A Quasistatic Electro-Viscoelastic Contact Problem with Adhesion

The aim of this paper is to study the process of contact with adhesion between a piezoelectric body and an obstacle, the so-called foundation. The material’s behavior is assumed to be electro-viscoelastic; the process is quasistatic, the contact is modeled by the Signorini condition. The adhesion process is modeled by a bonding field on the contact surface. We derive a variational formulation for the problem and then we prove the existence of a unique weak solution to the model. The proof is based on a general result on evolution equations with maximal monotone operators and fixed-point arguments.


Introduction
A piezoelectric body is one that produces an electric charge when a mechanical stress is applied (the body is squeezed or stretched). Conversely, a mechanical deformation (the body shrinks or expands) is produced when an electric field is applied. This  kind of materials appears usually in the industry as switches in radiotronics, electroacoustics, or measuring equipments. Piezoelectric materials for which the mechanical properties are elastic are also called electro-elastic materials, and those for which the mechanical properties are viscoelastic are also called electro-viscoelastic materials. Different models have been developed to describe the interaction between the electrical and mechanical fields( see, e.g., [2,14,[16][17][18][19][29][30][31] and the references therein). General models for elastic materials with piezoelectric effect, called electro-elastic materials, can be found in [2,4,14]. A static frictional contact problem for electricelastic materials was considered in [1,15], under the assumption that the foundation is insulated. Contact problems involving elasto-piezoelectric materials [1,15,28], viscoelastic piezoelectric materials [5,25] have been studied.
Adhesion may take place between parts of the contacting surfaces. It may be intentional, when surfaces are bonded with glue, or unintentional, as a seizure between very clean surfaces. The adhesive contact is modeled by a bonding field on the contact surface, denoted in this paper by β; it describes the pointwise fractional density of active bonds on the contact surface, and sometimes referred to as the intensity of adhesion. Following [10,11], the bonding field satisfies the restrictions 0 ≤ β ≤ 1; when β = 1 at a point of the contact surface, the adhesion is complete and all the bonds are active; when β = 0 all the bonds are inactive, severed, and there is no adhesion; when 0 < β < 1 the adhesion is partial and only a fraction β of the bonds is active. Basic modeling can be found in [10][11][12]. Analysis of models for adhesive contact can be found in [7,8] and in the monographs [24,27]. An application of the theory of adhesive contact in the medical field of prosthetic limbs was considered in [22,23]; there, the importance of the bonding between the bone-implant and the tissue was outlined, since debonding may lead to decrease in the persons ability to use the artificial limb or joint.
In this work we continue in this line of research, where we extend the result established in [3,20] for contact problem described with the Signorini conditions into contact problem described with the Signorini conditions with adhesion where the obstacle is a perfect insulator and the resistance to tangential motion is generated by the glue, in comparison to which the frictional traction can be neglected. Therefore, the tangential contact traction depends only on the bonding field and the tangential displacement.
The paper is structured as follows. In Sect. 2 we present the electro-viscoelastic contact model with adhesion and provide comments on the contact boundary conditions. In Sect. 3 we list the assumptions on the data and derive the variational formulation. In Sect. 4, we present our main existence and uniqueness result, Theorem 4.1, which states the unique weak solvability of the Signorini adhesive contact problem. The proof of the theorem is provided in Sect. 5, where it is carried out in several steps and is based on a general result on evolution equations with maximal monotone operators and fixed-point theorem.

The Model
We consider a body made of a piezoelectric material which occupies the domain ⊂ R d (d = 2, 3) with a smooth boundary ∂ = and a unit outward normal ν. The body is acted upon by body forces of density f 0 and has volume free electric charges of density q 0 . It is also constrained mechanically and electrically on the boundary. To describe these constraints we assume a partition of into three open disjoint parts 1 , 2 , and 3 , on the one hand, and a partition of 1 ∪ 2 into two open parts a and b , on the other hand. We assume that meas 1 > 0 and meas a > 0; these conditions allow the use of coercivity arguments in the proof of the unique solvability of the model. The body is clamped on 1 , and therefore, the displacement field vanishes there. Surface tractions of density f 2 act on 2 . We also assume that the electrical potential vanishes on a and a surface electrical charge of density q 2 is prescribed on b . On 3 the body is in adhesive contact with an insulator obstacle, the so-called foundation. The contact is frictionless and, since the foundation is assumed to be rigid, we model it with the Signorini condition. We are interested in the deformation of the body on the time interval [0 T ]. The process is assumed to be quasistatic, i.e., the inertial effects in the equation of motion are neglected. We denote by x ∈ ∪ and t ∈ [0 T ] the spatial and the time variable, respectively, and, to simplify the notation, we do not indicate in what follows the dependence of various functions on x and t. Here and everywhere in this paper, i, j, k, l = 1, . . . , d, summation over two repeated indices is implied, and the index that follows a comma represents the partial derivative with respect to the corresponding component of x. The dot above variable represents the time derivatives.
We denote by S d the space of second-order symmetric tensors on R d (d = 2, 3) and by ". , . the inner product and the norm on S d and R d , respectively, that is We also use the usual notation for the normal components and the tangential parts of vectors and tensors, respectively, given by υ ν = υ · ν, υ τ = υ − υ ν ν, σ ν = σ i j ν i ν j , and σ τ = σ ν − σ ν ν.
With these assumptions, the classical model for the process is the following.

