a piezoelectric contact problem with slip dependent coefficient of ...

A Piezoelectric Contact Problem with Slip Dependent Coefficient 231density q 0 . It is also submitted to mechanical and electric constraints on the boundary.To describe them, we consider a partition of Γ into three measurable parts Γ 1 , Γ 2 ,Γ 3 , on one hand, and on two measurable parts Γ a and Γ b , on the other hand, suchthat meas Γ 1 > 0 and meas Γ a > 0. We assume that the body is clamped on Γ 1and surfaces tractions of density f 2 act on Γ 2 . On Γ 3 the body is in frictional contactwith an obstacle, the so-called foundation. We model the contact with a version ofCoulomb’s law of dry friction, already used in [3] and [6], in which the normal stressis prescribed and the coefficient of friction depends on the slip. We also assume thatthe electrical potential vanishes on Γ a and a surface electric charge of density q 2is prescribed on Γ b . We denote by S d the space of second order symmetric tensorson R d or, equivalently, the space of symmetric matrices of order d. Also, below νrepresents the unit outward normal on Γ while “ · ” and ‖ · ‖ denote the inner productand the Euclidean norm on R d and S d , respectively.With the assumption above, the problem of equilibrium of the electroelastic bodyin frictional contact with a foundation is the following.Problem P . Find a displacement field u : Ω → R d , a stress field σ : Ω → S d , anelectric potential ϕ : Ω → R and an electric displacement field D : Ω → R d suchthatσ = Fε(u) − E T E(ϕ) in Ω, (2.1)D = Eε(u) + βE(ϕ) in Ω, (2.2)Div σ + f 0 = 0 in Ω, (2.3)div D = q 0 in Ω, (2.4)u = 0 on Γ 1 , (2.5)σν = f 2 on Γ 2 , (2.6)− σ ν = S on Γ 3 , (2.7){‖στ ‖ ≤ µ(‖u τ ‖)|S|,σ τ = −µ(‖u τ ‖)|S| ‖u uττ ‖ , if u τ ≠ 0on Γ 3 ,(2.8)ϕ = 0 on Γ a , (2.9)D · ν = q 2 on Γ b . (2.10)In (2.1) – (2.10) and below, in order to simplify the notation, we do not indicateexplicitly the dependence of various functions on the spatial variable x ∈ Ω ∪ Γ .Equations (2.1) and (2.2) represent the electroelastic constitutive law of the materialin which F is a given nonlinear function, ε(u) denotes the small strain tensor,E(ϕ) = −∇ϕ is the electric field, E represents the third order piezoelectric tensor,E T is its transposite and β denotes the electric permitivitty tensor. Details of thelinear version of the constitutive relations (2.1) and (2.2) can be find in [1, 2]. Equations(2.3) and (2.4) represent the equilibrium equations for the stress and electricdisplacementfields, respectively, (2.5) and (2.6) are the displacement and tractionboundary conditions, respectively, and (2.9), (2.10) represent the electric boundaryconditions.We now provide some comments on the frictional contact conditions (2.7) and(2.8), which are our main interest. Condition (2.7) states that the normal stress σ ν