A frictional contact problem with normal damped response conditions and thermal effects for a thermo-electro-viscoelastic material

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 formulatedas a systemthatincludesaparabolicequationof the firstkind for the temperature, an evolutionary ellipticquasivariationalinequalityforthedisplacementandavariationalellipticequalityfortheelectricstress. Theauthorestablishestheexistenceofauniqueweaksolutiontotheproblembyutilizingclassicalresultsforevolutionaryquasivariationalellipticinequalities,parabolicdifferentialequationsandfixedpointarguments. 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 – Theauthorprovestheexistenceofauniqueweaksolutiontotheproblemusingclassicalresultsfor 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 andalubricatedfoundationusingaversionofCoulomb ’ slawoffrictionincludingthenormaldampedresponse conditions and the heat exchange with a conductive foundation, which is original and requires a good understanding of modeling and mathematical tools.


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][2][3][4][5][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][10][11][12].For this, the new researches use coupled conditions between the mechanical, electrical and thermal behavior of the material see [13][14][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, thermomechanical 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 (1.1)

AJMS
This law is thermo-electro-viscoelastic Kelvin-Voigt model where A, G are nonlinear operators describing the purely viscous and the elastic properties of the material, respectively and E(w) 5 À∇w, E ¼ ðe ijk Þ, M, B, P are respectively electric field, piezoelectric, thermal expansion, electric permittivity, pyroelectric tensors, and E* is the transpose of E. Note also that when E ¼ 0 and D 5 0, (1.1)-(1.2) becomes the Kelvin-Voigt thermoviscoelastic constitutive relation used in [10].Moreover, when M ¼ 0 and P ¼ 0, the relations (1.1)-(1.2) becomes the Kelvin-Voigt electro-viscoelastic.The evolution of the temperature field obtained from the conservation of energy and defined with the following differential equation where θ is the temperature, K denotes the thermal conductivity tensor, M the thermal expansion tensor, 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, _ u ν and _ u τ denote the normal and tangential velocity field respectively.
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 w and the initial electric potential w 0 Moreover, for the thermal conductivity we use the following conditions on the contact surface where 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.
Frictional contact problem

The model
The physical setting is the following.A thermo-electro-viscoelastic body occupies a bounded domain Ω ⊂ R d ðd ¼ 2; 3Þ with outer Lipschitz surface Γ.This boundary is divided into three open disjoint parts Γ 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 3 (0, T ), so the displacement field vanishes there.A surface traction of density f 2 act on Γ 2 3 (0, T ).We also assume that the electrical potential vanishes on Γ a 3 (0, T ) and a surface electric charge of density q b is prescribed on Γ b 3 (0, T ).Moreover, we suppose that the temperature vanishes on Γ 1 ∪ Γ 2 ð Þ3 ð0; TÞ.Moreover, we suppose that the body forces and tractions vary slowly in time, and therefore, the accelerations in the system may be neglected.Neglecting the inertial terms in the equation of motion leads to a quasistatic approach to the process.
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 P. Find the displacement field u : Ω 3 ½0; T → R d , the stress field σ : Ω 3 ½0; T → S d , the electric potential w : Ω 3 ½0; T → R, the electric displacement field (2.9) 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 Þ .Next, (2.12) and (2.13) represent the electric boundary conditions for the electrical potential on Γ 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.

Variational formulation
In order to obtain the variational formulation of the Problem P, we use the following notations and preliminaries

