We consider a mathematical model which describes a static frictional contact between a piezoelectric body and a thermally conductive obstacle. The constitutive law is supposed to be thermo-electro-elastic and the contact is modeled with normal compliance and a version of Coulomb’s friction law. We derive a variational formulation of the problem and we prove the existence and uniqueness of its solution. The proof is based on some results of elliptic variational inequalities and fixed point arguments. Furthermore, a finite element approximation and a priori error estimates are obtained.

1. Introduction

Contact problems involving thermopiezoelectricity arise when there is an interaction between the mechanical, electrical, and thermal properties of the considered material. This kind of problems has a considerable interest in various fields of modern industries. Indeed, the thermal effects such as thermal deformations and pyroelectric effects are important to many smart ceramic materials. Thus, for some materials, it may be impossible to predict the electromechanical behavior without taking into account the thermal effects; see for example, [14]. There is a considerable interest in contact problems involving piezoelectric materials when thermal effects are considered. These so-called thermopiezoelectric materials can operate effectively as distributed sensors and actuators for controlling structural response. In sensor applications, mechanically or thermally induced disturbances can be determined from measurement of the induced electric potential difference (direct piezoelectric effect), whereas in actuator applications deformation or stress can be controlled through the introduction of an appropriate electric potential difference (converse piezoelectric effect). Among the numerous applications of these materials, we cite accelerometers, microphones, ultrasonic transducers, and so on. A good example is the use of this class of materials as sensors and actuators in microelectromechanical systems, for instance, the piezoelectric accelerometer which triggers an airbag in ten thousandths of a second during an accident.

Some general models and their analysis have been established for elastic-piezoelectric materials [58], for thermoelastic materials [911], and for electro/thermoviscoelastic bodies [12, 13]. The mathematical treatment of static contact process for thermopiezoelectric materials is recent. The reason lies in the considerable difficulties of the nonlinear evolutionary inequalities modeling the static contact problems present in the variational analysis. Existence and uniqueness results in the study of static contact problems can be found, for instance, in [1416].

The present paper is a continuation of this kind of models. It deals with a new and nonstandard mathematical model, which is in a form of a system coupling a nonlinear variational inequality for the displacement field and two nonlinear variational equations for the electric potential and the temperature field. The boundary conditions on the contact surface used in this paper is described with a general normal compliance law and the associated Coulomb’s friction law, taking into account the electrical and thermal conductivity of the foundation. This work serves two purposes. The first purpose is to provide the variational analysis of the mechanical problems and to show the existence of a unique solution to each model. The second is to study a discrete scheme based on finite element method for numerical solutions and to establish the unique solvability of the scheme and derive an optimal order error estimate.

The rest of the paper is structured as follows. In Section 2, we present the model of our frictional thermopiezoelectric contact problem and we derive its variational formulation, given as a coupled system for the displacement field, the electric potential, and the temperature fields. In Section 3, we provide the assumptions on the data and we state the main existence and uniqueness theorem together with its proof. In Section 4, we present the main error estimate results for the finite element approximation of the problem.

2. The Mathematical Model

We consider a thermopiezoelectric body that occupies a bounded domain in () with a Lipschitz continuous boundary . In the sequel, we decompose into three open and disjoint parts , , and such that and on the one hand and we consider a partition of into two sets and with disjoint, relatively open sets and such that on the other hand.

We assume that the body is fixed on where the displacement field vanishes and it is subject to a volume forces of density , a volume electric charges of density , and a heat source term per unit volume in . We assume also that a surface tractions of density and a strength of heat source act on and a surface electric charges of density acts on . Finally, we suppose that the electrical potential vanishes in and the temperature is maintained constant at the part of the boundary, set to be . Over the contact surface , the body comes in a frictional contact with a thermally conductive foundation.

Here and below, the indices , , , and run between and , the summation convention over repeated indices is adopted, and the index that follows a comma indicates a partial derivative with respect to the corresponding component of the spatial variable; for example, . In the sequel, let be the space of second order symmetric tensors on . The canonical inner products and norms on and are given by

To present the mathematical model which describes the physical setting above, we denote by , , , , , and the displacement field, the stress tensor, the electric potential field, the electric displacement field, the temperature field, and the heat flux vector, respectively. These are functions which depend on the spatial variable . Nevertheless, in what follows we do not indicate explicitly the dependence of these quantities on ; that is, we write instead of . Also, denotes the linearized strain tensor and is the electric field.