2)
Div (2.14) We now provide some comments on equations and conditions (2.1)-(2.14). First, equations (2.1) and (2.2) represent the electro-viscoelastic constitutive law in which σ = (σ i j ) is the stress tensor, ε(u) = (ε i j (u)) denotes the linearized strain tensor, E(ϕ) = −∇ϕ is the electric field, A and F are the viscosity and elasticity operators, respectively, E = (e i jk ) represents the third-order piezoelectric tensor, E * = (e * i jk ), where e * i jk = e ki j , is its transpose, B =(B i j ) denotes the electric permittivity tensor, and D = (D 1 , . . . , D d ) is the electric displacement vector. Details on the constitutive equations of the form (2.1) and (2.2) can be found, for instance, in [1,2,13,21] and the references therein.
Next, Eqs. (2.3) and (2.4) are the equilibrium equations for the stress and electric displacement fields, respectively, in which Div" and "div " denote the divergence operators for tensor and vector valued functions, respectively.
Conditions (2.5) and (2.6) are the displacement and traction boundary conditions, whereas (2.10) and (2.11) represent the electric boundary conditions. Note that we need to impose assumption (2.12) for physical reasons. Indeed, this condition models the case when the obstacle is a perfect insulator and was used in [1,9,15,25,26]. The evolution of the bonding field is governed by the differential Eq. (2.9) with given positive parameters γ ν and ε a where r + = max{0, r }.
Condition (2.7) represents the Signorini contact condition with adhesion, where u ν is the normal displacement, σ ν represents the normal stress, γ ν denotes a given adhesion coefficient, and R ν is the truncation operator defined by Here L > 0 is the characteristic length of the bond, beyond which it does not offer any additional traction (see [27]). We assume that the resistance to tangential motion is generated only by the glue, and is assumed to depend on the adhesion field and on the tangential displacement, but, again, only up to the bond length L (see (2.8)), where the truncation operator R τ is defined by Then, p τ (β) acts as the stiffness or spring constant, increasing with (β), and the traction is in the direction opposite to the displacement. The maximal modulus of the tangential traction is p τ (1)L.
Finally, (2.13) and (2.14) represent the initial conditions in which u 0 and β 0 are the prescribed initial displacement and bonding fields, respectively.