Notations and preliminaries
We present the notation we recall some preliminary material.For more details, we refer the reader to [23][24][25][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 S d the space of second order symmetric tensors on R d (d 5 2, 3).We recall that the canonical inner products and the corresponding norms on R d and S d , respectively are given by Let Ω ⊂ R d be a bounded domain with outer Lipschitz boundary Γ and let ν denote the unit outer normal on vΩ 5 Γ.We introduce the spaces Here ε : H 1 ðΩÞ d → H and Div : H 1 → H are the linearized deformation and divergence operators, respectively, defined by The spaces H, H, H We introduce the closed subspaces of H 1 (Ω) and H 1 (Ω) d defined by Since measΓ a > 0 and measΓ 1 > 0, the Korn's and Friedrichs-Poincar e inequalities hold, thus, where here and below 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 Z, we define the following inner products we also use the notation v to denote the trace map γv 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 C 1 ) tensor field σ : Ω → S d we define its normal and tangential components by where 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 0 of V. We write and we say that the inclusions above define a Gelfand triple.We denote by $ k k V , $ k k H , and $ k k V 0 , the norms on the spaces V, H and V 0 respectively, and we use $; $ ð Þ V 0 3 V for the duality pairing between V 0 and V.
and we recall the following Theorem Theorem 3.2.Let V ⊂ H ⊂ V 0 be a Gelfand triple.Assume that A: V → V 0 is a hemicontinuous and monotone operator that satisfies For some constants ω > 0, C > 0 and ζ ∈ R Then, given 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 Cð0; T; X Þ and C 1 ð0; T; X Þ the space of continuous and continuously differentiable functions from [0, T] to X, respectively, with the norms kfk Cð0;T;X Þ ¼ max 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 3 X 2 denotes the product Hilbert space endowed with the canonical inner product ($,$) X 1 3 X 2 .

Assumptions on the data
We now list the assumptions on the problem's data.The viscosity operator A : Ω 3 S d → S d satisfies ðaÞ There exists ðbÞ There exists m A > 0 such that for all ε 1 ; ε 2 ∈ S d ; a:e:x ∈ Ω: ðcÞ The mapping x ↦ Aðx; εÞ is Lebesgue measurable on Ω; for any ε ∈ S d : ðdÞ The mapping x ↦ Aðx; 0Þ belongs to H: The elasticity operator G : ðbÞ The mapping x ↦ Gðx; εÞ is Lebesgue measurable on Ω; for any ε ∈ S d : ðcÞ The mapping x ↦ Gðx; 0Þ belongs to H: The piezoelectric operator The thermal expansion operator M : ðbÞ The mapping x ↦ Mðx; θÞ is Lebesgue measurable on Ω; for any θ ∈ R: The nonlinear constitutive function ψ : ðbÞ The mapping x → ψðx; Mθ; uÞ is Lebesgue measurable on Ω for any θ ∈ R; for any u ∈ V : ðcÞ The mapping x → ψðx; 0; 0Þ ∈ L 2 ðΩÞ: The electric permittivity operator ðcÞ There exists a constant m B > 0 such that BE: The pyroelectric operator P : ðaÞ There exists a constant L P > 0 such that kPðx; θ 1 Þ À Pðx; θ 2 Þk ≤ L P kθ 1 À θ 2 k for all θ 1 ; θ 2 ∈ R d ; a:e:x ∈ Ω: ðbÞ The mapping x ↦ Pðx; 0Þ belongs to W: The thermal conductivity operator ðaÞ There exists a constant L K > 0 such that kKðx; r 1 Þ À Kðx; r 2 Þk ≤ L K kr 1 À r 2 k for all r 1 ; r 2 ∈ R d ; a:e:x ∈ Ω: The mapping x ↦ Sðx; 0; 0Þ belongs to L 2 ðΩÞ: The contact functions p s : ðaÞ There exists L s > 0 such that kp s ðx; w 1 Þ À p s ðx; w 2 Þk ≤ L s kr 1 À r 2 k for all r 1 ; r 2 ∈ R; a:e:x ∈ Γ 3 : ðbÞ The mapping x ↦ p s ðx; wÞ is Lebesgue measurable on Γ 3 ; for any r ∈ R: ðcÞ The mapping x ↦ p s ðx; 0Þ belongs to L 2 ðΓ 3 Þ: The electrical conductivity function H :

