimensional spaces
35K85	Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators
74M15	Contact in solid mechanics
74M10	Friction in solid mechanics
47J20	Variational and other types of inequalities involving nonlinear operators (general)
49J40	Variational inequalities

Keywords:

evolutionary variational inequalities; quasistatic frictional contact problems; linear elastic materials; Tresca’s friction law; Coulomb’s friction law

Citations:

Zbl 0977.47057