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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 161, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/161\hfil Antiplane frictional contact]
{Antiplane frictional contact of electro-viscoelastic cylinders}

\author[M. Dalah, M. Sofonea\hfil EJDE-2007/161\hfilneg]
{Mohamed Dalah, Mircea Sofonea} 


\address{Mohamed Dalah \newline
D\'{e}partement de Math\'ematiques, Facult\'e des Sciences\\
Universit\'e de Mentouri - Constantine, 25 000 Constantine
Alg\'erie}
\email{mdalah17@yahoo.fr}


\address{Mircea Sofonea \newline
Laboratoire de Math\'ematiques et Physique pour les Syst\`emes\\
Universit\'e de Perpignan, 52 Avenue  Paul Alduy, 66 860
Perpignan, France} 
\email{sofonea@univ-perp.fr}


\thanks{Submitted September 2, 2007. Published November 21, 2007.}
\subjclass[2000]{74M10, 74F15, 74G25, 49J40}
\keywords{Antiplane problem; electro-viscoelastic material; contact process;
\hfill\break\indent
 Tresca's friction law; evolutionary variational inequality; weak solution}

\begin{abstract}
 We study a mathematical model that describes the antiplane
 shear deformation of a cylinder in frictional contact with
 a rigid foundation. The material is assumed to be electro-viscoelastic,
 the process is quasistatic, friction is modelled with Tresca's law
 and the foundation is assumed to be electrically conductive.
 We derive a variational formulation of the model which is in
 a form of a system coupling a first order evolutionary variational
 inequality for the displacement field with a time-dependent
 variational equation for the electric potential field.
 Then, we prove the existence of a unique weak solution to the model.
 The proof is based on arguments of evolutionary variational
 inequalities and fixed points of operators.
 Also, we investigate the behavior of the solution as the viscosity
 converges  to zero and prove that it converges to the solution of
 the corresponding electro-elastic antiplane contact problem.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}\label{s:1}

Antiplane shear deformations are one of the simplest classes of
deformations that solids can undergo: in antiplane shear of a
cylindrical body, the displacement is parallel to the generators
of the cylinder and is independent of the axial coordinate. For
this reason, the antiplane problems play a useful role as pilot
problems, allowing for various aspects of solutions in Solid
Mechanics to be examined in a particularly simple setting.
Considerable attention has been paid to the modelling of such kind
of problems, see for instance \cite{Ho1,Ho2,Ho3} and the
references therein. In particular, the review article \cite{Ho1}
deals with modern developments for the antiplane shear model
involving linear and nonlinear solid materials, various
constitutive settings and applications. Antiplane frictional
contact problems were used in geophysics in order to describe
pre-earthquake evolution of the regions of hight tectonic
activity, see for instance \cite{CDI,CI} and the references
therein. The mathematical analysis of models for antiplane
frictional contact problems can be found in \cite{AMS,HM, IDC, IW,
MMS}.

Currently there is a considerable interest in frictional or
frictionless contact problems involving piezoelectric materials,
i.e. materials characterized by the coupling between the
mechanical and electrical properties. This coupling, in a
piezoelectric material, leads to the appearance of electric
potential when mechanical stress is present, and conversely,
mechanical stress is generated when electric potential is applied.
The first effect is used in mechanical sensors, and the reverse
effect is used in actuators, in engineering control equipments.
Piezoelectric materials for which the mechanical properties are
elastic are also called electro-elastic materials and
piezoelectric materials for which the mechanical properties are
viscoelastic are also called electro-viscoelastic materials.
General models for piezoelectric materials can be found in
\cite{BY, I, PK88}. Static frictional contact problems for
electro-elastic materials were studied in \cite{BLM, MB, M, SE1},
under the assumption that the foundation is insulated. Contact
problems with normal compliance for electro-viscoelastic materials
were investigated in \cite{LSS,SE2}. There, variational
formulations of the problems were considered and their unique
solvability was proved. Antiplane problems for piezoelectric
materials were considered in \cite{Ho4,SDA,ZWD}. We rarely
actually load piezoelectric bodies so as to cause them to deform
in antiplane shear; however, the governing equations and boundary
conditions for antiplane shear problems involving piezoelectric
materials are beautifully simple and the solution has many of the
features of the more general case and may help us to solve the
more complex problem too.


