\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 222, pp. 1--21.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/222\hfil A frictional  contact problem]
{Frictional contact problems for  electro-viscoelastic materials
with long-term memory, damage,  and adhesion}

\author[T. Hadj Ammar, B. Benabderrahmane, S. Drabla \hfil EJDE-2014/222\hfilneg]
{Tedjani Hadj Ammar, Benyattou Benabderrahmane, Salah Drabla} 

\address{Tedjani Hadj Ammar \newline
Departement of Mathematics,
Faculty of Sciences and Technology,
University of El-Oued, 39000 El-Oued,  Algeria}
\email{hat\_olsz@yahoo.com}

\address{Benyattou Benabderrahmane  \newline
Department of Mathematics, Faculty of Mathematics and Informatics, 
M'Sila University, Algeria}
\email{bbenyattou@yahoo.com}

\address{Salah Drabla \newline
 Department of Mathematics,
Faculty of Sciences,
University of S\'etif, 19000 S\'etif, Algeria}
\email{drabla\_s@yahoo.fr}

\thanks{Submitted May 20, 2014. Published October 21, 2014.}
\subjclass[2000]{49J40, 74H20, 74H25}
\keywords{Electro-viscoelastic material with long-term memory; damage;
\hfill\break\indent adhesion; friction contact; normal compliance;
variational inequality;  fixed point}

\begin{abstract}
 We consider a quasistatic contact problem  between two electro-viscoelastic
 bodies with long-term memory and damage. The contact is frictional and
 is modelled with a version of normal compliance condition and the associated
 Coulomb's law of friction in which the adhesion of contact surfaces is taken
 into account. We derive a variational formulation for the model and prove an
 existence and uniqueness result of the weak solution. The proof is based on
 arguments of evolutionary variational inequalities, a classical existence and
 uniqueness result on parabolic inequalities,  and Banach fixed point theorem.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{Introduction} \label{sec1}

The aim of this article is to study a quasistatic frictional  contact problem
with adhesion between two electro-viscoelastic  bodies.
We use the electro-viscoelastic constitutive law with long-term memory and
damage  given by
\begin{equation} \label{1.1}
 {\boldsymbol{\sigma}}^{\ell}
= \mathcal{A}^{\ell} \boldsymbol{\varepsilon}(\dot{\mathbf{u}}^{\ell})
+\mathcal{G}^{\ell} \boldsymbol{\varepsilon}(\mathbf{u}^{\ell})
+(\mathcal{E}^{\ell})^*\nabla\varphi^{\ell}
+ \int_0^t\mathcal{F}^{\ell}\big(t-s, \varepsilon({\mathbf{u}^{\ell}}(s)),
\zeta^{\ell}(s)\big)\,ds,
\end{equation}
where $ \mathbf{u}^{\ell} $   the displacement field,
 $ {\boldsymbol{\sigma}}^{\ell} $ and  $ \boldsymbol{\varepsilon}(\mathbf{u}^{\ell}) $  represent the stress and the linearized strain tensor, respectively. Here $ \mathcal{A}^{\ell} $ is a given nonlinear operator, $ \mathcal{F}^{\ell} $ is the relaxation operator, and $ \mathcal{G}^{\ell} $ represents the elasticity
operator$. E(\varphi^{\ell}) = - \nabla\varphi^{\ell}  $ is the electric field,
$ \mathcal{E}^{\ell} $ represents the third order piezoelectric tensor,
$ (\mathcal{E}^{\ell})^{*}$ is its transposition.  In \eqref{1.1} and everywhere
in this paper the dot above a variable represents derivative with respect
to the time variable $ t$. It follows from \eqref{1.1} that at each time moment,
the stress tensor    $ {\boldsymbol{\sigma}}^{\ell}(t) $ is split into  three
parts: $ {\boldsymbol{\sigma}}^{\ell}(t) = {\boldsymbol{\sigma}}_{V}^{\ell}(t)
+ {\boldsymbol{\sigma}}_{E}^{\ell}(t)+ {\boldsymbol{\sigma}}_{R}^{\ell}(t)$,
 where $ {\boldsymbol{\sigma}}_{V}^{\ell}(t)
= \mathcal{A}^{\ell} \boldsymbol{\varepsilon}(\dot{\mathbf{u}}^{\ell}(t)) $
represents the purely viscous part of the stress,
$ {\boldsymbol{\sigma}}_{E}^{\ell} (t)= (\mathcal{E}^{\ell})^*\nabla\varphi^{\ell}(t)$
represents the electric part of the stress  and
$ {\boldsymbol{\sigma}}_{R}^{\ell} (t)$  satisfies the \emph{rate-type}  elastic
relation
\begin{equation} \label{1.2}
{\boldsymbol{\sigma}}_{R}^{\ell}(t) =  \mathcal{G}^{\ell}
\boldsymbol{\varepsilon}(\mathbf{u}^{\ell}(t))
+ \int_0^t\mathcal{F}^{\ell}\big(t-s, \varepsilon({\mathbf{u}^{\ell}}(s)),
\zeta^{\ell}(s)\big)\,ds.
\end{equation}
Various results, example and mechanical interpretations in the study of elastic
materials of the form \eqref{1.2} can be found in \cite{CFR,T} and references
therein. Note also that  when $\mathcal{F}^{\ell}=0 $  the constitutive
law \eqref{1.1} becomes the Kelvin-Voigt electro-viscoelastic
constitutive relation
\begin{equation}\label{1.3}
{\boldsymbol{\sigma}}^{\ell}(t) = \mathcal{A}^{\ell}
\boldsymbol{\varepsilon}(\dot{\mathbf{u}}^{\ell}(t))
+\mathcal{G}^{\ell} \boldsymbol{\varepsilon}(\mathbf{u}^{\ell}(t))
+(\mathcal{E}^{\ell})^*\nabla\varphi^{\ell}(t).
\end{equation}
 Quasistatic contact problems with Kelvin-Voigt
materials of the form \eqref{1.3} can be found in     \cite{RSS1, RSS2, ZZBD}.
 The normal compliance contact condition was first considered in  \cite{MO}
in the study of dynamic problems with linearly elastic and viscoelastic materials
and then it was used in various references, see e.g. \cite{KO,RSS1}. This condition
allows the interpenetration of the body's surface into the obstacle and it was
justified by considering the interpenetration and deformation of surface asperities.

Processes of adhesion are important in many industrial settings where parts,
usually nonmetallic, are glued together. For this reason,  adhesive contact
between deformable bodies, when a glue is added to prevent relative motion of
the surfaces, has received recently increased attention in the mathematical
literature. Analysis of models for adhesive contact can be found in \cite{CMS,M,MS}
and recently in the monographs \cite{NH,OM}. The novelty in all these papers is
the introduction of a surface internal variable, the bonding field,
denoted in this paper by $\beta$. It describes the point wise fractional density
of adhesion of active bonds on the contact surface, and some times it is
called  \emph{the intensity of adhesion}. Following \cite{IP}, the bonding field
satisfies the restriction $0\leq \beta \leq 1, $  when $\beta = 1 $ at a point
of the contact surface,  the adhesion is complete and all the bonds are active,
 when $\beta = 0 $ all the bonds are inactive,  severed, and there is no adhesion,
when  $0 < \beta < 1  $  the adhesion is partial and only a fraction $  \beta $ of
the bonds is active. The damage is an extremely important topic in engineering,
since it affects directly the useful life of the designed structure or component.
There is a very large engineering literature on this topic. Models taking into
account the influence of internal damage of the
material on the contact process have been investigated mathematically.
General models for damage were derived in \cite{FN1,FN2}  from the virtual power
principle. Mathematical analysis
of one-dimensional problems can be found in \cite{FKS}. The three-dimensional case has been investigated in \cite{KS}. In all these papers the damage of the material is described with a damage function $\zeta^{\ell}$, restricted to have values between zero and one. When $\zeta^{\ell}=1, $  there is no damage in the material, when $\zeta^{\ell}=0, $  the material is completely damaged, when $0 < \zeta^{\ell}  < 1 $  there is partial damage and the system has a reduced load carrying capacity. Contact problems with damage have been investigated in \cite{FKS,RSS2,SS,SHS}. In this paper the inclusion used for the evolution of the damage field is
\begin{equation} \label{1.4}
\dot{\zeta}^{\ell}-\kappa^{\ell}\Delta\zeta^{\ell}
+ \partial\psi_{K^{\ell}}(\zeta^{\ell}) \ni\phi^{\ell}\big(\boldsymbol{\sigma}^{\ell}
-\mathcal{A}^{\ell}\boldsymbol{\varepsilon}(\dot{\mathbf{u}}^{\ell}) ,
\boldsymbol{\varepsilon}({\mathbf{u}}^{\ell} ), \zeta^{\ell} \big),
\end{equation}
where $K^{\ell}$ denotes the set of admissible damage functions defined by
\begin{equation}\label{1.5}
    K^{\ell}= \{ \xi\in H^{1}(\Omega^{\ell});
 0 \leq \xi \leq 1,\text{  a.e.  in } \Omega^{\ell}  \},
\end{equation}
$\kappa^{\ell} $ is a positive coefficient, $ \partial\psi_{K^{\ell}} $
represents the subdifferential of the indicator function of
the set $ K^{\ell } $ and $ \phi^{\ell} $ is a given constitutive function which
describes the sources of the damage in the system.
In this article we  consider a  mathematical frictional  contact problem between
two electro-viscoelastic    bodies with constitutive law with long-term memory
and damage. The contact is modelled with normal compliance where the adhesion of
the contact surfaces is taken into account
and is modelled with a surface variable, the bonding field.  We derive a variational
formulation of the problem and prove the existence of a unique weak solution.

This article is organized as follows.
In Section \ref{sec2} we describe the  mathematical models for the frictional
 contact problem between two electro-viscoelastics   bodies with long-term
memory and damage. The contact is modelled with normal compliance and  adhesion.
In Section \ref{sec3} we introduce some notation, list the assumptions on the
problem's data, and derive the variational formulation of the model.
We state our main result, the existence of a unique weak solution to the model
in Theorem \ref{thm4.1}.  The proof of the theorem is provided in Section \ref{sec4},
 where it is carried out in several steps and is based on arguments of
evolutionary variational inequalities, a classical existence and uniqueness
result on parabolic inequalities, differential equations and the Banach fixed
point theorem.

\section{Problem Statement}\label{sec2}

Let us consider two electro-viscoelastic  bodies with long-term memory occupying
two bounded domains  $\Omega^{1}$, $\Omega^2$ of the space
$\mathbb{R}^{d}(d=2,3)$. For each domain $\Omega^{\ell}$,
the boundary $\Gamma^{\ell }$ is assumed to be Lipschitz continuous, and is
partitioned into three disjoint measurable parts  $\Gamma_1^{\ell}$,
$\Gamma_2^{\ell}$ and $\Gamma_3^{\ell}$, on one hand, and on two measurable parts
$\Gamma_a^{\ell} $ and   $\Gamma_b^{\ell}$,  on the other hand, such that
$\operatorname{meas}\Gamma_1^{\ell} > 0$,
 $\operatorname{meas}\Gamma_a^{\ell} > 0$.
Let $ T > 0 $ and let $[0,T] $ be the time interval of interest. The body
$ \Omega^{\ell} $ is subjected to $\boldsymbol{f}^{\ell}_0$ forces and volume
electric charges of density $q^{\ell}_0$. The bodies are assumed to be
clamped on  $\Gamma^{\ell}_1 \times (0,T)$. The surface tractions
$\boldsymbol{f}^{\ell}_2$ act on  $\Gamma^{\ell}_2\times (0,T)$.
We also assume that the electrical potential vanishes on
$\Gamma^{\ell}_a\times (0,T)$   and a surface electric charge of density
$q^{\ell}_2$ is prescribed on $\Gamma^{\ell}_b\times (0,T)$.
The two bodies can enter in contact along the common part
$\Gamma_3^{1}=\Gamma_3^2=\Gamma_3$. The bodies are in adhesive contact
with an obstacle,  over the contact surface $ \Gamma_3$.
 With the assumption above, the classical formulation of the friction contact
problem with adhesion and damage between two electro-viscoelastics bodies
with long-term memory is following.

