## Abstract

### Purpose

The paper presents a mathematical problem involving quasistatic contact between a thermo-electro-viscoelastic body and a lubricated foundation, where the contact is described using a version of Coulomb’s law of friction that includes normal damped response conditions and heat exchange with a conductive foundation. The constitutive law for the material is thermo-electro-viscoelastic. The problem is formulated as a system that includes a parabolic equation of the first kind for the temperature, an evolutionary elliptic quasivariational inequality for the displacement and a variational elliptic equality for the electric stress. The author establishes the existence of a unique weak solution to the problem by utilizing classical results for evolutionary quasivariational elliptic inequalities, parabolic differential equations and fixed point arguments.

### Design/methodology/approach

The author establishes a variational formulation for the model and proves the existence of a unique weak solution to the problem using classical results for evolutionary quasivariational elliptic inequalities, parabolic difierential equations and fixed point arguments.

### Findings

The author proves the existence of a unique weak solution to the problem using classical results for evolutionary quasivariational elliptic inequalities, parabolic difierential equations and fixed point arguments.

### Originality/value

The author studies a mathematical problem between a thermo-electro-viscoelastic body and a lubricated foundation using a version of Coulomb’s law of friction including the normal damped response conditions and the heat exchange with a conductive foundation, which is original and requires a good understanding of modeling and mathematical tools.

## Keywords

## Citation

Djabi, A. (2023), "A frictional contact problem with normal damped response conditions and thermal effects for a thermo-electro-viscoelastic material", *Arab Journal of Mathematical Sciences*, Vol. ahead-of-print No. ahead-of-print. https://doi.org/10.1108/AJMS-08-2021-0174

## Publisher

:Emerald Publishing Limited

Copyright © 2023, Abdelmoumene Djabi

## License

Published in *the Arab Journal of Mathematical Sciences*. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) license. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this license may be seen at http://creativecommons.org/licences/by/4.0/legalcode

## 1. Introduction

Contact phenomena between deformable bodies or between a deformable body and a foundation are ubiquitous phenomena in everyday life. The contact of a wheel with the ground, the contact of the brake shoe with the wheel or the gradual sinking in a wheelchair during a seated posture, are just a few everyday examples, among many others. Some industrial processes such as metal stamping and metal extrusion lead to evolution problems where contact and friction conditions are decisive. These phenomena call upon sophisticated mathematical models, which are represented by systems of partial differential equations with boundary conditions describing complex contact processes (with or without friction). The mathematical theory of contact problems allows rigorous modeling of contact phenomena based on the principles of continuum mechanics as well as on variational analysis and numerical models.

Important developments concerning the mathematical study, numerical mechanics of the problems resulting from the mechanics of the contact were carried out during *XX*^{th} century. The first contact problem between a deformable body and a foundation was stated by Signorini and first solved by Fichera. Duvaut and Lions were the first to work on the mathematical theory of contact mechanics; They introduced variational formulations of contact problems and provided existence and uniqueness results. Subsequently, several new works have focused on the resolution of these variational problems such as the work of Refs [1–6]. However, mathematical theory of contact problems is a very broad field of study where many issues remain to be investigated.

The importance of the mathematical study of such problems leads to give coupled conditions for the material and the contact conditions.

Recent researches use coupled laws of behavior between mechanical and electric effects or between mechanical and thermal effects. For the case of coupled laws of behavior between mechanical and electric effects, numerous papers use different electro-mechanical conditions such as [2, 5, 7, 8]. For the case of coupled laws of behavior between mechanical and thermal effects, we can found several models in Refs [4, 6, 7, 9–12]. For this, the new researches use coupled conditions between the mechanical, electrical and thermal behavior of the material see [13–15].

The pyroelectric effect is characterized by a coupling between the electrical and thermal effects and does not produce mechanical effects. The pyroelectric effect used for fire alarm, pyroelectric detectors and sensors. Some pyroelectric applications can be found in Refs [9, 16, 17].

The piezoelectric effect is a coupling between the mechanical and electrical properties of the materials and does not produce heat effects. This coupling, leads to the appearance of electric field in the presence of a mechanical stress and conversely. A mechanical stress is generated when electric potential is applied. The first effect is used in sensors and the reverse effect is used in actuators. During the past few years, a lot of attention has been focused on the piezoelectric effects, such as [8, 18, 19].