The present paper represents a continuation of \cite{SDA}; there a
model for the antiplane contact of an  electro-elastic cylinder
was considered under the assumption that the foundation is
electrically conductive; the variational formulation of the model
was derived and the existence of a unique solution to the model
was proved by using arguments of evolutionary variational
inequalities.  Unlike \cite{SDA}, in the present paper  we deal
with an antiplane contact problem for an electro-viscoelastic
cylinder, which leads to a new mathematical model, different to
that presented in \cite{SDA}. Our interest is to describe a simple
physical process in which both frictional contact, viscosity and
piezoelectric effects are involved, and to show that the resulting
model leads to a well-posed mathematical problem.  Taking into
account the frictional contact between a viscous piezoelectric
body and an electrically conductive foundation in the study of an
antiplane problem leads to a new and interesting mathematical
model which has the virtue of relative mathematical simplicity
without loss of essential physical relevance.


Our paper is structured as follows. In Section \ref{s:2} we
present the model of the antiplane frictional contact of an
electro-viscoelastic cylinder. In Section \ref{s:3} we introduce
the notation, list the assumption on problem's data, derive the
variational formulation of the problem and state our main
existence and uniqueness result, Theorem \ref{t:1}. The proof of
the theorem is provided in Section \ref{s:4}; it is based on
arguments of evolutionary variational inequalities and fixed
point. Finally, in Section \ref{s:5} we investigate the behavior
of the solution as the viscosity converges  to zero and prove that
it converges to the solution of the corresponding electro-elastic
antiplane contact problem studied in \cite{SDA}.


\section{The model} \label{s:2}

We consider a piezoelectric body $\mathcal{B}$ identified with a
region in $\mathbb{R}^3$ it occupies in a fixed and undistorted reference
configuration. We assume that $\mathcal{B}$ is a cylinder with
generators parallel to the $x_3$-axes with a cross-section which
is a regular region $\Omega$ in the $x_1$, $x_2$-plane,
$Ox_1x_2x_3$ being a Cartesian coordinate system. The cylinder is
assumed to be sufficiently long so that the end effects in the
axial direction are negligible. Thus, $\mathcal{B}=\Omega\times
(-\infty,+\infty)$. The cylinder is acted upon by body forces of
density $\mathbf{f}_0$ and has volume  free electric charges of density
$q_0$. It is also constrained mechanically and electrically  on
the boundary. To describe the boundary conditions, we denote by
$\partial \Omega=\Gamma$ the boundary of $\Omega$ and we assume a
partition of $\Gamma$ into three open disjoint parts $\Gamma_1$,
$\Gamma_2$ and $\Gamma_3$, on the one hand, and a partition of
$\Gamma_1\cup\Gamma_2$ into two open parts $\Gamma_a$ and
$\Gamma_b$, on the other hand. We assume that the one-dimensional
measure of $\Gamma_1$ and $\Gamma_a$, denoted $\mathop{\rm meas}\Gamma_1$ and
$\mathop{\rm meas}\Gamma_a$, are positive. The cylinder is clamped on
$\Gamma_1\times (-\infty,+\infty)$ and therefore the displacement
field vanishes there. Surface tractions of density $\mathbf{f}_2$ act on
$\Gamma_2\times (-\infty,+\infty)$. We also assume that the
electrical potential vanishes on $\Gamma_a\times
(-\infty,+\infty)$ and a surface electrical charge of density
$q_b$ is prescribed on $\Gamma_b\times (-\infty,+\infty) $. The
cylinder is in contact over $\Gamma_3\times (-\infty,+\infty)$
with a conductive obstacle, the so called foundation. The contact
is frictional and is modeled with Tresca's law. We are interested
in the deformation of the cylinder on the time interval $[0,T]$.