\subsection*{Problem  P}
 For $\ell=1,2$, find  a displacement  field
$\mathbf{u}^{\ell}:\Omega^{\ell}\times(0,T)\to {\mathbb R}^{d}$,
a  stress field ${\boldsymbol{\sigma}}^{\ell}:\Omega ^{\ell
}\times(0,T)\to {\mathbb S}^{d}$, an electric potential
$\varphi^{\ell}:\Omega ^{\ell }\times(0,T)\to \mathbb{R}$,   a damage
$\zeta^{\ell}:\Omega ^{\ell}\times(0,T)\to \mathbb{R}$, a bonding
 $\beta:\Gamma_3\times(0,T)\to {\mathbb R}$ and an electric  displacement  field
$\boldsymbol{D}^{\ell }:\Omega
^{\ell }\times(0,T)\to {\mathbb R}^{d}$  such that
\begin{gather}\label{2.1}
\begin{gathered} 
{\boldsymbol{\sigma}}^{\ell} = \mathcal{A}^{\ell}
\boldsymbol{\varepsilon}(\dot{\mathbf{u}}^{\ell})
+\mathcal{G}^{\ell} \boldsymbol{\varepsilon}(\mathbf{u}^{\ell})
+(\mathcal{E}^{\ell})^*\nabla\varphi^{\ell}
+ \int_0^t\mathcal{F}^{\ell}\big(t-s, \varepsilon({\mathbf{u}^{\ell}}(s)),
\zeta^{\ell}(s)\big)\,ds,\\
  \text{in }   \Omega ^{\ell }\times(0,T),
\end{gathered} \\
 \label{2.2}
\boldsymbol{D}^{\ell } =\mathcal{E}^{\ell}\varepsilon(\mathbf{u}^{\ell}
)  - \mathcal{B}^{\ell } \nabla\varphi^{\ell} \quad  \text{in }
 \Omega ^{\ell }\times(0,T),  \\
 \label{2.3}
\dot{\zeta}^{\ell}-\kappa^{\ell}\Delta\zeta^{\ell}
+ \partial\psi_{K^{\ell}}(\zeta^{\ell}) \ni\phi^{\ell}\big(\boldsymbol{\sigma}^{\ell}
-\mathcal{A}^{\ell}\boldsymbol{\varepsilon}(\dot{\mathbf{u}}^{\ell}) ,
\boldsymbol{\varepsilon}({\mathbf{u}}^{\ell} ), \zeta^{\ell} \big)   \quad
 \text{in }    \Omega ^{\ell }\times(0,T),   \\
\label{2.4}
 \operatorname{Div}\boldsymbol{\sigma}^{\ell }+ \boldsymbol{f}_0^{\ell } =0
\quad   \text{in   }   \Omega ^{\ell }\times(0,T),   \\
\label{2.5}
 \operatorname{div}\boldsymbol{D}^{\ell } - q_0^{\ell } =0    \quad
\text{in   }    \Omega ^{\ell }\times(0,T),    \\
\label{2.6}
\mathbf{u}^{\ell } =0    \quad   \text{on }    \Gamma_1^{\ell }\times(0,T),  \\
\label{2.7}
\boldsymbol{\sigma}^{\ell }\boldsymbol{\nu}^{\ell } = \boldsymbol{f}_2^{\ell
}    \quad    \text{on }   \Gamma _2^{\ell }\times(0,T),  \\
\label{2.8} \left. \begin{gathered}
\sigma_{\nu}^{1}=  \sigma_{\nu}^2\equiv \sigma_{\nu}, \\
\sigma_{\nu}= - p_{\nu}([u_{\nu}]) + \gamma_{\nu}\beta^2R_{\nu}([u_{\nu}])
\end{gathered}  \right\} \quad  \text{on }     \Gamma _3\times(0,T),\\
\label{2.9}  \left.\begin{gathered}
\boldsymbol{\sigma}_{\tau}^{1}= - \boldsymbol{\sigma}_{\tau}^2 \equiv
\boldsymbol{\sigma}_{\tau}, \\
\| \boldsymbol{\sigma} _{\tau}
 + \gamma_{\tau}\beta^2 \boldsymbol{R}_{\tau}([\mathbf{u}_{\tau}]) \|
 \leq  \mu  p_{\nu}([u_{\nu}]),\\
\| \boldsymbol{\sigma}_{\tau}
 + \gamma_{\tau}\beta^2 \boldsymbol{R}_{\tau}([\mathbf{u}_{\tau}]) \|
 < \mu  p_{\nu}([u_{\nu}]) \Rightarrow [\mathbf{u}_{\tau}] = 0, \\
\| \boldsymbol{\sigma}_{\tau}
 + \gamma_{\tau}\beta^2 \boldsymbol{R}_{\tau}([\mathbf{u}_{\tau}]) \|
=  \mu  p_{\nu}([u_{\nu}])\\
\Rightarrow \exists \lambda \geq 0
\text{  such that }   \boldsymbol{\sigma}_{\tau}
 + \gamma_{\tau}\beta^2 \boldsymbol{R}_{\tau}([\mathbf{u}_{\tau}])
= - \lambda  [\mathbf{u}_{\tau}]
\end{gathered}  \right\} \quad     \text{on  }
   \Gamma _3\times(0,T), \\
\label{2.10}
\dot{\beta }= - \Big(\beta \big( \gamma_{\nu}(R_{\nu}([u_{\nu}]))^2
 + \gamma_{\tau}  | \boldsymbol{R}_{\tau}([\mathbf{u}_{\tau}]) |^2 \big)
 -\varepsilon_a\Big)_{+}   \quad   \text{on }   \Gamma _3\times(0,T), \\
\label{2.11}
    \varphi^{\ell } = 0    \quad   \text{on }    \Gamma_a^{\ell }\times(0,T),  \\
\label{2.12}
\boldsymbol{D}^{\ell}\cdot\boldsymbol{\nu}^{\ell} =q_2^{\ell }    \quad   \text{on }
  \Gamma_b^{\ell }\times(0,T),\\
\label{2.13}
    \frac{\partial\zeta^{\ell}}{\partial\nu^{\ell}} = 0 \quad   \text{on }
   \Gamma^{\ell}\times(0,T), \\
\label{2.14}
\mathbf{u}^{\ell}(0) = \mathbf{u}^{\ell}_0, \quad
\zeta^{\ell}(0) = \zeta^{\ell}_0   \quad   \text{in }    \Omega^{\ell}, \\
\label{2.15}\beta(0) = \beta_0     \quad   \text{on }     \Gamma_3.
\end{gather}
First, equations \eqref{2.1}  and \eqref{2.2}  represent the electro-viscoelastic
constitutive law with long
term-memory and damage, the evolution of the damage   is governed by the inclusion
of parabolic type given
by the relation \eqref{2.3}. Equations \eqref{2.4} and  \eqref{2.5} are the
equilibrium equations for the stress and electric-displacement fields,
respectively,  in which ``$\operatorname{Div}$'' and ``$\operatorname{div}$''
 denote the divergence operator for tensor and vector valued functions,
respectively. Next,  the equations \eqref{2.6} and  \eqref{2.7} represent the
displacement and traction boundary condition,  respectively.
 Condition  \eqref{2.8} represents the normal compliance conditions with
adhesion where $\gamma_{\nu}$ is a given adhesion coefficient,  $ p_{\nu} $
is a given positive function which will be described below
 and $ [u_{\nu}] = u_{\nu}^{1} + u_{\nu}^2 $  stands for the displacements
in normal direction,  in this condition the interpenetrability between two  bodies,
that is $ [u_{\nu}]$  can be positive on $\Gamma_3$. The contribution of the
 adhesive to the normal traction is represented by the term
$ \gamma_{\nu}\beta^2R_{\nu}([u_{\nu}]), $  the adhesive traction is tensile and
is proportional, with proportionality coefficient $ \gamma_{\nu}, $  to the square
of the intensity of adhesion and to the normal displacement,  but as long as
it does not exceed the bond length $L$. The maximal tensile traction is
 $ \gamma_{\nu}\beta^2L$. $R_{\nu}$  is the truncation operator defined by
\[
 R_{\nu}(s) = \begin{cases}
L & \text{if }  s<-L,\\
-s & \text{if } -L\le s\le 0,\\
0 & \text{if } s >0.
\end{cases}
\]
Here $L > 0$ is the characteristic length of the bond,  beyond which it does
not offer any additional traction.  The introduction of the operator $R_{\nu}$.
 together with the operator $  \boldsymbol{R}_{\tau}$ defined below,
is motivated by mathematical arguments but it is not
restrictive for physical point of view,  since no restriction on the size of
the parameter $L$ is made in what follows.  Condition  \eqref{2.9}
are a non local Coulomb's friction law conditions coupled with adhesive,
where $ [\mathbf{u}_{\tau}] = \mathbf{u}_{\tau}^{1} - \mathbf{u}_{\tau}^2 $
stands for the jump of the displacements in tangential direction.
$  \boldsymbol{R}_{\tau}$ is the truncation operator given by
\[
 \boldsymbol{R}_{\tau}(\mathbf{v})=\begin{cases}
 \mathbf{v}  &\text{if } |\mathbf{v}|\leq L,\\
L \frac{\mathbf{v}}{|\mathbf{v}|} & \text{if } |\mathbf{v}|> L.\end{cases}
\]
This condition shows that the shear on the contact surface depends on the
bonding field and on the tangential displacement, but as long as it does not exceed
the bond length $L$.

Next, the equation  \eqref{2.10} represents the ordinary differential equation which
describes the evolution of the bonding field and it was already used in \cite{CFSS},
see also \cite{SST, SHS} for more details. Here, besides $ \gamma_{\nu}$,
two new adhesion coefficients are involved,
$ \gamma_{\tau}$ and $\varepsilon_a$. Notice that in this model once debonding
occurs bonding cannot be
reestablished since,  as it follows from \eqref{2.10}, $\dot{\beta}\leq 0$.
\eqref{2.11} and \eqref{2.12} represent the electric boundary conditions.
The relation \eqref{2.13}  represents a homogeneous Neumann boundary condition
where $\frac{\partial\zeta^{\ell}}{\partial\nu^{\ell}}   $ is the normal derivative
of $\zeta^{\ell}$. \eqref{2.14} represents the initial displacement field and
the initial damage field. Finally, \eqref{2.15} represents the initial
condition in which $\beta_0$ is the given initial bonding field.


\section{Variational formulation and the main result}\label{sec3}

 In this section,  we list the assumptions on the data and derive a variational
formulation for the contact problem. To this end,  we need to introduce some
notation and preliminary material. Here and below,   $\mathbb{S}^{d}$ represent
the space of second-order symmetric tensors on $\mathbb{R}^{d}$. We recall that
the inner products and the corresponding norms on $\mathbb{S}^{d}$ and
$\mathbb{R}^{d}$ are given by
\begin{gather*}
\mathbf{u}^{\ell}.\mathbf{v}^{\ell} =
u^{\ell}_{i}.v^{\ell}_{i},\quad| \mathbf{v}^{\ell} |
 =(\mathbf{v}^{\ell}.\mathbf{v}^{\ell})^{\frac{1}{2}}, \quad
\forall \mathbf{u}^{\ell}, \mathbf{v}^{\ell} \in \mathbb{R}^{d},\\
\boldsymbol{\sigma}^{\ell}.\boldsymbol{\tau}^{\ell}
= \sigma^{\ell}_{ij}.\tau^{\ell}_{ij},\quad
|\boldsymbol{\tau}^{\ell}|=(\boldsymbol{\tau}^{\ell}\cdot
\boldsymbol{\tau}^{\ell})^{\frac{1}{2}}, \quad
\forall \boldsymbol{\sigma}^{\ell}, \boldsymbol{\tau}^{\ell} \in \mathbb{S}^{d}.
\end{gather*}
Here and below, the indices $i$ and $j$ run between $1$ and $d$ and the summation
convention over repeated indices is adopted.  Now, to proceed with the variational
formulation, we need the following function spaces:
 \begin{gather*}
 H^{\ell}= \{ \mathbf{v}^{\ell}=(v^{\ell}_{i});
   v^{\ell}_{i}\in  L^2(\Omega^{\ell})\},  \quad
 H^{\ell}_1= \{ \mathbf{v}^{\ell}=(v^{\ell}_{i});
  v^{\ell}_{i}\in  H^{1}(\Omega^{\ell})  \}, \\
\mathcal{H}^{\ell}=  \{\boldsymbol{\tau}^{\ell}=(\tau^{\ell}_{ij});
  \tau^{\ell}_{ij}=\tau^{\ell}_{ji}\in L^2(\Omega^{\ell})\},\quad
  \mathcal{H}^{\ell}_1=\{\boldsymbol{\tau}^{\ell}=(\tau^{\ell}_{ij})\in
 \mathcal{H}^{\ell};  \operatorname{div}\boldsymbol{\tau}^{\ell}\in H^{\ell}\}.
\end{gather*}
  The spaces $H^{\ell}$, $H_1^{\ell}$, $ \mathcal{H}^{\ell}$  and
$\mathcal{H}_1^{\ell}$ are real Hilbert spaces endowed with the canonical inner
products given by
  \begin{gather*}
( \mathbf{u}^{\ell}, \mathbf{v}^{\ell} )_{H^{\ell}}
= \int_{\Omega^{\ell}}\mathbf{u}^{\ell}.\mathbf{v}^{\ell}dx,   \quad
( \mathbf{u}^{\ell},\mathbf{v}^{\ell} )_{H_1^{\ell}}
= \int_{\Omega^{\ell}}\mathbf{u}^{\ell}.\mathbf{v}^{\ell}dx
+ \int_{\Omega^{\ell}}\nabla \mathbf{u}^{\ell}.\nabla \mathbf{v}^{\ell}dx,
\\
(\boldsymbol{\sigma}^{\ell}, \boldsymbol{\tau}^{\ell} )_{\mathcal{H}^{\ell}}
= \int_{\Omega^{\ell}}\boldsymbol{\sigma}^{\ell}.\boldsymbol{\tau}^{\ell}dx  , \quad
( \boldsymbol{\sigma}^{\ell}, \boldsymbol{\tau}^{\ell} )_{\mathcal{H}_1^{\ell}}
=\int_{\Omega^{\ell}}\boldsymbol{\sigma}^{\ell}.\boldsymbol{\tau}^{\ell}dx
+ \int_{\Omega^{\ell}}  \operatorname{div}\boldsymbol{\sigma}^{\ell}.
  \operatorname{Div}\boldsymbol{\tau}^{\ell}dx
 \end{gather*}
 and the associated norms $ \|\cdot\|_{H^{\ell}}$,
 $ \|\cdot\|_{H_1^{\ell}}$, $\|\cdot\|_{ \mathcal{H}^{\ell}}$,
 and $\|\cdot\|_{ \mathcal{H}^{\ell}_1}$   respectively.  Here and below
 we use the notation
\begin{gather*}
\nabla \mathbf{u}^{\ell} = (u^{\ell}_{i,j}), \quad
\varepsilon (\mathbf{u}^{\ell})=(\varepsilon_{ij}(\mathbf{u}^{\ell})), \quad
\varepsilon_{ij}(\mathbf{u}^{\ell}) =\frac{1}{2}(u^{\ell}_{i,j}+u^{\ell}_{j,i}),
\quad \forall u^{\ell} \in H^{\ell}_1, \\
\operatorname{Div}\boldsymbol{\sigma}^{\ell } = (\sigma^{\ell}_{ij,j}), \quad
\forall \boldsymbol{\sigma}^{\ell}\in \mathcal{H}_1^{\ell}.
\end{gather*}
For every element $\mathbf{v}^{\ell}\in H^{\ell}_1$,  we also use the
notation $\mathbf{v}^{\ell}$ for the trace  of $\mathbf{v}^{\ell}$ on
$\Gamma^{\ell}$ and   we denote by $v^{\ell}_{\nu }$ and
$\mathbf{v}^{\ell}_{\tau }$ the \emph{normal} and the \emph{tangential }
components of $\mathbf{v}^{\ell}$ on the boundary $\Gamma^{\ell} $ given by
\[
v^{\ell}_{\nu }= \mathbf{v}^{\ell}.\nu^{\ell} , \quad
\mathbf{v}^{\ell}_{\tau}=\mathbf{v}^{\ell}-v^{\ell}_{\nu}\boldsymbol{\nu}^{\ell}.
\]


Let $H'_{\Gamma^{\ell}}$ be the dual of
$H_{\Gamma^{\ell}}=H^{\frac{1}{2}}(\Gamma^{\ell})^{d}$   and let
 $ (\cdot,\cdot)_{-\frac{1}{2},\frac{1}{2},\Gamma^{\ell}}$ denote the duality pairing
between $H'_{\Gamma^{\ell}}$ and $H_{\Gamma^{\ell}}$.
For every element $\boldsymbol{\sigma}^{\ell}\in \mathcal{H}_1^{\ell}$ let
$ \boldsymbol{\sigma}^{\ell}\boldsymbol{\nu}^{\ell} $ be the element of
$ H'_{\Gamma^{\ell}} $ given by
\[
 (\boldsymbol{\sigma}^{\ell}\boldsymbol{\nu}^{\ell},
\mathbf{v}^{\ell} )_{-\frac{1}{2},\frac{1}{2},\Gamma^{\ell}}
= ( \boldsymbol{\sigma}^{\ell}, \varepsilon(\mathbf{v}^{\ell})
)_{\mathcal{H}^{\ell}} + (  \operatorname{Div}\boldsymbol{\sigma}^{\ell},
 \mathbf{v}^{\ell} )_{H^{\ell}} \quad \forall
\mathbf{v}^{\ell}\in H_1^{\ell}.
\]

Denote by $\sigma^{\ell}_{\nu }$ and $\boldsymbol{\sigma}^{\ell}_{\tau}$ the
\emph{normal} and the \emph{tangential }traces of
 $\boldsymbol{\sigma}^{\ell} \in \mathcal{H}^{\ell}_1$, respectively.
If $\boldsymbol{\sigma}^{\ell}$    is   continuously differentiable on
 $\Omega^{\ell}\cup \Gamma^{\ell}$, then
\begin{gather*}
\sigma^{\ell}_{\nu }=(\boldsymbol{\sigma}^{\ell} \boldsymbol{\nu}^{\ell} )\cdot
\boldsymbol{\nu}^{\ell},\quad
\boldsymbol{\sigma}^{\ell}_{\tau}=\boldsymbol{\sigma}^{\ell}\boldsymbol{\nu}^{\ell}
-\sigma^{\ell}_{\nu}\boldsymbol{\nu}^{\ell},  \\
 (\boldsymbol{\sigma}^{\ell}\boldsymbol{\nu}^{\ell},
\mathbf{v}^{\ell})_{-\frac{1}{2},\frac{1}{2},\Gamma^{\ell}}
= \int_{\Gamma^{\ell} }\boldsymbol{\sigma}^{\ell}
\boldsymbol{\nu}^{\ell}\cdot\mathbf{v}^{\ell}da
\end{gather*}
fore all $ \mathbf{v}^{\ell}\in H^{\ell}_1$, where $da$ is the surface measure
element.

