Analysis of a contact problem with wear and unilateral constraint

Abstract

This paper represents a continuation of our previous work, where a mathematical model which describes the equilibrium of an elastic body in frictional contact with a moving foundation was considered. An existence and uniqueness result was proved, together with a convergence result. The proofs were carried out by using arguments of elliptic variational inequalities. In this current paper, we complete our model by taking into account the wear of the foundation. This makes the problem evolutionary and leads to a new and nonstandard mathematical model, which couples a time-dependent variational inequality with an integral equation. We provide the unique weak solvability of the model by using a fixed point argument, among others. Then, we penalize the unilateral contact condition and prove that the penalized problem has a unique solution which converges to the solution of the original problem, as the penalization parameter converges to zero.

Keywords:

• Elastic material
• frictional contact
• normal compliance
• unilateral constraint
• wear
• weak solution
• penalization

AMS Subject Classifications:

• 74M15
• 74G25
• 74G30
• 49J40

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of the first two authors has been partially supported by project LEA MATH MODE 2014/2015. The work of the second author has been partially supported by project POSDRU/159/1.5/S/132400: Young successful researchers-professional development in an international and interdisciplinary environment at Babeş-Bolyai University.