Below in this paper the indices $i$ and $j$ denote components of
vectors and tensors and run from 1 to 3, summation over two
repeated indices is implied, and the index that follows a comma
represents the partial derivative with respect to the
corresponding spatial variable; also, a dot above represents the
time derivative. We use $\mathcal{S}^3$ for the linear space of
second order symmetric tensors on $\mathbb{R}^3$ or, equivalently,  the
space of symmetric matrices of order $3$, and $``\cdot"$,
$\|\cdot\|$ will represent the inner products and the Euclidean
norms on $\mathbb{R}^3$ and $\mathcal{S}^3$; we have:
\begin{gather*}
\mathbf{u} \cdot \mathbf{v} = u_i v_i, \quad
\|\mathbf{v} \| =(\mathbf{v}\cdot\mathbf{v})^{1/2} \quad
  \text{for all }\mathbf{u}=(u_i),\; \mathbf{v} =(v_i)\in\mathbb{R}^3,\\
\boldsymbol{\sigma}\cdot\boldsymbol{\tau} = \sigma_{ij} \tau_{ij},
\quad\ \| \boldsymbol{\tau} \| = (\boldsymbol{\tau}\cdot\boldsymbol{\tau}
)^{1/2} \quad \text{for  all } \boldsymbol{\sigma}=(\sigma_{ij}),\;
\boldsymbol{\tau}=(\tau_{ij}) \in \mathcal{S}^3.
\end{gather*}
We assume that
\begin{gather}
\label{2.1}\mathbf{f}_0=(0,0,f_0) \quad \mbox{with }\
f_0=f_0(x_1,x_2,t):\Omega\times[0,T]\to\mathbb{R},\\
\label{2.2} \mathbf{f}_2=(0,0,f_2) \quad\text{with }\
f_2=f_2(x_1,x_2,t):\Gamma_2\times[0,T]\to\mathbb{R},\\
\label{2.1n}q_0=q_0(x_1,x_2,t):\Omega\times[0,T]\to\mathbb{R},\\
\label{2.2n}q_2=q_2(x_1,x_2,t):\Gamma_b\times[0,T]\to\mathbb{R}.
\end{gather}
 The forces (\ref{2.1}), (\ref{2.2}) and the electric
charges (\ref{2.1n}), (\ref{2.2n}) would be expected to give rise
to deformations and to electric charges of the piezoelectric
cylinder corresponding to a displacement $\mathbf{u}$ and to an electric
potential field $\varphi$ which are independent on $x_3$ and have
the form
\begin{gather}
\label{2.3}\mathbf{u}=(0,0,u) \quad\text{with }
u=u(x_1,x_2,t):\Omega\times[0,T]\to\mathbb{R},\\
\label{2.3n}\varphi= \varphi(x_1,x_2,t):\Omega\times[0,T]\to\mathbb{R}.
\end{gather}
Such kind of deformation, associated to a displacement
field of the form (\ref{2.3}),  is called an {\it antiplane
shear}, see for instance \cite{Ho1,Ho3} for details.


We denote by
$\boldsymbol{\varepsilon}(\mathbf{u})=(\varepsilon_{ij}(\mathbf{u}))$
the strain tensor and by $\boldsymbol{\sigma}=(\mathbf{\sigma}_{ij})$
the stress tensor; we also denote by
$\mathbf{E}(\varphi)=(E_i(\varphi))$ the electric field and by
$\mathbf{D}=(D_i)$ the electric displacement field. Here and
below, in order to simplify the notation, we do not indicate the
dependence of various functions on $x_1$, $x_2$, $x_3$ or $t$ and
we recall that
\[
\varepsilon_{ij}(\mathbf{u})=\frac{1}{2}\,(u_{i,j}+u_{j,i}),\quad
E_i(\varphi)=-\varphi,_i.
\]

The material's behavior  is modelled by an electro-viscoelastic
constitutive law of the form
\begin{gather}
\label{2.4} \boldsymbol{\sigma}=
2\theta\boldsymbol{\varepsilon}(\dot{\mathbf{u}})+
\zeta\,\mbox{tr}\,\boldsymbol{\varepsilon}(\dot{\mathbf{u}})\,\mathbf{I}
+2\mu\boldsymbol{\varepsilon}
(\mathbf{u})+\lambda\,\mbox{tr}\,\boldsymbol{\varepsilon}(\mathbf{u})\,\mathbf{I}-\mathcal{
E}^*\mathbf{E}(\varphi),\\
\label{2.4n}
\mathbf{D}=\mathcal{E}\boldsymbol{\varepsilon}(\mathbf{u})+\beta\mathbf{E}(\varphi),
\end{gather}
where $\zeta$ and $\theta$ are viscosity coefficients, $\lambda$
and $\mu$ are the Lam\'e coefficients, $\mathop{\rm
tr}\,\boldsymbol{\varepsilon}(\mathbf{u})=\varepsilon_{ii}(\mathbf{u})$,
$\mathbf{I}$ is the unit tensor in $\mathbb{R}^3$, $\beta$ is the
electric permittivity constant, $\mathcal{E}$ represents the
third-order piezoelectric tensor and $\mathcal{E}^*$ is its
transpose.
 We assume that
\begin{equation}\label{e}
\mathcal{E}\boldsymbol{\varepsilon}=
\begin{pmatrix}
e(\varepsilon_{13}+\varepsilon_{31})\\
e(\varepsilon_{23}+\varepsilon_{32})\\
e\varepsilon_{33}
\end{pmatrix} \quad\forall\boldsymbol{\varepsilon}=(\varepsilon_{ij})\in\mathcal{S}^3,
\end{equation}
where $e$ is a piezoelectric coefficient. We also assume that the
coefficients $\theta$, $\mu$, $\beta$ and $e$  depend on the
spatial variables $x_1$, $x_2$, but are independent on the spatial
variable $x_3$. Since
$\mathcal{E}\boldsymbol{\varepsilon}\cdot\mathbf{v}=
\boldsymbol{\varepsilon}\cdot\mathcal{E}^*\mathbf{v}$ for all
$\boldsymbol{\varepsilon}\in \mathcal{S}^3$,
$\mathbf{v}\in\mathbb{R}^3$, it follows from (\ref{e}) that
\begin{equation}\label{en}
\mathcal{E}^*\mathbf{v}=\begin{pmatrix}
0&0&ev_1\\
0&0&ev_2\\
ev_1&ev_2&ev_3
\end{pmatrix} \quad\forall\mathbf{v}=(v_i)\in\mathbb{R}^3.
\end{equation}