To obtain the variational formulation of the problem \eqref{2.1}--\eqref{2.15},
we introduce for the bonding field the set
\[
\mathcal{Z} = \big\{ \theta \in  L^{\infty}\big(0,T; L^2(\Gamma_3)\big);
 0 \leq \theta(t) \leq 1\, \forall t \in [0,T],  \text{ a.e.  on \ } \Gamma_3 \big\},
\]
and for the displacement field we need the closed subspace of $H_1^{\ell}$
defined by
\[
V^{\ell }  =   \big\{ \mathbf{v}^{\ell } \in  H_1^{\ell}  ;
\mathbf{v}^{\ell } = 0   \text{ on }    \Gamma_1^{\ell }  \big\}.
\]
 Since $\operatorname{meas}\Gamma_1^{\ell} > 0$,  the following   Korn's
inequality holds:
  \begin{equation} \label{3.1}
\|\varepsilon(\mathbf{v}^{\ell})\|_{\mathcal{H}^{\ell}}
\geq c_{K}\|\mathbf{v}^{\ell}\|_{H_1^{\ell}}  \quad  \forall \mathbf{v}^{\ell}\in
V^{\ell},
\end{equation}
where the constant $c_{K}$ denotes a positive  constant which may depends only
on $\Omega^{\ell}$, $ \Gamma_1^{\ell}$ (see \cite{NH}). Over the space
 $ V^{\ell}$    we
consider the inner product given by
\begin{equation}\label{3.2}
 ( \mathbf{u}^{\ell} , \mathbf{v}^{\ell}  )_{V^{\ell} }
 =  ( \varepsilon (\mathbf{u}^{\ell}),
\varepsilon(\mathbf{v}^{\ell }))_{\mathcal{H}^{\ell }}, \quad \forall
\mathbf{u}^{\ell} , \mathbf{v}^{\ell}  \in  V^{\ell},
 \end{equation}
and  let $\|\cdot\|_{ V^{\ell} }$  be  the associated norm.
It follows from  Korn's  inequality  \eqref{3.1}  that the norms
 $\|\cdot\|_{H_1^{\ell}} $ and $\|\cdot\|_{V^{\ell}}$ are equivalent  on
$ V^{\ell}$. Then $( V^{\ell},\|\cdot\|_{V^{\ell}})$ is a real Hilbert space.
 Moreover, by the Sobolev trace theorem and \eqref{3.2}, there exists a
constant $c_0>0, $  depending only on $\Omega^{\ell}$, $ \Gamma_1^{\ell} $
 and $ \Gamma_3$ such that
\begin{equation}\label{3.3}
\|\mathbf{v}^{\ell}\|_{L^2(\Gamma_3)^{d}} \leq c_0 \|\mathbf{v}^{\ell}\|_{V^{\ell}}
\quad \forall  \mathbf{v}^{\ell}   \in  V^{\ell}.
 \end{equation}
We also introduce the spaces
 \begin{gather*}
E_0^{\ell} =  L^2(\Omega^{\ell}), \quad
E_1^{\ell} =  H^{1}(\Omega^{\ell}), \quad
 W^{\ell }=   \{  \psi^{\ell } \in E_1^{\ell}  ;   \psi^{\ell } = 0   \text{ on }
  \Gamma_a^{\ell } \}, \\
\mathcal{W}^{\ell } =  \{ \boldsymbol{D}^{\ell}=(D^{\ell}_{i}) ;
 D^{\ell}_{i}\in L^2(\Omega^{\ell}),
    \operatorname{div}\boldsymbol{D}^{\ell }\in L^2(\Omega^{\ell})  \}.
\end{gather*}
Since $ \operatorname{meas}\Gamma_a^{\ell}>0$, the following Friedrichs-Poincar\'{e}
  inequality holds:
\begin{equation}\label{3.4}
\|\nabla\psi^{\ell }\|_{W^{\ell } } \geq c_{F} \|\psi^{\ell }
\|_{H^{1}(\Omega^{\ell})} \quad  \forall \psi^{\ell } \in W^{\ell },
\end{equation}
where $ c_{F} > 0 $ is a constant which depends only on $\Omega^{\ell}$,
$\Gamma_a^{\ell}$.
In the space $ W^{\ell },  $ we consider the inner product
 \begin{equation}\label{3.5}
 (\varphi^{\ell }, \psi^{\ell })_{W^{\ell }}
=  \int_{\Omega^{\ell}} \nabla\varphi^{\ell }\cdot\nabla\psi^{\ell }dx
\end{equation}
and let $\|\cdot \|_{W^{\ell }} $ be the associated norm.
It follows from \eqref{3.4} that
 $ \| \cdot \|_{H^{1}(\Omega^{\ell}) } $ and $ \| \cdot \|_{W^{\ell } } $
 are equivalent norms on $  W^{\ell } $ and therefore
$ (W^{\ell }, \| \cdot \|_{W^{\ell }}) $ is areal Hilbert space.
 Moreover, by the Sobolev trace theorem, there exists a constant $\mathbf{c}_0, $
 depending only on $\Omega^{\ell}, $  $ \Gamma_a^{\ell}  $
 and $ \Gamma_3$, such that
\begin{equation}\label{3.6}
\|\zeta^{\ell }\|_{L^2(\Gamma_3)} \leq  \mathbf{c}_0 \|\zeta^{\ell }\|_{W^{\ell}} \quad  \forall \zeta^{\ell } \in W^{\ell }.
\end{equation}
The space $ \mathcal{W}^{\ell } $  is real Hilbert space with the inner product
\[
  (\boldsymbol{D}^{\ell}, \boldsymbol{\Phi}^{\ell})_{\mathcal{W}^{\ell }}
=  \int_{\Omega^{\ell}} \boldsymbol{D}^{\ell}\cdot\boldsymbol{\Phi}^{\ell}dx
 +  \int_{\Omega^{\ell}} \operatorname{div}\boldsymbol{D}^{\ell}\cdot
\operatorname{div}\boldsymbol{\Phi}^{\ell}dx,
\]
where $ \operatorname{div}\boldsymbol{D}^{\ell} = (\boldsymbol{D}^{\ell}_{i,i})$,
 and the associated norm $ \|\cdot\|_{\mathcal{W}^{\ell } }$.

To simplify notation, we define the product spaces
\begin{gather*}
\boldsymbol{V}  = V^{1}\times V^2, \quad
 H  = H^{1} \times H^2, \quad 
H_1 = H_1^{1} \times H_1^2,   \\\
 \mathcal{H}  = \mathcal{H}^{1}   \times \mathcal{H}^2, \quad
\mathcal{H}_1 = \mathcal{H}_1^{1} \times \mathcal{H}_1^2,\quad
E_0 = E_0^{1}\times E_0^2,\\
E_1 = E_1^{1}\times E_1^2,\quad
 W =W^{1 } \times W^2,  \quad 
\mathcal{W}   = \mathcal{W}^{1 } \times \mathcal{W}^2.
\end{gather*}
The spaces  $\boldsymbol{V}$, $E_1$, $W$   and $ \mathcal{W} $  are real Hilbert
 spaces endowed with the canonical inner products denoted by
$(\cdot,\cdot)_{\boldsymbol{V}}$,  $(\cdot,\cdot)_{E_1}, $  $(\cdot,\cdot)_{W}
$ and $(\cdot,\cdot)_{\mathcal{W}}$. The associate norms will be denoted by
$\|\cdot\|_{\boldsymbol{V}}$,  $\|\cdot\|_{E_1} $, $\|\cdot\|_{W}  $ and
 $ \|\cdot\|_{\mathcal{W}},  $ respectively.

Finally, for any real Hilbert space $ X, $ we use the classical notation for
the spaces $ L^{p}(0,T;X)$, $ W^{k,p}(0,T;X)$, where $ 1\leq p \leq \infty$,
 $ k\geq 1$. We denote by $ C(0,T;X)   $ and  $ C^{1}(0,T;X)   $
the space of continuous and continuously differentiable functions from
$ [0,T] $ to $ X$, respectively,  with the norms
 \begin{gather*}
\|f\|_{C(0,T; X)} = \max_{t\in [0,T]} \|f(t)\|_{X},\\
 \|f\|_{C^{1}(0,T; X)} = \max_{t\in [0,T]} \|f(t)\|_{X}
+ \max_{t\in [0,T]} \|\dot{f}(t)\|_{X},
  \end{gather*}
respectively. Moreover, we use the dot above to indicate the derivative with respect
to the time variable and, for areal number $ r, $ we use $ r_{+} $ to represent
its positive part,
that is $ r_{+} = \max\{0,r \}$. For the convenience of the reader, we recall
the following version of the classical theorem  of Cauchy-Lipschitz
(see, \cite[p.48]{SHS}).

\begin{theorem}\label{thm3.1}
Assume that $ (X,\|\cdot\|_{X}) $ is a real Banach space and $ T > 0 $.
Let $ F(t,\cdot):   X \to X $ be an operator defined a.e. on $ (0,T) $
satisfying the following conditions:
\begin{enumerate}
\item There exists a constant $ L_{F} > 0 $ such that
\[
\|F(t,x) - F(t,y)\|_{X} \leq L_{F} \| x  -  y \|_{X}\quad \forall x, y \in X,
 \textnormal{ a.e. } t \in (0,T).
\]

\item There exists $ p \geq 1 $ such that $ t \mapsto F(t,x) \in L^{p}(0,T;X)$
 for all $x \in X$.
\end{enumerate}
Then for any $ x_0 \in X, $ there exists a unique function
$ x \in W^{1,p}(0,T; X) $ such that
\begin{gather*}
\dot{x}(t)=F(t,x(t)), \quad   \text{ a.e. }  t \in (0,T), \\
x(0) = x_0.
\end{gather*}
\end{theorem}

This theorem will be used in section\ref{sec4} to prove the unique
 solvability of the intermediate problem involving the bonding field.

In the study of the Problem P, we consider the
following assumptions:

The  \emph{ viscosity function } $ \mathcal{A}^{\ell}: \Omega^{\ell}\times \mathbb{S}^{d}\to
\mathbb{S}^{d} $ satisfies:
\begin{equation}
\parbox{10cm}{
(a) There exists $L_{\mathcal{A}^{\ell}}  > 0$ such that
$|\mathcal{A}^{\ell}(\boldsymbol{x}, \boldsymbol{\xi}_1)
- \mathcal{A}^{\ell}(\boldsymbol{x}, \boldsymbol{\xi}_2)|  \le
  L_{\mathcal{A}^{\ell}}  |\boldsymbol{\xi}_1 - \boldsymbol{\xi}_2|$
for all $\boldsymbol{\xi}_1, \boldsymbol{\xi}_2 \in\mathbb{S}^{d}$,  a.e.
$\boldsymbol{x}\in\Omega^{\ell}$.  \\
(b) There exists $m_{\mathcal{A}^{\ell}} > 0$  such that
 $(\mathcal{A}^{\ell}(\boldsymbol{x}, \boldsymbol{\xi}_1)-\mathcal{
A}^{\ell}(\boldsymbol{x}, \boldsymbol{\xi}_2))\cdot(\boldsymbol{\xi}_1-
\boldsymbol{\xi}_2)\geq  m_{\mathcal{A}^{\ell}}
|\boldsymbol{\xi}_1-\boldsymbol{\xi}_2 |^2$
for all $\boldsymbol{\xi}_1, \boldsymbol{\xi}_2\in
\mathbb{S}^{d}$,  a.e. $\boldsymbol{x} \in \Omega^{\ell}$.\\
 (c)  The mapping  $\boldsymbol{x}\mapsto \mathcal{A}^{\ell}( \boldsymbol{x},
\boldsymbol{\xi})$  is Lebesgue measurable on $\Omega^{\ell}$,\\
 for any $\boldsymbol{\xi}\in \mathbb{S}^{d}$.\\
 (d) The mapping $\boldsymbol{x}\mapsto \mathcal{A}^{\ell}( \boldsymbol{x},
\boldsymbol{0})$ is  continuous on $\mathbb{S}^{d}$,  a.e.
$\boldsymbol{x} \in \Omega^{\ell}$.
}    \label{3.7}
\end{equation}
The \emph{elasticity operator}
$ \mathcal{G}^{\ell}: \Omega^{\ell}\times \mathbb{S}^{d}\to
\mathbb{S}^{d} $ satisfies:
\begin{equation}
\parbox{10cm}{
(a) There exists  $L_{\mathcal{G}^{\ell}}>0$ such that
$|\mathcal{G}^{\ell}(\boldsymbol{x},
\boldsymbol{\xi}_1)-\mathcal{G}^{\ell}(\boldsymbol{x},
\boldsymbol{\xi}_2) | \le
  L_{\mathcal{G}^{\ell}}  |\boldsymbol{\xi}_1-\boldsymbol{\xi}_2 |$
 for all $\boldsymbol{\xi}_1, \boldsymbol{\xi}_2\in\mathbb{S}^{d}$, a.e.
$\boldsymbol{x}\in\Omega^{\ell}$.  \\
 (b)  The mapping $\boldsymbol{x}\mapsto \mathcal{G}^{\ell}( \boldsymbol{x},
\boldsymbol{\xi})$  is Lebesgue measurable on $\Omega^{\ell}$,
 for any $\boldsymbol{\xi}\in \mathbb{S}^{d}$.\\
(c) The mapping $\boldsymbol{x}\mapsto \mathcal{G}^{\ell}( \boldsymbol{x},
\boldsymbol{0})$ belongs to $\mathcal{H}^{\ell}$.
}   \label{3.8}
\end{equation}
The \emph{relaxation function} $ \mathcal{F}^{\ell}:
\Omega^{\ell}\times (0,T)\times \mathbb{S}^{d}\times \mathbb{R }\to
\mathbb{S}^{d} $ satisfies:
\begin{equation}\label{3.9}
\parbox{10cm}{
 (a) There exists $L_{\mathcal{F}^{\ell}}  > 0$  such that
$|\mathcal{F}^{\ell}(\boldsymbol{x}, t, \boldsymbol{\xi}_1, d_1)
- \mathcal{F}^{\ell}(\boldsymbol{x},t, \boldsymbol{\xi}_2, d_2)|  \le
  L_{\mathcal{F}^{\ell}}  \big(  |\boldsymbol{\xi}_1 - \boldsymbol{\xi}_2|
 + |d_1 - d_2|  \big)$,
 for  all  $t \in (0,T)$,
$\boldsymbol{\xi}_1, \boldsymbol{\xi}_2 \in\mathbb{S}^{d}$,
$d_1, d_2 \in\mathbb{R}$,  a.e. $\boldsymbol{x}\in\Omega^{\ell}$.  \\
(b)  The mapping  $\boldsymbol{x}\mapsto \mathcal{F}^{\ell}( \boldsymbol{x},
t, \boldsymbol{\xi},d)$   is Lebesgue measurable in $\Omega^{\ell}$,
 for any $t \in (0, T)$,  $\boldsymbol{\xi}\in \mathbb{S}^{d}$,
$d \in \mathbb{R}$.\\
(c)  The mapping  $t \mapsto \mathcal{F}^{\ell}( \boldsymbol{x}, t,
\boldsymbol{\xi},d)$   is continuous in $(0, T)$,
 for any  $ \boldsymbol{\xi}\in \mathbb{S}^{d}$,  $d \in \mathbb{R}$,   a.e.
$\boldsymbol{x}\in\Omega^{\ell}$. \\
 (d) The mapping $\boldsymbol{x}\mapsto \mathcal{F}^{\ell}( \boldsymbol{x}, t,
\boldsymbol{0}, 0)$  belongs  to  $\mathcal{H}^{\ell}$,   for  all  $t\in (0,T)$.
}
\end{equation}
The \emph{damage source function}
$ \phi^{\ell}:  \Omega^{\ell}\times \mathbb{S}^{d}\times \mathbb{S}^{d}\times
\mathbb{R}\to \mathbb{R} $ satisfies:
\begin{equation}\label{3.10}
\parbox{10cm}{
 (a) There exists  $L_{\phi^{\ell}}  > 0$ such that
$|\phi^{\ell}(\boldsymbol{x}, \boldsymbol{\eta}_1,
\boldsymbol{\xi}_1, \alpha_1)- \phi^{\ell}(\boldsymbol{x},\boldsymbol{\eta}_2,
\boldsymbol{\xi}_2, \alpha_2)|  \le
  L_{\phi^{\ell}}  \big( |\boldsymbol{\eta}_1 - \boldsymbol{\eta}_2|
+ |\boldsymbol{\xi}_1 - \boldsymbol{\xi}_2|
+ |\alpha_1 - \alpha_2|  \big)$,
 for all $\boldsymbol{\eta}_1, \boldsymbol{\eta}_2,
 \boldsymbol{\xi}_1, \boldsymbol{\xi}_2 \in\mathbb{S}^{d}$ and
$\alpha_1, \alpha_2 \in\mathbb{R}$   a.e.
$\boldsymbol{x}\in\Omega^{\ell}$,   \\
 (b)  The mapping $\boldsymbol{x}\mapsto \phi^{\ell}( \boldsymbol{x},
\boldsymbol{\eta}, \boldsymbol{\xi}, \alpha)$
 is Lebesgue measurable on $\Omega^{\ell}$,
 for any $\boldsymbol{\eta}, \boldsymbol{\xi}\in \mathbb{S}^{d}$ and
$\alpha \in \mathbb{R}$, \\
(c) The mapping $\boldsymbol{x}\mapsto \phi^{\ell}( \boldsymbol{x},
\boldsymbol{0}, \boldsymbol{0}, 0)$  belongs  to  $L^2(\Omega^{\ell})$, \\
(d)  $\phi^{\ell}( \boldsymbol{x}, \boldsymbol{\eta}, \boldsymbol{\xi}, \alpha)$
 is  bounded  for  all $\boldsymbol{\eta}, \boldsymbol{\xi}\in \mathbb{S}^{d}$,
$ \alpha \in \mathbb{R}$  a.e.
$\boldsymbol{x}\in\Omega^{\ell}$.
}
\end{equation}
The \emph{piezoelectric tensor }
$\mathcal{E}^{\ell}   : \Omega^{\ell}\times\mathbb{S}^d\to \mathbb{R}^d  $ satisfies:
\begin{equation}\label{3.11}
\parbox{10cm}{
(a)  $\mathcal{E}^{\ell}(\boldsymbol{x},\tau)
= (e_{ijk}^{\ell}(\boldsymbol{x})\tau_{jk})$ for all
$\tau =(\tau_{ij}) \in  \mathbb{S}^d$ a.e.
$\boldsymbol{x}\in\Omega^{\ell}$.\\
(b)   $e_{ijk}^{\ell} = e_{ikj}^{\ell}\in L^{\infty}(\Omega^{\ell})$,
$1\leq i,j,k\leq d$.
}
\end{equation}
Recall also that the transposed operator $ (\mathcal{E}^{\ell})^{*} $
is given by $ (\mathcal{E}^{\ell})^{*}=(e_{ijk}^{\ell,*}) $ where
$ e_{ijk}^{\ell,*}= e_{kij}^{\ell }  $ and the following equality hold
\[
    \mathcal{E}^{\ell} \sigma.\mathbf{v}
= \sigma\cdot(\mathcal{E}^{\ell})^{*}\mathbf{v}  \quad \forall  \sigma
\in \mathbb{S}^d, \; \forall  \mathbf{v} \in \mathbb{R}^d.
\]
The \emph{electric permittivity operator }
$ \mathcal{B}^{\ell} = (b_{ij}^{\ell})
:\Omega^{\ell}\times\mathbb{R}^d \to \mathbb{R}^d $ satisfies:
\begin{equation}
\parbox{10cm}{
 (a)  $\mathcal{B}^{\ell}(\boldsymbol{x},\mathbf{E})
=(b_{ij}^{\ell}(\boldsymbol{x})E_{j})$
for all $\mathbf{E}=(E_i)\in\mathbb{R}^d$, a.e.
$\boldsymbol{x}\in\Omega^{\ell}$.\\
(b) $b_{ij}^{\ell} = b_{ji}^{\ell}$,
$b_{ij}^{\ell}\in L^{\infty}(\Omega^{\ell})$, $1\leq i,j\leq d$.\\
(c) There exists $m_{\mathcal{B}^{\ell}} > 0$, such
that $\mathcal{B}^{\ell}\mathbf{E}\cdot\mathbf{E}
\geq m_{\mathcal{B}^{\ell}} |\mathbf{E}|^2$
for all $\mathbf{E}=(E_i)\in {\mathbb R}^d$, a.e.
$\boldsymbol{x} \in \Omega^{\ell}$.
}\label{3.12}
\end{equation}
The \emph{normal compliance function}
$ p_\nu : \Gamma_3\times{\mathbb R}\to {\mathbb R_+}$  satisfies:
\begin{equation}\label{3.13}
\parbox{10cm}{
 (a) There exists $L_\nu>0$  such that
$|p_\nu(\boldsymbol{x}, r_1)-p_\nu(\boldsymbol{x}, r_2)|\leq L_\nu |r_1 - r_2|$
for all $r_1, r_2  \in {\mathbb R}$, a.e. $\boldsymbol{x}\in\Gamma_3$.\\
(b) $(p_\nu(\boldsymbol{x}, r_1)-p_\nu(\boldsymbol{x}, r_2))(r_1-r_2)\geq 0$
for all $r_1,r_2  \in {\mathbb R}$, a.e. $\boldsymbol{x}\in\Gamma_3$.\\
(c) The mapping  $\boldsymbol{x}\mapsto p_\nu(\boldsymbol{x},r)$ is
measurable on $\Gamma_3$ for all $r\in {\mathbb R}$.\\
(d) $p_\nu(\boldsymbol{x},r) = 0$  for
all $r \leq  0$, a.e. $\boldsymbol{x}\in\Gamma_3$.
}
\end{equation}
The following regularity is assumed on the density of volume forces,
traction, volume electric charges and
surface electric charges:
\begin{equation}  \label{3.14}
\begin{gathered}
{\mathbf{f}}_0^{\ell}\in C(0,T;L^2(\Omega^{\ell})^d),  \quad
  {\mathbf{f}}_2^{\ell}  \in C(0,T;L^2(\Gamma_2^{\ell})^d), \\
q_0^{\ell}\in  C(0,T;L^2(\Omega^{\ell})),  \quad
  q_2^{\ell}\in C(0,T;L^2(\Gamma_b^{\ell})).
\end{gathered}
\end{equation}
The adhesion coefficients $ \gamma_{\nu},   \gamma_{\tau} $ and
$  \varepsilon_a $ satisfy the conditions
\begin{equation}\label{3.15}
\gamma_{\nu},   \gamma_{\tau} \in L^{\infty}(\Gamma_3), \quad
\varepsilon_a \in L^2(\Gamma_3), \quad
\gamma_{\nu},   \gamma_{\tau},   \varepsilon_a \geq 0, \quad   \text{a.e.  on }
 \Gamma_3.
\end{equation}
The microcrack diffusion coefficient satisfies
\begin{equation}\label{3.16}
    \kappa^{\ell} > 0.
\end{equation}
Finally,  the friction coefficient and  the initial data satisfy
\begin{gather}
\mu \in L^{\infty}(\Gamma_3), \quad \mu(x)\geq 0 \quad \text{ a.e.  on }
  \Gamma_3 \label{3.17} \\
\label{3.18}
\mathbf{u}^{\ell}_0\in \boldsymbol{V}^{\ell},   \quad
   \zeta_0^{\ell}\in K^{\ell},   \quad   \beta_0\in L^2(\Gamma_3),
\quad   0 \leq \beta_0 \leq 1, \text{  a.e.  on } \Gamma_3.
\end{gather}
where $K^{\ell} $ is the set of admissible damage functions
defined in \eqref{1.5}.