Recent modeling, analysis and numerical simulations of electro-mechanical, thermo-mechanical and thermo-electro-mechanical contact problems with friction can be found in Refs [2, 4, 5, 7, 10, 11, 14]. General models of energy can be found in Refs [1]. a generalized Coulomb friction version is given in Refs [3, 20]. Indeed, the authors used the normal damped response conditions for a lubricated foundation; see, for instance [21, 22].

Nowadays, there are increasing efforts to investigate coupled-field problems. In this respect, electro-thermo-mechanical coupling is one particular application, which occurs, for example, in Car fan or Computer fan. In this paper we use mixed conditions between electrical, thermal and mechanical conditions. The law of behavior used is given by

This law is thermo-electro-viscoelastic Kelvin-Voigt model where *E*(*φ*) = −∇*φ*, ** B**,

**D**= 0, (1.1)–(1.2) becomes the Kelvin-Voigt thermo-viscoelastic constitutive relation used in [10]. Moreover, when

The evolution of the temperature field obtained from the conservation of energy and defined with the following differential equation

*θ*is the temperature,

*q*

_{th}is the density of volume heat sources and

*ψ*is a nonlinear function, assumed here depends on thermal expansion tensor and the displacement field.

Processes of contact are present in numerous domestic and industrial applications which may change from body to body depending on the constitutive law of the body studied. In this paper we use mechanical, thermal and electrical contact conditions.

For the mechanical contact conditions, the Coulomb friction is one of the most useful friction laws and known from the literature. This law has two basic ingredients namely the concept of friction threshold and its dependence on the normal stress. We use normal damped response conditions associated with the Coulomb’s law of dry friction given by:

This condition models frictional contact between the body and lubricated foundation where *p*_{ν} and *p*_{τ} represent given contact functions,

On the other hand in the study of this problem, we make the assumption that the foundation is thermo-electrical conductive, the electrical conductivity assumed depends on the linear function *H* defined as:

Here, we assume that the electrical conductivity *H* depends only on the electric potential *φ* and the initial electric potential *φ*_{0}

Moreover, for the thermal conductivity we use the following conditions on the contact surface

*k*

_{e}is the heat exchange coefficient between the body and the obstacle,

*θ*

_{F}is the temperature of the foundation.

The paper is organized as follows. In Section 2 we present the model. In Section 3 we introduce the notations, some preliminaries results, list of the assumptions on the data and we give the variational formulation of the problem. In Section 4 we state our main existence and uniqueness result theorem 4.1. The proof of the theorem is based on evolutionary elliptic variational inequalities, ordinary differential equations and fixed point arguments.

## 2. The model

The physical setting is the following. A thermo-electro-viscoelastic body occupies a bounded domain _{1}, Γ_{2} and Γ_{3}, on one hand and a partition of Γ_{1} ∪ Γ_{2} into two open parts Γ_{a} and Γ_{b}, on the other hand. We assume that *meas*(Γ_{1}) > 0 and *meas*(Γ_{a}) > 0. Let *T* > 0 and [0, *T*] be the time interval of interest. The body is subjected to the action of body forces of density **f**_{0}, volume electric charges of density *q*_{0} and a heat source of constant strength *q*_{th}.The body is clamped on Γ_{1} × (0, *T*), so the displacement field vanishes there. A surface traction of density **f**_{2} act on Γ_{2} × (0, *T*). We also assume that the electrical potential vanishes on Γ_{a} × (0, *T*) and a surface electric charge of density *q*_{b} is prescribed on Γ_{b} × (0, *T*). Moreover, we suppose that the temperature vanishes on

In the reference configuration, the body is in contact with a foundation, over the contact surface Γ_{3}. The model of the contact is frictional specified by the normal damped response conditions and it is associated with the Coulomb’s law of dry friction for the mechanical contact, an associated temperature boundary condition for the thermal contact and electrical conditions modeling electric potential exchange between the body and the conductive foundation.

The classical formulation of the mechanical problem is as follows.

**Problem**

Equations (2.1) and (2.2) represent the thermo-electro-viscoelastic constitutive law, the evolution of the temperature field is governed by differential equation given by the relation (2.3) where *ψ* is the mechanical source of the temperature growth, assumed to be rather general function of the strains. Next equations (2.4) and (2.5) are the steady equations for the stress and electric-displacement field, conditions (2.6) and (2.7) are the displacement and traction boundary conditions. Equation (2.11) means that the temperature vanishes on _{a} and the electric charges on Γ_{b} respectively. Equation (2.14) represents the initial displacement field and the initial temperature field where the initial displacement is **u**_{0}, and *θ*_{0} is the initial temperature.