In the antiplane context (\ref{2.3}), (\ref{2.3n}), using the
constitutive equations (\ref{2.4}), (\ref{2.4n}) and equalities
(\ref{e}), (\ref{en}) it follows that the stress field and the
electric displacement field are given by
\begin{gather} \label{2.5}
\boldsymbol{\sigma}= \begin{pmatrix}
0&0&\theta\dot{u},_1+\mu u,_1+e\varphi,_1\\
0&0&\theta\dot{u},_2+\mu u,_2+e\varphi,_2\\
\theta\dot{u},_1+\mu u,_1+e\varphi,_1&\theta\dot{u},_2\mu
u,_2+e\varphi,_2&0
\end{pmatrix}, \\
 \label{2.5n}
\mathbf{D}= \begin{pmatrix}
eu,_1-\beta\varphi,_1\\
eu,_2-\beta\varphi,_2\\
0
\end{pmatrix}.
\end{gather}
We assume that the process is mechanically quasistatic and
electrically static and therefore is governed by the equilibrium
equations
\[
\mathop{\rm Div}
\boldsymbol{\sigma}+\mathbf{f}_{0}=\boldsymbol{0},\quad
D_{i,i}-q_0=0\quad\text{in }\mathcal{B}\times (0,T),
\]
where $\mathop{\rm Div}\boldsymbol{\sigma}=(\sigma_{ij,j})$
represents the divergence of the tensor field $\boldsymbol{\sigma}$.
Taking into account (\ref{2.5}), (\ref{2.5n}), (\ref{2.3}),
(\ref{2.3n}), (\ref{2.1}) and (\ref{2.1n}), the equilibrium
equations above reduce to the following scalar equations
\begin{gather}
\label{1}\mathop{\rm div}(\theta\nabla\dot{u}+\mu\nabla
u+e\nabla\varphi)+f_0=0\quad
\text{in }\Omega\times (0,T),\\
\label{2}\mathop{\rm div}(e\nabla
u-\beta\nabla\varphi)=q_0\quad\mbox{in }\Omega\times (0,T).
\end{gather}
Here and below we use the notation
\begin{gather*}
\mathop{\rm div} \boldsymbol{\tau}=
\tau_{1,1}+\tau_{1,2}\quad\text{for } \boldsymbol{\tau}=
(\tau_1(x_1,x_2,t),\tau_2(x_1,x_2,t)),\\
\nabla v= (v_{,1},v_{,2}),\quad
\partial_{\nu}v=v,_1\,\nu_1+v,_2\,\nu_2\quad\text{for }
v=v(x_1,x_2,t).
\end{gather*}

We now describe the boundary conditions. During the process the
cylinder is clamped on $\Gamma_1\times (-\infty,+\infty)$ and the
electric potential vanish on $\Gamma_1\times (-\infty,+\infty)$;
thus, (\ref{2.3}) and (\ref{2.3n}) imply that
\begin{gather}
\label{3}u=0\quad \text{on } \Gamma_1\times (0,T),\\
\label{4}\varphi=0\quad \text{on } \Gamma_a\times (0,T).
\end{gather}

