A class of hemivariational inequalities for electroelastic contact problems with slip dependent friction

In this paper we deal with a class of inequality problems for static frictional 
contact between a piezoelastic body and a foundation. The constitutive 
law is assumed to be electrostatic and involves a nonlinear elasticity operator. 
The friction condition is described by the Clarke subdifferential relations of nonmonotone 
and multivalued character in the tangential directions on the boundary. 
We derive a variational formulation which is a coupled system of a hemivariational 
inequality and an elliptic equation. The existence of solutions to the model 
is a consequence of a more general result obtained from the theory 
of pseudomonotone mappings.

1. Introduction. In this paper we consider a mathematical model which describes the stationary contact problem with friction between a piezoelectric body and a rigid foundation. The body is assumed to be elasto-piezoelectric with a nonlinear elasticity operator. We first provide an abstract formulation in the form of operator inclusion of subdifferential type. For it we obtain the existence of solutions by using an approach based on a surjectivity result for a suitable operator in Banach spaces. Then we apply the result to static contact problem with friction for an elasto-piezoelectric body and we prove the existence of weak solutions. As far as the mechanical problem is concerned, the mathematical results have been delivered in [2,12,9,22]. To our knowledge, except [15], there is no result in the literature dealing with hemivariational inequality for piezoelectric frictional contact problems. In contrast to [22], we do not assume that the elasticity operator is strongly monotone and lipschitzean. We also relax the assumption on the friction coefficient.
In this paper we propose an approach based on a general result for pseudomonotone operators. The main feature of the mechanical problem is a nonmonotone multidimensional and multivalued friction boundary condition. It is expressed as the Clarke subdifferential of a locally Lipschitz potential. Such formulation leads in a natural way to the study of a class of hemivariational inequalities. The novelty of this paper is that the friction subdifferential boundary condition has a nonmonotone character since it comes from a nonconvex and nondifferentiable potential. The friction boundary condition is supposed to depend on the displacement field, so in particular, the friction coefficient is allowed to be slip-dependent (see examples in Section 4). We mention that the result on a hemivariational inequality for viscoelastic problems with slip-dependent friction were considered in [16]. In the framework of variational inequalities the contact phenomena for piezoelectric bodies has been considered in [22]. On the other hand, the hemivariational inequalities modelling static and dynamic frictional contact problems without piezoelectric effects have been widely studied in recent years, cf. e.g. [19,20,7,13,14,16,15]. Finally, we remark that an extension of our result to a coupled dynamic system of piezoelectrics with nonmonotone friction boundary conditions seems to be, in a general case, an open problem. For a result on dynamic bilateral contact problem for viscoelastic piezoelectric materials with adhesion, we refer to [17]. We hope to report more on our efforts in this direction in a forthcoming paper.
The paper is organized as follows. In Section 2 we recall some notation and present some auxiliary material. The result on an abstract inclusion is given in Section 3. In Section 4 we state the mechanical problem, describe the classical model for the process, derive its variational formulation and prove existence of the weak solution to the system.

2.
Preliminaries. In this section we introduce the notation and recall some definitions needed in the sequel.
We denote by S d the linear space of second order symmetric tensors on R d (d = 2, 3), or equivalently, the space R d×d s of symmetric matrices of order d. We define the inner products and the corresponding norms on R d and S d by for all σ, τ ∈ S d . The summation convention over repeated indices is used, all indices take values in {1, . . . , d}.
Let Ω ⊂ R d be a bounded domain with a Lipschitz boundary Γ and let n denote the outward unit normal vector to Γ. We use the following spaces and Div : H 1 → L 2 (Ω; R d ) denote the deformation and the divergence operators, respectively, given by ε(u) = (ε ij (u)), ε ij (u) = 1 2 (u i,j + u j,i ), Div σ = (σ ij,j ) and the index following a comma indicates a partial derivative. The spaces H, H, H 1 and H 1 are Hilbert spaces equipped with the inner products The associated norms in H, H, H 1 and H 1 are denoted by · H , · H , · H1 and · H1 , respectively. For every v ∈ H 1 we denote by v its trace γv on Γ, where γ : Given v ∈ H 1/2 (Γ; R d ) we denote by v N and v T the usual normal and the tangential components of v on the boundary For a normed space (X, · X ), if U ⊂ X, then we write U X = sup{ x X : x ∈ U }. Given a reflexive Banach space Y , we denote by ·, · Y * ×Y (or simply ·, · ) the pairing between Y and its dual Y * . Following [23,18,4] we recall some definitions.
Remark 1. The condition (ii) of Definition 1 is equivalent (still under condition (i)) to the following one (ii) ′ if u n → u weakly in Y and lim sup T u n , u n − u ≤ 0, then T u n → T u weakly in Y * and lim T u n , u n − u = 0.