We assume that the process is static, then the equations of stress equilibrium, the equation of the quasistationary electric field, and the heat conduction equation arewhere and are the divergence operator for tensor and vector field. The material is assumed to be thermopiezoelectric and satisfies the following constitutive laws [1, 4]:where is the elasticity tensor, is the third-order piezoelectric tensor, represents the electric permittivity tensor, is the thermal expansion, and denotes the pyroelectric tensor. Moreover, is the transpose of and it satisfiesFor the heat flux, we adopt the following Fourier-type law:where denotes the thermal conductivity tensor. We use classical decomposition in the normal and the tangential components of the displacement and of the stress on ; that is,where is the outward unit normal vector on and the physical setting to complete our model with the following boundary conditions:We model the frictional contact on with the following reduced normal compliance condition:where , , , and are material interface parameters and represents the penetration approach; for more detail see [1719]. Here (15) is the normal compliance power law and (16) is a variant of Coulomb’s friction law. Furthermore, the thermoelectric contact is described with the following regularized conditions (see [13, 14]):where is the thermal conductance function, supposed to be zero for and positive otherwise, nondecreasing, and Lipschitz continuous, and is the foundation temperature. The truncate function is defined bywhere is a large positive constant. At last, we note that condition (17) describes the fact that the foundation is supposed to be a perfect electrical insulator. Under all these conditions, the classical formulation of our problem is as follows.

Problem (). Find a displacement field , an electric potential , and a temperature field such that (2)–(7) and (9)–(18) hold.

Note that once the triplet which solve is known, then the stress tensor , the electric displacement field , and the heat flux can be obtained from (5) and (7). In order to derive the variational formulation of the problem , we need the following Hilbert spaces:endowed with the following inner products:Next, we note that, by the Sobolev trace theorem, we can define the trace of a function on such that if . For simplicity, for an element we still denote by its trace on . Let and , then the trace operator is a linear continuous operator; that is, there exists a positive constant , depending only on , such thatKeeping in mind condition (9), we introduce the closed subspace of given bySince , Korn’s inequality holds; there exists a positive constant which depends only on and such that (see, e.g., [7, 20])We define over the space , the following inner product, and its associated norm:It comes from (24) that the norms and are equivalent on and therefore is a real Hilbert space. Finally, note that from (22) and (24), we deduce that there exists a positive constant depending on , , and such thatAccording to the boundary conditions (11) and (13), the electric potential field and the temperature field are, respectively, to be found in the closed subspace and of given bySince , the Friedrichs-Poincaré inequality holds and thus there exists a positive constant depending only on and such thatWe introduce on the inner product and its associated norm . It follows from (28) that the norms and are equivalent on and therefore, the space is a real Hilbert. Since , we can prove in an analogous way that the norm associated with the inner product is equivalent to the usual norm on and thus, is a real Hilbert space. Using the Sobolev trace theorem, we get that there exists a positive constant depending only on , , and such thatand a positive constant which depends only on , , and such that

In the study of the mechanical problem , the following assumptions will be needed:(h1)The elasticity operator , the electric permittivity tensor , and the thermal conductivity tensor satisfy the usual properties of symmetry, boundedness, and ellipticityand there exist positive constants , , and such that(h2)The piezoelectric tensor , the thermal expansion tensor , and the pyroelectric tensor satisfy(h3)The thermal conductance satisfies the following conditions:(h4)The function is a Lipschitz function on for all ; that is,(h5)The forces, the traction, the charges densities, and the strength of the heat source satisfy(h6)The foundation temperature satisfies(h7)The friction bound function and the coefficient of friction satisfy(h8)The material interface parameters satisfyUsing the Riesz representation theorem, we can define , , and as follows:We consider the functionals and defined byIt follows from (h3) and (h5)–(h8) that the integrals above are well defined. We note that if and are sufficiently regular, the following Greens formulas hold:Using the previous Greens formulas, it is straightforward to see that if are sufficiently regular functions which satisfy (2)–(7) and (9)–(18), thenfor all , , and . We use (5), (7), and the notation to obtain the following variational formulation of our problem.

Problem . Find a displacement field an electric potential and a temperature field such that

3. Existence and Uniqueness of Weak Solution

The main existence and uniqueness result in the study of the problem are as follows.

