Existence and Uniqueness of Weak Solutions to Frictionless-Antiplane Contact Problems

Abstract

We investigate a quasi-static-antiplane contact problem, examining a thermo-electro-visco-elastic material with a friction law dependent on the slip rate, assuming that the foundation is electrically conductive. The mechanical problem is represented by a system of partial differential equations, and establishing its solution involves several key steps. Initially, we obtain a variational formulation of the model, which comprises three systems: a hemivariational inequality, an elliptic equation, and a parabolic equation. Subsequently, we demonstrate the existence of a unique weak solution to the model. The proof relies on various arguments, including those related to evolutionary inequalities, techniques for decoupling unknowns, and certain results from differential equations.

1. Introduction

Over the past few decades, phenomena related to the contact between deformable bodies have had a significant impact on society. The contact of brakes with wheels, a ball with the ground, and the application of a force to cutting objects, are just a few everyday instances, among many other examples. Consequently, research in this area is growing across various fields, particularly in engineering and mathematical literature [1,2].

A comprehensive study of mechanical problems involves mathematical modeling. The modeling of mechanical phenomena is determined by a set of hypotheses that influence the system of partial differential equations. These hypotheses cover aspects such as the nature of the mechanical process (static, quasi-static, or dynamic), the behavior of the material (electromechanical, thermomechanical, elastic, etc.), and the boundary conditions on the contact surface (friction, adhesion, etc.) [3,4].

In 1933, notable advancements in the mathematical and mechanical exploration of contact-mechanics problems took place. Signorini seems to have been the first to delve into this subject when he formulated the challenge of contact between a deformable body and a foundation. The solution to this problem eventually emerged in 1964, courtesy of Fichera, who employed emerging mathematical techniques. The initial findings concerning the existence and uniqueness of contact problems were established by Duvaut and Lions. Following this, a multitude of subsequent works have concentrated on resolving these variational problems [5,6].

Mathematics plays a pivotal role in the field of contact mechanics by providing diverse contact conditions. Contemporary research in this field also encompasses behavior laws that establish connections between mechanical and electrical effects, known as piezoelectricity [7,8]. This area is extensively explored in engineering structures, due to its temperature-dependent characteristics. Several articles have tackled the thermo-piezoelectric contact problem with friction [5,9,10,11,12,13], thermo-elasto-visco-plasticity [14,15,16], and thermo-visco-elasticity [17,18]. These studies are rooted in variations in constitutive laws and contact conditions [19,20,21,22,23,24,25].

Here we propose a novel mathematical model designed to address the frictional-antiplane contact problem between a thermo-piezoelectric body and a conductive foundation. The innovation within our model lies in the introduction of a novel frictional contact tailored for such materials. This model incorporates a slip-rate-dependent friction law and introduces modifications to the electrical and thermal conditions used to describe the contact. We specifically focus on the case of antiplane-shear deformation.

The manuscript is organized as follows. In Section 2, we recall some notations and definitions used in the sequel. In Section 3, we introduce our mathematical model for the quasi-static thermo-electro-visco-elastic-antiplane contact problem. Given some consistency assumptions about the data, the variational formulation of the model is then obtained in Section 4. Finally, in Section 5, we prove the existence and uniqueness result of the formulated problem.

2. Preliminaries

In this brief section, we review the fundamental notations and definitions that are employed throughout the work. For more details, we refer the interested reader to [25].

The space of second-order symmetric tensors on ${\mathbb{R}}^3$ is denoted by ${\mathcal{S}}^3$. The inner product and the norm on ${\mathbb{R}}^3$ are given as follows:

$$u^h·v^h=u_i^h{v}_i^h,\parallel v^h\parallel ={(v^h·v^h)}^{\frac{1}{2}}\mathrm{for}\mathrm{all}{u}^h=\left(u_i^h\right),v^h=\left(v_i^h\right)\in {\mathbb{R}}^3,1\le i,j\le 3.$$

The inner product on ${\mathcal{S}}^3$ is defined by

$${\sigma }^h·{\tau }^h={\sigma }_{i,j}^h{\tau }_{i,j}^h,\mathrm{for}\mathrm{all}{\sigma }^h=\left({\sigma }_{i,j}^h\right),{\tau }^h={\tau }_{i,j}^h\in {\mathcal{S}}^3,1\le i,j\le 3,$$

while its corresponding norm is given by

$$\parallel {\tau }^h\parallel ={\left({\tau }^h·{\tau }^h\right)}^{\frac{1}{2}}.$$

Next, we consider the following function spaces:

$$V^h=\{v^h\in H^1\left({\mathrm{\Omega }}^h\right),v^h=0\mathrm{on}{\mathrm{\Gamma }}_1^h\},$$

$$W^h=\{{\psi }^h\in H^1\left({\mathrm{\Omega }}^h\right),{\psi }^h=0\mathrm{on}{\mathrm{\Gamma }}_1^h\},$$

$$E^h=\{{\theta }^h\in H^1\left({\mathrm{\Omega }}^h\right),{\theta }^h=0\mathrm{on}{\mathrm{\Gamma }}_1^h\cup {\mathrm{\Gamma }}_2^h\}.$$

It is an established fact that $V^h$, $W^h$, and $E^h$ are real Hilbert spaces with the inner products

$${(u^h,v^h)}_{V^h}={\int }_{{\mathrm{\Omega }}^h}\nabla u^h·\nabla v^{h}dx,\forall u^h,v^h\in V^h,$$

$${({\phi }^h,{\psi }^h)}_{W^h}={\int }_{{\mathrm{\Omega }}^h}\nabla {\phi }^h·\nabla {\psi }^{h}dx,\forall {\phi }^h,{\psi }^h\in W^h,$$

$${({\theta }^h,{\varphi }^h)}_{E^h}={\int }_{{\mathrm{\Omega }}^h}\nabla {\theta }^h·\nabla {\varphi }^{h}dx,\forall {\theta }^h,{\varphi }^h\in E^h.$$

Furthermore, the associated norms

$$\parallel v^h{\parallel }_{V^h}=\parallel \nabla v^h{\parallel }_{L^2{\left({\mathrm{\Omega }}^h\right)}^2},\forall v^h\in V^h,$$

(1)

$$\parallel {\psi }^h{\parallel }_{W^h}=\parallel \nabla {\psi }^h{\parallel }_{L^2{\left({\mathrm{\Omega }}^h\right)}^2},\forall {\psi }^h\in W^h,$$

(2)

$$\parallel {\theta }^h{\parallel }_{E^h}=\parallel \nabla {\theta }^h{\parallel }_{L^2{\left({\mathrm{\Omega }}^h\right)}^2},\forall {\theta }^h\in E^h$$

(3)

on $V^h$, $W^h$, and $E^h$ are equivalent to the usual norm ${\parallel ·\parallel }_{H^1\left({\mathrm{\Omega }}^h\right)}$. In the light of Sobolev’s trace theorem, we deduce that

$$\exists c_{V^h}>0:{\parallel v^h\parallel }_{L^2\left({\mathrm{\Gamma }}_3^h\right)}\le c_V^h{\parallel v^h\parallel }_{V^h},\forall v^h\in V^h,$$

(4)

$$\exists c_{W^h}>0:{\parallel v^h\parallel }_{L^2\left({\mathrm{\Gamma }}_3^h\right)}\le c_W^h{\parallel {\psi }^h\parallel }_{W^h},\forall \psi \in W^h,$$

(5)

$$\exists c_{E^h}>0:{\parallel v^h\parallel }_{L^2\left({\mathrm{\Gamma }}_3^h\right)}\le c_E^h{\parallel {\theta }^h\parallel }_{E^h},\forall \theta \in E^h.$$

(6)

In a real Banach space $(X^h,\parallel ·{\parallel }_{X^h})$, we denote the spaces of continuous and continuously differentiable functions on $[0,T]$ into $X^h$ by $C(0,T,X^h)$ and $C^1(0,T,X^h)$, respectively. These spaces are equipped with their respective norms,

$$\parallel x^h{\parallel }_{C(0,T,X^h)}=\underset{t\in [0,T]}{max}{\parallel x^h\parallel }_{X^h}$$

and

$$\parallel x^h{\parallel }_{C^1(0,T,X^h)}=\underset{t\in [0,T]}{max}\parallel x^h{\parallel }_{X^h}+\underset{t\in [0,T]}{max}{\parallel {\dot{x}}^h\parallel }_{X^h}.$$

Below, we shall use the notations for the Lebesgue space $L^2(0,T,X^h)$ as well as the Sobolev space $W^{1,2}(0,T,X^h)$. Recall that the norm on $L^2(0,T,X^h)$ is given by

$$\parallel u^h{\parallel }_{L^2(0,T,X^h)}^2={\int }_0^T{\parallel u^h\left(t\right)\parallel }_{X^h}^{2}dt.$$

Moreover,

$$\parallel u^h{\parallel }_{W^{1,2}(0,T,X^h)}^2={\int }_0^T\parallel u^h{\left(t\right)\parallel }_{X^h}^{2}dt+{\int }_0^T{\parallel {\dot{u}}^h\left(t\right)\parallel }_{X^h}^{2}dt$$

defines a norm on $W^{1,2}(0,T,X^h)$.

