In this paper, we consider a mathematical model of a contact problem in thermo-electro-viscoelasticity with the normal compliance conditions and Tresca’s friction law. We present a variational formulation of the problem, and we prove the existence and uniqueness of the weak solution. We also study the numerical approach using spatially semidiscrete and fully discrete finite element schemes with Euler’s backward scheme. Finally, we derive error estimates on the approximate solutions.

1. Introduction

In the recent years, piezoelectric contact problems have been of great interest to modern engineering. General models of electroelastic characteristics of piezoelectric materials can be found in [1, 2]. The problems of piezo-viscoelastic materials have been studied with different contact conditions within linearized elasticity in [35] and within nonlinear viscoelasticity in [68]. The modeling of these problems does not take into account the thermic effect. Mindlin [9] was the first to propose the thermo-piezoelectric model. The mathematical model which describes the frictional contact between a thermo-piezoelectric body and a conductive foundation is already addressed in the static case in [10, 11].

Sofonea et al. considered in [12] the modeling of quasistatic viscoelastic problem with normal compliance friction and damage; they proved the existence and uniqueness of the weak solution, and they derived error estimates on the approximate solutions.

In the article [13], we find the recent result of a new quasistatic mathematical model which describes the chosen thermo-electro-viscoelastic body behavior and the contact by Signorini condition with nonfrictional and nonconductive foundation; also, the variational formulation of this problem is derived and its unique weak solvability is established.

In this paper, we consider a quasi-static contact problem with Tresca’s friction between a thermo-electro viscoelastic body and an electrically and thermally conductive rigid foundation. The novelty in this model, which can be considered as the generalization of the model presented in [13], lies in the use of the penalized normal compliance contact condition:

This means that we allow a weak interpenetration between the body and the foundation. On the contact zone, we consider the following regularized electrical and thermal conditions:which describe both the thermal and electrical conductivities of the foundation. This leads to nonlinear coupling between the mechanical displacement and thermal and electrical fields and hence more complexities on the model.

Since the friction conditions are inequalities, we derive a quasivariational formulation of this problem and we prove the existence and uniqueness of the weak solution based on arguments variational inequalities, Galerkin method, compactness method, and Banach fixed point theorem. We derive error estimates for the numerical approximations based on discrete schemes.

The paper is structured as follows. In Section 2, we present the model of equilibrium process of the thermo-electro-viscoelastic body in frictional contact with a conductive rigid foundation and we introduce the notations and assumptions on the problem data. We also derive the variational formulation of the problem. We state the main results concerning the existence and the uniqueness of a weak solution. We present a spatially semidiscrete scheme and a fully discrete scheme to approximate the contact problem. We then use the finite element method to discretize the domain and Euler’s forward scheme to discretize the time derivatives. Finally, the proofs are established in Section 3.

2. Formulation and Main Results

2.1. Problem Setting

We consider a body of a piezoelectric material which occupies the domain in the reference configuration which will be supposedly bounded with a smooth boundary . This boundary is divided into three open disjoint parts , , and on one hand and a partition of into two open parts and on the other hand, such that and . Let be the time interval, where .

The body is submitted to the action of body forces of density , a volume electric charge of density , and a heat source of constant strength . It is also submitted to mechanical, electrical, and thermal constants on the boundary. Indeed, the body is assumed to be clamped in , and therefore, the displacement field vanishes there. Moreover, we assume that a density of traction forces, denoted by , acts on the boundary part . We also assume that the electrical potential vanishes on , and a surface electrical charge of density is prescribed on . We consider that the temperature is prescribed on the surface .

In the reference configuration, the body may come in contact over with an electrically thermally conductive foundation. Assume that its potential and temperature are maintained at and . The contact is frictional, and there may be electrical charges and heat transfer on the contact surface. The normalized gap between and the rigid foundation is denoted by .