Frictional contact problem
ðaÞ There exists m H e > 0 such that 2 for all w 1 ; w 2 ∈ R; a:e:x ∈ Ω: ðbÞ The mapping x ↦ H ðx; wÞ is Lebesgue measurable on Ω; for any w ∈ R: ðcÞ The mapping x ↦ H ðx; 0Þ belongs to L 2 ðΩÞ: 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 and the initial temperature field satisfies 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, w ∈ Z, f ∈ W and a.e t ∈ 0; T ð Þgiven as follows,

Existence and uniqueness of a solution
Now, we propose our existence and uniqueness result.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 η ∈ Cð0; T; V Þ, χ ∈ L 2 ð0; T; Z 0 Þ we consider the following three auxiliary problems:

Problem PV η
Find a displacement field u η : [0, T] → V and a stress field σ η : ½0; T → H such that σ η ðtÞ ¼ Aðεð _ We begin with an auxiliary result on the properties of the functionals j : V 3 V → R and G 1 : Z 3 Z → R defined by (3.39) and (3.42), respectively.Lemma 4.2.Under the hypotheses (3.18)-(3.32), the functionals j and G 1 satisfy jðu; :Þ is convex and lower semicontinuous on V ; (4.16) 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 which implies 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 where We have the following result for Problem PV η .

Now, we use Riesz Representation Theorem to define the operators
for all u, v ∈ V and t ∈ 0; T ½ .
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 i.e. that A: V → V is a strongly monotone operator on V.Moreover, using (4.26) and (3.19(a)) we find if (4.1) is satisfied, since 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 η ∈ Cð0; T; V Þ we use the definitions 3.44 and (4.27) to show that f η ∈ Cð0; T; V Þ we deduce from classical results for evolutionary elliptic variational inequalities (see for example [27]) that there exists a unique function u η ∈ C 1 ð0; T; V Þ.Moreover, for u i ¼ u η i solutions of the Problem PV ηi for i 5 1, 2, then We have Recent modeling Using (4.31) the inequality (4.32) becomes Next, we apply Gronwall's inequality to deduce  Moreover, if θ i are the solutions of Problem PV χ i , corresponding χ ¼ χ i ∈ Cð0; T; Z 0 Þ for i 5 1,

2, then
Proof [of Lemma (4.4)].The inclusion mapping of Z; : ð Þ is continuous and its range is dense.We can write the Gelfand triple The problem (4.12)-(4.13)may be written as where, K : Z → Z 0 and Q : 0; T ½ → Z 0 are defined as ðKτ; τ:wda; Which shows that is K a strongly monotone operator.Choosing w ¼ 0 Z in (4.39), we obtain Next, by (4.38) we deduce that Kτ This inequality implies that K satisfies condition (3.17).
Moreover, for χðtÞ ∈ L 2 ð0; T; Z 0 Þ and q th ðtÞ ∈ L 2 0; T; It follows now from Theorem 3.2 that there exists a unique function θ χ ∈ L 2 ð0; T; ZÞ \ Cð0; T; L 2 ðΩÞÞ, which satisfies the Problem PV χ .Now, to provide the estimate (4.35), we take the substitution χ 5 χ 1 and χ 5 χ 2 in (4.12) and subtracting the two obtained equations, we deduce by choosing w We use the definition on the functional H given in (1.5) to obtain AJMS Using the inequality (3.8), we get We use Riesz representation theorem to define the operator F: W → W by Let w 1 , w 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 where C E and C 1 are a positives constants.Thus, Thus, by (4.44) and (4.45) we conclude that F t ð Þ is a strongly monotone and Lipschitz continuous operator on W and, therefore, there exists a unique element w η ∈ W such that FðtÞw η ðtÞ ¼ q e þ Pθ; ∀w η ∈ W : (4.46) Let η 1 ; η 2 ∈ Cð0; T; V Þ.Using the last equality, we get Moreover, we use the assumption (3.30), to obtain Using (3.2) and (3.30), we get Frictional contact problem which implies Then, for every u η ∈ C 1 ð0; T; V Þ, the previous inequality and the regularity of q e imply that w η ∈ C 0; T; W ð Þ .We now use (3.41) and definition of the divergence operator div to see that We have the following result.

,
For the Problem PV χ we have the following result.