Theorem 1. Assume (h1)–(h3) and (h5)–(h8) hold. Then one has the following:(1)The problem has at least one solution.(2)Under (h4), there exists a constant such that ifthen the problem has a unique solution.Here the norms of the tensors and are given by

The proof will be carried out in several steps and it is based on arguments of variational inequalities and fixed point techniques. We assume in what follows that (h1)–(h3) and (h5)–(h8) hold and for every where , , we defineFor all supposed to be known, we consider the following auxiliary problem.

Problem . Find the elements , , and such thatIn the study of this problem, the two following results will be needed.

Lemma 2. Let be a positive real number. For all , one hasand consequently for all , one has

Proof. This lemma can be obtained by examining the two cases and .

Theorem 3. Let be open bounded set of with a Lipschitz boundary and . The trace operator satisfies the following results:(1)If , the map is compact for any . Then, there exists such that(2)If , the map is compact for any and then, there exists such that

Proof. For the proof of the trace Theorem 3, we can refer to [21, 22].

Lemma 4. Under the assumptions (h1)–(h3) and (h5)–(h8), the problem has a unique solution which depends Lipschitz continuously on .

Proof. We use Riesz’s representation theorem to define the element of and the operator such thatThus equation (55) will beUsing the assumptions of (see (h1)), we can deduce that is a linear symmetric and positive definite operator. Hence, is linear continuous and invertible operator on and let denote its inverse. Thus, by the Lax-Milgram theorem, we get that problem (62) has a unique solutionIt follows from the properties of the operators , , and that and are linear continuous operators. Moreover, we apply Riesz’s representation theorem to define the element and the operators and as follows:Let be the adjoint operator of and then, it comes from (66) thatNext, we replace (64), (65), and (66) in (54) to obtainKeeping in mind the properties of (see (h1)), we get that is a linear symmetric and positive definite operator. Hence, the operator is invertible and let be its inverse. We haveIt comes from (63), (69) that inequality (53) is equivalent to find such thatThe variational problem above is equivalent to the following minimization problem:where the functional is defined as follows:We consider the function whose derivative isApplying Lemma 2, we get for all and of thatHence the function is convex. In plus, it follows from the strict convexity of that the functional is strictly convex on . Moreover, since is coercive and , we deduce that is coercive. Consequently, the minimization problem has a unique solution . Therefore, keeping in mind (63) and (69), we conclude that the variational problem has a unique solution of
For the second part of Lemma 4, let and be two given elements of the reflexive space such that and . We consider , , the unique solution of the problems , , respectively. Then, the variational inequality (53) leads toWe take in the first inequality and in the second to getand by adding the two induced inequalities, we obtainMoreover, the definition of and Lemma 2 imply thatThen, we haveIt follows from (26) and the definition of thatIn addition, the variational equation (54) leads toAfter taking in (81) and in (81), we add the induced equationsUsing (79), (80), (83), (30), and (6), the strong monotonicity of , and the ellipticity of and after some algebra, we find that there exists a constant such thatFrom (60), (51), and (30), we obtainand from (61), (62), (85), and the ellipticity of , we find that there exists such thatFinally, we combine (84) and (87) to find that there exists a constant such thatHence the second part of Lemma 4 is proved.

Remark 5. The second part of Lemma 4 implies that the function where the triplet is the solution of is a continuous function from to .

Lemma 6. If the triplet is a solution of problem , then there exists a positive constant such that

Proof. Taking in the variational equation (55), we getUsing (43), (30), and the ellipticity of the operator , we findMoreover, if we take in (53) and in (54), we haveKeeping in mind (6), the ellipticity of and , the positivity of and , and the properties of and , we deduce that there exists a constant such thatWe combine the two inequalities (91) and (93) to obtainwhich finishes the proof.

Remark 7. Using the same argument as in the proof of in Lemma 6, we have that if the triplet is a solution of problem , thenwhere is the same constant as in (94).

In this step, we consider the operator defined byNow, we shall prove that the operator has a unique fixed point . For that, we need to introduce the two closed convex subsets of where and will be defined below.

Lemma 8. For a specified values of and , the operator has at least one fixed point.

Proof. Let . We have and and thenOn another hand, it follows from the definition of thatUsing (h3), (h8), and Theorem 3, we deduce that there exists a constant such thatSince , it becomes from Lemma 6 that there exists a constant such thatIf we choose and , we getHence is an operator of into itself. Since is a nonempty, convex, and closed subset of the reflexive space , then is weakly compact. Using the continuity of the functions and and Remark 5, we deduce that