We turn to the contact conditions (2.8)–(2.10) describe a mixed contact on the potential contact surface Γ_{3}. The relation (2.8) describes a normal damped response conditions with the Coulomb’s law of dry friction (2.9) represents an associated temperature boundary condition on contact surface. Finally, (2.10) shows models the electric conductivity.

## 3. Variational formulation

In order to obtain the variational formulation of the Problem

### 3.1 Notations and preliminaries

We present the notation we recall some preliminary material. For more details, we refer the reader to [23–26]. In what follows the indices *i* and *j* run from 1 to *d*, the summation convention over repeated indices is used and the index that follows a comma indicates a partial derivative with respect to the corresponding component of the independent variable. We denote by *d* = 2, 3). We recall that the canonical inner products and the corresponding norms on

Let *ν* denote the unit outer normal on *∂*Ω = Γ. We introduce the spaces

*H*,

*H*

^{1}(Ω)

^{d}and

_{H},

*H*

^{1}(Ω) and

*H*

^{1}(Ω)

^{d}defined by

Since *meas*Γ_{a} > 0 and *meas*Γ_{1} > 0, the Korn’s and Friedrichs-Poincaré inequalities hold, thus,

*C*

_{0},

*C*

_{1}and

*C*

_{2}are positive constants that depend on the problem data and are independent of the solutions.

On the spaces *V*, *W* and

It follows from (3.1) and (3.4) that ‖._{V} are equivalent norms on *V*, (3.2) and (3.5) follows that ‖._{W} are equivalent norms on *W* and from (3.3) and (3.6) we deduce that ‖.*V*, (⋅,⋅)_{V}), (*W*, (⋅,⋅)_{W}) and *C*_{0}, *C*_{1} and *C*_{2}, three positive constants, such that

Let **v** ∈ *H*^{1}(Ω)^{d}, we also use the notation **v** to denote the trace map *γ***v** of **v** on Γ, and we denote by *v*_{ν} and **v**_{τ} the *normal* and *tangential* components of **v** on Γ given by

Similarly, for a regular (say *normal* and *tangential* components by

**θ**∈

*H*

^{1}(Ω)

^{d}and

We recall the following definition of an Gelfand triple.

**Definition 3.1.**

Let *V* and *H* be real Hilbert spaces such that *V* is dense in *H* and the injection map is continuous. The space *H* is identified with its own dual and with a subspace of the dual *V*′ of *V*. We write

*V*,

*H*and

*V*′ respectively, and we use

*V*′ and

*V*. Note that if

*f*∈

*H*then

and we recall the following Theorem

**Theorem 3.2.**

*Let* *V* ⊂ *H* ⊂ *V*′ *be a Gelfand triple. Assume that* *A*: *V* → *V*′ *is a hemicontinuous and monotone operator that satisfies*

*For some constants*

*ω*> 0,

*C*> 0

*and*

*Then, given*

**u**

_{0}∈

*H*

*and*

*, there exists a unique function*

*satisfies*

The proof of this abstract result may be found in [3, p. 141], and will be used in the study of thermal problem presented in Section 5.

Finally, for any real Hilbert space *X*, we use the classical notation for the spaces *L*^{p}(0, *T*; *X*) and *W*^{k,p}(0, *T*; *X*), where 1 ≤ *p* ≤ *∞* and *k* > 1. For *T* > 0 we denote by *T*] to *X*, respectively, with the norms

Moreover, we use the dot above to indicate the derivative with respect to the time variable and if *X*_{1} and *X*_{2} are real Hilbert spaces then *X*_{1} × *X*_{2} denotes the product Hilbert space endowed with the canonical inner product (⋅,⋅)

### 3.2 Assumptions on the data

We now list the assumptions on the problem’s data.

The *viscosity operator*

The *elasticity operator*

The *piezoelectric operator*

The *thermal expansion operator*

The *nonlinear constitutive function*

The *electric permittivity operator*

The *pyroelectric operator*

The *thermal conductivity operator*

The *contact functions* *p*_{s}:

The *electrical conductivity function*

The density of volume forces, traction, volume electric charges, surface electric charges and the temperature evolution increase satisfy

The initial displacement, the potential of the foundation, the initial temperature and the temperature of the foundation fields satisfy

Using the above notation and Green’s formulas given by (3.12)–(3.14), we obtain the variational formulation of the mechanical problem (2.1)–(2.14) for all functions **v** ∈ *V*, *ϕ* ∈ *W* and a.e

### 3.3 Problem P V

Find the displacement field **u**: [0, *T*] → *V*, the stress field *φ*: [0, *T*] → *W*, the electric displacement field **D:** [0, *T*] → *H* and the temperature *θ*: [0, *T*] → *V* such that

**u**,

**v**∈

*V*,

*ϕ*∈

*W*and

*t*∈ [0,

*T*]. We note that the definitions of

**f**and

*q*

_{e}are based on the Riesz representation theorem. Moreover, conditions (3.28) and (3.29) imply that

## 4. Existence and uniqueness of a solution

Now, we propose our existence and uniqueness result.