Definition 2.
A multivalued operator T : Y → 2 Y * is said to be pseudomonotone if the following conditions hold: (i) the set T v is nonempty, bounded, closed and convex for all v ∈ Y ; (ii) T is usc from each finite dimensional subspace of Y into Y * endowed with the weak topology; The following result is well known, cf. e.g. [23,5].
is a generalized pseudomonotone operator which is bounded and has nonempty, closed and convex values, then T is pseudomonotone.
The generalized gradient of h at x, denoted by ∂h(x), is a subset of a dual space E * given by ∂h( Let Ω be an open bounded subset of R d , d ≥ 1, with Lipschitz continuous boundary Γ. Let X be a closed subspace of H 1 (Ω; R s ), s ≥ 1 and let Z = H δ (Ω; R s ) with a fixed δ ∈ (1/2, 1). Let A : X → X * be an operator, N : Z → 2 Z * be a multivalued map and g ∈ X * . We consider the following problem find u ∈ X such that g ∈ Au + N u. (1) We say that an element u ∈ X is a solution to (1) if and only if there exists z ∈ Z * such that g = Au + z and z ∈ N u.
The following hypotheses are needed in the sequel.
, H(N ) hold and g ∈ X * , then problem (1) has a solution.
Proof. Let g ∈ X * . We claim that the folowing operator F : Then the result follows from the main surjectivity theorem of nonlinear analysis, cf. Section 32.4 of [23] or Theorem 1.3.70 of [5].
To establish that F is pseudomonotone, it is enough (see Proposition 1) to show that F is generalized pseudomonotone operator (since F is bounded with nonempty closed convex values).
We have v * n = Av n + w n with w n ∈ N v n . By the boundedness of N (cf. H(N )(iii)), by passing to a subsequence if necessary, we obtain w n → w weakly in Z * with w ∈ Z * while the compactness of the embedding Exploiting the pseudomonotonicity of A (cf. Remark 1), we get Av n → Av weakly in X * and Av n , v n − v X * ×X → 0.

Passing to the limit in the equation
For the coercivity of F , it is enough to observe that Thus F is surjective which means that the problem (1) admits a solution.

Formulation of a class of piezoelectric contact problems with friction.
In this section we deal with a class of frictional problems for piezoelectric bodies that can be studied by employing Theorem 1. We now state the contact problem under consideration and give its variational formulation.
Consider an elastic piezoelectric body which initially occupies an open bounded subset Ω in R d , d = 2, 3. The boundary Γ = ∂Ω is assumed to be Lipschitz continuous. The body may come in frictional contact with an obstacle, the fixed foundation. We consider two partitions of Γ. A first partition is given by three mutually disjoint open parts Γ D , Γ N and Γ C such that m(Γ D ) > 0. The second one consists of two disjoint open parts Γ a and Γ b such that m(Γ a ) > 0. The body is subjected to volume forces of density f 1 and volume electric charges of density q 1 . The body is clamped on Γ D and a surface tractions of density f 2 act on Γ N . Moreover, the electric potential vanishes on Γ a and the surface electric charge of density q 2 is applied on Γ b . On Γ C the body may come into contact with a foundation.
We denote by u : Ω → R d the displacement field, by ε(u) = (ε ij (u)), ε ij (u) = 1 2 (u i,j + u j,i ), i, j = 1, . . . , d the strain tensor, by σ : Ω → S d , σ = (σ ij ) the stress tensor and by D : Ω → R d , D = (D i ) the electric displacement field. We also denote E(ϕ) = (E i (ϕ)) the electric vector field, where ϕ : Ω → R is an electric potential such that E i (ϕ) = − ∂ϕ ∂xi . We begin with the strong formulation of the problem of static deformation of an elastic piezoelectric body. The governing equations consist (cf. [10,2,12,9,1]) of the equilibrium equations given by − div D = q 1 in Ω, and the stress-charge form of piezoelectric constitutive relations which have the form We assume that F : Ω × S d → S d is a nonlinear elasticity operator, P : Ω × S d → R d and P ⊤ : Ω × R d → S d is a linear piezoelectric operator and its transpose, respectively and β : Ω × R d → R d is a linear electric permittivity operator. The operators are represented by } are the transpose to p(x) and β(x) = {β ij (x)} are dielectric coefficients, i, j ∈ {1, . . . , d} (second order tensor field). We use here the notation p ⊤ to denote the transpose of the tensor p given p ε · ξ = ε : p ⊤ ξ for ε ∈ S d and ξ ∈ R d .
When the elasticity operator F (x, ·) is linear, then F (x, ε) = C(x)ε with the elasticity coefficients C(x) = {c ijkl (x)}, i, j, k, l = 1, . . . , d (fourth order tensor field) which may be functions of position in a nonhomogeneous material. We also remark that the decoupled state (purely elastic and purely electric deformations) can be obtained by setting p ijk = 0.
To complete the mechanical model, according to the description of the physical setting, we have On the contact surface Γ C , we consider the subdifferential boundary condition in a tangential direction − σ N = S on Γ C , where S is the normal load imposed on Γ C , h : Γ C × R d → R is prescribed and ∂j represents the Clarke subdifferential of the function j : Γ C × R d → R which is locally Lipschitz in its second variable. The strong formulation of the problem consists in finding the displacement u : Ω → R d and the electric potential ϕ : Ω → R such that (2)-(8) hold. To give a variational formulation of this problem we need the following hypotheses.
(v) either j(x, ·) or −j(x, ·) is regular in the sense of Clarke for a.e.
x ∈ Γ C . where ∂j denotes the Clarke subdifferential of j with respect to the variable ξ.
Remark 2. The regularity hypothesis in H(j)(v) is satisfied, for instance, for a function which is represented as the difference of convex functions. More precisely, let us consider the following condition (for simplicity we omit the x-dependence). H(d.c.): The function j : R d → R is locally Lipschitz and of d.c.-type, i.e. j(ξ) = j 1 (ξ) − j 2 (ξ) for ξ ∈ R d , where j k : R d → R, k = 1, 2 are convex functions and one of the convex subdifferentials ∂j k is assumed to be a singleton for every ξ ∈ R d and the growth conditions hold η ≤ c 0 (1 + ξ ) for η ∈ ∂j k (ξ) for all ξ ∈ R d , k = 1, 2 with c 0 > 0. Under the hypothesis H(d.c.) either j or −j is regular in the sense of Clarke and We shortly comment on the friction condition (8). For a detailed description of the examples, we refer to [11,14,13,16,8,21] and the references therein. Example 1. (Contact with nonmonotone friction laws) Consider first, the simple case, when h = 1. This is a case of nonmonotone friction laws which are independent of the slip displacement. The friction law (8) takes the form −σ T ∈ ∂j(x, u T ) on Γ C . This law appears in the tangential direction of an adhesive interface and describes the partial cracking and crushing of the adhesive bonding material. We refer to Section 2.4 of [20] for several examples of the zig-zag friction laws which can be formulated in this form. As a model example we can consider a nonconvex function j : R → R given by j(r) = min{j 1 (r), j 2 (r)}, where j 1 (r) = ar 2 , j 2 (r) = a 2 (r 2 + 1), a > 0. For more details, see [20,14,13].
By making a suitable choice of function h and the convex (hence Clarke's regular) function j(x, ξ) = ξ in the contact boundary conditions (7), (8), we obtain a number of well known monotone friction laws. Two of them we recall below.