 Using the Riesz representation theorem, we define the linear mappings
 $\mathbf{ f }=(\mathbf{ f}^{1}, \mathbf{ f}^2) : [0,T]\to \boldsymbol{V} $
and $ q = (q^{1}, q^2): [0,T]\to W $  as follows:
\begin{gather}
(\mathbf{f}(t), \mathbf{v})_{\boldsymbol{V}}
= \sum_{\ell=1}^2 \int_{\Omega^{\ell}}
\mathbf{f}_0^{\ell}(t)\cdot\mathbf{v}^{\ell}\,dx
 + \sum_{\ell=1}^2\int_{\Gamma_2^{\ell}} \mathbf{f}_2^{\ell}(t)
\cdot\mathbf{v}^{\ell}\,da \quad \forall \mathbf{v} \in \boldsymbol{V},  \label{3.19}
 \\
(q(t), \zeta)_W= \sum_{\ell=1}^2\int_{\Omega^{\ell}}
q_0^{\ell}(t)\zeta^{\ell}\,dx
- \sum_{\ell=1}^2\int_{\Gamma^{\ell}_b} q_2^{\ell}(t)\zeta^{\ell}\,da \quad
\forall \zeta \in W.  \label{3.20}
\end{gather}
Next,  we define the  mappings $ a:E_1 \times E_1  \to \mathbb{R}$,
$ j_{ad} : L^2(\Gamma_3)\times \boldsymbol{V}\times \boldsymbol{V}\to \mathbb {R}$,
$ j_{\nu c} :   \boldsymbol{V}\times \boldsymbol{V}\to \mathbb {R} $ and
$ j_{fr} :   \boldsymbol{V}\times \boldsymbol{V}\to \mathbb {R}$, respectively,
by
\begin{gather}   \label{3.21}
    a(\zeta, \xi)
= \sum_{\ell=1}^{^2}\kappa^{\ell}\int_{\Omega^{\ell}}\nabla\zeta^{\ell}\cdot
\nabla\xi^{\ell} dx, \\
\label{3.22}
j_{ad}(\beta, \mathbf{u}, \mathbf{v})
= \int_{\Gamma_3}\Big(-\gamma_{\nu}\beta^2R_{\nu}([u_{\nu}])[v_{\nu}]
+ \gamma_{\tau}\beta^2 \boldsymbol{R}_{\tau}([\mathbf{u}_{\tau}])\cdot
[\mathbf{v}_{\tau}]\Big)\, da,\\
\label{3.23}
j_{\nu c}(\mathbf{u}, \mathbf{v})= \int_{\Gamma_3}p_{\nu}([u_{\nu}])[v_{\nu}]\, da, \\
\label{3.24}
j_{fr}(\mathbf{u}, \mathbf{v})= \int_{\Gamma_3}\mu p_{\nu}([u_{\nu}])
\big\|[v_{\tau}]\big\|\, da
\end{gather}
for all $ \mathbf{u}, \mathbf{v} \in \boldsymbol{V}$   and $ t \in  [0, T]$.
We note that conditions \eqref{3.14}   imply
\begin{equation} \label{3.25}
\mathbf{f}  \in C(0,T; \boldsymbol{V}), \quad  q    \in  C(0,T; W).
\end{equation}
By a standard procedure based on Green's formula, we derive the following
variational formulation of the
mechanical \eqref{2.1}--\eqref{2.15}.

\subsection*{Problem PV}
Find  a displacement  field
 $\mathbf{u}=(\mathbf{u}^{1}, \mathbf{u}^2): [0,T] \to \boldsymbol{V}$,
  a  stress field
${\boldsymbol{\sigma}}=({\boldsymbol{\sigma}}^{1},{\boldsymbol{\sigma}}^2)
:  [0,T] \to \mathcal{ H}$,
an electric potential   $\varphi = (\varphi^{1}, \varphi^2)  : [0,T] \to W$,
a damage  $\zeta= (\zeta^{1}, \zeta^2): [0,T] \to E_1$,  a bonding
$\beta: [0,T] \to L^{\infty}(\Gamma_3) $  and an electric  displacement  field
$\boldsymbol{D}= (\boldsymbol{D}^{1}, \boldsymbol{D}^2) : [0,T] \to \mathcal{ W} $
 such that
\begin{gather}\label{3.26}
\begin{gathered}
  {\boldsymbol{\sigma}}^{\ell}
= \mathcal{A}^{\ell} \boldsymbol{\varepsilon}(\dot{\mathbf{u}}^{\ell})
+\mathcal{G}^{\ell} \boldsymbol{\varepsilon}(\mathbf{u}^{\ell})
+(\mathcal{E}^{\ell})^*\nabla\varphi^{\ell}
+ \int_0^t\mathcal{F}^{\ell}
\big(t-s, \varepsilon({\mathbf{u}^{\ell}}(s)), \zeta^{\ell}(s)\big)\,ds,\\
  \text{in } \  \Omega ^{\ell }\times(0,T),
\end{gathered} \\
 \label{3.27} \boldsymbol{D}^{\ell }
=\mathcal{E}^{\ell}\varepsilon(\mathbf{u}^{\ell}
)  - \mathcal{B}^{\ell } \nabla\varphi^{\ell}  \quad   \text{in }
  \Omega ^{\ell }\times(0,T),  \\
 \label{3.28}
\begin{aligned}
&\sum_{\ell=1}^2({\boldsymbol{\sigma}}^{\ell},
\varepsilon(\mathbf{v}^{\ell})-\boldsymbol{\varepsilon}
(\dot{\mathbf{u}}^{\ell}(t)))_{\mathcal{H}^{\ell}}
+  j_{ad}(\beta(t), \mathbf{u}(t), \mathbf{v}
- \dot{\mathbf{u}}(t))    \\
&+  j_{\nu c}(\mathbf{u}(t), \mathbf{v}-\dot{\mathbf{u}}(t))
 + j_{fr}(\mathbf{u}(t), \mathbf{v}) -
j_{fr}(\mathbf{u}(t),   \dot{\mathbf{u}}(t))\\
&\geq (\mathbf{f}(t), \mathbf{v}-\dot{\mathbf{u}}(t))_{\boldsymbol{V}  } \quad
 \forall \mathbf{v} \in \boldsymbol{V},  \quad\text{a.e. } t \in (0,T),
 \end{aligned}\\
\label{3.29}
\begin{aligned}
&\quad  \zeta(t)\in K, \\
&\sum_{\ell=1}^2(\dot{\zeta}(t),  \xi-\zeta(t))_{L^2(\Omega^{\ell})} +
a(\zeta(t), \xi-\zeta(t))\\
&\geq \sum_{\ell=1}^2\Big(\phi^{\ell}\big(\boldsymbol{\sigma}^{\ell}(t)
-\mathcal{A}^{\ell}\boldsymbol{\varepsilon}(\dot{\mathbf{u}}^{\ell}(t)) ,
\boldsymbol{\varepsilon}({\mathbf{u}}^{\ell}(t) ),
\zeta^{\ell}(t) \big), \xi^{\ell}-\zeta^{\ell}(t) \Big)_{L^2(\Omega^{\ell})},\\
&\quad  \forall \xi \in K,\text{a.e. }t\in (0,T), \\
 \end{aligned} \\
\begin{gathered}
\sum_{\ell=1}^2(\mathcal{B}^{\ell} \nabla\varphi^{\ell}(t),
\nabla\phi^{\ell} )_{H^{\ell}}
-  \sum_{\ell=1}^2(  \mathcal{E}^{\ell}
\varepsilon(\mathbf{u}^{\ell}(t)),\nabla\phi^{\ell} )_{H^{\ell}}
= (q(t),\phi)_{W}, \\
   \forall \phi\in W,  \text{ a.e. }  t\in (0,T),
\end{gathered}  \label{3.30}  \\
\label{3.31}
\dot{\beta}(t)=-\Big(\beta(t)\big(\gamma_{\nu}(R_{\nu}([u_{\nu}(t)]))^2
+\gamma_{\tau}| \boldsymbol{R}_{\tau}([\mathbf{u}_{\tau}(t)])|^2 \big)
-\varepsilon_a\Big)_{+} \quad\text{a.e. } (0,T),
\\
\mathbf{u}(0)= \mathbf{u}_0, \quad  \zeta(0)=\zeta_0,\quad
 \beta(0)= \beta_0.  \label{3.32}
\end{gather}
We notice that the variational Problem  PV  is formulated in terms
of a displacement field, a  stress field, an electrical potential, a damage,
 a bonding   and an electric  displacement  field. The existence of the unique
solution of Problem  PV is stated and proved in the next section.

\begin{remark}\label{rmq3.1} \rm
We note that, in  Problem {\rm  P}  and in Problem  PV,
we do not need to impose explicitly the restriction $0 \leq \beta \leq 1$.
Indeed, equation \eqref{3.31} guarantees that
$\beta(x,t) \leq \beta_0(x)$ and, therefore, assumption \eqref{3.18} shows
that $\beta(x,t) \leq 1$ for $t \geq 0, $
a.e. $x \in \Gamma_3$. On the other hand, if $ \beta(x,t_0)=0 $ at time $ t_0$,
then it follows from \eqref{3.31}
that $\dot{\beta}(x,t)=0$ for all $t \geq t_0$  and therefore, $\beta(x,t)=0$
for all $t \geq t_0$, a.e. $x \in \Gamma_3$.  We conclude that
$0 \leq \beta(x,t) \leq 1$ for all $t \in [0,T]$, a.e. $x \in \Gamma_3$.
\end{remark}