3. Formulation of the Problem

We begin by making explicit the physical setting under investigation. Let us consider a thermo-electro-visco-elastic body ${\mathcal{B}}^h$ with friction. Assume that ${\mathcal{B}}^h\in {\mathbb{R}}^3$ is a cylinder with generators parallel to the $x_3^h$-axes having a regular region in its cross-section ${\mathrm{\Omega }}^h$ in the $x_1^h,x_2^h$-plane, $O{x}_1^h{x}_2^h{x}_3^h$ being a Cartesian-coordinate system. The cylinder is presumptively long enough for the end effects in the axial direction to be insignificant. Thus, ${\mathcal{B}}^h={\mathrm{\Omega }}^h\times (-\infty ,+\infty )$. Let $\mathsf{\partial }{\mathrm{\Omega }}^h={\mathrm{\Gamma }}^h$. We suppose that ${\mathrm{\Gamma }}^h$ is divided into three disjoint measurable parts: ${\mathrm{\Gamma }}_1^h$, ${\mathrm{\Gamma }}_2^h$, and ${\mathrm{\Gamma }}_3^h$. One has a partition of ${\mathrm{\Gamma }}_1^h\cup {\mathrm{\Gamma }}_2^h$ into two open parts, ${\mathrm{\Gamma }}_a^h$ and ${\mathrm{\Gamma }}_b^h$, such that the measures of ${\mathrm{\Gamma }}_a^h$ and ${\mathrm{\Gamma }}_b^h$ are positive. For $T>0$, we denote by $[0,T]$ the time interval of interest. The ${\mathcal{B}}^h$ body is affected to time-dependent forces ${\mathbf{f}}_{\mathbf{0}}^{\mathbf{h}}$ and has a volume with free electric charges of density $q_0^h$. The body is clamped on ${\mathrm{\Gamma }}_1^h\times (-\infty ,+\infty )$ and, therefore, the displacement field disappears there. The surface tractions of density ${\mathbf{f}}_{\mathbf{2}}^{\mathbf{h}}$ act on ${\mathrm{\Gamma }}_2^h\times (-\infty ,+\infty )$. We also assume that the electrical potential vanishes on ${\mathrm{\Gamma }}_a^h\times (-\infty ,+\infty )$ and that a surface electrical charge of density $q_2^h$ is prescribed on ${\mathrm{\Gamma }}_b^h\times (-\infty ,+\infty )$. The body is in contact with a rigid foundation ${\mathrm{\Gamma }}_3^h\times (-\infty ,+\infty )$ [26,27]. We assume that

$$\begin{array}{cc}\hfill q_0^h& =q_0^h(x_1^h,x_2^h,t):{\mathrm{\Omega }}^h\times [0,T]⟶\mathbb{R},\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill q_2^h& =q_2^h(x_1^h,x_2^h,t):{\mathrm{\Gamma }}_b^h\times [0,T]⟶\mathbb{R},\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill {\mathbf{f}}_0^h=(0,0,f_0^h),& \mathrm{with}{f}_0^h=f_0^h(x_1^h,x_2^h,t):{\mathrm{\Omega }}^h\times [0,T]⟶\mathbb{R},\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {\mathbf{f}}_2^h=(0,0,f_2^h),& \mathrm{with}{f}_2^h=f_2^h(x_1^h,x_2^h,t):{\mathrm{\Gamma }}_2^h\times [0,T]⟶\mathbb{R}.\hfill \end{array}$$

(10)

The electric charges (7) and (8) and the forces (9) and (10) are expected to give rise to electric charges and to deformations of the piezoelectric cylinder corresponding to an electric-potential field ${\phi }^h$ and to a displacement ${\mathbf{u}}^h$, which are independent of $x_3^h$ and have the form

$$\begin{array}{cc}\hfill {\phi }^h=& {\phi }^h(x_1^h,x_2^h,t):{\mathrm{\Omega }}^h\times [0,T]⟶\mathbb{R},\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill {\mathbf{u}}^h=& u^h(0,0,u^h),\mathrm{with}{u}^h=u^h(x_1^h,x_2^h,t):{\mathrm{\Omega }}^h\times [0,T]⟶\mathbb{R}.\hfill \end{array}$$

(12)

We denote by ${\theta }^h$ the temperature field, which is of the form

$${\theta }^h={\theta }^h(x_1^h,x_2^h,t):{\mathrm{\Omega }}^h\times [0,T]⟶\mathbb{R},$$

(13)

by ${\epsilon }^h\left(u^h\right)=\left({\epsilon }_{i,j}^h\left(u^h\right)\right)$ the infinitesimal strain tensor, by ${\sigma }^h=\left({\sigma }_{ij}^h\right)$ the stress field, by $E^h\left({\phi }^h\right)=\left(E_i^h\left({\phi }^h\right)\right)$ the electric field, and by $D^h=\left(D_i^h\right)$ the electric displacement field, where

$$\begin{array}{cc}\hfill {\epsilon }_{i,j}^h\left(u^h\right)& =\frac{1}{2}(u_{i,j}^h+u_{j,i}^h),\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill E_i^h\left({\phi }^h\right)& =-{\phi }_{,i}^h.\hfill \end{array}$$

(15)

The material is modeled by a thermo-electro-visco-elastic constitutive law of the form

$$\begin{array}{cc}\hfill {\sigma }^h& =2{\alpha }^h{\epsilon }^h\left({\dot{u}}^h\right)+{\zeta }^{h}tr{\epsilon }^h\left(\dot{u^h}\right)I+2{\mu }^h{\epsilon }^h\left(u^h\right)+{\lambda }^{h}tr\epsilon \left(u^h\right)I-{\mathcal{E}}^{h*}{E}^h\left({\phi }^h\right)-M_e^h{\theta }^h,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill D^h& ={\mathcal{E}}^{h*}{\epsilon }^h\left(u^h\right)+{\beta }^h{E}^h\left({\phi }^h\right),\hfill \end{array}$$

(17)

where ${\zeta }^h$ and ${\alpha }^h$ are viscosity coefficients, ${\lambda }^h$ and ${\mu }^h$ are the Lame coefficients, tr${\epsilon }^h\left(u^h\right)={\epsilon }_{ii}^h\left(u^h\right)$, I is the unit tensor in ${\mathbb{R}}^3$, ${\beta }^h$ is the electric permittivity constant, ${\mathcal{E}}^h$ represents a third-order piezoelectric tensor, and ${\mathcal{E}}^{h*}$ is its transpose. We assume that

$${\mathcal{E}}^h{\epsilon }^h=\left(\begin{array}{c}{e}^h({\epsilon }_{13}^h+{\epsilon }_{31}^h)\\ e^h({\epsilon }_{23}^h+{\epsilon }_{32}^h)\\ e^h\left({\epsilon }_{33}^h\right)\end{array}\right),\forall {\epsilon }^h=\left({\epsilon }_{i,j}^h\right)\in {\mathcal{S}}^3,$$

(18)

where $e^h$ is the piezoelectric coefficient. Additionally, $M_e^h=\left(M_{ij}^h\right)$ represents the thermal expansion tensor, which takes the form

$$M_e^h=\left(\begin{array}{ccc}0& 0& M_{e_1}\\ 0& 0& M_{e_2}\\ M_{e_1}& M_{e_2}& 0\end{array}\right).$$

(19)

Conversely, we assume that the coefficients $M_e^h$, ${\alpha }^h$, ${\mu }^h$, ${\beta }^h$, and $e^h$ depend on $x_1^h$ and $x_2^h$. However, they are independent on $x_3^h$. Owing to ${\mathcal{E}}^h{\epsilon }^h·v^h={\epsilon }^h·{\mathcal{E}}^{h*}{v}^h$ for all ${\epsilon }^h\in {\mathcal{S}}^3$ and $v^h\in {\mathbb{R}}^3$, it follows from that (18)

$${\mathcal{E}}^{h*}{v}^h=\left(\begin{array}{ccc}0& 0& e^h{v}_1^h\\ 0& 0& e^h{v}_2^h\\ e^h{v}_1^h& e^h{v}_2^h& e^h{v}_3^h\end{array}\right),\forall v^h=\left(v_i^h\right)\in {\mathbb{R}}^3.$$