Example 3. (A version of Coulomb's friction law)
We consider a contact problem modeled with a version of Coulomb's law of dry friction on Γ C : Here S ∈ L ∞ (Γ C ), S ≥ 0 is a given normal stress and the coefficient of friction µ satisfies H(µ) of Example 2. This law was used in [22] for a piezolectric contact problem and in [6,19,8] for elastic and viscoelastic contact problems. If we take h(x, ξ) = S(x)µ(x, ξ T ) and j(x, ξ) = ξ , then the above Coulomb friction law has the form of (7) and (8). Since ∂ ξ equals to B(0, 1) if ξ = 0 and ξ ξ if ξ = 0, the condition (8) is equivalent to Example 4. (Contact with the Tresca law) Consider the static version of the Tresca law (cf. e.g. [6,8]) on Γ C : where g ≥ 0 represents the friction bound, i.e. the magnitude of force at which slipping begins. This law can be put in the form (8) with h(x, ξ) = g and j(x, ξ) = ξ .
We now pass to the variational formulation of the problem (2)- (8). We introduce the spaces for the displacement and the electric potential: which are closed subspaces of H 1 and H 1 (Ω). On V we consider the inner product and the corresponding norm given by u, v V = ε(u), ε(v) H and v V = ε(v) H for u, v ∈ V . Then (V, · V ) is a Hilbert space. On Φ we consider the inner product (ϕ, ψ) Φ = (ϕ, ψ) H 1 (Ω) for ϕ, ψ ∈ Φ. Then (Φ, · Φ ) is also a Hilbert space.
Assuming sufficient regularity of the functions involved in the problem, using the Green formula, the constitutive relations and the equality ΓC σn · v dΓ = ΓC (σ N v N + σ T · v T ) dΓ, we obtain the following variational formulation of the problem (2)-(8): find u ∈ V and ϕ ∈ Φ such that where This weak formulation represents a coupled system of a hemivariational inequality and an elliptic equation. The existence theorem for (9) will be a consequence of a more general result which we prove below.
The main existence result is as follows Proof. The idea is to apply Theorem 1 and show the existence of a solution to the inclusion (1) with suitable data X, A, N and g. Then we establish that every solution to this inclusion is also a solution to (9). First let X = V × Φ ⊂ H 1 (Ω; R d+1 ) be a Hilbert space endowed with the inner product (y, z) X = (u, v) V + (ϕ, ψ) Φ for y, z ∈ X, y = (u, ϕ), z = (v, ψ). The corresponding norm is denoted by · X . We observe that under the assumption (H 0 ) the elements f and q satisfy f ∈ V * and q ∈ Φ * , i.e. g = (f, q) ∈ X * .
The following two lemmata show the properties of the operators A and N , respectively.  For the proof of Lemma 1 we refer to Lemma 1 of [15]. The proof of Lemma 2 is based on the following result (cf. Lemma 11 in [16]).