Below in this section $\beta, \beta_1, \beta_2 $ denote elements of
$L^2(\Gamma_3)$ such that $0\leq \beta, \beta_1$, $\beta_2 \leq 1$ a.e.
 $x\in \Gamma_3, $ $\mathbf{u}_1$, $ \mathbf{u}_2$ and $\mathbf{v}$
represent elements of $\boldsymbol{V}$ and $C > 0$ represents generic
constants which may depend on $ \Omega^{\ell}, $ $\Gamma_3$, $p_{\nu}$,
 $ \gamma_{\nu}$, $ \gamma_{\tau}$ and $L$.
First, we note that the functional $ j_{ad } $ and $ j_{\nu c } $ are linear
with respect to the last argument and, therefore,
\begin{equation}\label{3.33}
\begin{gathered}
j_{ad}(\beta, \mathbf{u},-\mathbf{v})
= -j_{ad}(\beta, \mathbf{u}, \mathbf{v}), \\
j_{\nu c}(\mathbf{u},-\mathbf{v})  = -j_{\nu c}(\mathbf{u},\mathbf{v}).
  \end{gathered}
\end{equation}
  Next,  using   \eqref{3.23} and  \eqref{3.13}(b)   implies
\begin{equation} \label{3.34}
  j_{\nu c}(\mathbf{u}_1, \mathbf{v}_2)-  j_{\nu c}(\mathbf{u}_1,
\mathbf{v}_1)+ j_{\nu c}(\mathbf{u}_2, \mathbf{v}_1)- j_{\nu c}(\mathbf{u}_2,
\mathbf{v}_2)   \leq 0, \quad   \forall  \mathbf{u}_1, \mathbf{u}_2,
\mathbf{v}_1, \mathbf{v}_2  \in \boldsymbol{V},
\end{equation}
and use \eqref{3.24},   \eqref{3.13}(a), keeping in mind \eqref{3.3},  we obtain
\begin{equation} \label{3.35}
\begin{split}
& j_{fr}(\mathbf{u}_1, \mathbf{v}_2)
 -  j_{fr}(\mathbf{u}_1, \mathbf{v}_1)+ j_{fr}(\mathbf{u}_2, \mathbf{v}_1)
 - j_{fr}(\mathbf{u}_2, \mathbf{v}_2) \\
& \leq c_0^2 L_{\nu} \|\mu\|_{L^{\infty}(\Gamma_3)} \|\mathbf{u}_1
 -\mathbf{u}_2\|_{\boldsymbol{V} }  \|\mathbf{v}_1-\mathbf{v}_2\|_{\boldsymbol{V} }
  \quad  \forall  \mathbf{u}_1, \mathbf{u}_2, \mathbf{v}_1, \mathbf{v}_2
 \in \boldsymbol{V}.
\end{split}
\end{equation}
Inequalities  \eqref{3.33}--\eqref{3.35}   will be used in various places in the rest
of this article.  Our main existence and uniqueness result that we state now
and prove in the next section is the following.

\begin{theorem}\label{thm3.2}
Assume that \eqref{3.7}--\eqref{3.18} hold. Then there exists a
unique solution of Problem  PV.   Moreover, the solution satisfies
\begin{gather}
\mathbf{u}\in   C^{1}(0,T; \boldsymbol{V}),   \label{3.36}\\
\boldsymbol{\sigma} \in C(0,T;\mathcal{H}_1),  \label{3.37}\\
\varphi\in C(0,T;W),  \label{3.38}\\
\zeta\in H^{1}(0,T;E_0)\cap  L^2(0,T;E_1), \label{3.39}\\
\beta\in W^{1,\infty}(0,T;L^2(\Gamma_3))\cap \mathcal{Z},   \label{3.40}\\
\boldsymbol{D}  \in C(0,T;\mathcal{W}).  \label{3.41}
\end{gather}
\end{theorem}
 The functions $ \mathbf{u}$, $  \boldsymbol{ \sigma}$,
 $ \varphi$, $ \zeta$, $ \beta  $ and $  \boldsymbol{ D}  $   which satisfy
 \eqref{3.26}-\eqref{3.32} are called a weak solution of the contact
 Problem  P.  We conclude that, under the assumptions \eqref{3.7}-- \eqref{3.18},
the mechanical problem \eqref{2.1}--\eqref{2.15} has a unique weak solution
satisfying \eqref{3.36}--\eqref{3.41}.

\section{Proof of Theorem \ref{thm3.2}}\label{sec4}

 The proof of Theorem \ref{thm3.2} is carried out in several steps and is
based on the following abstract result for evolutionary variational
inequalities.

 Let $ X $ be a real Hilbert space with the inner product $(\cdot,\cdot)_{X}$
 and the associated norm $\|\cdot\|_{X}, $ and consider the problem of finding
$ \mathbf{u}: [0,T] \to X $ such that
\begin{equation}\label{4.1}
\begin{gathered}
 \begin{aligned}
&(A\dot{\mathbf{u}}(t), \mathbf{v}-\dot{\mathbf{u}}(t))_{X}
+  (B\mathbf{u}(t), \mathbf{v}-\dot{\mathbf{u}}(t))_{X}
+ j(\mathbf{u}(t), \mathbf{v}) - j(\mathbf{u}(t), \dot{\mathbf{u}}(t)) \\
 &  \geq (f(t), \mathbf{v} - \dot{\mathbf{u}}(t))_{X} \quad
 \forall \mathbf{v} \in X, \; t\in [0, T],
\end{aligned} \\
 \mathbf{u}(0) = \mathbf{u}_0.
\end{gathered}
\end{equation}
To study problem \eqref{4.1} we need the following assumptions:
The operator $ A: X \to X $ is Lipschitz continuous and strongly monotone,  i.e.,
\begin{equation} \label{4.2}
\parbox{10cm}{
 (a) There exists a positive constant $L_{A}$ such that
$$
\|A\mathbf{u}_1-A\mathbf{u}_2\|_{X} \leq
  L_{A} \|\mathbf{u}_1-\mathbf{u}_2\|_{X}
\quad\forall \mathbf{u}_1, \mathbf{u}_2\in X,
$$
(b) There exists a positive constant  $m_{A}$ such that
$$
(A\mathbf{u}_1- A\mathbf{u}_2, \ \mathbf{u}_1-
\mathbf{u}_2)_{X}\geq  m_{A}\|\mathbf{u}_1-\mathbf{u}_2\|_{X}
  \quad  \forall \mathbf{u}_1, \mathbf{u}_2  \in X.
$$
}
\end{equation}
 The nonlinear operator $B : X \to X $ is Lipschitz continuous, i.e.,
there exists a positive constant  $L_{B}$ such that
\begin{equation}\label{4.3}
    \|B\mathbf{u}_1- B\mathbf{u}_2\|_{X} \le
  L_{B} \|\mathbf{u}_1-\mathbf{u}_2\|_{X}
    \quad \forall \mathbf{u}_1,\mathbf{u}_2\in X.
\end{equation}
The functional $ j: X \times X \to \mathbb{R} $   satisfies:
\begin{equation}
\parbox{10cm}{
 (a) $j(\mathbf{u}, \cdot)$ is  convex  and  I.S.C.  on   $X$  for   all
$u \in  X$.\\
 (b) There exists $m_{j} > 0 $  such that
\begin{align*}
&j(\mathbf{u}_1, \mathbf{v}_2)-
j(\mathbf{u}_1, \mathbf{v}_1)+
j(\mathbf{u}_2, \mathbf{v}_1)-
j(\mathbf{u}_2, \mathbf{v}_2) \\
&\leq m_{j}\|\mathbf{u}_1-\mathbf{u}_2\|_{X}\|\mathbf{v}_1-\mathbf{v}_2\|_{X}
  \quad \forall \mathbf{u}_1, \mathbf{u}_2, \mathbf{v}_1, \mathbf{v}_2  \in X.
\end{align*}
}    \label{4.4}
\end{equation}
Finally, we assume that
 \begin{gather}\label{4.5}
    f\in C(0, T; X) , \\
\label{4.6}
\mathbf{u}_0 \in X.
 \end{gather}
The following existence, uniqueness result and regularity was proved in
  \cite{HS1} and may be found in \cite[p.230--234]{HS2}.

 \begin{theorem}\label{thm4.1}
Let \eqref{4.7}--\eqref{4.6} hold. Then:
 \begin{enumerate}
   \item  There exists a unique solution  $ \mathbf{u}  \in C^{1}(0, T; X) $
of Problem \eqref{4.1}.

\item  If, moreover, $ \mathbf{u}_1 $ and $ \mathbf{u}_2 $ are two solutions
of  \eqref{4.1} corresponding to the data $f_1, f_2\in C(0, T; X)$, then there exists
  $ c > 0 $ such that
  \begin{equation}\label{4.7}
\|\dot{\mathbf{u}}_1(t) -\dot{\mathbf{u}}_2(t)\|_{X}
\leq c \big( \|f_1(t) -f_2(t)\|_{X}
+\| \mathbf{u}_1(t) - \mathbf{u}_2(t)\|_{X}\big),
  \end{equation}
  for all $t \in [0, T]$.
 \end{enumerate}
  \end{theorem}

  We turn now to the proof of Theorem \ref{thm3.2} which will be carried
out in several steps and is based on arguments of nonlinear equations
with monotone operators, a classical existence and uniqueness
result on parabolic inequalities and fixed-point arguments.
 To this end,  we assume in what follows that  \eqref{3.7}--\eqref{3.18}  hold,
and we consider that $C$ is a generic positive constant which depends on
$ \Omega^{\ell}$, $\Gamma_1^{\ell}$, $\Gamma_1^{\ell}$, $\Gamma_3$,
 $p_{\nu}, p_{\tau}$, $\mathcal{A}^{\ell}$, $\mathcal{B}^{\ell}$,
 $\mathcal{G}^{\ell}$, $\mathcal{F}^{\ell}$,  $\mathcal{E}^{\ell}$,
 $ \gamma_{\nu}$, $ \gamma_{\tau}$, $ \phi^{\ell}$, $ \kappa^{\ell}$,   and
$ T$. but does not depend on $t$ nor of the rest of input data,
and whose value may change from place to place.  Let a
$ \eta=(\eta^{1}, \eta^2) \in C(0,T; \boldsymbol{V}) $   be given.
In the first step we consider the following variational problem.

\subsection*{Problem  PV$_{\eta}^{u}$}
Find  a displacement  field
  $\mathbf{u}_{\eta}=(\mathbf{u}_{\eta}^{1}, \mathbf{u}_{\eta}^2)
: [0,T] \to \boldsymbol{V}  $   such that
\begin{gather}\label{4.8}
\begin{aligned}
&\sum_{\ell=1}^2(\mathcal{A}^{\ell} \boldsymbol{\varepsilon}
(\dot{\mathbf{u}}_{\eta}^{\ell}),
\varepsilon(\mathbf{v}^{\ell})-\boldsymbol{\varepsilon}
(\dot{\mathbf{u}}_{\eta}^{\ell}(t)))_{\mathcal{H}^{\ell}}+
\sum_{\ell=1}^2(\mathcal{G}^{\ell} \boldsymbol{\varepsilon}
(\mathbf{u}_{\eta}^{\ell}),
 \varepsilon(\mathbf{v}^{\ell})-\boldsymbol{\varepsilon}
(\dot{\mathbf{u}}_{\eta}^{\ell}(t)))_{\mathcal{H}^{\ell}}   \\
&+ j_{\nu c}(\mathbf{u}_{\eta}(t),
\mathbf{v}-\dot{\mathbf{u}}_{\eta}(t))
+ j_{fr}(\mathbf{u}_{\eta}(t), \mathbf{v}) - j_{fr}(\mathbf{u}_{\eta}(t),
\dot{\mathbf{u}}_{\eta}(t))
+  ( \eta(t),  \mathbf{v}  - \dot{\mathbf{u}}_{\eta}(t))_{\boldsymbol{V} }\\
&\geq (\mathbf{f}(t), \mathbf{v}-\dot{\mathbf{u}}_{\eta}(t))_{\boldsymbol{V} }
\quad \forall \mathbf{v} \in \boldsymbol{V},  \;   t \in (0,T),
\end{aligned} \\
 \label{4.9}
\mathbf{u}_{\eta}(0) = \mathbf{u}_0.
\end{gather}

We have the following result for the problem  PV$_{\eta}^{u}$.

\begin{lemma}\label{lem4.2}
\begin{itemize}
\item[(1)]   There exists a unique solution $\mathbf{u}_\eta \in
C^1(0,T ;\boldsymbol{V})$ to the problem  \eqref{4.8} and \eqref{4.9}.

\item[(2)] If $ \mathbf{u}_1$ and $ \mathbf{u}_2$ are two solutions of
\eqref{4.8} and \eqref{4.9} corresponding to the data
$\eta_1$, $ \eta_2\in C(0,T ;  \boldsymbol{V})$, then there exists $c>0$ such
that
\begin{equation} \label{4.10}
\|\dot{\mathbf{u}}_1(t)-\dot{\mathbf{u}}_2(t)\|_{\boldsymbol{V}}\leq
c\,\big(\| \eta_1(t)- \eta_2(t)\|_{\boldsymbol{V}}+\|
\mathbf{u}_1(t)- \mathbf{u}_2(t)\|_{\boldsymbol{V}}\big)\quad \forall
t\in[0,T].
\end{equation}
 \end{itemize}
\end{lemma}

\begin{proof} We apply Theorem \ref{thm4.1}  where
$X= \boldsymbol{V}$, with the inner product $(\cdot,\cdot)_{\boldsymbol{V}}$
 and the associated norm $\|\cdot\|_{\boldsymbol{V}}$.
We   use the Riesz representation theorem to
define the operators $A: \boldsymbol{V} \to \boldsymbol{V}$, and
$B: \boldsymbol{V} \to \boldsymbol{V} $  by
\begin{gather} \label{4.11}
(A\mathbf{u},\mathbf{v})_{\boldsymbol{V}}
= \sum_{\ell=1}^2(\mathcal{A}^{\ell}\boldsymbol{\varepsilon}(\mathbf{u}^{\ell}),
\boldsymbol{\varepsilon}
(\mathbf{v}^{\ell}))_{\mathcal{H}^{\ell}},  \\
\label{4.12} (B\mathbf{u},\mathbf{v})_{\boldsymbol{V}}
=\sum_{\ell=1}^2(\mathcal{G}^{\ell}\boldsymbol{\varepsilon}(\mathbf{u}^{\ell}),
\boldsymbol{\varepsilon}
(\mathbf{v}^{\ell}))_{\mathcal{H}^{\ell}},
 \end{gather}
for all  $\mathbf{u},  \mathbf{v}\in \boldsymbol{V}$, and define
the functions $\mathbf{f}_\eta : [0,T]\to \boldsymbol{V}$,
 $j: \boldsymbol{V}\times \boldsymbol{V}\to \mathbb{R}  $ by
\begin{gather} \label{4.13}
\mathbf{f}_\eta (t)= \mathbf{f}(t) -\boldsymbol{\eta}(t)\quad \forall t \in [0, T],\\
\label{4.14}
 j(\mathbf{u}, \mathbf{v})= j_{\nu c}(\mathbf{u}, \mathbf{v}) + j_{fr}(\mathbf{u},
\mathbf{v}), \quad \forall  \mathbf{u},  \mathbf{v}\in \boldsymbol{V}.
\end{gather}
Assumptions  \eqref{3.7} and \eqref{3.8} imply  that the operators $A $ and $B $
satisfy conditions \eqref{4.2} and \eqref{4.3}, respectively.

It follows from  \eqref{3.13}, \eqref{3.17},  \eqref{3.23} and \eqref{3.24}
that the functional $ j$,  \eqref{4.14},
satisfies condition   \eqref{4.4}(a). We use again
 \eqref{3.34},  \eqref{3.35} and  \eqref{4.14} to find
\begin{equation} \label{4.15}
\begin{split}
& j(\mathbf{u}_1, \mathbf{v}_2)-  j(\mathbf{u}_1, \mathbf{v}_1)
 + j(\mathbf{u}_2, \mathbf{v}_1)- j(\mathbf{u}_2, \mathbf{v}_2) \\
 & \leq c_0^2 L_{\nu} \|\mu\|_{L^{\infty}(\Gamma_3)}
\|\mathbf{u}_1-\mathbf{u}_2\|_{\boldsymbol{V} }
\|\mathbf{v}_1-\mathbf{v}_2\|_{\boldsymbol{V} }
  \quad  \forall  \mathbf{u}_1, \mathbf{u}_2, \mathbf{v}_1, \mathbf{v}_2
\in \boldsymbol{V},
\end{split}
\end{equation}
which shows that the functional $ j $ satisfies condition \eqref{4.4}(b) on
$ X = \boldsymbol{V}$.
Moreover, using  \eqref{3.25}    and, keeping in mind that
$\boldsymbol{\eta} \in C(0,T ; \boldsymbol{V})$,
we deduce from   \eqref{4.13} that $  \mathbf{f}_\eta  \in C(0,T ;  \boldsymbol{V})$,
 i.e., $\mathbf{f}_\eta$ satisfies  \eqref{4.5}.
 Finally, we note that  \eqref{3.18}  shows that condition   \eqref{4.6}
 is satisfied.   Using now
 \eqref{4.11}--\eqref{4.14}  we find that  Lemma \ref{lem4.2} is a
direct consequence of Theorem \ref{thm4.1}.
\end{proof}

In the second step,   we use the displacement field $ \mathbf{u}_{\eta} $
obtained in Lemma \ref{lem4.2} and we consider the following variational
problem.

