APPLICATIONES MATHEMATICAE 36,1 (2009), pp. 107–127 Arezki Touzaline (Bab Ezzouar) A QUASISTATIC UNILATERAL AND FRICTIONAL CONTACT PROBLEM WITH ADHESION FOR ELASTIC MATERIALS Abstract. We consider a quasistatic contact problem between a linear elastic body and a foundation. The contact is modelled with the Signorini condition and the associated non-local Coulomb friction law in which the adhesion of the contact surfaces is taken into account. The evolution of the bonding field is described by a first order differential equation. We de- rive a variational formulation of the mechanical problem and prove exis- tence of a weak solution if the friction coefficient is sufficiently small. The proofs employ a time-discretization method, compactness and lower semi- continuity arguments, differential equations and the Banach fixed point the- orem. 1. Introduction. Contact problems involving deformable bodies are quite frequent in industry as well as in daily life and play an important role in structural and mechanical systems. Because of the importance of such processes a considerable effort has been put into their modelling and numerical simulations. A first study of frictional contact problems within the framework of variational inequalities was made in [8]. The mathematical, mechanical and numerical state of the art can be found in [15]. In this paper we study a quasistatic unilateral contact problem with a non-local Coulomb friction law and adhesion between a linear elastic body and an obstacle, the so-called foundation. Models for dynamic or quasistatic processes of frictionless adhesive contact between a deformable body and a foundation have been studied in [3, 4, 11, 20]. As in [10, 11] we use the bonding field β as an additional state variable, defined on the contact surface of the boundary. 2000 Mathematics Subject Classification : 74M10, 74M15, 47J20, 49J40. Key words and phrases : elastic materials, adhesion, quasistatic, time-discretization, fric- tion, fixed point, weak solution. DOI: 10.4064/am36-1-8 [107] c  Instytut Matematyczny PAN, 2009