Variational and Numerical Analysis of a Static Thermo-Electro-Elastic Problem with Friction

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.


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, [1][2][3][4].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 [5][6][7][8], for thermoelastic materials [9][10][11], 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 [14][15][16].
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.

The Mathematical Model
We consider a thermopiezoelectric body that occupies a bounded domain Ω in R  ( = 2,3) with a Lipschitz continuous boundary Γ.In the sequel, we decompose Γ into three open and disjoint parts Γ 1 , Γ 2 , and Γ 3 such that Γ = Γ 1 ∪ Γ 2 ∪ Γ 3 and meas(Γ 1 ) > 0 on the one hand and we consider a partition of Γ 1 ∪ Γ 2 into two sets Γ  and Γ  with disjoint, relatively open sets Γ  and Γ  such that meas(Γ  ) > 0 on the other hand.
We assume that the body is fixed on Γ 1 where the displacement field vanishes and it is subject to a volume forces of density  0 , a volume electric charges of density  0 , and a heat source term per unit volume  0 in Ω.We assume also that a surface tractions of density  2 and a strength of heat source  2 act on Γ 2 and a surface electric charges of density  2 acts on Γ  .Finally, we suppose that the electrical potential vanishes in Γ  and the temperature is maintained constant at the part Γ 1 of the boundary, set to be   .Over the contact surface Γ 3 , the body comes in a frictional contact with a thermally conductive foundation.
Here and below, the indices , , , and  run between 1 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 S  be the space of second order symmetric tensors on R  .The canonical inner products and norms on R  and S  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 are where Div  =  , and div  =  , 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 F is the elasticity tensor, E is the third-order piezoelectric tensor,  represents the electric permittivity tensor, M is the thermal expansion, and P denotes the pyroelectric tensor.Moreover, E * is the transpose of E and it satisfies For the heat flux, we adopt the following Fourier-type law: where K denotes the thermal conductivity tensor.We use classical decomposition in the normal and the tangential components of the displacement  and of the stress  on Γ; that is, where ] is the outward unit normal vector on Γ and the physical setting to complete our model with the following boundary conditions: We model the frictional contact on Γ 3 with the following reduced normal compliance condition: ≤   ( ] − ) where   ,   ,   , and   are material interface parameters and ( ] − ) + = max ( ] − , 0) represents the penetration approach; for more detail see [17][18][19].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  < 0 and positive otherwise, nondecreasing, and Lipschitz continuous, and   is the foundation temperature.The truncate function   is defined by where  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.
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 V of a function For simplicity, for an element V ∈  1 we still denote by V its trace V on Γ.Let  Γ =  1/2 (Γ, R  ) and  2 (Γ)  =  2 (Γ, R  ), then the trace operator  :  1 →  Γ ⊂  2 (Γ)  is a linear continuous operator; that is, there exists a positive constant , depending only on Ω, such that Keeping in mind condition (9), we introduce the closed subspace  of  1 given by Since meas(Γ 1 ) > 0, Korn's inequality holds; there exists a positive constant   which depends only on Ω and Γ 1 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 ‖ ⋅ ‖  1 are equivalent on  and therefore (, ‖V‖  ) is a real Hilbert space.Finally, note that from (22) and (24), we deduce that there exists a positive constant  0 depending on Ω, Γ 1 , and Γ 3 such that According 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  1 (Ω) given by In the study of the mechanical problem (), the following assumptions will be needed: (h 1 ) The elasticity operator F = (  ) : Ω × S  → S  , the electric permittivity tensor  = (  ) : Ω × R  → R  , and the thermal conductivity tensor K = (  ) : Ω × R  → R  satisfy the usual properties of symmetry, boundedness, and ellipticity and there exist positive constants  F ,   , and  K such that The piezoelectric tensor E = (  ) : Ω × S  → R  , the thermal expansion tensor M = (  ) : Ω × R  → R  , and the pyroelectric tensor P = (  ) : Ω → R  satisfy The forces, the traction, the charges densities, and the strength of the heat source satisfy (h 6 ) The foundation temperature satisfies The friction bound function and the coefficient of friction satisfy (h 8 ) The material interface parameters satisfy Using the Riesz representation theorem, we can define  ∈ ,  ∈ , and  ∈  as follows: We consider the functionals   ,   :  ×  → R and  :  ×  ×  → R defined by (, , ) = ∫ It follows from (h 3 ) and (h 5 )-(h 8 ) that the integrals above are well defined.We note that if  ∈ H 1 and Ψ ∈ W 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), then for all V ∈ ,  ∈ , and  ∈ .We use ( 5), (7), and the notation () = −∇ to obtain the following variational formulation of our problem.