\subsection*{Problem PV$^{\varphi}_{\eta}$}
  Find the electric potential   $ \varphi_{\eta} : [0,T] \to  W   $ such that
\begin{equation}\label{4.16}
 \sum_{\ell=1}^2(\mathcal{B}^{\ell} \nabla\varphi_{\eta}^{\ell}(t),
\nabla\phi^{\ell} )_{H^{\ell}} -  \sum_{\ell=1}^2(  \mathcal{E}^{\ell}
\varepsilon(\mathbf{u}_{\eta}^{\ell}(t)), \nabla\phi^{\ell} )_{H^{\ell}}
= (q(t), \phi)_{W}
\end{equation}
 for all $\phi\in W$,  a.e.  $t\in (0,T)$.
 We have the following result.

\begin{lemma}\label{lem4.3}
 Problem {\rm PV}$^{\varphi}_{\eta}$  has a unique solution $ \varphi_{\eta} $
 which satisfies the regularity \eqref{3.38}.
\end{lemma}

\begin{proof}
We define a bilinear form: $ b(\cdot,\cdot): W \times W \to \mathbb{ R } $
such that
\begin{equation} \label{4.17}
 b(\varphi, \phi) = \sum_{\ell=1}^2(\mathcal{B}^{\ell} \nabla\varphi^{\ell},
\nabla\phi^{\ell} )_{H^{\ell}}  \quad    \forall \varphi, \phi \in W.
\end{equation}
We use   \eqref{3.4}, \eqref{3.5}, \eqref{3.12} and \eqref{4.17} to show that
the bilinear form $ b(\cdot,\cdot) $ is continuous,  symmetric and coercive on
$ W$,  moreover using \eqref{3.20} and the Riesz representation
Theorem we may define an element
$ q_{\eta} : [0, T]\to W $ such that
\begin{gather*}
( q_{\eta}(t), \phi)_{W} =  (q(t), \phi)_{W} +
\sum_{\ell=1}^2(  \mathcal{E}^{\ell} \varepsilon(\mathbf{u}_{\eta}^{\ell}(t)),
\nabla\phi^{\ell} )_{H^{\ell}}
 \quad     \forall   \phi \in W, t\in (0,T).
\end{gather*}
We apply the Lax-Milgram Theorem to deduce that there exists a unique element
 $ \varphi_{\eta}(t) \in W $
such that
\begin{equation} \label{4.18}
b(\varphi_{\eta}(t), \phi) =  ( q_{\eta}(t), \phi)_{W}
 \quad    \forall  \phi \in W.
\end{equation}
We conclude that $ \varphi_{\eta}   $  is a solution of Problem
 PV$^{\varphi}_{\eta}$. Let  $ t_1,  t_2 \in [0, T]$, it follows from
\eqref{4.16} that
\begin{equation}\label{4.19}
 \|\varphi_{\eta}(t_1)-\varphi_{\eta}(t_2)\|_{W}  \leq C\big(
\|\mathbf{u}_{\eta}(t_1) - \mathbf{u}_{\eta}(t_2)\|_{\boldsymbol{V}}
+ \|q(t_1)-q(t_2)\|_{W}  \big).
\end{equation}
We also note that assumptions \eqref{3.25}  and
$ \mathbf{u}_{\eta}\in C^{1}(0,T; \boldsymbol{V})$, inequality \eqref{4.19}
implies that $ \varphi_{\eta} \in C(0, T; W)$.
 \end{proof}

In the third step, we use the displacement field  $ \mathbf{u}_{\eta} $
obtained in Lemma\ref{lem4.2} and we consider the following initial-value problem.

\subsection*{Problem PV$^{\beta}_{\eta}$}
 Find  the adhesion   $\beta_{\eta}: [0,T] \to L^2(\Gamma_3)$ such that
\begin{gather}
\begin{gathered}
\dot{\beta}_{\eta}(t)= - \Big(\beta_{\eta}(t)
\big(\gamma_{\nu}(R_{\nu}([u_{\eta\nu}(t)]))^2
+ \gamma_{\tau}|  \boldsymbol{R}_{\tau}([\mathbf{u}_{\eta\tau}(t)])|^2 \big)
- \varepsilon_a\Big)_{+},\\
\text{a.e. } t\in (0,T),
\end{gathered} \label{4.20} \\
\beta_{\eta}(0)= \beta_0.   \label{4.21}
\end{gather}
We have the following result.

\begin{lemma} \label{lem4.4}
There exists a unique solution
 $\beta_{\eta}\in W^{1,\infty}(0,T;L^2(\Gamma_3)) \cap \mathcal{Z} $
to Problem  {\rm PV}$^{\beta}_{\eta}$.
\end{lemma}

\begin{proof}
 For  simplicity we suppress the dependence of various functions on $\Gamma_3$,
and note that the equalities and inequalities below are valid a.e. on
$\Gamma_3$. Consider the
mapping $F_{\eta}: [0,T]\times L^2(\Gamma_3) \to L^2(\Gamma_3) $ defined by
\begin{gather*}
F_{\eta}(t,\beta) = - \Big(\beta \big[\gamma_{\nu}(R_{\nu}([u_{\eta\nu}(t)]) )^2
+ \gamma_{\tau}|  \boldsymbol{R}_{\tau}([\mathbf{u}_{\eta\tau}(t)])|^2 \big]
-\varepsilon_a\Big)_{+},
\end{gather*}
for all $t \in [0,T]$ and $ \beta \in L^2(\Gamma_3)$.
It follows from the properties of the truncation operator $ R_{\nu } $ and
$ \boldsymbol{R}_{\tau } $ that $ F_{\eta} $ is Lipschitz continuous with
respect to the second variable, uniformly in time. Moreover, for all
$ \beta \in L^2(\Gamma_3)$,   the mapping $t \to  F_{\eta}(t,\beta)$
 belongs to $L^{\infty}(0,T;L^2(\Gamma_3))$. Thus using the Cauchy-Lipschitz
theorem given in Theorem \ref{thm3.1} we deduce that there exists a unique
function $\beta_{\eta}\in W^{1,\infty}(0,T;L^2(\Gamma_3))  $ solution to
 Problem PV$^{\beta}_{\eta}$. Also,  the arguments used in
 Remark \ref{rmq3.1} show that $0 \leq  \beta_{\eta}(t) \leq  1 $  for all
$t \in [0,T]$, a.e. on $\Gamma_3$. Therefore, from the definition of the
 set $\mathcal{Z}$, we find that $\beta_{\eta}  \in  \mathcal{Z}$,
 which concludes the proof of the lemma.
\end{proof}

In the forth  step we let  $ \theta \in C(0,T; E_0) $ be given and consider
the following variational problem for the damage.

\subsection*{Problem  PV$_\theta^{\zeta}$}
Find a damage  $\zeta_\theta = (\zeta_\theta^{1},\zeta_\theta^2):[0,T]\to E $
such that $\zeta_\theta(t)\in K$ and
\begin{equation} \label{4.22}
\begin{split}
&\sum_{\ell=1}^2(\dot{\zeta}^{\ell}_\theta(t),
 \xi^{\ell}-\zeta^{\ell}_\theta(t))_{L^2(\Omega^{\ell})} +
a(\zeta_\theta(t), \xi-\zeta_\theta(t)) \\
&\geq  \sum_{\ell=1}^2\big( \theta^{\ell}(t),
\xi^{\ell} -\zeta^{\ell}_\theta(t) \big)_{L^2(\Omega^{\ell})}, \quad
 \forall  \xi \in K, \quad\text{a.e. } t\in (0,T), \quad
 \end{split}
\end{equation}
where $ K = K^{1} \times K^2$. The following abstract result for parabolic
 variational inequalities (see, e.g., \cite[p.47]{SHS}).

\begin{theorem}\label{thm4.5}
Let $X\subset Y= Y' \subset X'$ be a Gelfand triple. Let $F$ be a nonempty,
closed, and convex set of $ X$. Assume that
$ a(\cdot,\cdot) : X\times X \to \mathbb{R} $ is a continuous
and symmetric bilinear form such that for some constants $ \alpha > 0 $ and $ c_0$,
\[
    a(v,v) + c_0\|v\|_{Y}^2\geq \alpha \|v\|_{X}^2\quad \forall v \in X.
\]
Then, for every $ u_0 \in F $ and $f \in L^2(0, T ;Y)$, there exists a unique
function $u \in H^{1}(0, T ;Y)\cap L^2(0, T ;X) $ such that $ u(0)= u_0$,
  $ u(t) \in F $  for all $t \in [0, T]$, and
\[
     (\dot{u}(t), v-u(t))_{X'\times X} + a(u(t),v-u(t))\geq (f(t), v-u(t))_{Y}
\quad \forall v \in F   \text{ a.e. } t\in (0,T).
\]
\end{theorem}

We prove next the unique solvability of Problem  PV$_\theta^{\zeta}$.

\begin{lemma}\label{lem4.6}
There exists a unique solution $ \zeta_{\theta} $  of Problem {\rm PV}$_\theta^{\zeta}$
and it satisfies
\[
  \zeta_{\theta}\in H^{1}(0,T;E_0)\cap  L^2(0,T;E_1).
\]
\end{lemma}

 \begin{proof} The inclusion mapping of  $(E_1, \|\cdot\|_{E_1}) $ into
$ (E_0, \|\cdot\|_{E_0})$  is continuous
and its range is dense. We denote by $ E_1' $ the dual space of $ E_1  $ and,
identifying
the dual of $ E_0 $ with itself, we can write the Gelfand triple
\[
    E_1\subset E_0 = E_0' \subset E_1'.
\]
We use the notation $ (\cdot,\cdot)_{E_1'\times E_1} $ to represent the
duality pairing between $ E'$ and $E_1$. We have
\[
    (\zeta, \xi)_{E_1'\times E_1} =  (\zeta, \xi)_{E_0} \quad \forall
\zeta \in E_0, \xi \in E_1,
\]
and we note that $K$ is a closed convex set in $ E_1$. Then, using
 \eqref{3.16},   \eqref{3.21}  and the fact that $ \zeta_0 \in K $ in \eqref{3.18},
it is easy to see that
Lemma \ref{lem4.6}  is a straight consequence of Theorem \ref{thm4.5}.
\end{proof}

Finally as a consequence of these results and using the properties of the
 operator $ \mathcal{E}^{\ell},  $ the operator $ \mathcal{F}^{\ell}$,
the functional $ j_{ad}  $  and  the functional $ \phi^{\ell}$,
for $ t \in [0, T], $ we consider the element
\begin{equation}\label{4.23}
\Lambda( \eta, \theta)(t)= \big(\Lambda^{1}( \eta, \theta)(t),
 \Lambda^2( \eta, \theta)(t)\big)\in  \boldsymbol{V} \times E_0,
\end{equation}
defined by the equations
\begin{gather}\label{4.24}
\begin{aligned}
&(\Lambda^{1}( \eta, \theta)(t),\mathbf{v})_{\boldsymbol{V}}\\
& = \sum_{\ell=1}^2\Big( \int_0^t \mathcal{F}^{\ell}\big(t-s,
 \varepsilon({\mathbf{u}_{\eta}^{\ell}}(s)), \zeta^{\ell}_{\theta}(s)\big)\,ds,
  \varepsilon(\mathbf{v}^{\ell})\Big)_{\mathcal{H}^{\ell}}\\
&\quad + \sum_{\ell=1}^2\big((\mathcal{E}^{\ell})^*\nabla\varphi_{\eta}^{\ell} ,
 \varepsilon(\mathbf{v}^{\ell})\big)_{\mathcal{H}^{\ell}}
+ j_{ad}(\beta_{\eta}(t), \mathbf{u}_{\eta}(t), \mathbf{v} ),  \quad
    \forall \mathbf{v} \in \boldsymbol{V},
 \end{aligned} \\
\label{4.25}
 \Lambda^2( \eta, \theta)(t)   =
 \Big(\phi^{1}\big(\boldsymbol{\sigma}^{1}_{\eta\theta}(t),
\boldsymbol{\varepsilon}(\mathbf{u}_{\eta}^{1}(t) ),
\zeta_{\theta}^{1}(t) \big), \  \phi^2\big(\boldsymbol{\sigma}^2_{\eta\theta}(t),
 \boldsymbol{\varepsilon}(\mathbf{u}_{\eta}^2(t) ),
\zeta_{\theta}^2(t) \big)\Big).
  \end{gather}
Here, for every $ (\eta, \theta) \in C(0,T; \boldsymbol{V}\times E_0)$,
 $ \mathbf{u}_{\eta}$,  $ \varphi_{\eta}$, $ \beta_{\eta} $ and
$ \zeta_{\theta} $  represent the displacement field,
the potential electric field  and bonding field obtained in Lemmas \ref{lem4.2},
  \ref{lem4.3},   \ref{lem4.4} and \ref{lem4.6} respectively, and
$ \boldsymbol{\sigma}^{\ell}_{\eta\theta} $ denote by
\begin{equation}\label{4.26}
    \boldsymbol{\sigma}^{\ell}_{\eta\theta} (t)
=    \mathcal{G}^{\ell} \boldsymbol{\varepsilon}(\mathbf{u}_{\eta}^{\ell}(t))
+   (\mathcal{E}^{\ell})^*\nabla\varphi_{\eta}^{\ell}
+     \int_0^t\mathcal{F}^{\ell}\big(t-s, \varepsilon({\mathbf{u}_{\eta}^{\ell}}(s)),
\zeta_{\theta}^{\ell}(s)\big)\,ds,
\end{equation}
in $\Omega ^{\ell }\times(0,T)$.
We have the following result.

