Variational inequality models arising in the study of viscoelastic materials

Abstract

In this paper, we present a synthesis of different results obtained recently in the papers (Chau and Awbi, Appl. Anal. 83(6):635-648, 2004; Addi et al., Discrete Continuous Dyn. Syst. 31(4):1039-1051, 2011; Adly et al., Numer. Algebra Control Optim. 2(1):89-101, 2012; Adly and Chau, Math. Program. Ser. B, 2014; Chau et al., Int. J. Appl. Math. Mech., 2012). It concerns the study of contact problems for viscoelastic materials with possible thermal effects. We first describe a general thermo-viscoelastic model involving a thermo-viscoelastic Kelvin-Voigt constitutive law, a temperature field governed by the heat equation and a subdifferential surface contact condition. Then we study a model which describes the frictional contact between a short memory thermo-viscoelastic body and a given rigid foundation. The free boundary contact problem for a long memory viscoelastic material is also considered. Finally, we provide numerical simulations for different fundamental examples of thermal contact problems.