(20)

In the antiplane context, having in mind (11)–(13), and given the constitutive Equations (16) and (17) and equalities (18) and (20), we obtain the stress field and the electric displacement field as

$${\sigma }^h=\left(\begin{array}{ccc}0& 0& {\sigma }_{13}^h\\ 0& 0& {\sigma }_{23}^h\\ {\sigma }_{31}^h& {\sigma }_{32}^h& 0\end{array}\right),{\mathbf{D}}^h=\left(\begin{array}{c}e{u}_{,1}^h-{\beta }^h{\phi }_{,1}^h\\ e{u}_{,2}^h-{\beta }^h{\phi }_{,2}^h\\ 0\end{array}\right),$$

(21)

where

$${\sigma }_{13}^h={\sigma }_{31}^h={\alpha }^h{\dot{u}}_{,1}^h+{\mu }^h{u}_{,1}^h+e^h{\phi }_{,1}^h-M_{e_1}{\theta }^h,$$

$${\sigma }_{23}^h={\sigma }_{32}^h={\alpha }^h{\dot{u}}_{,2}^h+{\mu }^h{u}_{,2}^h+e^h{\phi }_{,2}^h-M_{e_2}{\theta }^h.$$

We presume that the process is electrically static and mechanically quasi-static. Therefore, the equilibrium equations that govern it are given by

$$\begin{array}{cc}\hfill \mathrm{div}{\sigma }^h+{\mathbf{f}}_0^h& =0,\hfill \\ \hfill {\mathit{D}}_{i,i}^h-q_0^h& =0\mathrm{in}{\mathcal{B}}^h\times (0,T),\hfill \end{array}$$

(22)

where $\mathrm{div}{\sigma }^h=\left({\sigma }_{ij,j}^h\right)$ is the divergence of the tensor field ${\sigma }^h$. Accordingly, taking into account (7), (9), (11)–(13), and (21), the equilibrium Equation (22) condenses into the following scalar equations:

$$\begin{array}{cc}\hfill \mathrm{div}({\alpha }^h\nabla {\dot{u}}^h+{\mu }^h\nabla u^h)+\mathrm{div}(e^h\nabla {\phi }^h)-M_e^h\mathrm{div}\left({\theta }^h\right)+f_0^h& =0,\mathrm{in}{\mathrm{\Omega }}^h\times (0,T),\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill \mathrm{div}(e^h\nabla u^h)-\mathrm{div}({\beta }^h\nabla {\phi }^h)& =q_0^h,\mathrm{in}{\mathrm{\Omega }}^h\times (0,T),\hfill \end{array}$$

(24)

with

$$M_e^h=\left(\begin{array}{c}{M}_{e_1}\\ M_{e_2}\\ 0\end{array}\right).$$

(25)

In the sequel, we use the notation

$$\mathrm{div}{\tau }^h={\tau }_{1,1}^h+{\tau }_{1,2}^h\mathrm{for}{\tau }^h=({\tau }_1^h(x_1^h,x_2^h,t),{\tau }_2^h(x_1^h,x_2^h,t)),$$

$$\nabla v^h=(v_{,1}^h,v_{,2}^h),{\mathsf{\partial }}_{\nu }^h{v}^h=v_{,1}^h{\nu }_1^h+v_{,2}^h{\nu }_2^h\mathrm{for}{v}^h=v^h(x_1^h,x_2^h,t).$$

Keeping in mind that the cylinder is clamped on ${\mathrm{\Gamma }}_1^h\times (-\infty ,+\infty )$, the electrical potential vanishes on ${\mathrm{\Gamma }}_a^h\times (-\infty ,+\infty )$, and from (7) and (12) we find that

$$\begin{array}{cc}\hfill u^h=& 0\mathrm{on}{\mathrm{\Gamma }}_1^h\times (0,T),\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill {\phi }^h=& 0\mathrm{on}{\mathrm{\Gamma }}_a^h\times (0,T).\hfill \end{array}$$

(27)

Note that ${\nu }^h$ is the unit normal on ${\mathrm{\Gamma }}^h\times (-\infty ,+\infty )$, where

$${\nu }^h=({\nu }_1^h,{\nu }_2^h,0),\mathrm{with}{\nu }_i^h={\nu }_i^h(x_1^h,x_2^h):{\mathrm{\Gamma }}^h⟶\mathbb{R},\mathrm{for}i=1,2.$$

(28)

We denote by $v_{\nu }^h$ and ${\mathbf{v}}_{\tau }^h$ the normal and tangential components, respectively, of ${\mathbf{v}}^h$ on the boundary—that is,

$$v_{\nu }^h={\mathbf{v}}^h·{\nu }^h,{\mathbf{v}}_{\tau }^h={\mathbf{v}}^h-v_{\nu }^h{\nu }^h.$$

(29)

Therefore, we denote by ${\sigma }_{\nu }^h$ and ${\sigma }_{\tau }^h$ the normal and the tangential components, respectively, of ${\sigma }^h$ on the boundary, meaning that

$${\sigma }_{\nu }^h=\left({\sigma }^h{\nu }^h\right)·{\nu }^h,{\sigma }_{\tau }^h={\sigma }^h{\nu }^h-{\sigma }_{\nu }^h{\nu }^h.$$

(30)

From (21) and (28), we conclude that the Cauchy stress vector and the normal component of the electric displacement field are given by

$${\sigma }^h{\nu }^h=(0,0,{\alpha }^h{\mathsf{\partial }}_{{\nu }^h}{\dot{u}}^h+{\mu }^h{\mathsf{\partial }}_{{\nu }^h}{u}^h+{\mathsf{\partial }}_{{\nu }^h}{\phi }^h-M_e^h{\theta }^h·{\nu }^h),D^h·{\nu }^h=e^h{\mathsf{\partial }}_{{\nu }^h}{u}^h-{\beta }^h{\mathsf{\partial }}_{{\nu }^h}{\phi }^h,$$

(31)

respectively. Following this, we utilize the following notations:

$${\mathsf{\partial }}_{{\nu }^h}{u}^h=u_1^h{\nu }_1^h+u_2^h{\nu }_2^h,$$

$${\mathsf{\partial }}_{{\nu }^h}{\phi }^h={\phi }_1^h{\nu }_1^h+{\phi }_2^h{\nu }_2^h.$$

Taking into account the traction boundary condition on ${\mathrm{\Gamma }}_2^h\times (-\infty ,+\infty )$ and the electric condition on ${\mathrm{\Gamma }}_b^h\times (-\infty ,+\infty )$, it follows from (8), (10), and (31) that

$$\begin{array}{c}\hfill {\alpha }^h{\mathsf{\partial }}_{{\nu }^h}{\dot{u}}^h+{\mu }^h{\mathsf{\partial }}_{{\nu }^h}{u}^h+e^h{\mathsf{\partial }}_{{\nu }^h}{\phi }^h-M_e^h{\theta }^h·{\nu }^h=f_2^h\mathrm{on}{\mathrm{\Gamma }}_2^h\times (0,T),\end{array}$$

(32)

$$\begin{array}{c}\hfill e^h{\mathsf{\partial }}_{{\nu }^h}{u}^h-{\beta }^h{\mathsf{\partial }}_{{\nu }^h}{\phi }^h=q_2^h\mathrm{on}{\mathrm{\Gamma }}_b^h\times (0,T).\end{array}$$

(33)

We now describe the frictional contact condition on ${\mathrm{\Gamma }}_3^h\times (-\infty ,+\infty )$. First, we remark that from (12), (28), and (29), we find that $u_{\nu }=0$, which indicates that the contact is bilateral. Thus, the contact is kept during the whole process. Now, using (12) and (28)–(31), we obtain

$$\begin{array}{cc}\hfill u_{\tau }^h& =(0,0,u^h),\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill {\sigma }_{\tau }^h& =(0,0,{\sigma }_{\tau }^h),\hfill \end{array}$$

(35)

where

$${\sigma }_{\tau }^h={\alpha }^h{\mathsf{\partial }}_{\nu }^h{\dot{u}}^h+{\mu }^h{\mathsf{\partial }}_{\nu }^h{u}^h+e^h{\mathsf{\partial }}_{\nu }^h{\phi }^h-M_e^h{\theta }^h·{\nu }^h.$$

(36)

We suppose that the friction is invariant concerning the $x_3^h$ axis, being modeled with a slip-rate-dependent friction law, where the strict inequality is satisfied in the stick zone and the equality in the slip zone—that is,

$$\left\{\begin{array}{c}|{\sigma }_{\tau }^h|\le r(|{\dot{u}}_{\tau }^h\left|\right),\mathrm{on}{\mathrm{\Gamma }}_3^h\times [0,T],\hfill \\ {\sigma }_{\tau }^h=-r\left(\right|{\dot{u}}^h\left|\right)\frac{{\dot{u}}_{\tau }^h}{|{\dot{u}}_{\tau }^h|},\mathrm{on}{\mathrm{\Gamma }}_3^h\times [0,T].\hfill \end{array}\right.$$