\begin{lemma}\label{lem4.7}
There exists a unique $(\eta^{*},\theta^{*} )\in C(0,T;  \boldsymbol{V}
\times E_0)   $ such that $ \Lambda( \eta^{*}, \theta^{*})
=   ( \eta^{*}, \theta^{*})$.
\end{lemma}

\begin{proof}
Let $(\eta_1, \theta_1)$,  $( \eta_2, \theta_2)  \in C(0,T;
 \boldsymbol{V}\times E_0) $ and
denote by $\mathbf{u}_{i}$,
$ \varphi_{i}$, $ \beta_{i}$,
$ \zeta_{i} $ and
$ \boldsymbol{\sigma}_{i}$ the functions obtained  in Lemmas
\ref{lem4.2},   \ref{lem4.3},   \ref{lem4.4},  \ref{lem4.6}  and
the relation \eqref{4.26} respectively, for $(\eta, \theta)=(\eta_{i}, \theta_{i})$,
$ i=1,2$. Let $ t \in [0, T]$.
  We use \eqref{3.9}, \eqref{3.10}, \eqref{3.11},   \eqref{3.22} and
 the definition of $R_{\nu}$, $ \boldsymbol{R}_{\tau}$,  we have
\begin{align*}
&\|\Lambda^{1}(\eta_1, \theta_1)(t)- \Lambda^{1}(\eta_2,
\theta_1)(t)\|^2_{\boldsymbol{V}  }\\
& \leq \sum_{\ell=1}^2  \|(\mathcal{E}^{\ell})^*\nabla\varphi_1^{\ell}(t)
- (\mathcal{E}^{\ell})^*\nabla\varphi_2^{\ell}(t)\|_{\mathcal{H}^{\ell}}^2 \\
&\quad +      \sum_{\ell=1}^2 \int_0^t \big\|\mathcal{F}^{\ell}
\big(  t-s, \varepsilon({\mathbf{u}_1^{\ell}}(s)), \zeta^{\ell}_1(s)\big)
-    \mathcal{F}^{\ell} \big( t-s, \varepsilon({\mathbf{u}_2^{\ell}}(s)),
\zeta^{\ell}_2(s)\big)\big\|_{\mathcal{H}^{\ell}}^2\,ds \\
&\quad  + C\|\beta_1^2(t)R_{\nu}([u_{1\nu}(t)])
 - \beta_2^2(t)R_{\nu}([u_{2\nu}(t)]) \|_{L^2(\Gamma_3)}^2 \\
&\quad + C\|\beta_1^2(t) \boldsymbol{R}_{\tau}([\mathbf{u}_{1\tau}(t)])
 - \beta_2^2(t)  \boldsymbol{R}_{\tau}([\mathbf{u}_{2\tau}(t)]) \|_{L^2(\Gamma_3)}^2.
\end{align*}
  Therefore,
\begin{equation} \label{4.27}
\begin{aligned}
&\|\Lambda^{1}(\eta_1, \theta_1)(t)- \Lambda^{1}(\eta_2,
 \theta_1)(t)\|^2_{\boldsymbol{V} }\\
&\leq C \Big(  \int_0^t \| \mathbf{u}_1(s)
 - \mathbf{u}_2(s))\|_{\boldsymbol{V}}^2\,ds
 + \int_0^t \| \zeta_1(s) - \zeta_2(s))\|_{E_0}^2\,ds \\
&\quad  +\|\varphi_1(t) -  \varphi_2(t) \|_{W}^2 + \|\beta_1(t)
 -  \beta_2(t) \|^2_{L^2(\Gamma_3)}\Big).
\end{aligned}
  \end{equation}
 Recall that  $ u_{\eta\nu}^{\ell} $ and $ \mathbf{u}_{\eta\tau}^{\ell} $
denote the normal and the tangential component of the
function  $ \mathbf{u}_{\eta}^{\ell} $  respectively.
By similar arguments, from \eqref{4.25}, \eqref{4.26}  and \eqref{3.10}
 it follows that
\begin{equation}
\begin{aligned}
&\|\Lambda^2(\eta_1, \theta_1)(t)- \Lambda^2(\eta_2, \theta_1)(t)\|^2_{E_0 }\\
 &\leq C \Big(\| \mathbf{u}_1(t) - \mathbf{u}_2(t) \|^2_{\boldsymbol{V} }
+   \int_0^t \| \mathbf{u}_1(s) - \mathbf{u}_2(s))\|_{\boldsymbol{V}}^2\,ds\\
&\quad  + \| \zeta_1(t) - \zeta_2(t))\|_{E_0}^2
+ \int_0^t \| \zeta_1(s) - \zeta_2(s))\|_{E_0}^2\,ds
+ \|\varphi_1(t) -  \varphi_2(t) \|_{W}^2 \Big).
\end{aligned} \label{4.28}
  \end{equation}
It follows now from  \eqref{4.27} and \eqref{4.28} that
\begin{equation}
\begin{aligned}
&\|\Lambda(\eta_1, \theta_1)(t)- \Lambda(\eta_2,
 \theta_1)(t)\|^2_{\boldsymbol{V} \times E_0 }\\
&\leq C \Big(\| \mathbf{u}_1(t) - \mathbf{u}_2(t) \|^2_{\boldsymbol{V} }
 +   \int_0^t \| \mathbf{u}_1(s) - \mathbf{u}_2(s))\|_{\boldsymbol{V}}^2\,ds
 + \| \zeta_1(t) - \zeta_2(t))\|_{E_0}^2\\
&\quad + \int_0^t \| \zeta_1(s) - \zeta_2(s))\|_{E_0}^2\,ds
 + \|\varphi_1(t) -  \varphi_2(t) \|_{W}^2
 + \|\beta_1(t) -  \beta_2(t) \|^2_{L^2(\Gamma_3)}\Big).
\end{aligned} \label{4.29}
\end{equation}
Also, since
\[
\mathbf{u}_{i}^{\ell}(t) = \int_0^{t}\dot{\mathbf{u}}^{\ell}_{i}(s)ds
+ \mathbf{u}_0^{\ell}(t), \quad t\in [0,T], \; \ell =1,2,
\]
we have
\[
    \| \mathbf{u}_1(t) - \mathbf{u}_2(t) \|_{\boldsymbol{V} }\leq
     \int_0^t \| \dot{\mathbf{u}}_1(s) -\dot{\mathbf{u}}_2(s))\|_{\boldsymbol{V}}\,ds
\]
and using this inequality in  \eqref{4.10} yields
\begin{equation}\label{4.30}
\| \mathbf{u}_1(t) - \mathbf{u}_2(t) \|_{\boldsymbol{V} }\leq C\Big(
   \int_0^t \| \eta_1(s) - \eta_2(s))\|_{\boldsymbol{V}} \,ds +
   \int_0^t \| \mathbf{u}_1(s) - \mathbf{u}_2(s))\|_{\boldsymbol{V}}\,ds \Big).
\end{equation}
Next, we apply Gronwall's inequality to deduce
\begin{equation}\label{4.31}
      \| \mathbf{u}_1(t) - \mathbf{u}_2(t))\|_{\boldsymbol{V}}
\leq   C \int_0^t   \|\eta_1(s) - \eta_2(s)\|_{\boldsymbol{V}}\,ds \quad
 \forall t \in [0, T].
 \end{equation}
 On the other hand,  from the Cauchy problem \eqref{4.20}--\eqref{4.21}
we can write
\[
\beta_{i}(t)=\beta_0-\int_0^{t}\Big(\beta_{i}(s)
\big(\gamma_{\nu}(R_{\nu}([u_{i\nu}(s)]))^2+\gamma_{\tau}
| \boldsymbol{R}_{\tau}([\mathbf{u}_{i\tau}(s)])|^2\big)
-\varepsilon_a\Big)_{+}ds 
\]
and then
\begin{align*}
&\big\|\beta_1(t)-\beta_2(t)\big\|_{L^2(\Gamma_3)}\\
&\leq C \int_0^{t}\big\|\beta_1(s)
 R_{\nu}([u_{1\nu}(s)])^2-\beta_2(s) R_{\nu}([u_{2\nu}(s)])^2
\big\|_{L^2(\Gamma_3)}ds   \\
&\quad +  C \int_0^{t}\big\|\beta_1(s)| \boldsymbol{R}_{\tau}
([\mathbf{u}_{1\tau}(s)])|^2- \beta_2(s)|\boldsymbol{R}_{\tau}
([\mathbf{u}_{2\tau}(s)])  |^2\big\|_{L^2(\Gamma_3)}ds.
\end{align*}
Using the definition of  $R_{\nu}$ and $ \boldsymbol{R}_{\tau}$  and writing
 $ \beta_1  =  \beta_1 - \beta_2 + \beta_2 $,  we obtain
\begin{equation} 
\begin{aligned}
 &\big\|\beta_1(t) - \beta_2(t)\big\|_{L^2(\Gamma_3)}   \\
& \leq  C \Big( \int_0^{t}\|\beta_1(s) - \beta_2(s)  \|_{L^2(\Gamma_3)}ds
 + \int_0^{t}\big\|\mathbf{u}_1(s) - \mathbf{u}_2(s)  \big\|_{L^2(\Gamma_3)^{d}}ds
\Big).
 \end{aligned} \label{4.32}
\end{equation}
 Next, we apply Gronwall's inequality to deduce
\[
  \|\beta_1(t) - \beta_2(t)\|_{L^2(\Gamma_3)}
 \leq   C   \int_0^{t}\|\mathbf{u}_1(s)
 - \mathbf{u}_2(s)  \|_{L^2(\Gamma_3)^{d}}ds.
\]
 and from the relation \eqref{3.3} we obtain
\begin{align}\label{4.33}
  \| \beta_1(t) -  \beta_2(t)\|^2_{L^2(\Gamma_3)}
 \leq   C   \int_0^{t}\|\mathbf{u}_1(s) - \mathbf{u}_2(s)  \|^2_{V}ds.
\end{align}
We use now \eqref{4.16}, \eqref{3.4},   \eqref{3.11} and \eqref{3.12} to find
\begin{equation}\label{4.34}
  \| \varphi_1(t) -  \varphi_2(t)\|^2_{W}
\leq   C    \|\mathbf{u}_1(t) - \mathbf{u}_2(t)  \|^2_{V}.
\end{equation}
 From  \eqref{4.22} we deduce that
\[
 (\dot{\zeta}_1-\dot{\zeta}_2,  \zeta_1- \zeta_2)_{E_0} +
a(\zeta_1- \zeta_2,\zeta_1- \zeta_2)
\leq  \big(\theta_1- \theta_2, \zeta_1- \zeta_2\big)_{E_0}, \quad
 \text{ a.e. } t\in (0,T).
\]
Integrating the previous inequality with respect to time,
 using the initial conditions  $ \zeta_1(0) = \zeta_2(0)= \zeta_0  $
and inequality $ a(\zeta_1- \zeta_2,\zeta_1- \zeta_2) \geq 0$, we find
\[
 \frac{1}{2}\| \zeta_1(t)- \zeta_2(t)\|^2_{E_0}  \leq \int_0^{t}
  \big(\theta_1(s)- \theta_2(s), \zeta_1(s)- \zeta_2(s)\big)_{E_0}\,ds,
\]
 which implies that
\[
  \| \zeta_1(t)- \zeta_2(t)\|^2_{E_0}  \leq \int_0^{t}
  \| \theta_1(s)- \theta_2(s)\|^2_{E_0} \,ds + \int_0^{t}
  \| \zeta_1(s)- \zeta_2(s)\|^2_{E_0} \,ds.
\]
 This inequality  combined with Gronwall's inequality  leads to
\begin{equation}\label{4.35}
  \| \zeta_1(t)- \zeta_2(t)\|^2_{E_0}  \leq C\int_0^{t}
  \| \theta_1(s)- \theta_2(s)\|^2_{E_0} \,ds  \quad \forall  t \in [0,T].
\end{equation}
We substitute \eqref{4.31}, \eqref{4.33}, \eqref{4.34} and \eqref{4.35}
in \eqref{4.29} to obtain
\[
    \|\Lambda(\eta_1, \theta_1)(t)- \Lambda(\eta_2, \theta_1)(t)
\|^2_{\boldsymbol{V}\times E_0 }
\leq C   \int_0^{t} \|(\eta_1, \theta_1)(s)-  (\eta_2, \theta_1)(s)
\|^2_{\boldsymbol{V} \times E_0 }\,ds.
\]
 Reiterating this inequality  $ m $ times we obtain
\[
\|\Lambda^{m}(\eta_1, \theta_1)- \Lambda^{m}(\eta_2, \theta_1)
\|^2_{C(0,T; \boldsymbol{V} \times E_0 )}
\leq   \dfrac{C^{m}T^{m}}{m! }    \| (\eta_1, \theta_1)
-  (\eta_2, \theta_1 )\|^2_{C(0,T; \boldsymbol{V} \times E_0)}.
\]
Thus, for $ m $ sufficiently large, the operator $ \Lambda^{m}(\cdot,\cdot) $
is a contraction on the Banach space $ C(0,T; \boldsymbol{V}\times E_0)$,
  and so $ \Lambda(\cdot,\cdot) $ has a unique fixed point.
\end{proof}

Now, we have all the ingredients to prove Theorem \ref{thm3.2}.