Existence and Uniqueness of Weak Solution
The main existence and uniqueness result in the study of the problem () are as follows.
(2) Under (h 4 ), there exists a constant  * > 0 such that if then the problem () has a unique solution.
Here the norms of the tensors P = (  ) and M = (  ) 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 (h 1 )-(h 3 ) and (h 5 )-(h 8 ) hold and for every  = ( 1 ,  2 ) where  1 ,  2 ≥ 0 ∈  2 (Γ 3 ), we define For all  supposed to be known, we consider the following auxiliary problem.
Problem (  ).Find the elements   ∈ ,   ∈ , and   ∈  such that In the study of this problem, the two following results will be needed.

Lemma 2.
Let  be a positive real number.For all ,  ∈ R, one has and consequently for all  ∈ R, one has Proof.This lemma can be obtained by examining the two cases  ≥ 0 and  ≤ 0.
Proof.We use Riesz's representation theorem to define the element and the operator T :  →  such that Thus equation (55) will be Find   ∈  such that (T  , )  = (  , )  , ∀ ∈ . (62) Using the assumptions of K (see (h 1 )), we can deduce that T is a linear symmetric and positive definite operator.Hence, T is linear continuous and invertible operator on  and let C denote its inverse.Thus, by the Lax-Milgram theorem, we get that problem (62) has a unique solution It follows from the properties of the operators T, M, and P that MC and PC 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) that Next, we replace (64), (65), and (66) in (54) to obtain Keeping in mind the properties of  (see (h 1 )), we get that  is a linear symmetric and positive definite operator.Hence, the operator  is invertible and let  −1 be its inverse.We have It comes from ( 63), (69) that inequality (53) is equivalent to find   ∈  such that The variational problem above is equivalent to the following minimization problem: where the functional   :  → R is defined as follows: We consider the function j Applying Lemma 2, we get for all V and  of  that Hence the function j is convex.In plus, it follows from the strict convexity of   − j that the functional   = (  − j ) + j is strictly convex on .Moreover, since   − j is coercive and   (V) ≥   (V) − j (V), 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  = ( 1 ,  2 ) and   = (  1 ,   2 ) be two given elements of the reflexive space  2 (Γ 3 ) 2 such that  2 ≥ 0 and   2 ≥ 0. We consider (  ,   ,   ), (   ,    ,    ), the unique solution of the problems (  ), (   ), respectively.Then, the variational inequality (53) leads to We take V =    in the first inequality and V =   in the second to get and by adding the two induced inequalities, we obtain Moreover, the definition of   and Lemma 2 imply that Then, we have It follows from (26) and the definition of  2 that In addition, the variational equation (54) leads to After taking  =   −    in (81) and  =    −   in (81), we add the induced equations Using (79), (80), ( 83), (30), and ( 6), the strong monotonicity of F, and the ellipticity of  and after some algebra, we find that there exists a constant   > 0 such that From ( 60), (51), and (30), we obtain Proof.Taking  =   in the variational equation (55), we get Using ( 43), (30), and the ellipticity of the operator K, we find Moreover, if we take V = 0 ∈  in (53) and  =   ∈  in (54), we have where c1 is the same constant as in (94).
Lemma 8.For a specified values of  1 and  2 , the operator Λ has at least one fixed point. Proof.
On another hand, it follows from the definition of Λ that Using (h 3 ), (h 8 ), and Theorem 3, we deduce that there exists a constant c > 0 such that Since ‖ 1 ‖  2 (Γ 3 ) ≤  1 , it becomes from Lemma 6 that there exists a constant c1 > 0 such that Hence Λ is an operator of K 1 × K 2 into itself.Since K 1 × K 2 is a nonempty, convex, and closed subset of the reflexive space  2 (Γ 3 )× 2 (Γ 3 ), then K 1 ×K 2 is weakly compact.Using the continuity of the functions   and   and Remark 5, we deduce that Λ is a continuous operator.Hence, by Schauder's fixed point theorem the operator Λ has a fixed point.
Proof of Theorem 1.
Existence.Let  * be the fixed point of the operator Λ obtained in Lemma 8. We denote by ( * ,  * ,  * ) the solution of the problem (  ) for  =  * .It follows from the definition of Λ and (  ) that ( * ,  * ,  * ) is a solution of () which concludes the proof of the existence part.
Uniqueness.Here, we will show that there exists a positive constant  * such that () has a unique solution if   + ‖P‖ + ‖M‖ +    +   <  * .We consider ( 1 ,  1 ,  1 ) and ( 2 ,  2 ,  2 ) two solutions of ().From (46), we have After taking V =  2 in the first inequality and V =  1 in the second and adding the resulting inequalities, it follows from the positivity of   ( 1 −  2 ,  1 −  2 ) that Plus the variational equation (47) leads to We take  =  1 −  2 in the first equation and  =  2 −  1 in the second to obtain Hence, the addition of ( 104) and (106) implies that Moreover, we use the variational equation (48) to deduce We substitute  by  1 −  2 and we subtract the two induced equations to obtain Let us consider Then, we have where Taking in mind (26), (30), and the assumptions (h Using the following mathematical inequalities (see [23]) Then, we obtain Keeping in mind Theorem 3 and Remark 7, we obtain where 107), (114), and (118), after some algebra it follows that there exists C > 0 such that where Choose  * = 1/ C.Then, if   + ‖P‖ + ‖M‖ +    +   <  * holds, we will conclude that  1 =  2 ,  1 =  2 , and  1 =  2 which leads to the uniqueness part of Theorem 1.