(37)

Here, $r:{\mathrm{\Gamma }}_3^h\times \mathbb{R}⟶{\mathbb{R}}^3$ is a given function, the friction bound, and ${\dot{u}}_{\tau }^h$ is the tangential velocity on the contact boundary: see [28,29] for details. By (34) and (36), we ascertain that the conditions in (37) are given by

$$\left\{\begin{array}{c}|{\alpha }^h{\mathsf{\partial }}_{\nu }^h{\dot{u}}^h+{\mu }^h{\mathsf{\partial }}_{\nu }^h{u}^h+e^h{\mathsf{\partial }}_{\nu }^h{\phi }^h-M_e^h{\theta }^h·{\nu }^h|\le r(|{\dot{u}}^h\left|\right),\hfill \\ {\alpha }^h{\mathsf{\partial }}_{\nu }^h{\dot{u}}^h+{\mu }^h{\mathsf{\partial }}_{\nu }^h{u}^h+e^h{\mathsf{\partial }}_{\nu }^h{\phi }^h-M_e^h{\theta }^h·{\nu }^h=-r\left(\right|{\dot{u}}^h\left|\right)\frac{{\dot{u}}^h}{|{\dot{u}}^h|},on{\mathrm{\Gamma }}_3^h\times [0,T].\hfill \end{array}\right.$$

(38)

Finally, we present the initial conditions:

$$u^h\left(0\right)=u_0^h,{\phi }^h\left(0\right)={\phi }_0^h,{\theta }^h\left(0\right)={\theta }_0^h\mathrm{in}{\mathrm{\Omega }}^h,$$

(39)

where $u_0^h$, ${\phi }_0^h$, and ${\theta }_0^h$ are given functions on ${\mathrm{\Omega }}^h$. Putting this all together, we obtain the problem under investigation.

Problem 1.

Determine a displacement field $u^h:{\mathrm{\Omega }}^h\times \left[0,T\right]\to \mathbb{R}$, the electric field ${\phi }^h:{\mathrm{\Omega }}^h\times \left[0,T\right]\to \mathbb{R}$, and a temperature field ${\theta }^h:{\mathrm{\Omega }}^h\times \left[0,T\right]\to {\mathbb{R}}_{+}$, such that

$$\mathrm{div}({\alpha }^h\nabla {\dot{u}}^h+{\mu }^h\nabla u^h)+\mathrm{div}(e^h\nabla {\phi }^h)-M_e^h\mathrm{div}\left({\theta }^h\right)+f_0^h=0,\mathit{in}{\mathrm{\Omega }}^h\times (0,T),$$

(40)

$$\mathrm{div}(e^h\nabla u^h)-\mathrm{div}({\beta }^h\nabla {\phi }^h)=q_0^h,\mathit{in}{\mathrm{\Omega }}^h\times (0,T),$$

(41)

$${\dot{\theta }}^h-\mathrm{div}(k\nabla {\theta }^h)=-M_e^h\nabla ({\dot{u}}^h)+p\left(t\right),\mathit{in}{\mathrm{\Omega }}^h\times (0,T),$$

(42)

$$u^h=0,\mathit{on}{\mathrm{\Gamma }}_1^h\times (0,T),$$

(43)

$${\alpha }^h{\mathsf{\partial }}_{{\nu }^h}{\dot{u}}^h+{\mu }^h{\mathsf{\partial }}_{{\nu }^h}{u}^h+e^h{\mathsf{\partial }}_{{\nu }^h}{\phi }^h-M_e^h{\mathsf{\partial }}_{{\nu }^h}{\theta }^h=f_2^h,\mathit{on}{\mathrm{\Gamma }}_2^h\times (0,T),$$

(44)

$$\left\{\begin{array}{c}|{\alpha }^h{\mathsf{\partial }}_{{\nu }^h}{\dot{u}}^h+{\mu }^h{\mathsf{\partial }}_{{\nu }^h}{u}^h+{\mathsf{\partial }}_{\nu }^h{\phi }^h-M_e^h{\theta }^h·{\nu }^h|\le r(|{\dot{u}}^h\left|\right),\hfill \\ {\alpha }^h{\mathsf{\partial }}_{{\nu }^h}{\dot{u}}^h+{\mu }^h{\mathsf{\partial }}_{{\nu }^h}{u}^h+{\mathsf{\partial }}_{\nu }^h{\phi }^h-M_e^h{\theta }^h·{\nu }^h=-r\left(\right|{\dot{u}}^h\left|\right)\frac{{\dot{u}}^h}{|{\dot{u}}^h|},\mathit{on}{\mathrm{\Gamma }}_3^h\times (0,T),\hfill \end{array}\right.$$

(45)

$${\phi }^h=0,\mathit{on}{\mathrm{\Gamma }}_a^h\times (0,T),$$

(46)

$$e^h{\mathsf{\partial }}_{{\nu }^h}{u}^h-{\beta }^h{\mathsf{\partial }}_{{\nu }^h}{\phi }^h=q_2^h,\mathit{on}{\mathrm{\Gamma }}_b^h\times (0,T),$$

(47)

$${\theta }_0^h=0,\mathit{on}{\mathrm{\Gamma }}_1^h\cup {\mathrm{\Gamma }}_2^h\times (0,T),$$

(48)

$$-{\mathcal{K}}_{i,j}^h\frac{\mathsf{\partial }{\theta }^h}{\mathsf{\partial }{x}_j^h}{v}_i^h={\mathcal{K}}_e^h({\theta }^h-{\theta }_R^h),\mathit{on}{\mathrm{\Gamma }}_3^h\times (0,T),$$

(49)

$$u^h\left(0\right)=u_0^h,{\phi }^h\left(0\right)={\phi }_0^h,{\theta }^h\left(0\right)={\theta }_0^h,\mathit{in}{\mathrm{\Omega }}^h\times (0,T).$$

(50)

It should be noted that the problem we consider here is different from that of [20]. Indeed, in [20] they studied electro-visco-elastic material, meaning its behavior was influenced solely by electrical and mechanical factors. The focus was purely on the electrical impact without thermal interference, represented by equation (2.13) of [20]. By contrast, in our paper we add the factor of temperature, which plays a significant role in altering electromechanical properties through heating and cooling. For instance, the electro-rheological frictional force may vary depending on the temperature, thereby impacting the rheological properties of the material. Therefore, we can say that the material’s behavior in our study is influenced by mechanical, electrical, and thermal factors, represented by Equation (16), followed by the associated conditions boundary in the Problem 1. On the other hand, the addition of the temperature factor also provides us with an opportunity to explore and search for a different and simpler mathematical-solution method (the decoupling-of-unknowns method). Additionally, there is a difference between the friction conditions. Indeed, in [20] the authors used Tresca’s friction law (equation (2.25) there), which is characterized by setting a maximum limit for slip. It relies solely on the maximum force without considering time rates. This was what the equation described there, observing that the friction bound was constant and had one condition, making it easier to solve the posed problem. By contrast, in our paper we use the slip-rate-dependent friction law (37): here, the friction level depends on the rate of slip changes over time. This implies that variations in the slip rate impact the friction level (the friction being higher when the rate of change is greater and lower when the rate of change is smaller). This behavior is described by Equation (37), where we represent friction bound by the variable r.