Variational Formulation and Preliminaries
In this section, we list the assumptions on the data and derive a variational formulation for the contact problem. To this end we need to introduce some notation and preliminaries.
Everywhere below, we use the classical notation for L p and Sobolev spaces associated to and . Moreover, we use the notation L 2 ( ) d , H 1 ( ) d , H, and H 1 for the following spaces The spaces L 2 ( ) d , H 1 ( ) d , H, and H 1 are real Hilbert spaces endowed with the canonical inner products given by and the associated norms · L 2 ( ) d , · H 1 ( ) d , · H , and · H 1 , respectively. Here and below we use the notation For every element υ ∈ H 1 ( ) d we also write υ for the trace of υ on and we denote by υ ν and υ τ the normal and tangential components of υ on .
We now list the assumptions on the problem's data. The viscosity operator A and the elasticity operator F are assumed to satisfy the conditions The piezoelectric tensor E and the electric permittivity tensor B satisfy (3.4) As in [8] we assume that the tangential contact function satisfies The forces, tractions, volume, and surface-free charge densities satisfy The adhesion coefficient γ ν and the limit bound ε a satisfy the conditions Also, we assume that the initial bonding field satisfies Moreover, the tensor E and its transpose E * satisfy the equality Let now consider the closed subspace of H 1 ( ) d defined by Since meas ( 1 ) > 0 and the viscosity tensor satisfies assumption (3.1), it follows that V is a real Hilbert space endowed with the inner product (3.11) and let · V be the associated norm. We also introduce the following spaces Since meas ( a ) > 0 it is well known that W is a real Hilbert space endowed with the inner product (ϕ, ψ) W = (∇ϕ, ∇ψ) L 2 ( ) d , and the associated norm · W . Also we have the following Friedrichs-Poincaré inequality where c F > 0 is a constant which depends only on and a . The space W is a real Hilbert space endowed with the inner product and the associated norm · W . Moreover, by the Sobolev trace theorem, there exist two positive constants c 0 andc 0 such that for all υ ∈ V, ψ ∈ W , and t ∈ [0, T ]. We note that the definitions of f and q are based on the Riesz representation theorem. Moreover, it follows from assumptions (3.6) and (3.7) that For Signorini problem, we use the convex subset of admissible displacements fields given by U ad = {υ ∈ V / υ ν ≤ 0 on 3 } , and we make the regularity assumption on the initial data. Also, we introduce the set a.e. on 3 }.
Let X be a real Hilbert space with the inner product (·, ·) X and the associated norm · X , and let A : The operator A : (ii) maximal monotone if A is monotone and there is no monotone operator B : X −→ 2 X such that Gr(A) is a proper subset of Gr(B), which is equivalent to the following implication For a function φ : X −→] − ∞, +∞] we use the notation D(φ) and ∂φ for the effective domain and the subdifferential of φ, i.e., Finally, let φ K : X →] − ∞, +∞] denote the indicator function of the set K , i.e., It can be shown that the subdifferential of the indicator function ∂φ K : X −→ 2 X of a closed convex K of the space X is a maximal monotone operator. We can also show that the sum of a maximal monotone operator and a single-valued monotone Lipschitz continuous operator is a maximal monotone operator.
Finally, we use the usual notation for the Lebesgue spaces L p (0, T ; X ) and Sobolev spaces W k, p (0, T ; X ) where 1 ≤ p ≤ ∞ and k ∈ N. We will need the following result for existence and uniqueness proofs.

Theorem 4.2 Let X be a real Hilbert space and let A : D(A)
⊂ X −→ 2 X be a multivalued operator such that the operator A + ωI X is maximal monotone for some real ω. Then, for every f ∈ W 1,1 (0, T ; X ) and u 0 ∈ D(A), there exists a unique function u ∈ W 1,∞ (0, T ; X ) which satisfieṡ A proof of Theorem 4.2 may be found in ( [6], p. 32). Here and below I X is the identity map on X .

Proof of Theorem 4.1
We assume in the following that the conditions of Theorem 4.1 hold and below we denote by c a generic positive constant which is independent of time and whose value may change from place to place.
By the Riesz representation theorem we can define the following operators G : W −→ W and R : V −→ W , respectively, by We can show that G is a linearly continuous symmetric positive definite operator. Therefore, G is an invertible operator on W . We can also prove that R is a linear continuous operator on V . Let R * the adjoint of R. Thus, from (3.10) we can write where we obtain for all t ∈ [0, T ]. On the other hand, G is invertible where the previous equality gives us Now, using (5.3), (5.5), and (3.22) we obtain Let η ∈ W 1,∞ (0, T ; V ) be given. In the first step we prove the following existence and uniqueness result for the displacement field.

Lemma 5.1 There exists a unique function u
Proof Let now the operator L : V → V defined by Using the properties of the operators G, R, and R * we deduce that L is a continuous linear operator on V . Thus we have By the Riesz representation theorem we can define an operator G : Now, taking into account (3.1), (3.2), (3.11) and (5.11) it follows that is, G is a Lipschitz continuous operator. Moreover, the operator is a monotone Lipschitz continuous operator on V .
Let the function f : [0 T ] −→ V given by Keeping in mind that η ∈ W 1,∞ (0, T ; V ), using (3.16), (3.17) and the fact that R * G −1 is linearly continuous, it follows from (5.13) that Let φ U ad : V → − ∞, +∞ denote the indicator function of the set U ad and let ∂φ U ad be the subdifferential of φ U ad . Since U ad is a nonempty, convex, closed part of V , it follows that ∂φ U ad is a maximal monotone operator on V and D(∂φ U ad ) = U ad . Moreover, the sum is a maximal monotone operator. Thus, conditions (3.18) and (5.14 ) allow us to apply Theorem 4.
Since for any elements u, g ∈ V , the following equivalence holds the differential inclusion (5.15) is equivalent to the following variational inequality We use now (5.17), (5.11), (3.11) to see that u η satisfies the following inequality It follows now from (5.18), (5.13), (5.9), and (5.16) that u η satisfies (5.7) and (5.8), which concludes the proof of Lemma 5.1.
In the second step we use the displacement field u η obtained in Lemma 5.1 to obtain the following existence and uniqueness result for the electric potential field.
In the third step, we use again the displacement field u η obtained in Lemma 5.1 and we consider the following initial value problem.
We obtain the following result.