Quasistatic Adhesive Contact of Piezoelectric Cylinders

We consider two mathematical models which describe the antiplane shear deformation of a piezoelectric cylinder in adhesive contact with a rigid foundation. The material is assumed to be electro-viscoelastic in the first model and electro-elastic in the second one. In both models the process is quasistatic, the foundation is electrically conductive and the adhesion is described with a surface variable, the bonding field. We derive a variational formulation of the models which is given by a system coupling two variational equations for the displacement and the electric potential fields, respectively, and a differential equation for the bonding field. Then we prove the existence of a unique weak solution to each model. We also investigate the behavior of the solution of the electro-viscoelastic problem as the viscosity converges to zero and prove that it converges to the solution of the corresponding electro-elastic problem.


Introduction
The present paper is devoted to the study of quasistatic antiplane contact problems with adhesion for piezoelectric cylinders.Our interest is to present two problems in which both antiplane shear, contact, adhesion and piezoelectric effect are involved, to prove their unique solvability, and to study their link by providing a convergence result.
Antiplane shear deformations are one of the simplest examples of deformations that solids can undergo: in antiplane shear of a cylindrical body, the displacement is parallel to the generators of the cylinder and is independent of the axial coordinate.For this reason, considerable attention has been paid to the modelling of such kind of problems, see for instance [1][2][3].Antiplane frictional contact problems were used in geophysics in order to describe pre-earthquake evolution of the regions of hight tectonic activity, see for instance [4,5] and the references therein.The mathematical analysis of various models for antiplane frictional contact problems can be found in [6][7][8] and in the recent monograph [9].
Piezoelectric materials are characterized by the coupling between the mechanical and electrical properties, see [10][11][12] and the references therein.This coupling leads to the appearance of electric potential when mechanical stress is present and, conversely, mechanical stress is generated when electric potential is applied.Piezoelectric materials for which the mechanical properties are elastic are called electro-elastic materials and those for which the mechanical properties are viscoelastic are called electro-viscoelastic materials.Antiplane contact problems for piezoelectric materials were considered in [13][14][15][16].In [13,15,16] the contact was assumed to be frictional and in [14] is was assumed to be adhesive.
Processes of adhesion are important in many industrial settings where parts, usually nonmetallic, are glued together.For this reason the adhesive contact between bodies has recently received increased attention in the literature.General models can be found in [17,18] and the mathematical analysis of various adhesive contact problems can be found in [19][20][21][22][23]. Existence and uniqueness results in the study of mathematical models which describes the adhesive contact of piezoelectric materials were obtained recently in [24,25], in the three-dimensional framework.
The present paper represents a continuation of [14].There, a mathematical model which describes the antiplane shear deformation of a piezoelectric cylinder in adhesive contact with a rigid foundation was considered.The material was assumed to be electroviscoelastic and the process was assumed to be mechanically dynamic.An existence and uniqueness result of the solution to the model was obtained by using arguments of evolution equations with monotone operators and fixed point.Unlike [14], in the present paper we model the material's behavior by an electro-viscoelastic constitutive law or by an electro-elastic constitutive law; also, we neglect the inertial term in the equation of motion and, therefore, we assume that the process is mechanically quasistatic.This leads to consider two mathematical models, different from that studied in [14], which represents the first trait of novelty of this paper.We derive the variational formulation of the models and then we prove the existence of a unique weak solution, for each model.In addition, we study the link of the two models and provide a converge result, which consists the second trait of novelty of this paper.
The rest of the paper is structured as follows.In Section 2 we present the models for the antiplane adhesive contact of piezoelectric cylinders.Then we introduce the notation, list the assumptions on problem's data and derive the variational formulation of each model.In Section 3 we study the electro-viscoelastic problem for which we state and prove an existence and uniqueness result, Theorem 1.In Section 4 we state and prove an existence and uniqueness result for the electro-elastic problem, Theorem 2. The proof of both theorems are carried out in several steps by constructing intermediate problems for the displacement field, the electric potential and the bonding field.We prove the unique solvability of the intermediate problems, then we consider a contraction mapping whose unique fixed point leads us to construct the solution of the original problem.Finally, in Section 5 we provide a convergence result, Theorem 3. It states that the solution of the electro-viscoelastic problem converges to the solution of the electro-elastic problem as the viscosity converges to zero.