In the following sections, we use to denote the space of second-order symmetric tensors on while “” and will represent the inner product and the Euclidean norm on and , that is

We denote as the displacement field, and the stress tensor, the temperature, and the heat flux vector, and and the electric displacement field. We also denote as the electric vector field, where is the electric potential. Moreover, let denote the linearized strain tensor given by , and “” and “” denote the divergence operators for tensor and vector valued functions, respectively, i.e., and . We shall adopt the usual notation for normal and tangential components of displacement vector and stress: , and , where n denotes the outward normal vector on .

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

Here, the equations (4) and (5) represent the thermo-electro-viscoelastic constitutive law of the material in which denotes the stress tensor, is the linearized strain tensor, is the electric field. , , , , , and are, respectively, elastic, piezoelectric, thermal expansion, electric permittivity, pyroelectric tensor, and (fourth-order) viscosity tensor. is the transpose of given by

The constant represents the reference temperature.

Fourier’s law of heat conduction is given bywhere denotes the thermal conductivity tensor.

Equations (6)–(8) represent the equilibrium equations for the stress. Relations (9) and (10), (11) and (12), and (13) represent the mechanical, the electrical, and the thermal boundary conditions. The unilateral boundary condition (16) represents the normal compliance condition and (17)–(19) represent Tresca’s friction law in which S is the coefficient of friction.

Following [14], the contact conditions (20) and (21) on are obtained as follows:

When there is no contact at a point on the surface, there is no free electrical charges on the surface and no thermal transfer; that is

If the contact holds, i.e., , the normal component of the electric displacement field or the free charge (resp., thermal transfer) is assumed to be proportional to the difference between the potential of foundation and the body’s surface potential (resp., to the difference between the temperature of foundation and the body’s surface temperature). Thus,

We combine (25) and (26) to obtainwhere is the characteristic function of the interval defined by

Equation (27) represents the regularization electrical contact condition and the heat flux condition on , whereand where , and L is a large positive constant, is a small parameter, is supposed to be zero for and positive, otherwise nondecreasing and Lipschitz continuous.

Remark 1. We note that when , equality (20) becomeswhich models the case when the foundation is a perfect electric insulator.
Similarly, in equality (21), we have

2.2. Weak Formulation and Uniqueness Result

To obtain a variational formulation of Problem , we need additional notations and need to recall some definitions in the sequel.

We use the following functional Hilbert spaces:endowed with the canonical inner product given byand the associated norms , , , and , respectively.

Let V, W, and Q be the closed subspaces of given byand the set of admissible displacements

It is known that V, W, and Q are real Hilbert spaces with the inner products , , and , respectively.

Moreover, the associated norm is equivalent on V to the usual norm and and are equivalent on W and Q, respectively, with the usual norms .

By Sobolev’s trace theorem, there exists three positive constants , , and depending on , , , , , and :

Since and Korn’s inequality holdwhere in a nonnegative constant depending only on and . Notice also that since , the following Friedrichs–Poincaré inequalities hold:where and are the positive constants which depend only on , , , and .

For a real Banach space X and , we consider the Banach spaces and of continuous and continuously differentiable functions from to X with the norms

To simplify the writing, we denote by , , , and the following bilinear and symmetric applications:and by , , and , the following bilinear applications:

In the study of mechanical Problem , we make the following assumptions: The elasticity operator , the electric permittivity tensor , the viscosity tensor , and the thermal conductivity tensor satisfy the usual properties of symmetry, boundedness, and ellipticity:and there exists that such that The piezoelectric tensor , the thermal expansion tensor , and the pyroelectric tensor satisfyand there exist the positive constants , and such that The surface electrical conductivity and the thermal conductance satisfy the following hypothesis for such that , a.e., and is measurable on for all and is zero for all .The function is a Lipschitz function on for all ., where is a positive constant. The forces, the traction, the volume, the surfaces charge densities, and the strength of the heat source are as follows:

The potential and the temperature satisfy

The initial conditions, the friction-bounded function, and the gap function satisfy

Using Riesz’s representation theorem, we define , , and by the following:

We define the mappings , , , and a.e., by respectively.

