Analysis of a history-dependent frictional contact problem

Abstract

We consider a mathematical model which describes the quasistatic contact between a viscoelastic body and a foundation. The material’s behaviour is modelled with a constitutive law with long memory. The contact is frictional and is modelled with normal compliance and memory term, associated to the Coulomb’s law of dry friction. We present the classical formulation of the problem, list the assumptions on the data and derive a variational formulation of the model. Then we prove the unique weak solvability of the problem. The proof is based on arguments of history-dependent variational inequalities. We also study the dependence of the weak solution with respect to the data and prove a convergence result.

Keywords:

• viscoelastic material
• frictional contact
• history-dependent variational inequality
• weak solution

AMS Subject Classifications:

• 74M15
• 74M10
• 74G25
• 74G30
• 49J40

Acknowledgments

The work of the second author was supported within the Sectoral Operational Programme for Human Resources Development 2007–2013, co-financed by the European Social Fund, under the project POSDRU/107/1.5/ S/76841 entitled Modern Doctoral Studies: Internationalization and Interdisciplinarity, at University Babeş-Bolyai, Cluj-Napoca, Romania.