A Quasistatic Contact Problem for Viscoelastic Materials with Slip‐Dependent Friction and Time Delay

Abstract

A mathematical model which describes an explicit time-dependent quasistatic frictional contact problem between a deformable body and a foundation is introduced and studied, in which the contact is bilateral, the friction is modeled with Tresca’s friction law with the friction bound depending on the total slip, and the behavior of the material is described with a viscoelastic constitutive law with time delay. The variational formulation of the mathematical model is given as a quasistatic integro-differential variational inequality system. Based on arguments of the time-dependent variational inequality and Banach's fixed point theorem, an existence and uniqueness of the solution for the quasistatic integro-differential variational inequality system is proved under some suitable conditions. Furthermore, the behavior of the solution with respect to perturbations of time-delay term is considered and a convergence result is also given.

Quasistatic bilateral contact problem with time delay for viscoelastic materials

“…Suppose that assumptions (17) to (22) hold and m A .m 0 c 2 e kgk 2 : Then Problem 11 has a unique solution u 2 V:…”

“…In paper [16] Sofonea et al studied quasistatic viscoelastic problems with bilateral contact and Tresca’s friction, which is modelled by an evolutionary variational inequality. This work was generalized by Yao and Huang [17]. They studied a model which described a time-dependent quasistatic frictional contact problem between a deformable body and a foundation, in which the contact is bilateral, the friction is modelled with Tresca’s friction law with the friction bound depending on the total slip, and the behaviour of the material is described with a viscoelastic constitutive law with time delay.…”

A Dynamic Contact Problem with History-Dependent Operators

“…The research that we present in this paper concerns the study on existence and uniqueness of solutions to the hemivariational inequalities with history-dependent operators, and the investigation on the behaviour of solutions to such inequalities with respect to perturbations of operators and functions. We note that the history-dependent operators have been considered for quasistatic and evolutionary contact problems by several authors, for example, by Sofonea et al in [8,21,[26][27][28] and [29], Migórski et al in [5,15,19] and [20], Ogorzały in [23], Yao et al in [31], and Zhu in [33]. The normal compliance contact condition was introduced in [13,14,22] and it was used in many papers, see, e.g., [3,9,25] and [29].…”

“…Another novel feature of this paper is the analysis of the dynamics. In contrast to other contributions in the field, cf., e.g., [1,12,17,27] and [31], we treat a dynamic contact problem for which the mathematical techniques are less developed than for quasistatic evolutionary models. We underline that there are no results on existence, uniqueness and convergence of solutions to the dynamic hemivariational inequality in Problem 17, which models the contact problem under consideration.…”

A Class of Quasistatic Contact Problems for Viscoelastic Materials With Nonlocal Coulomb Friction and Time-Delay

“…However, to the best of our knowledge, there are only a few papers to study contact problems for viscoelastic materials with time-delay. In the recent paper [24], Yao and Huang introduced and studied a mathematical model which describes an explicit time-dependent quasistatic frictional contact problem between a deformable body and a foundation, in which the contact is bilateral, the behavior of the material is described with a viscoelastic constitutive law with time-delay, and the friction is modeled with Tresca's friction law with the friction bound depending on the total slip.…”