\begin{proof}[Proof of Existence] 
Let  $ ( \eta^{*}, \theta^{*})  \in C(0,T; \boldsymbol{V}\times E_0)   $ 
 be the fixed point of $ \Lambda(\cdot,\cdot) $  and denote
\begin{gather}
\mathbf{u}_{*}  = \mathbf{u}_{\eta^{*}}, \quad
\varphi_{*} = \varphi_{\eta^{*}}, \quad
\zeta_{*} = \zeta_{\theta^{*}}, \quad
\beta_{*} = \beta_{\eta^{*}}, \label{4.36} \\
{\boldsymbol{\sigma}}_{*}^{\ell} 
= \mathcal{A}^{\ell} \boldsymbol{\varepsilon}(\dot{\mathbf{u}}_{*}^{\ell}) 
+\mathcal{G}^{\ell} \boldsymbol{\varepsilon}(\mathbf{u}_{*}^{\ell}) 
+(\mathcal{E}^{\ell})^*\nabla\varphi_{*}^{\ell} 
+  \int_0^t\mathcal{F}^{\ell}\big(t-s, \varepsilon({\mathbf{u}_{*}^{\ell}}(s)),
 \zeta^{\ell}_{*}(s)\big)\,ds,   \label{4.37}  \\
\boldsymbol{D}_{*}^{\ell } =\mathcal{E}^{\ell}\varepsilon(\mathbf{u}_{*}^{\ell}
)  - \mathcal{B}^{\ell } \nabla\varphi_{*}^{\ell}.   \label{4.38}
\end{gather}
We prove that the  $ \{\mathbf{u}_{*}, {\boldsymbol{\sigma}}_{*}, \varphi_{*},
  \zeta_{*}, \beta_{*}, \boldsymbol{D}_{*} \} $ satisfies \eqref{3.26}--\eqref{3.32} 
and the regularities \eqref{3.36}--\eqref{3.41}.  
Indeed, we write \eqref{4.8}  for $ \eta = \eta^{*}$ and use  \eqref{4.36}  to find
\begin{equation}\label{4.39}
\begin{aligned}
&\sum_{\ell=1}^2(\mathcal{A}^{\ell} \boldsymbol{\varepsilon}
(\dot{\mathbf{u}}_{*}^{\ell}),  
\varepsilon(\mathbf{v}^{\ell})-\boldsymbol{\varepsilon}
 (\dot{\mathbf{u}}_{*}^{\ell}(t)))_{\mathcal{H}^{\ell}}
+\sum_{\ell=1}^2(\mathcal{G}^{\ell} \boldsymbol{\varepsilon}(\mathbf{u}_{*}^{\ell}),
\varepsilon(\mathbf{v}^{\ell})-\boldsymbol{\varepsilon}
 (\dot{\mathbf{u}}_{*}^{\ell}(t)))_{\mathcal{H}^{\ell}}\\
&+ j_{\nu c}(\mathbf{u}_{*}(t), \mathbf{v}-\dot{\mathbf{u}}_{*}(t))
 + j_{fr}(\mathbf{u}_{*}(t), \mathbf{v}) - j_{fr}(\mathbf{u}_{*}(t),
 \dot{\mathbf{u}}_{*}(t))
 +  ( \eta^{*}(t),  \mathbf{v}   - \dot{\mathbf{u}}_{*}(t)))_{\boldsymbol{V}}\\
&\geq (\mathbf{f}(t), \mathbf{v}-\dot{\mathbf{u}}_{*}(t))_{\boldsymbol{V} } 
\quad \forall \mathbf{v} \in \boldsymbol{V},  \text{ a.e. } t\in [0,T].
\end{aligned}
\end{equation}
We use equalities   $\Lambda^{1}(\eta^{*}, \theta^{*}) = \eta^{*} $ and 
$\Lambda^2(\eta^{*}, \theta^{*}) = \theta^{*} $  it follows from 
\eqref{4.24} and \eqref{4.25} that
\begin{gather} \label{4.40}
\begin{aligned}
 (\eta^{*}(t), \mathbf{v})_{\boldsymbol{V}}
& =  \sum_{\ell=1}^2\big((\mathcal{E}^{\ell})^*\nabla\varphi_{*}^{\ell}(t),  
 \varepsilon(\mathbf{v}^{\ell})\big)_{\mathcal{H}^{\ell}} 
 +  j_{ad}(\beta_{*}(t), \mathbf{u}_{*}(t), \mathbf{v} )\\
&\quad +  \sum_{\ell=1}^2\Big( \int_0^t \mathcal{F}^{\ell}\big(t-s, 
\varepsilon({\mathbf{u}_{*}^{\ell}}(s)), \zeta^{\ell}_{*}(s)\big)\,ds, 
 \varepsilon(\mathbf{v}^{\ell})\Big)_{\mathcal{H}^{\ell}},\\
 &\quad  \forall \mathbf{v} \in \boldsymbol{V},  \text{ a.e. } t\in (0,T),
\end{aligned}  \\
\label{4.41}
  \theta^{\ell}_{*}(t)  
 = \phi^{\ell}\big(\boldsymbol{\sigma}^{\ell}_{*}(t)
 - \mathcal{A}^{\ell} \boldsymbol{\varepsilon}(\dot{\mathbf{u}}^{\ell}_{*}(t))
 , \boldsymbol{\varepsilon}(\mathbf{u}_{*}^{\ell}(t) ), 
\zeta_{*}^{\ell}(t) \big), \quad\text{a.e. } t\in (0,T),   \; \ell=1,2.
\end{gather}
We now substitute \eqref{4.40} in \eqref{4.39}  to obtain
 \begin{equation}
\begin{aligned}
&\sum_{\ell=1}^2(\mathcal{A}^{\ell} \boldsymbol{\varepsilon}
 (\dot{\mathbf{u}}_{*}^{\ell})(t),   \varepsilon(\mathbf{v}^{\ell})
 -\boldsymbol{\varepsilon}(\dot{\mathbf{u}}_{*}^{\ell}(t)))_{\mathcal{H}^{\ell}}
+ \sum_{\ell=1}^2(\mathcal{G}^{\ell} \boldsymbol{\varepsilon}
 (\mathbf{u}_{*}^{\ell})(t),  \ \varepsilon(\mathbf{v}^{\ell})
 -\boldsymbol{\varepsilon}(\dot{\mathbf{u}}_{*}^{\ell}(t)))_{\mathcal{H}^{\ell}}
\\   
&+  \sum_{\ell=1}^2\Big( \int_0^t \mathcal{F}^{\ell}\big(t-s,  
\varepsilon({\mathbf{u}_{*}^{\ell}}(s)), \zeta^{\ell}_{*}(s)\big)\,ds, 
 \varepsilon(\mathbf{v}^{\ell})-\boldsymbol{\varepsilon}
(\dot{\mathbf{u}}_{*}^{\ell}(t))\Big)_{\mathcal{H}^{\ell}}  \\  
&+ j_{ad}(\beta_{*}(t), \mathbf{u}_{*}(t), \mathbf{v} 
- \dot{\mathbf{u}}_{*}(t)) + j_{\nu c}(\mathbf{u}_{*}(t), \mathbf{v}
 -\dot{\mathbf{u}}_{*}(t))  +    j_{fr}(\mathbf{u}_{*}(t), \mathbf{v})  \\
&- j_{fr}(\mathbf{u}_{*}(t),   \dot{\mathbf{u}}_{*}(t))
 + \sum_{\ell=1}^2\big((\mathcal{E}^{\ell})^*\nabla\varphi_{*}^{\ell}(t),  
 \varepsilon(\mathbf{v}^{\ell})
 -\boldsymbol{\varepsilon}(\dot{\mathbf{u}}_{*}^{\ell}(t))
 \big)_{\mathcal{H}^{\ell}}  \\
& \geq (\mathbf{f}(t), \mathbf{v}-\dot{\mathbf{u}}_{*}(t))_{\boldsymbol{V} } 
\quad \forall \mathbf{v} \in \boldsymbol{V}  \text{ a.e. } t\in [0,T],  
\end{aligned}  \label{4.42}
\end{equation}
and we substitute \eqref{4.41} in \eqref{4.22} to have $\zeta_{*}(t)\in K$ and
\begin{equation} \label{4.43}  
\begin{split}
&\sum_{\ell=1}^2(\dot{\zeta}^{\ell}_{*}(t),  
\xi^{\ell}-\zeta^{\ell}_{*}(t))_{L^2(\Omega^{\ell})} +
a(\zeta_{*}(t), \xi-\zeta_{*}(t))\\
&\geq  \sum_{\ell=1}^2\Big(\phi^{\ell}\big(\boldsymbol{\sigma}_{*}^{\ell}(t) 
-\mathcal{A}^{\ell}\boldsymbol{\varepsilon}(\dot{\mathbf{u}}_{*}^{\ell}(t)) ,
\boldsymbol{\varepsilon}({\mathbf{u}}_{*}^{\ell}(t) ), 
\zeta_{*}^{\ell}(t) \big), \xi^{\ell}
-\zeta_{*}^{\ell}(t) \Big)_{L^2(\Omega^{\ell})}, 
 \end{split}
\end{equation}
for all  $\xi \in K$, a.e. $t\in (0,T)$. 
We write now \eqref{4.16} for $\eta=\eta^{*}  $ and use  
 \eqref{4.36} to see that
\begin{equation}
 \sum_{\ell=1}^2(\mathcal{B}^{\ell} \nabla\varphi_{*}^{\ell}(t), 
\nabla\phi^{\ell} )_{H^{\ell}} -  \sum_{\ell=1}^2(  \mathcal{E}^{\ell} 
\varepsilon(\mathbf{u}_{*}^{\ell}(t)), \nabla\phi^{\ell} )_{H^{\ell}} 
= (q(t), \phi)_{W}   \label{4.44}
\end{equation}
for all $\phi\in W$, $t\in [0,T]$.
Additionally,  we use $ \mathbf{u}_{\eta^{*}} $ in \eqref{4.20} and \eqref{4.36}  
 to find
\begin{equation} \label{4.45}  
\dot{\beta}_{*}(t)= - \Big(\beta_{*}(t)\big(\gamma_{\nu}(R_{\nu}([u_{*\nu}(t)]))^2
+ \gamma_{\tau}|  \boldsymbol{R}_{\tau}([\mathbf{u}_{*\tau}(t)]) |^2 \big) 
- \varepsilon_a\Big)_{+},
\end{equation}
 a.e.  $t \in [0,T]$.
Relations \eqref{4.36}, \eqref{4.37}, \eqref{4.38},    \eqref{4.42},    \eqref{4.43},
     \eqref{4.44}  and \eqref{4.45}  allow us to conclude now that 
 $ \{ \mathbf{u}_{*},       {\boldsymbol{\sigma}}_{*}, \varphi_{*},  
\zeta_{*}, \beta_{*}, \boldsymbol{D}_{*} \} $  
satisfies \eqref{3.26}--\eqref{3.31}. Next, \eqref{3.32} and the 
regularity \eqref{3.36}, \eqref{3.38}--\eqref{3.40} follow from Lemmas 
\ref{lem4.2}, \ref{lem4.3},  \ref{lem4.4}    and \ref{lem4.6}.  
Since $  \mathbf{u}_{*}$, $ \varphi_{*} $ and $  \zeta_{*} $ 
satisfies \eqref{3.36}, \eqref{3.38} and  \eqref{3.39}, respectively,  
it follows from  \eqref{4.37} that
\begin{equation}\label{4.46}
         \boldsymbol{\sigma}_{*} \in C(0,T;\mathcal{H}).
\end{equation}
For $\ell=1,2,  $ we choose $  \mathbf{v} = \dot{\mathbf{u}} \pm \phi $  
in \eqref{4.42}, with $ \phi=(\phi^{1}, \phi^2)$,   
$  \phi^{\ell}\in D(\Omega^{\ell})^{d}  $ and $ \phi^{3-\ell}=0$, to obtain
       \begin{equation} \label{4.47}
 \operatorname{Div}\boldsymbol{\sigma}_{*}^{\ell} (t) 
= -  \boldsymbol{f}_0^{\ell}(t) \quad \forall t \in [0, T],  \quad \ell=1,2,   
\end{equation} 
where $   D(\Omega^{\ell})   $ is the space of infinitely differentiable
real functions with a compact support in $\Omega^{\ell}$. 
The regularity \eqref{3.37}
follows from   \eqref{3.14}, \eqref{4.46} and \eqref{4.47}.   
Let now $ t_1, t_2 \in [0, T]$,  by \eqref{3.11},  \eqref{3.12}, \eqref{3.4} 
and \eqref{4.38}, we deduce that
\[
      \|\boldsymbol{D}_{*}(t_1)-  \boldsymbol{D}_{*}(t_2) \|_{H}  
\leq C \left(\|\varphi_{*}(t_1) -  \varphi_{*}(t_2) \|_{W} 
+ \|\mathbf{u}_{*}(t_1) -  \mathbf{u}_{*}(t_2) \|_{\boldsymbol{V}}  \right).
\] 
The regularity of $ \mathbf{u}_{*} $ and $ \varphi_{*} $ given by \eqref{3.36} 
 and \eqref{3.38} implies
\begin{equation}\label{4.48}
    \boldsymbol{D}_{*} \in C(0,T;  H).
\end{equation} 
For $ \ell=1,2, $ we choose $\phi = (\phi^{1}, \phi^2)$ with 
$ \phi^{\ell}  \in D(\Omega^{\ell})^{d} $ and $ \phi^{3-\ell} =0 $ 
in \eqref{4.44}  and using \eqref{3.20}   we find
\begin{equation}\label{4.49}
     \operatorname{div}\boldsymbol{D}_{*}^{\ell}(t) = q_0^{\ell}(t) \quad 
 \forall t \in [0,T],   \quad \ell=1, 2.
\end{equation} 
Property \eqref{3.41} follows from \eqref{3.14}, \eqref{4.48} and 
\eqref{4.49}.  
\end{proof}
Finally we conclude that the weak solution 
$ \{ \mathbf{u}_{*}, {\boldsymbol{\sigma}}_{*}, \varphi_{*},  \zeta_{*},   
\beta_{*}, \boldsymbol{D}_{*} \} $ of the piezoelectric contact
Problem  PV  has the regularity \eqref{3.36}--\eqref{3.41},  
 which concludes the existence part of Theorem \ref{thm3.2}.

\begin{proof}[Proof of Uniqueness] 
The uniqueness of the solution is a consequence of the uniqueness
of the fixed point of the operator $ \Lambda(\cdot,\cdot) $  
defined by \eqref{4.24}-\eqref{4.25} and the unique solvability of the 
Problems  PV$^{u}_{\eta}, $ PV$^{\varphi}_{\eta}, $ PV$^{\beta}_{\eta}, $ and
PV$^{\zeta}_{\eta}$. 
\end{proof}

\begin{thebibliography}{00}

\bibitem{BS}
R.\,C.\,Batra and J.\,S.\,Yang;
Analysis and numerical approach of a piezoelectric contact problem,
 \emph{Ann. Aca. Rom.  Sci } \textbf{1}(2009), 7--30.

\bibitem{CFR} M.\,Campo, \, J.\,R.\,Fernandez and  A.\,Rodriguez-Aros;
A quasistatic contact problem with normal compliance and damage
involving viscoelastic materials with long memory,
 \emph{Applied Numerical Mathematics } \textbf{58}(2008), 1274--1290.

\bibitem{CFSS}
O.\,Chau, J.\,R.\,Fernandez, M.\,Shillor and M.\,Sofonea;  
Variational and numerical analysis
of a quasistatic viscoelastic contact problem with adhesion, 
 \emph{Journal of Computational and Applied Mathematics } 
\textbf{159}(2003),431--465.

\bibitem{CMS}
C.\,Ciulcu, D.\,Motreanu and M.\,Sofonea;
Analysis of an elastic contact problem with slip dependent coefficient 
of friction,  \emph{Mathematical Inequalities \& Applications} 
\textbf{4}  (2001), 465--479.

\bibitem{FN1} M. Fr\'{e}mond, B. Nedjar;
Damage in concrete: The unilateral phenomenon,  \emph{Nucl. Eng. Des. }
\textbf{156} (1995)323--335.

\bibitem{FN2} M. Fr\'{e}mond, B. Nedjar;
Damage, gradient of damage and principle of virtual work, \emph{Int. J.
Solids Stuct }   \textbf{33}(8) (1996) 1083--1103.

\bibitem{FKS} M. Fr\'{e}mond, K. L. Kuttler, and M. Shillor; 
Existence and uniqueness of solutions for a one-dimensinal damage model,
\emph{J. Math. Anal. Appl.}
\textbf{229} (1999)  271--294.

\bibitem{HS1} W. Han, M. Sofonea;
 Evolutionary Variational inequalities
arising in viscoelastic contact problems, \emph{SIAM Journal of Numerical
Analysis} \textbf{38}, (2000) 556--579.

\bibitem{HS2} W. Han, M. Sofonea;
\emph{Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity,}
Studies in Advanced Mathematics 30, American Mathematical Society,
Providence, RI - Intl. Press, Sommerville, MA, 2002.

\bibitem{IP} I.\,R.\,Ionescu and J.\,C.\,Paumier;
On the contact problem with slip displacement dependent friction in elastostatics,
 \emph{Int. J. Engng. Sci.} \textbf{34}  (1996), 471--491.

\bibitem{KO}  N.\,Kikuchi and  J.\,T.\,Oden;
\emph{Contact Problems in Elasticity, A Study of Variational Inequalities
 and Finite Element Methods}, SIAM, Philadelphia,   1988.

\bibitem{KS} K. L. Kuttler and M. Shillor;
\emph{Existence for models of damage,} preprint  2001.

\bibitem{LSS} Z.\,Lerguet, M.\,Shillor and  M.\,Sofonea;
 A Frictional Contact Problem for an electro-viscoelastic Boby,
  \emph{Electronic Journal of Differential Equations, } \textbf{170} (2007) 1--16.

\bibitem{MO} J.\,A.\,C.\,Martins, J.\,T.\,Oden;
 Existence and uniqueness results for dynamic contact problems with 
nonlinear normal and friction interface laws,
\emph{Nonlinear Anal. TMA } {\bf11} (1987) 407--428.

\bibitem{M} R.\,D.\,Mindlin;
 Elasticity, piezoelasticity and crystal lattice dynamics, 
\emph{Journal of Elasticity} \textbf{4} (1972), 217--280.

\bibitem{MS} D.\,Motreanu and M.\,Sofonea;
Quasivariational inequalities and applications in frictional
contact problems with normal compliance, 
\emph{Adv. Math. Sci. Appl. } \textbf{10} (2000), 103--118.

\bibitem{NH} J.\,Ne\v{c}as,  and I.\,Hlav\'{a}\v{c}ek;
Mathematical Theory of Elastic and Elastico-Plastic Bodies: 
An Introduction, \emph{Elsevier Scientific Publishing Company},
 Amsterdam, Oxford, New York, 1981.

\bibitem{OM} J.\,T.\,Oden and J.\, A.\, C.\, Martins;
Models and computational methods for dynamic friction phenomena,
\emph{Computer Methods in Applied Mechanics and Engineering } 
\textbf{52 } (1985), 527--634.

\bibitem{RSS1} M.\,Rochdi,  M.\,Shillor  and  M.\,Sofonea;
 A quasistatic viscoelastic contact problem with normal compliance and friction,
\emph{J. Elasticity  } \textbf{51} (1998) 105--126.

\bibitem{RSS2} M.\,Rochdi,  M.\,Shillor  and  M.\,Sofonea;
Analysis of a quasistatic viscoelastic problem with friction
and damage,{ \em Adv. Math. Sci. Appl.} \textbf{10 } (2002) 173--189.

\bibitem{SS} M.\,Selmani,\, L. Selmani;
 A Dynamic Frictionless Contact Problem with Adhesion and Damage,
 \emph{Bull. Pol. Acad. Sci., Math} {\bf55} (2007), 17--34.

\bibitem{SST} M.\,Shillor, M.\,Sofonea and J.\,J.\,Telega;
\emph{Models and Variational Analysis of Quasistatic Contact},
Lecture Notes Phys.  655, Springer, Berlin, 2004.

\bibitem{SHS} M.\,Sofonea, W.\,Han and M.\,Shillor;
\emph{Analysis and Approximation of Contact Problems with Adhesion or Damage}, 
Pure and Applied Mathematics. 276, Chapman-Hall/CRC Press, New York, 2006.

\bibitem{T} A.\, Touzaline;
\emph{Analysis of a viscoelastic frictionless contact problem with adhesion}, 
 Rev. Roum. Math. Pur. Appl. \textbf{55} (2010), 411--430.

\bibitem{ZZBD} L.\,Zhor, Z.\,Zellagui, H.\,Benseridi and S.\,Drabla;
\emph{Variational analysis of an electro viscoelastic contact problem with friction}, 
J.  Assoc. Arab.  Univ. Bas.  and Appl.  (2013), 1--8.

\end{thebibliography}

\end{document}