Statement of the problems
We consider a piezoelectric body which occupies a region B ⊂ R 3 , in a fixed and undistorted reference configuration.We assume that B is a cylinder with generators parallel to the x 3 -axes with a cross-section which is a regular domain Ω in the x 1 , x 2 plane, Ox 1 x 2 x 3 being a cartesian coordinate system.The cylinder is assumed to be sufficiently long so that the end effects in the axial direction are negligible.Thus, B = Ω × (−∞, +∞).The cylinder is acted upon by body forces and electric charges.It is also constrained mechanically and electrically on the boundary.To describe the boundary conditions we denote by ∂Ω = Γ the boundary of Ω and we assume a partition of Γ into three open disjoint parts Γ 1 , Γ 2 and Γ 3 , on the one hand, and a partition of Γ 1 ∪ Γ 2 into two open parts Γ a and Γ b , on the other hand, such that the one-dimensional measure of Γ 1 and Γ a , denoted by meas Γ 1 and meas Γ a , are positive.The cylinder is clamped on Γ 1 × (−∞, +∞) and therefore the displacement field vanishes there.We assume that surface tractions act on Γ 2 × (−∞, +∞), the electrical potential vanishes on Γ a × (−∞, +∞) and a surface electrical charge is prescribed on Γ b × (−∞, +∞).Also, the cylinder is in contact over Γ 3 × (−∞, +∞) with a conductive obstacle, the so called foundation; the contact is adhesive and it is modelled with a surface internal variable, the bonding field.We assume that the process is mechanically quasistatic, i.e. we neglect the inertial term in the equation of motion; moreover, we consider the antiplane context described in [14], in which the evolution of the cylinder's state does not depend on the axial coordinate and is described by functions defined on the x 1 , x 2 plane.
We denote by T > 0 the time interval of interest; everywhere in this paper the dot above represents the derivative with respect to the time, i.e. u = ∂u ∂t , and the index that follows a comma represents the partial derivative with respect to the corresponding spatial variable, i.e. u ,i = ∂u ∂xi , i = 1, 2. We denote by ν 1 , ν 2 the components of the unit normal on Γ and we use the notation For the first problem we assume that the material is electro-viscoelastic.Then, following the arguments in [14], it follows that the problem can be formulated as follows.
We now describe problem (1)-( 11) and provide a brief explanation of the equations and the boundary conditions.More details can be found in [14] where the dynamic version of Problem P was considered.
Equations ( 1) and ( 2) represent the balance equations in which θ is a viscosity coefficient, µ is the the Lamé coefficient, α is the electric permittivity constant and e is a piezoelectric coefficient.Here f 0 and q 0 represent the axial component of the body force and the electric charge density, respectively.We note that equation ( 1) is obtained from the equation of motion by neglecting the inertial term and we use it since the process is assumed to be mechanically quasistatic.Conditions (3) and (4) represent the boundary conditions for the displacement and the electrical potential field and prescribe that these variables vanish on Γ 1 and Γ a , respectively, during the process.Conditions ( 5) and ( 6) represent the traction and electrical condition on Γ 2 and Γ b , respectively, in which f 2 and q b represent the densities of the axial component of the traction force and the electric charge, respectively.Condition (7) represents the traction condition on the contact surface Γ 3 and we use it since the contact is adhesive.Here p is a given function and R is the real valued function defined by with L > 0 being a characteristic length of the bonds, see e.g.[18].It follows from (7) that the shear of the contact surface depends on the bonding field and on the tangential displacement, but only up to the bond length L. The frictional tangential traction is assumed to be much smaller than the adhesive one and, therefore, omitted.Condition ( 8) represents the electrical conductivity on the contact surface, where ϕ F represents the electric potential of the foundation and k is the electric conductivity coefficient.This condition shows that the normal component of the electric displacement field is proportional to the difference between the potential on the foundation and the body's surface.We use it since the foundation is electrically conductive and the shear is antiplane, which implies that there is no loss of the contact during the process.