4. Variational Formulation

To study our Problem 1, we assume that the viscosity coefficient, the electric permittivity coefficient, the Lame coefficient, and the piezoelectric coefficient, satisfy

$$\begin{array}{cc}\hfill & {\alpha }^h\in L^{\infty }\left({\mathrm{\Omega }}^h\right)\mathrm{and}\mathrm{there}\mathrm{exists}{\alpha }^{h*}>0,\mathrm{such}\mathrm{that}{\alpha }^h\left(x^h\right)\ge {\alpha }^{h*},\mathrm{a}.\mathrm{e}.,x^h\in {\mathrm{\Omega }}^h,\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill & {\beta }^h\in L^{\infty }\left({\mathrm{\Omega }}^h\right)\mathrm{and}\mathrm{there}\mathrm{exists}{\beta }^{h*}>0,\mathrm{such}\mathrm{that}{\beta }^h\left(x^h\right)\ge {\beta }^{h*},\mathrm{a}.\mathrm{e}.,x^h\in {\mathrm{\Omega }}^h,\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill & {\mu }^h\in L^{\infty }\mathrm{and}{\mu }^h\left(x^h\right)>0,\mathrm{a}.\mathrm{e}.,x^h\in {\mathrm{\Omega }}^h,\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill & e^h\in L^{\infty },\hfill \end{array}$$

(54)

respectively. We also assume that the thermal tensors $M_e^h=\left(M_{ij}\right):{\mathrm{\Omega }}^h\to {\mathcal{S}}^3$ satisfy

$$M_{ij}=M_{ji}\in L^{\infty }\left({\mathrm{\Omega }}^h\right),1⩽i,j⩽3.$$

(55)

The thermal conductivity tensor $\widetilde{{\mathcal{K}}^h}=\left({\mathcal{K}}_{ij}^h\right):{\mathrm{\Omega }}^h\to {\mathcal{S}}^3$ satisfies

$$\left\{\begin{array}{c}\left(\mathrm{i}\right){\mathcal{K}}_{ij}^h={\mathcal{K}}_{ji}^h\in L^{\infty }\left({\mathrm{\Omega }}^h\right),1⩽i,j⩽3,\hfill \\ \left(\mathrm{ii}\right)\mathrm{there}\mathrm{exists}{m}_{{\mathcal{K}}^h}>0,\mathrm{such}\mathrm{that}\hfill \\ {\mathcal{K}}^{h}s·s\ge m_{{\mathcal{K}}^h}{\left|s\right|}^2,\mathrm{a}.\mathrm{e}.,x\in {\mathrm{\Omega }}^h,\forall s\in {\mathbb{R}}^3.\hfill \end{array}\right.$$

(56)

The boundary thermal data satisfy

$$p^h\in W^{1,2}(0,T;L^2\left({\mathrm{\Omega }}^h\right)),{\theta }_R^h\in W^{1,2}(0,T;L^2\left({\mathrm{\Gamma }}_3^h\right)),K_e^h\in L^{\infty }\left({\mathrm{\Omega }}^h,{\mathbb{R}}_{+}\right).$$

(57)

The forces, tractions, volume, and surface free charge densities have the following regularity:

$$\begin{array}{c}\hfill f_0^h\in W^{1,2}(0,T,L^2\left({\mathrm{\Omega }}^h\right)),f_2^h\in W^{1,2}(0,T,L^2\left({\mathrm{\Gamma }}_2^h\right)),\end{array}$$

(58)

$$\begin{array}{c}\hfill q_0^h\in W^{1,2}(0,T,L^2\left({\mathrm{\Omega }}^h\right)),\end{array}$$

(59)

$$\begin{array}{c}\hfill q_2^h\in W^{1,2}(0,T,L^2\left({\mathrm{\Gamma }}_b^h\right)),q_2^h=0,\mathrm{a}.\mathrm{e}.,x^h\in {\mathrm{\Gamma }}_b^h,\end{array}$$

(60)

respectively. The friction bound satisfies

$$\left\{\begin{array}{c}a)r:{\mathrm{\Gamma }}_3^h\times \mathbb{R}⟶{\mathbb{R}}^{+},\hfill \\ b)\exists L_r>0,\mathrm{such}\mathrm{that}|r(x^h,s_1)-r(x^h,s_2)|\le L_r|s_1-s_2|,\hfill \\ c)x^h⟶r(x^h,s)\mathrm{is}\mathrm{Lebesgue-measurable}\mathrm{on}{\mathrm{\Gamma }}_3^h,\forall s\in \mathbb{R},\hfill \\ d)\mathrm{the}\mathrm{mapping}{x}^h⟶r(x^h,0)\mathrm{belongs}\mathrm{to}{L}^2\left({\mathrm{\Gamma }}_3^h\right).\hfill \end{array}\right.$$

(61)

We define the functional $j:v^h\times v^h⟶\mathbb{R}$ by

$$j({\dot{u}}^h,v^h)={\int }_{{\mathrm{\Gamma }}_3^h}r\left(\right|{\dot{u}}^h\left|\right)|v^h|da,\forall v^h\in V^h.$$

(62)

Let ${\eta }_1$, ${\eta }_2$, $v_1$, $v_2\in V$. By using (61) and (62), we find that

$$j({\eta }_1,v_2^h)-j({\eta }_1,v_1^h)+j({\eta }_2,v_1^h)-j({\eta }_2,v_2^h)| \le L_r\parallel {\eta }_1-{\eta }_2{\parallel }_{L^2\left({\mathrm{\Gamma }}_3^h\right)}{\parallel v_1-v_2\parallel }_{L^2\left({\mathrm{\Gamma }}_3^h\right)},$$

and applying the norm mentioned in (4), one obtains

$$L_r\parallel {\eta }_1-{\eta }_2{\parallel }_{L^2\left({\mathrm{\Gamma }}_3^h\right)}\parallel v_1-v_2{\parallel }_{L^2\left({\mathrm{\Gamma }}_3^h\right)}\le c_V^2{L}_r\parallel {\eta }_1-{\eta }_2{\parallel }_{V^h}{\parallel v_1^h-v_2^h\parallel }_{V^h}.$$

Hence, we conclude that

$$\begin{array}{c}\hfill j({\eta }_1,v_2^h)-j({\eta }_1,v_1^h)+j({\eta }_2,v_1^h)-j({\eta }_2,v_2^h)\le c_V^2{L}_r\parallel {\eta }_1-{\eta }_2{\parallel }_{V^h}{\parallel v_1^h-v_2^h\parallel }_{V^h}.\end{array}$$

(63)

The initial data verify

$$u_0^h\in V^h,{\theta }_0^h\in L^2\left({\mathrm{\Omega }}^h\right).$$

(64)

We use functions $f^h:[0,T]⟶V^h$ and $q^h:[0,T]⟶W^h$ as

$$\begin{array}{c}\hfill {(f^h,v^h)}_{V^h}={\int }_{{\mathrm{\Omega }}^h}{f}_0^h{v}^{h}d{x}^h+{\int }_{{\mathrm{\Gamma }}_2^h}{f}_2^h{v}^{h}da,\forall v^h\in V^h,\end{array}$$

(65)

$$\begin{array}{c}\hfill {(q^h,{\psi }^h)}_{V^h}={\int }_{{\mathrm{\Gamma }}_b^h}{q}_2^h{\psi }^{h}d{x}^h+{\int }_{{\mathrm{\Omega }}^h}{q}_0^h{\psi }^{h}da,\forall {\psi }^h\in W^h.\end{array}$$

(66)

The definitions of $f^h$ and $q^h$ are based on Riesz’s representation theorem. Therefore, by using assumptions (65) and (66), we ascertain that the above integrals are well defined and that

$$\begin{array}{c}\hfill f^h\in W^{1,2}(0,T,V^h),\end{array}$$

(67)

$$\begin{array}{c}\hfill q^h\in W^{1,2}(0,T,W^h).\end{array}$$

(68)

Then, the function $P:[0,T]⟶{\left(E^h\right)}^{\prime }$ and the operators $\widetilde{{\mathcal{K}}^h}:E⟶{\left(E^h\right)}^{\prime },\widetilde{{\mathcal{M}}^h}:V^h⟶{\left(E^h\right)}^{\prime }$ are defined by

$${\displaystyle {\langle P\left(t\right),{\mu }^h\rangle }_{{\left(E^h\right)}^{\prime }\times E^h}={\int }_{{\mathrm{\Omega }}^h}{p}^h{\mu }^{h}d{x}^h+{\int }_{{\mathrm{\Gamma }}_3^h}{\mathcal{K}}_e^h{\theta }_R^h{\mu }^{h}ds,}$$

$${\displaystyle {\langle \widetilde{{\mathcal{K}}^h}\tau ,{\mu }^h\rangle }_{{\left(E^h\right)}^{\prime }\times E^h}={\int }_{{\mathrm{\Gamma }}_3^h}{\mathcal{K}}_e^h\tau {\mu }^{h}ds+\sum _{i,j=1}^d{\int }_{{\mathrm{\Omega }}^h}{\mathcal{K}}_{i,j}^h\frac{\mathsf{\partial }{\mu }^h}{\mathsf{\partial }{x}_j^h}\frac{\mathsf{\partial }{\mu }^h}{\mathsf{\partial }{x}_i^h}dx,}$$

$${\langle \widetilde{{\mathcal{M}}^h}{v}^h,{\mu }^h\rangle }_{{\left(E^h\right)}^{\prime }\times E^h}=-{\int }_{{\mathrm{\Omega }}^h}(M_e^h\nabla v^h){\mu }^{h}dx+{\int }_{{\mathrm{\Gamma }}_3^h}{h}_{\tau }\left(\right|v_{\tau }^h\left|\right){\mu }^{h}ds$$

for all $v^h\in V^h,\tau \in E^h,{\mu }^h\in E^h$. Moreover, we define the bilinear forms $a_{{\alpha }^h}:V^h\times V^h⟶\mathbb{R}$, $a_{{\mu }^h}:V^h\times V^h⟶\mathbb{R}$, $a_{e^h}:V^h\times W^h⟶\mathbb{R}$, $a_{e^h}:W^h\times V^h⟶\mathbb{R}$, $a_{{\beta }^h}:W^h\times W^h⟶\mathbb{R}$ and $a_{M^h}:E^h\times V^h⟶\mathbb{R}$ by

$$\begin{array}{cc}\hfill a_{{\alpha }^h}(u^h,v^h)& {\displaystyle ={\int }_{{\mathrm{\Omega }}^h}{\alpha }^h\nabla u^h·\nabla v^{h}dx,}\hfill \end{array}$$