**Theorem 4.1.**

*Assume that* (3.18)–(3.32) *hold. Then there exists a constant* *α*_{0}* which depends only on* Ω, Γ_{1}, Γ_{3} *and* *such that if*

*where*

*such that*

*is defined in*(3.18)

*and*

*C*

_{0}

*defined by*(3.7)

*.Then there exists a unique solution*

*to problem*

*Moreover, the solution satisfies*

The proof of Theorem 4.1 is carried in several steps. It is based on results of evolutionary variational inequalities, ordinary differential equations and fixed point arguments.

To prove the theorem we consider the following three auxiliary problems for given

### 4.1 Problem P V η

Find a displacement field **u**_{η}: [0, *T*] → *V* and a stress field

**u**

_{η},

**v**∈

*V*and

### 4.2 Problem P V χ

Find the temperature

*t*∈ (0,

*T*),

### 4.3 Problem P V φ

Find an electrical potential *φ*: [0, *T*] → *W*,

*φ*,

*ϕ*∈

*W*,

We begin with an auxiliary result on the properties of the functionals

**Lemma 4.2.**

*Under the hypotheses* (3.18)–(3.32)*, the functionals* *j* *and* *G*_{1} *satisfy*

*Proof (Lemma 4.2*). We use the assumption (3.26) and inequality (3.7) to see that the functional *j* defined by (3.39) is a seminorm on *V* and moreover,

Thus, the seminorm *j* is continuous on *V* and, therefore, (4.16) hold.

From the definition of the functional *j* given by (3.39), we have

Using (3.39), the last equality becomes

Next, we use the following inequalities

The inequality (4.20) becomes

Using (3.7) and (4.1), we conclude

Moreover, the functional *G*_{1} defined in (3.42) by

Thus by the assumption (3.32) and inequality (4.21), we get

From the inequality (3.8), we obtain

Thus, we can write

We have the following result for Problem

**Lemma 4.3.**

*Under the hypotheses* (3.18)*–*(3.32)*, for every* *, Problem* *has a unique weak solution* *, such that*

*Moreover, if*

*are the solutions of Problem*

*, corresponding*

*for*

*i*= 1, 2

*,then*

*Proof [of* *Lemma (4.3)**]*. Choosing

Using the definition (3.40) for **f**, we deduce

With the regularity assumption (3.28) on **f**_{0}, we see that *Div**σ*_{η}(*t*) ∈ *H*. Therefore,

Now, we use Riesz Representation Theorem to define the operators *A*: *V* → *V*, *B*: *V* → *V* and the function

**u, v**∈

*V*and

It follows from (4.25) and (3.18(*a*)) that

Which shows that *A*: *V* → *V* is Lipschitz continuous. Now, by (4.25) and (3.18(*b*)) we find

*A*:

*V*→

*V*is a strongly monotone operator on

*V*. Moreover, using (4.26) and (3.19(

*a*)) we find

*A*is a strongly monotone and Lipschitz continuous operator on

*V*and

*B*is Lipschitz continuous operator on

*V*,

*j*(

**u**, .) satisfies conditions (4.16) and (4.17),

**u**

_{0}satisfies the assumption (3.31), and we note that for any fixed

*i*= 1, 2, then

Since

We have

Recent modeling Using (4.31) the inequality (4.32) becomes

Next, we apply Gronwall’s inequality to deduce

□

For the Problem

**Lemma 4.4.**

*Under the hypotheses* (3.18)*–*(3.32)*, for every* *, Problem* *has a unique weak solution such that*

*Moreover, if θ_{i} are the solutions of Problem *

*Proof* [*of Lemma (4.4)*]. The inclusion mapping of

The problem (4.12)–(4.13) may be written as

It follows from the definition of the operator *K*, and (3.15) the assumption (3.25(b)) that

Now, by (4.36) and (3.25(c)), we find

Which shows that is *K* a strongly monotone operator. Choosing

Thus, *K* satisfies condition (3.16) with

Next, by (4.38) we deduce that

This inequality implies that *K* satisfies condition (3.17).