The differential equation ( 9) describes the evolution of the bonding field in which γ and ǫ a are given adhesion coefficients, R is defined by (12) and r + = max {r, 0}.In ( 9) and below we use the simplified notation R(u) 2 for the square of R(u), i.e.R(u) 2 = (R(u)) 2 .We note that the adhesive process described by ( 9) is irreversible; indeed, once debonding occurs, bonding cannot be reestablished, since β ≤ 0. Considering a condition which allows the adhesive process for rebonding would represent an important extension of the results in this paper.
Finally, ( 10) and ( 11) represent the initial conditions in which u 0 and β 0 are the prescribed initial displacement and bonding fields, respectively.
For the second problem we assume that the material is electro-elastic, i.e. the viscosity coefficient vanishes.Therefore, we remove the initial condition for the displacement field and take θ = 0 in Problem P, to obtain the following problem.
We turn now to the variational formulation of the Problems P and Q.To this end we introduce the function spaces where, here and below, we write w for the trace on Γ of a function w ∈ H 1 (Ω).Since meas Γ 1 > 0 and meas Γ a > 0, it is well known that V and W are real Hilbert spaces with the inner products Moreover, the associated norms are equivalent with the usual norm • H 1 (Ω) .Also, by Sobolev's trace theorem we deduce that there exists positive constants For a real Banach space (X, • X ) we use the usual notation for the spaces L p (0, T ; X) and W k,p (0, T ; X) where 1 ≤ p ≤ ∞, k = 1, 2, . ..; we also denote by C([0, T ]; X) and C 1 ([0, T ]; X) the spaces of continuous and continuously differentiable functions on [0, T ] with values in X, respectively, with the norms Finally, we use the set and we recall that if X is reflexive, then W 1,∞ (0, T ; X) is the space of Lipschitz continuous functions defined on [0, T ] with values in X.
We list now the assumptions on the problem's data.We assume that the viscosity coefficient and the electric permittivity coefficient satisfy θ ∈ L ∞ (Ω) and there exists θ * > 0 such that θ(x) ≥ θ * a.e.x ∈ Ω, (25) α ∈ L ∞ (Ω) and there exists α * > 0 such that α(x) ≥ α * a.e.x ∈ Ω. (26) We also assume that the Lamé coefficient and the piezoelectric coefficient satisfy The tangential function p is such that The adhesion coefficients γ and ǫ a satisfy the conditions The forces, tractions, volume and surface free charges densities have the regularity and the electric conductivity coefficient satisfies Finally, we assume that the electric potential of the foundation and the initial data are such that Next, we define bilinear forms a θ : for all u, v ∈ V , ϕ, ψ ∈ W. Assumptions ( 25)-( 28) and (34) imply that the integrals above are well defined and, using ( 23) and (24), it follows that the forms a θ , a µ , a e , a * e and a α are continuous; moreover, the forms a θ , a µ and a α are symmetric and, in addition, the form a θ is V -elliptic and the form a α is W -elliptic, i.e.
We also define the mappings respectively, by The definition of f and q is based on Riesz's representation theorem; moreover, it follows from (32)-( 35) that Next, we perform integrals par parts and use notation ( 38)-( 41), ( 44)-( 46) to obtain the following variational formulation of the electro-viscoelastic Problem P. and Similar arguments lead to the following variational formulation of the electro-elastic Problem Q. and Well-posedness of the variational Problems P V and Q V will be proved in Theorems 1 and 2 below.We conclude by these theorems the existence of a unique weak solution to Problems P and Q, respectively.

Study of the electro-viscoelastic problem
Our main existence and uniqueness result in the study Problem P V is the following.
Theorem 1. Assume that (25)-(37) hold.Then, there exists a unique solution of Problem (49)-(52).Moreover, the solution satisfies The proof of Theorem 1 will be carried out in several steps and is based on arguments similar to those used in [14].The modifications arise mainly in the treatment of the variational equation (49) since, unlike [14], here the process is assumed to be mechanically quasistatic.The treatment of the variational equation (50) as well as that of the differential equation ( 51) is similar to that in [14] and, for this reason, we omit the corresponding details.We assume in what follows that ( 25)-(37) hold and below in this section we denote by c a generic positive constant which may depend on Ω, Γ 1 , Γ 2 , Γ 3 , Γ a , Γ b , θ, µ, e, α p, L and T , but does not depend on the time, nor on the rest of the input data, and whose value may change from place to place.