(69)

$$\begin{array}{cc}\hfill a_{{\mu }^h}(u^h,v^h)& {\displaystyle ={\int }_{{\mathrm{\Omega }}^h}{\mu }^h\nabla u^h·\nabla v^{h}dx,}\hfill \end{array}$$

(70)

$$\begin{array}{cc}\hfill a_{M^h}(u^h,v^h)& {\displaystyle =-{\int }_{{\mathrm{\Omega }}^h}{M}_e^h{u}^h·\nabla v^{h}dx,}\hfill \end{array}$$

(71)

$$\begin{array}{cc}\hfill a_{e^h}(u^h,{\phi }^h)& {\displaystyle ={\int }_{{\mathrm{\Omega }}^h}{e}^h\nabla u^h·\nabla v^{h}dx=a_{e^h}(v^h,u^h),}\hfill \end{array}$$

(72)

$$\begin{array}{cc}\hfill a_{{\beta }^h}({\phi }^h,{\psi }^h)& {\displaystyle ={\int }_{{\mathrm{\Omega }}^h}{\beta }^h\nabla {\phi }^h·\nabla {\psi }^{h}dx}\hfill \end{array}$$

(73)

for all $u^h,v^h\in V^h$, ${\phi }^h,{\psi }^h\in W^h$. We note that by assumptions (51)–(54), the above integrals are well defined. Using the definition of norms (1)–(5), we see that the forms $a_{{\alpha }^h}$, $a_{{\mu }^h}$, $a_{e^h}$, and $a_{{\beta }^h}$, are continuous and that the forms $a_{{\alpha }^h}$, $a_{{\mu }^h}$, and $a_{{\beta }^h}$ are symmetric. Furthermore, the form $a_{{\alpha }^h}$ is V-elliptic, i.e.,

$$a_{{\alpha }^h}(u^h,v^h)\le \parallel {\alpha }^h{\parallel }_{L^{\infty }\left({\mathrm{\Omega }}^h\right)}\parallel u^h{\parallel }_{V^h}{\parallel v^h\parallel }_{V^h},\forall u^h,v^h\in V^h,$$

(74)

$$a_{\alpha }^h(v^h,v^h)\ge {\alpha }^{h*}{\parallel v^h\parallel }_{V^h}^2\forall v^h\in V^h.$$

(75)

Now, we can state the variational formulation of our Problem 1.

Problem 2

(Variational Problem). Determine a displacement field $u^h:[0,T]⟶V^h$, an electric-potential field ${\phi }^h:[0,T]⟶W^h$, and a temperature field ${\theta }^h:[0,T]⟶E^h$, such that

$$a_{{\alpha }^h}({\dot{u}}^h\left(t\right),v^h-{\dot{u}}^h\left(t\right))+a_{{\mu }^h}(u^h\left(t\right),v^h-{\dot{u}}^h\left(t\right))+a_{e^h}(\phi \left(t\right),v-{\dot{u}}^h\left(t\right))+a_{M^h}({\theta }^h\left(t\right),v^h-{\dot{u}}^h\left(t\right))$$

$$+j({\dot{u}}^h\left(t\right),v^h)-j({\dot{u}}^h\left(t\right),{\dot{u}}^h\left(t\right))\ge {(f^h,v^h-{\dot{u}}^h)}_{V^h},\forall v^h\in V^h,t\in [0,T],$$

(76)

$$a_{{\beta }^h}({\phi }^h,{\psi }^h)-a_e(u^h,{\psi }^h)={(q^h,\psi )}_W^h,\forall {\psi }^h\in W^h,$$

(77)

$${\dot{\theta }}^h\left(t\right)+\widetilde{{\mathcal{K}}^h}{\theta }^h\left(t\right)=\widetilde{{\mathcal{M}}^h}{\dot{u}}^h\left(t\right)+P\left(t\right),\mathit{in}{E}^h,$$

(78)

$$u^h\left(0\right)=u_0^h,{\phi }^h\left(0\right)={\phi }_0^h,{\theta }^h\left(0\right)={\theta }_0^h.$$

(79)

5. Existence and Uniqueness Result

In this section, we use the variational formulation to prove the existence and uniqueness of a weak solution to Problem 1. For that, we make use of some auxiliary problems and lemmas.

Problem 3

(Auxiliary Problem). Determine a displacement field $u^h:[0,T]⟶V^h$, such that

$$\begin{array}{cc}\hfill a^h(u^h\left(t\right),v^h-{\dot{u}}^h\left(t\right))+b^h({\dot{u}}^h\left(t\right),v^h-{\dot{u}}^h\left(t\right))& + j^h({\dot{u}}^h\left(t\right),v^h)-j^h({\dot{u}}^h\left(t\right),{\dot{u}}^h\left(t\right))\hfill \\ \hfill & \ge {(F\left(t\right),v^h-{\dot{u}}^h\left(t\right))}_{V^h},\forall v^h\in V^h,t\in [0,T],\hfill \end{array}$$

(80)

$$u^h(0)=u_0^h.$$

(81)

In order to study Auxiliary Problem 3, we assume that

$$\left\{\begin{array}{c}{a}^h:V^h\times V^h⟶\mathbb{R}\mathrm{is}\mathrm{a}\mathrm{bilinear}\mathrm{form}\mathrm{and}\mathrm{there}\mathrm{exists}{M}^h>0,\mathrm{such}\mathrm{that}\hfill \\ |a^h(u^h,v^h)| \le M^h\parallel u^h{\parallel }_{V^h}{\parallel v^h\parallel }_{V^h},\forall u^h,v^h\in V^h;\hfill \end{array}\right.$$

(82)

$$\left\{\begin{array}{c}{b}^h:V^h\times V^h⟶\mathrm{is}\mathrm{a}\mathrm{bilinear}\mathrm{symmetric}\mathrm{form}\mathrm{verifying}:\hfill \\ \left(a\right)\mathrm{There}\mathrm{exists}{M}^{h^{\prime }}>0,\mathrm{such}\mathrm{that}|b^h(u^h,v^h)|\infty \le M^{h^{\prime }}\parallel u^h{\parallel }_{V^h}{\parallel v^h\parallel }_{V^h},\forall u^h,v^h\in X^h,\hfill \\ \left(b\right)\mathrm{there}\mathrm{exists}{m}^{h^{\prime }}>0,\mathrm{such}\mathrm{that}{b}^h(v^h,v^h)\ge m^{h^{\prime }}{\parallel v^h\parallel }_{V^h}^2,\forall v^h\in V^h;\hfill \end{array}\right.$$

(83)

$$\left\{\begin{array}{c}{j}^h:V^h\times V^h⟶\mathbb{R}\mathrm{satisfies}:\hfill \\ \left(a\right)\mathrm{For}\mathrm{all}\eta \in V^h,j^h(\eta ,·)\mathrm{is}\mathrm{convex}\mathrm{and}\mathrm{l}.\mathrm{s}.\mathrm{c}.\mathrm{on}{V}^h,\hfill \\ \left(b\right)\mathrm{there}\mathrm{exists}{\alpha }^{h^{\prime }}\ge 0,\mathrm{such}\mathrm{that}\mathrm{for}\mathrm{all}{\eta }_1,{\eta }_2,v_1^h,v_2^h\in V^h\mathrm{we}\mathrm{have}\hfill \\ |j^h({\eta }_1,v_2^h)-j^h({\eta }_1,v_1^h)+j^h({\eta }_2,v_1^h)-j^h({\eta }_2^h,v_2^h)| \le {\alpha }^{h^{\prime }}\parallel {\eta }_1-{\eta }_2{\parallel }_{V^h}{\parallel v_1^h-v_2^h\parallel }_{V^h};\hfill \end{array}\right.$$

(84)

$$u_0^h\in V^h;$$

(85)

$$F\in W^{1,2}(0,T,V^h).$$

(86)

From this point onward, we denote a generic constant as $c^h>0$.