Moreover, for

It follows now from Theorem 3.2 that there exists a unique function

Now, to provide the estimate (4.35), we take the substitution *χ* = *χ*_{1} and *χ* = *χ*_{2} in (4.12) and subtracting the two obtained equations, we deduce by choosing

Then integrating the last property over

For the last Problem

**Lemma 4.5.**

*Under the hypotheses* (3.18)*–*(3.32)*, for every* *, Problem* *has a unique weak solution* *such that*

*Moreover, if*

*are the solutions of*

*p*

*roblem*

*, corresponding*

*for*

*i*= 1, 2

*, then*

*Proof [of* *Lemma (4.5)*]. First, for the functional

Let *φ*_{1}, *φ*_{2} ∈ *W*, we find that

We use the definition on the functional *H* given in (1.5) to obtain

Using the inequality (3.8), we get

We use Riesz representation theorem to define the operator *F*: *W* → *W* by

Let *φ*_{1}, *φ*_{2} ∈ *W*. Using the assumption (3.23) and (3.27), we find that

On the other hand, using the assumptions (3.20), (3.23) and the inequality (4.42), we have

*C*

_{1}are a positives constants. Thus,

Thus, by (4.44) and (4.45) we conclude that *W* and, therefore, there exists a unique element *φ*_{η} ∈ *W* such that

Let

Moreover, we use the assumption (3.30), to obtain

We conclude that

Using (3.2) and (3.30), we get

which implies

Then, for every *q*_{e} imply that *φ*_{η} ∈

We now use (3.41) and definition of the divergence operator div to see that

This shows that

Let *i* = 1, 2, We use (4.15) and arguments similar to those used in the proof of (4.47) to obtain (4.41) □

Finally, as a consequence of these results and using the properties of the operators *ψ* for

We have the following result.

**Lemma 4.6.**

*Let* (4.1) *be satisfied. Then for* *,**the function* *is continuous, and there is a unique element* **.** *Such that* Λ(*η**, *χ**) = (*η**, *χ**)

*Proof [of Lemma 4.6]*. Let

The last inequality and (4.18), implies

Using Hölder’s inequality, we get

For the electric potential field, we use (4.33) and (4.41), we obtain

For the displacement, we use (4.23) to get

Moreover, using the inequality (4.35) obtained in Lemma 4.4 for the temperature.

Applying Young’s, Hölder’s inequalities, the increases (4.35), (4.54) and (4.55), then the inequality (4.53) becomes

Thus, for *m* sufficiently large, Λ^{m} is a contraction on

Now, we have all the ingredients to prove Theorem 4.1.

*Proof [of Theorem (4.1)].* **Existence**

Let

Let _{1}(*η**, *χ**) = *η** and Λ_{2}(*η**, *χ**) = *χ** combined with 4.49–4.51 show that 3.33–3.38 are satisfied. Next, the regularity 4.2–4.6 follows from Lemmas 4.3, 4.4 and 4.5. □

**Uniqueness**

*Proof.* The uniqueness part of solution is a consequence of the uniqueness of the fixed point of the operator Λ defined by (4.49)–(4.51) and the unique solvability of the problems

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## Further reading

28.Zeng S, Bai Y, Gasi´nski L, Leszek, Winkert P. Convergence analysis for double phase obstacle problems with multivalued convection term. Adv Nonlinear Anal. 2021; 10: 659-72.

29Barbu V. Nonlinear semigroups and differential equations in Banach spaces. Leyden: Editura Academiei, Bucharest-Noordhoff; 1976. 164-167.

30.Bowen CR, Taylor J, LeBoulbar E, Zabek D, Chauhan A, Vaish R. Pyroelectric materials and devices for energy harvesting applications. Energ Environ Sci. 2014; 7(12): 3836-56.

## Acknowledgements

The author would like to thank the reviewers for the important comments. This article presents an extension of sources [2, 11]. The article is based on the work of the abovementioned sources by extending the mathematical model to a multiphysical thermoelectromechanical law with three elements, introducing complex boundary conditions of different physical types and models.

For the mathematical model, the existence of a unique weak solution to the problem is demonstrated using results on quasivariational elliptical inequalities, parabolic differential equations and fixed point arguments.

These contributions represent an important step forward in the field of boundary problems in contact mechanics.