Now, by a standard variational technique, it is straightforward to see that if satisfies the conditions (4)–(21), a.e. , then

We assume that the initial conditions and satisfy the following compatibility condition: there exists such that

This nonlinear problem, has a unique solution , by using the fixed point theorem.

Using all of these assumptions, notations, and , we obtain the following variational formulation of the Problem .

Problem : find a displacement field , an electric potential , and a temperature field , a.e., such that

We present now the existence and the uniqueness of solution to Problem .

Theorem 1. Assume that the assumptions , (37)–(42), for ,and the conditionshold. Then, Problem has a unique solution as follows:

2.3. Spatially Semidiscrete Approximation

In this paragraph, we consider a semidiscrete approximation of the Problem by discretizing the spatial domain, using the finite element method. Let be a regular finite element partition of the domain compatible with the boundary partition . We then define a finite element space for the approximates of the displacement field u, for the electric potential φ, and for the temperature θ defined by

A spatially semidiscrete scheme can be formed as the following problem:

Problem .Find , , and , a.e., such that for , , and

Here, , , and are appropriate approximations of , , and , respectively.

Using the same argument presented in Section 2, it can be shown that Problem has a unique solution , , and .

In this paragraph, we are interested in obtaining estimates for the errors , , and .

Using the initial value condition, we have

Theorem 2. Assume that the assumptions stated in Theorem 1 are hold, for . Then, under the conditionsthe semi-discrete solution of converges as follows:

2.4. Fully Discrete Approximation

In this paragraph, we consider a fully discrete approximation of Problem . We use the finite element spaces , , and introduced in Section 2.3. We introduce a partition of the time interval . We denote the step size for and let be the maximal step size. For a sequence , we denote .

The fully discrete approximation method is based on the backward Euler scheme, and it has the following form.

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

Remark 2. The choice of and instead of and is motivated by the fixed point method in the proof of the existence and uniqueness. Otherwise, we may get another different condition for the uniqueness of the solution of fixed iteration problem (79)–(82). In addition, this choice will be helpful for the application of discrete Grönwall’s lemma in the next.

To simplify again the notation, we introduce the velocity

This problem has a unique solution, and the proof is similar to that used in Theorem 1.

We now derive the following convergence result.

Theorem 3. Assuming that the initial values , , and are chosen in such a way thatunder the condition stated in Theorem 1 and for , the fully discrete solution converges, i.e.,

3. Proof of Main Results

In this section, we prove the theorems presented in the previous section.

3.1. Proof of Theorem 1

The proof of Theorem 1 is based on fixed point argument, Galerkin method, and compactness method, similar to those used in [14, 15] but with a different choice of the operators.

We turn now the following existence and uniqueness result.

Let given by

In the first step, we consider the intermediate Problem .

Problem . Find for, a.e., such that

We have the following result for .

Lemma 1. For all and for, a.e., , the Problem has a unique solution .

Proof. Using Riesz’s representation theorem, we define the operator and the element byThen, Problem can be written in the following form:For all , there exists a constant which depends only on , , and ϵ such thatThe assumption and imply that and by , the operator is continuous and coercive.
We use now the result presented in pp. 61–65 in [16], and we conclude Problem has a unique solution .
Next, we use the displacement field obtained in the first step and we consider the following lemma proved in [15].

Lemma 2. (a)For all and, a.e., , the problemhas a unique solution .(b)For all and for, a.e., , the problemhas a unique solution .

In the last step, for , and are the functions obtained in Lemma 2 and we consider the operator defined byfor all and for, a.e., .

For the operator , we have the following result obtained in [15].

Lemma 3. There exists a unique such that .

We now turn to a proof of Theorem 1.

Existence. Let be the fixed point of the operator and us denote the solution of variational Problem , for ; the definition of and Problem prove that is a solution of Problem .

Uniqueness. The uniqueness of the solution follows from the uniqueness of the fixed point of the operator .

3.2. Proof of Theorem 2

To prove Theorem 2, we need the following result

Lemma 4. Assume that . Then, we have the estimate as follows:for a positive constant .

Proof. Take in (64) and add the inequality to (71), we haveThen,where