Lemma 1.

Assume that (82)–(86) holds. If $m^{h^{\prime }}>{\alpha }^{h^{\prime }}$, then there exists a unique solution $u^h\in W^{1,2}(0,T,V^h)$ to Auxiliary Problem 3.

Proof. 

By using (80), for any $t_1$ and $t_2\in [0,T]$ we find that

$$\begin{array}{cc}\hfill a^h(u^h\left(t_1\right),v^h-{\dot{u}}^h\left(t_1\right))& + b^h({\dot{u}}^h\left(t_1\right),v^h-{\dot{u}}^h\left(t_1\right))+j^h({\dot{u}}^h\left(t_1\right),v^h)-j^h({\dot{u}}^h\left(t_1\right),{\dot{u}}^h\left(t_1\right))\\ \hfill & \ge {(F\left(t_1\right),v^h-{\dot{u}}^h\left(t_1\right))}_{V^h},\forall v^h\in V^h,\hfill \end{array}$$

(87)

$$\begin{array}{cc}\hfill a^h(u^h\left(t_2\right),v^h-{\dot{u}}^h\left(t_2\right))& + b^h({\dot{u}}^h\left(t_2\right),v^h-{\dot{u}}^h\left(t_2\right))+j^h(\dot{u}\left(t_2\right),v)-j^h({\dot{u}}^h\left(t_2\right),{\dot{u}}^h\left(t_2\right))\hfill \\ \hfill & \ge {(F\left(t_2\right),v^h-{\dot{u}}^h\left(t_2\right))}_{V^h},\forall v^h\in V^h.\hfill \end{array}$$

(88)

We take $v^h={\dot{u}}^h\left(t_2\right)$ in (87) and $v^h={\dot{u}}^h\left(t_1\right)$ in (88) and, adding the two inequalities, we obtain

$$\begin{array}{cc}\hfill & b^h({\dot{u}}^h\left(t_1\right)-{\dot{u}}^h\left(t_2\right),{\dot{u}}^h\left(t_1\right)-{\dot{u}}^h\left(t_2\right))+j({\dot{u}}^h\left(t_1\right),{\dot{u}}^h\left(t_2\right))-j^h({\dot{u}}^h\left(t_1\right),{\dot{u}}^h\left(t_2\right))+j^h({\dot{u}}^h\left(t_2\right),{\dot{u}}^h\left(t_1\right))\hfill \\ \hfill & -j^h({\dot{u}}^h\left(t_2\right),{\dot{u}}^h\left(t_2\right))\le a^h(u^h\left(t_1\right)-u^h\left(t_2\right),{\dot{u}}^h\left(t_2\right)-{\dot{u}}^h\left(t_1\right))+(F\left(t_1\right)-F\left(t_2\right),{\dot{u}}^h\left(t_2\right)-{\dot{u}}^h\left(t_1\right)).\hfill \end{array}$$

(89)

We now use assumptions (82)–(84) to obtain

$$\parallel {\dot{u}}^h\left(t_1\right)-{\dot{u}}^h\left(t_2\right){\parallel }_{V^h} \le c^h(\parallel u^h\left(t_1\right)-u^h\left(t_2\right){\parallel }_{V^h}+\parallel F\left(t_1\right)-F\left(t_2\right){\parallel }_{V^h}),$$

(90)

where $c^h=max\left\{{\displaystyle \frac{M^h}{m^{h^{\prime }}-{\alpha }^{h^{\prime }}},\frac{1}{m^{h^{\prime }}-{\alpha }^{h^{\prime }}}}\right\}$. This inequality, combined with the regularity $u^h\in C^1(0,t,V^h)$, shows that ${\dot{u}}^h:[0,T]⟶V^h$ is an absolutely continuous function. Furthermore, it should be noted that

$$\parallel {\ddot{u}}^h{\left(t\right)\parallel }_{V^h}\le c^h(\parallel {\dot{u}}^h{\left(t\right)\parallel }_{V^h}+{\parallel F\left(t\right)\parallel }_{V^h})\mathrm{a}.\mathrm{e}.,t\in [0,T].$$

We conclude that $u^h\in W^{2,2}(0,T,V^h)$. □

Using the solution $u^h$ asserted by Lemma 1, we proceed by considering a second auxiliary problem.

Problem 4

(Auxiliary Problem). Determine a function ${\theta }_{\eta }^h:[0,T]⟶E^h$, such that

$$\left\{\begin{array}{c}{\dot{\theta }}_{\eta }^h\left(t\right)+\widetilde{{\mathcal{K}}^h}{\theta }_{\eta }^h\left(t\right)=P\left(t\right)+\widetilde{{\mathcal{M}}^h}{\dot{u}}_{\eta }^h\left(t\right),\\ {\theta }_{\eta }^h\left(0\right)={\theta }_0^h,t\in [0,T].\end{array}\right.$$

(91)

Lemma 2.

For all $\eta \in C(0,T,V^h)$, there exists a unique solution to Auxiliary Problem 4 with

$${\theta }_{\eta }^h\in L^2\left(0,T,E^h\right)\cap C\left(\left[0,T\right],L^2\left({\mathrm{\Omega }}^h\right)\right)\cap W^{1,2}(0,T,{\left(E^h\right)}^{\prime }).$$

(92)

Moreover, there exists $c^h>0$, such that, for all ${\eta }_1,{\eta }_2\in C(0,T,V^h)$, we have

$$\parallel {\theta }_{{\eta }_1}^h\left(t\right)-{\theta }_{{\eta }_2}^h{\left(t\right)\parallel }_{E^h}\le c^h{\int }_0^t{\parallel {\dot{u}}_{{\eta }_1}^h\left(s\right)-{\dot{u}}_{{\eta }_2}^h\left(s\right)\parallel }_{V^h}^{2}ds,\forall t\in [0,T].$$

(93)

Proof. 

The existence and uniqueness result to (91) follows by considering the Gelfand evolution (see, e.g., [30]):

$$E^h\subset F^h\equiv {\left(F^h\right)}^{\prime }\subset {\left(E^h\right)}^{\prime },$$

and verifying that the operator $\widetilde{{\mathcal{K}}^h}:E^h⟶{\left(E^h\right)}^{\prime }$ is strongly monotonic and linearly continuous. As inferred from the expression of the operator $\widetilde{{\mathcal{M}}^h}$, we have

$${\dot{u}}_{\eta }^h\left(t\right)\in W^{1,2}(0,T;V^h)⟹\widetilde{{\mathcal{M}}^h}{\dot{u}}_{\eta }^h\left(t\right)\in W^{1,2}(0,T,F^h),$$

and

$$P\left(t\right)\in W^{1,2}(0,T,E^h)⟹\widetilde{{\mathcal{M}}^h}{\dot{u}}_{\eta }^h\left(t\right)+P\left(t\right)\in W^{1,2}(0,T,{\left(E^h\right)}^{\prime }).$$

For ${\eta }_1,{\eta }_2\in C(0,T,V^h)$, we have

$$\begin{array}{cc}\hfill & {\left({\dot{\theta }}_{{\eta }_1}^h\left(t\right)-{\dot{{\theta }_{{\eta }_2}}}^h\left(t\right),{\theta }_{{\eta }_1}^h\left(t\right)-{\theta }_{{\eta }_2}^h\left(t\right)\right)}_{{\left(E^h\right)}^{\prime }\times E^h}+{\left(\widetilde{{\mathcal{K}}^h}{\theta }_{{\eta }_1}^h\left(t\right)-\widetilde{{\mathcal{K}}^h}{\theta }_{{\eta }_2}^h\left(t\right),{\theta }_{{\eta }_1}^h\left(t\right)-{\theta }_{{\eta }_2}^h\left(t\right)\right)}_{E^h}\hfill \\ \hfill & ={\left(\widetilde{{\mathcal{M}}^h}{\dot{u}}_{{\eta }_1}^h\left(t\right)-\widetilde{{\mathcal{M}}^h}{\dot{u}}_{{\eta }_2}^h\left(t\right),{\theta }_{{\eta }_1}^h\left(t\right)-{\theta }_{{\eta }_2}^h\left(t\right)\right)}_{L^2\left({\mathrm{\Omega }}^h\right)},t\in [0,T],\hfill \end{array}$$

(94)

and, by integrating the last property over $(0,t)$ and utilizing the Lipschitz continuity of $\widetilde{{\mathcal{M}}^h}:V^h⟶{\left(E^h\right)}^{\prime }$, as well as the strong monotonicity of $\widetilde{{\mathcal{K}}^h}$, we deduce that (93) holds for $t\in [0,T]$. □

We are now ready to prove our main result: the existence of a unique solution to our Problem 1.

Theorem 1.

