In this chapter we present notation and preliminary material which is necessary in the study of the boundary value problems presented in Chapters 5–7. We start by introducing some function spaces that will be relevant in the study of contact problems. Then we provide a general description of the mathematical modelling of the processes involved in contact between an elastic, viscoelastic or viscoplastic body and an obstacle, say a foundation. We describe the physical setting, the variables which determine the state of the system, the material's behavior which is reflected in the constitutive law, the input data, the equation of equilibrium for the state of the system and the boundary conditions for the system variables. Finally, we extend this description to the contact of piezoelectric bodies for which we consider both the case when the foundation is conductive and the case when it is an insulator.

Everywhere in the rest of the book we assume that Ω ⊂ ℝd (d = 1, 2, 3) is open, connected, bounded, and has a Lipschitz continuous boundary Γ. We denote by = Ω ∪ Γ the closure of Ω in ℝd. We use bold face letters for vectors and tensors, such us the outward unit normal on Γ, denoted by ν. A typical point in ℝd is denoted by x = (xi). The indices i, j, k, l run between 1 and d, and, unless stated otherwise, the summation convention over repeated indices is used.