Let η ∈ C([0, T ]; V ) be given.The first step of the proof is given by the following existence and uniqueness result for the displacement field.

Lemma 1. There exists a unique function u
Proof.We use the properties of the bilinear form a θ and the Lax-Milgram lemma to see that, for all t ∈ [0, T ], there exists a unique element v η (t) ∈ V such that Consider now t 1 , t 2 ∈ [0, T ]; using (62) and (42) we find that We note that regularity of f and η combined with (63) imply that v η ∈ C([0, T ]; V ).Let u η : [0, T ] → V be the function defined by It follows from ( 62) and (64) that u η is a solution of the problem (60)-( 61) and it satisfies u η ∈ C 1 ([0, T ]; V ).This concludes the existence part of Lemma 1.The uniqueness part follows from the uniqueness of the solution of the variational equation (62), at any t ∈ [0, T ].
In the next two steps we use the displacement field u η obtained in Lemma 1 to obtain the following existence and uniqueness result for the electric potential field and the bonding field, respectively.Lemma 2. There exists a unique function ϕ η ∈ W 1,∞ (0, T ; W ) such that Lemma 3.There exists a unique function The proof of Lemma 2 is based on arguments similar to those used in the proof of Lemma 1, see also [14].The proof of Lemma 3 can be found in [14]; it is based on a version of the Cauchy-Lipschitz theorem, see for instance [23, p. 48] .Now, for η ∈ C([0, T ]; V ) we denote by u η , ϕ η and β η the functions obtained in Lemmas 1, 2 and 3, respectively.We use Riesz's representation theorem to define the function Λη : [0, T ] → V by for all w ∈ V and t ∈ [0, T ].We have the following result.
Proof.Let η ∈ C([0, T ]; V ) and let t 1 , t 2 ∈ [0, T ].Using (68), the continuity of the bilinear forms a µ and a * e and (46), we obtain Now, keeping in mind (24), assumptions on the function p, the inequality 0 ≤ β η ≤ 1 and the properties of the operator R we find that Since In what follows we use the notation Using arguments similar to those in the proof of (69) we find that On the other hand, from (65), ( 66) and (67), it was proved in [14] that We combine now the inequalities (70), ( 71) and ( 72) to obtain Also, since u 1 and u 2 have the same initial value it follows that We use now the last two inequalities to obtain Next, (62) and the properties of the form a θ yield and, using this inequality in (73), we deduce that Reiterating this inequality m times yields which implies that for m sufficiently large a power Λ m of Λ is a contraction in the Banach space C([0, T ]; V ).Therefore, there exists a unique element η * ∈ C([0, T ]; V ) such that and Λη * = η * .The regularity η * ∈ W 1,∞ (0, T ; V ) follows from the regularity Λη * ∈ W 1,∞ (0, T ; V ), which concludes the proof.Now, we have all the ingredients necessary to prove Theorem 1.
Uniqueness.The uniqueness of the solution follows from the uniqueness of the fixed point of Λ and the uniqueness part in Lemmas 1-3.

Study of the electro-elastic problem
The proof of the unique solvability of the electro-elastic Problem Q V could be obtained by using arguments similar to those used in the proof of Theorem 1.However, since the viscosity term is missing, in this case the corresponding inequality (74) would not contain an integral term.As a consequence, the use of the Banach fixed point arguments would require a smallness assumption on the problem's data and therefore would restrict the solvability of the problem.To avoid this restriction, in the study of electro-elastic Problem Q V we shall use a method which is different from that used in the study of the electro-viscoelastic Problem P V .We start by reinforcing assumption (27) as follows: µ ∈ L ∞ (Ω) and there exists µ * > 0 such that µ(x) ≥ µ * a.e.x ∈ Ω.
We note that in this case the bilinear form a µ is V -elliptic, since it safisfies Our main result concerning the unique solvability of Problem Q V is the following.
Theorem 2. Assume that (26)-( 37) and (75) hold.Then, there exists a unique solution of Problem (53)-(56).Moreover, the solution satisfies The proof of Theorem 2 will be carried out in several steps.We assume in what follows that (26)-( 37) and (75) hold; below in this section c will denote a generic positive constant which may depend on Ω, Γ 1 , Γ 2 , Γ 3 , Γ a , Γ b , µ, e, α p, L and T , but does not depend on t, nor on the rest of the input data, and whose value may change from place to place.