Assume that (51)–(68) are satisfied. Then, there exists $Z_0$, which depends on ${\mathrm{\Omega }}^h$, ${\mathrm{\Gamma }}_1^h$, ${\mathrm{\Gamma }}_2^h$, and ${\mathrm{\Gamma }}_3^h$, such that if $L_r<Z_0$ then there exists a unique solution $(u^h,{\phi }^h,{\theta }^h)$ to (VP), satisfying

$$u^h\in W^{2,2}(0,T,V^h),$$

(95)

$${\theta }^h\in L^2\left(0,T,E^h\right)\cap C\left(\left[0,T\right],L^2\left({\mathrm{\Omega }}^h\right)\right)\cap W^{1,2}(0,T,{\left(E^h\right)}^{\prime }),$$

(96)

$${\phi }^h\in W^{1,2}(0,T,W^h).$$

(97)

Proof. 

From (77) we ascertain that

$${({\beta }^h{\phi }^h,{\psi }^h)}_{W^h}-{(e^h{u}^h,{\psi }^h)}_{W^h}={(q^h,{\psi }^h)}_{W^h},$$

(98)

while the use of (98) gives

$${\beta }^h{\phi }^h\left(t\right)=e^h{u}^h\left(t\right)+q^h.$$

Hence, we deduce that

$${\phi }^h\left(t\right)={\displaystyle \frac{e^h}{{\beta }^h}}{u}^h\left(t\right)+{\displaystyle \frac{q^h}{{\beta }^h}}.$$

(99)

From (78) and (79), we obtain the solution to

$${\displaystyle {\theta }^h\left(t\right)={\int }_0^t\widetilde{{\mathcal{M}}^h}{e}^{-{\int }_s^t\widetilde{{\mathcal{K}}^h}dx}{\dot{u}}^h\left(s\right)ds+{\int }_0^t{e}^{-{\int }_s^t\widetilde{{\mathcal{K}}^h}dx}P\left(s\right)ds+{\theta }_0^h{e}^{-{\int }_0^t\widetilde{{\mathcal{K}}^h}ds}.}$$

(100)

Taking (99) and (100) and substituting them into (76), we obtain

$$\begin{array}{cc}\hfill a_{{\theta }^h}({\dot{u}}^h\left(t\right),v^h-{\dot{u}}^h\left(t\right))+a_{{\mu }^h}(u^h\left(t\right),v^h-{\dot{u}}^h\left(t\right))+& a_{e^h}\left({\displaystyle \frac{e^h}{{\beta }^h}}{u}^h\left(t\right)+{\displaystyle \frac{q^h}{{\beta }^h}},v^h-\dot{u^h}\left(t\right)\right)\hfill \\ \hfill +a_{M^h}({\int }_0^t\widetilde{{\mathcal{M}}^h}{e}^{-{\int }_s^t\widetilde{{\mathcal{K}}^h}dx}{\dot{u}}^h\left(s\right)ds& +{\int }_0^t{e}^{-{\int }_s^t\widetilde{{\mathcal{K}}^h}dx}P\left(s\right)ds+{\theta }_0^h{e}^{-{\int }_0^t\widetilde{{\mathcal{K}}^h}ds},v^h-{\dot{u}}^h\left(t\right))\hfill \\ \hfill +j^h({\dot{u}}^h\left(t\right),v^h)-j^h({\dot{u}}^h\left(t\right),{\dot{u}}^h\left(t\right))& \ge {(f^h\left(t\right),v^h-{\dot{u}}^h)}_{V^h},\forall v^h\in V^h,t\in [0,T],\hfill \\ \hfill u^h\left(0\right)& =u_0^h.\hfill \end{array}$$

Next, we define the bilinear forms $a^h:V^h\times V^h⟶\mathbb{R}$ and $b^h:V^h\times V^h⟶\mathbb{R}$ as follows:

$$\begin{array}{cc}\hfill a^h(u^h\left(t\right),v^h-{\dot{u}}^h\left(t\right))& =a_{{\mu }^h}(u^h\left(t\right),v^h-{\dot{u}}^h\left(t\right))+a_{e^h}\left(\frac{e^h}{{\beta }^h}{u}^h\left(t\right),v^h-{\dot{u}}^h\left(t\right)\right),\hfill \\ \hfill & + a_{M^h}\left({\int }_0^t\widetilde{{\mathcal{M}}^h}{e}^{-{\int }_s^t\widetilde{{\mathcal{K}}^h}dx}{\dot{u}}^h\left(s\right)ds,v^h-{\dot{u}}^h\left(t\right)\right),\hfill \end{array}$$

(101)

$$b^h({\dot{u}}^h\left(t\right),v^h-{\dot{u}}^h\left(t\right))=a_{{\theta }^h}({\dot{u}}^h\left(t\right),v^h-{\dot{u}}^h\left(t\right)).$$

(102)

We also consider the function $F:[0,T]⟶V^h$, defined by

$$\begin{array}{cc}\hfill {(F\left(t\right),v^h-{\dot{u}}^h)}_{V^h}=& {(f^h\left(t\right),v^h-{\dot{u}}^h)}_{V^h}-a_{e^h}\left({\displaystyle \frac{q^h}{{\beta }^h}},v^h-{\dot{u}}^h\left(t\right)\right)\hfill \\ \hfill & -a_{M^h}\left({\displaystyle {\int }_0^t{e}^{-{\int }_s^t\widetilde{{\mathcal{K}}^h}dx}P\left(s\right)ds+{\theta }_0^h{e}^{-{\int }_0^t\widetilde{{\mathcal{K}}^h}ds},v^h-{\dot{u}}^h\left(t\right)}\right).\hfill \end{array}$$

(103)

From (101)–(103), we obtain the following results:

• the bilinear form $a^h$ and the initial data $u_0^h$ satisfy conditions (82) and (85);
• $f^h\in W^{1,2}(0,T,V^h)$, $q^h\in W^{1,2}(0,T,W^h)$, and $P\in W^{1,2}(0,T,E^h)$, combined with the definition of $F(·)$ in (103), satisfy (86);
• for all $\eta \in V^h$, the functional $j^h(\eta ,·):V^h⟶\mathbb{R}$ is a continuous seminorm on $V^h$, thus satisfying the condition (84)-(a) (recall that $j^h$ also fulfills inequality (63), demonstrating that condition (84)-(b) holds with ${\alpha }^{h^{\prime }}=c_V{L}_r$);
• from (74), it is evident that the bilinear form $b^h$ satisfies condition (84) with $m^{h^{\prime }}={\alpha }^{h*}$.

Now, we choose ${\displaystyle Z_0=\frac{{\alpha }^{h*}}{c_V^2}}$, which depends on ${\mathrm{\Omega }}^h$, ${\mathrm{\Gamma }}_1^h$, ${\mathrm{\Gamma }}_2^h$, ${\mathrm{\Gamma }}_3^h$, and ${\alpha }^{h^{\prime }}$. Then, if $L_r<Z_0$, we have $m^{h^{\prime }}>{\alpha }^{h^{\prime }}$. Therefore, (95) and (96) are a direct consequence of Lemmas 1 and 2. For the proof of (97), let ${\phi }^h:[0,T]⟶W^h$ be the electrical-potential field defined by (99). We have

$$\left\{\begin{array}{c}{u}^h\in W^{2,2}(0,T,V^h)\hfill \\ q^h\in W^{1,2}(0,T,W^h)\hfill \end{array}\right.\Rightarrow {\phi }^h\in W^{1,2}(0,T,W^h),$$

which completes the proof. □

6. Conclusions

In this paper, we theoretically investigated a quasi-static-antiplane contact problem with a slip-rate-dependent friction law involving a thermo-electro-visco-elastic body. The slip-rate-dependent friction law was interesting to investigate because the friction level depends on the rate at which the slip changes over time. This implies that variations in the slip rate impact the friction level, with the friction force increasing as the slip-rate speed increases, and decreasing as the slip-rate speed decreases. This is more reasonable because it considers both the quasi-static equilibrium conditions and the localized dynamic interactions that may occur within the contact interface.

We used Green’s formula to derive the variational formulation of our problem. Following that, we demonstrated the existence and uniqueness of the weak solution, using various techniques, including the time-dependent variational equation, the variational-evolution equation, the differential equation, and the decoupling of unknowns. The advantages of the developed model include both the quasi-static equilibrium conditions and the localized dynamic interactions that may occur within the contact interface. This approach provides a more comprehensive and realistic representation of the material response under varying thermal and loading conditions. One of the drawbacks we faced was the time consumption and the difficulty of providing practical examples without resorting to numerical analysis and simulation.

We have numerous prospects ahead. In particular, we intend to conduct work on numerical analysis and simulation based on the results obtained for the studied problem.