Discrete Approximation
This section deals with the discrete approximation of the problem ().We assume that the conditions (h 1 )-(h 8 ) hold, then the problem () has a unique solution (, , ) ∈  × ×.Let T ℎ = (  ) ∈G ℎ be a family of regular triangulations of the polygonal domain Ω such that Here and below ℎ > 0 is a discretization parameter.We define the following finite dimensional subspaces  ℎ ,  ℎ , and  ℎ which approximate, respectively, the spaces , , and  by where P 1 (  ) denotes the space of polynomials of a degree lower or equal to one on   .Then, the discrete approximation of the problem () is as follows.
Problem ( ℎ ).Find the displacement  ℎ ∈  ℎ , the electric potential  ℎ ∈  ℎ , and the temperature  ℎ ∈  ℎ such that Under the assumptions of Theorem 1 and with the same arguments, we can prove that the discrete problem ( ℎ ) has a unique solution ( ℎ ,  ℎ ,  ℎ ) in  ℎ ×  ℎ ×  ℎ .Now, we proceed to derive some error estimates for the discrete solution.In the sequel,  denotes positive constants which are independent of the discretization parameter ℎ.
As a result of the previous theorem, the following corollary is about the convergence order error estimates for the fully discrete approximations with the previous subspaces  ℎ ,  ℎ , and  ℎ .Let Π ℎ be, as usual, the interpolation operator This corollary gives an estimation of the numerical errors of the problem () and its proof is based on the above approximation properties of the finite element spaces  ℎ ,  ℎ , and  ℎ .Now, we investigate the particular case where   = 0 and 1 ≤   < 2. Indeed, under some conditions on   and   , we can prove the following result.

Conclusion
In this work, we presented a model for the static process of frictional contact between a piezoelectric body and an electrically thermally conductive foundation.The constitutive relation of the material is assumed to be thermo-electroelastic.The contact was modeled with the normal compliance condition and the associated Coulomb's friction law, including the electrical and thermal conductivity conditions.The existence of the unique weak solution for the problem was established by using arguments from the theory of variational inequalities and a fixed point theorem.A discrete scheme by finite element method was used to approach the problem and an optimal order error estimate was derived.A numerical validation of the convergence result included in this method will be provided in a forthcoming paper.