Let β ∈ Z be given.The first step of the proof is given by the following existence and uniqueness result.Lemma 5.There exists a unique couple Proof.We consider the product space X = V × W together with the inner product and the associated norm • X .Let t ∈ [0, T ] be given; we define the operator A β (t) : X → X and the element h(t) ∈ X by It is easy to see that equalities (80) and ( 81) hold if and only if the element x β (t) = (u β (t), ϕ β (t)) ∈ X satisfies the following equation in X: In order to solve (85), we investigate the properties of the operator A β (t).First, we use ( 46), ( 29) and (12) to see that for all u 1 , u 2 , v ∈ V .Next, we use the previous two inequalities, (83), (76), ( 43) and (82) to find that A β (t) satisfies for all x 1 , x 2 ∈ X.We conclude that A β (t) is a strongly monotone Lipschitz continuous operator and therefore, using a standard existence and uniqueness result, we obtain the existence of a unique element x β (t) ∈ X which solves (85).We conclude from above that there exists a unique couple of functions (u β (t), ϕ β (t)) which solve (80) and (81), at any t ∈ [0, T ].
Next, we let t 1 , t 2 ∈ [0, T ] and use the notation We use standard arguments in (80) and (81) to find Then, we add these equalities and use ( 76) and (43) to obtain We use again ( 46), ( 29) and (12) to see that and therefore, combining (86) and (87), after some algebra we find This inequality and the regularity of the functions f , q and β show that u β ∈ C([0, T ]; V ) and ϕ β ∈ C([0, T ]; W ). Thus, we conclude the existence part in Lemma 5 and we note that the uniqueness of the solution follows from of the unique solvability of ( 80) and (81), at any t ∈ [0, T ].
In the next step we use the displacement field u β obtained in Lemma 5 and the arguments used in the proof of Lemma 3 to obtain the following existence and uniqueness result for the bonding field.Lemma 6.There exists a unique function Proof.Suppose that β 1 and β 2 are two functions in Z and denote by u i , ϕ i and ξ i the functions obtained in Lemmas 5 and 6 for β = β i , i = 1, 2. Let t ∈ [0, T ].We use arguments similar to those used in the proof of (88) to deduce that On the other hand, (91) and the estimate (72) obtained for the Cauchy problem (89)-(90) leads to We now combine (93) and (92) to see that and, by reiterating this last inequality m times, we obtain Recall that Z is a nonempty closed set in the Banach space C([0, T ]; L 2 (Γ 3 )) and note that (94) shows that for m sufficiently large the operator Λ m : Z → Z is a contraction.
Uniqueness.The uniqueness of the solution follows from the uniqueness of the fixed point of Λ and the uniqueness part in Lemmas 5-3.

A convergence result
In this section we investigate the behavior of the weak solution of the electro-viscoelastic Problem P V as the viscosity converges to zero.In order to outline the dependence on the viscosity coefficient θ, we reformulate Problem P V as follows.
Next, note that Theorem 3 shows that the convergences (103)-(105) hold under the assumption (100), whatever is the choice of the initial displacement of the electro-viscoelastic Problem P θ V .It also shows that, if the initial displacement u 0 is chosen to be the displacement of the corresponding electro-elastic Problem Q at t = 0 and (100) is replaced by the stronger assumption (101), then the convergences (103)-( 105) can be reinforced by the convergences (106)-(108).
Finally, consider the case of homogeneous viscosity, i.e. the case when assumption ( 25) is replaced by the assumption θ(x) = θ > 0 a.e.x ∈ Ω, where θ is given.In this case θ L ∞ (Ω) = θ, θ * = θ and therefore the convergences (100) and ( 101) are equivalent to θ → 0. Therefore, by Theorem 3 we conclude that the solution to the electro-viscoelastic Problem P V may be approached by the weak solution to the electro-elastic Problem Q V , as the viscosity is small enough.In addition to the mathematical interest of this result, it is important from the mechanical point of view, since it shows that the electro-elasticity with adhesion can be considered as a limit case of electro-viscoelasticity with adhesion as the viscosity decreases.
Then we use the Banach fixed point theorem to conclude the proof.Now, we have all the ingredients needed to prove Theorem 2.