 Let $\boldsymbol{\nu}$ denote the unit normal on
$\Gamma\times(-\infty,+\infty)$. We have
\begin{equation} \label{2.6}
\boldsymbol{\nu}=(\nu_1,\nu_2,0)\quad\text{with }\
\nu_i=\nu_i(x_1,x_2):\Gamma\to\mathbb{R},\quad i=1,2.
\end{equation}
For a vector $\mathbf{v}$ we denote by ${v}_\nu$ and ${\mathbf{v}}_\tau$ its
normal and  tangential components on the boundary, given by
\begin{equation} \label{2.7}
{v}_\nu =\mathbf{v}\cdot\boldsymbol{\nu},\quad {\mathbf{v}}_\tau=
\mathbf{v}-{v}_{\nu}\boldsymbol{\nu}.
\end{equation}
For a given stress field $\mathbf{\sigma}$ we denote by $\sigma_\nu$ and
$\mathbf{\sigma}_\tau$ the normal and the tangential components on the
boundary, that is
\begin{equation} \label{2.8}
\sigma_\nu=(\boldsymbol{\sigma}\boldsymbol{\nu})\cdot\boldsymbol{\nu},\quad
\boldsymbol{\sigma}_{\tau}=\boldsymbol{\sigma}\boldsymbol{\nu}-\sigma_{\nu}\boldsymbol{\nu}.
\end{equation}
 From (\ref{2.5}), (\ref{2.5n}) and (\ref{2.6}) we deduce that the
Cauchy stress vector and the normal component of the electric
displacement field are given by
\begin{equation} \label{2.9}
\boldsymbol{\sigma}\boldsymbol{\nu}=(0,0,\theta\partial_\nu\dot{u}+\mu\partial_\nu
u+e\partial_\nu\varphi),\quad
\mathbf{D}\cdot\boldsymbol{\nu}=e\partial_\nu
u-\beta\partial_\nu\varphi.
\end{equation}

Taking into account (\ref{2.2}), (\ref{2.2n}) and (\ref{2.9}), the
traction condition on $\Gamma_2\times(-\infty,\infty)$ and the
electric conditions on $\Gamma_b\times(-\infty,\infty)$ are given
by
\begin{gather}
\label{5}
\theta\partial_\nu\dot{u}+\mu\partial_\nu u+e\partial_\nu\varphi
=f_2 \quad\text{on }\Gamma_2\times(0,T),\\
\label{6}e\partial_\nu u-\beta\partial_\nu\varphi=q_b
\quad\text{on }\Gamma_b\times(0,T).
\end{gather}


We now describe the frictional contact condition and the electric
conditions on $\Gamma_3\times(-\infty,+\infty)$. First, from
(\ref{2.3}) and (\ref{2.6}) we infer that the normal displacement
vanishes, $u_\nu=0$, which shows that the contact is bilateral,
that is, the contact is kept during all the process. Using now
(\ref{2.3}), (\ref{2.5}), (\ref{2.6})--(\ref{2.8}) we conclude
that
\begin{equation}\label{2.10}
\mathbf{u}_\tau=(0,0,u),\quad
\boldsymbol{\sigma}_\tau=(0,0,\theta\partial_\nu\dot{u}+\mu\partial_\nu
u+e\partial_\nu\varphi).
\end{equation}
We assume that the friction is invariant with respect to the $x_3$
axis and is modeled with Tresca's friction law, that is
\begin{equation}\label{2.11}
\|\boldsymbol{\sigma}_\tau\|\leq g,\quad
\boldsymbol{\sigma}_\tau=-\,g\,\frac{\dot{\mathbf{u}}_\tau}{\|\dot{\mathbf{u}}_\tau\|}
\quad\mbox{if}\quad\dot{\mathbf{u}}_\tau\neq\boldsymbol{0}\quad\mbox{on
} \Gamma_3\times (0,T).
\end{equation}
Here $g:\Gamma_3\to\mathbb{R}_+$ is a given function, the
friction bound, and $\dot{\mathbf{u}}_\tau$ represents the tangential
velocity on the contact boundary, see \cite{HS,SST} for details.
Using now (\ref{2.10}) it is straightforward to see that  the
friction law (\ref{2.11}) implies
\begin{equation} \label{7}
\begin{gathered}
|\theta\partial_\nu\dot{u}+\mu\partial_\nu
u+e\partial_\nu\varphi|\leq g,\\
\theta\partial_\nu\dot{u}+\mu\,\partial_\nu
u+e\partial_\nu\varphi=-g\,\frac{\dot{u}}{|\dot{u}|}
\quad\text{if }\dot{u}\neq 0
\end{gathered}
\end{equation}
on $\Gamma_3\times(0,T)$.


Next, since the foundation is electrically conductive and the
contact is bilateral, we assume that the normal component of the
electric displacement field or the free charge is proportional to
the difference between the potential on the foundation and the
body's surface. Thus,
\[
\mathbf{D}\cdot\boldsymbol{\nu}=k\,(\varphi-\varphi_F)\quad\mbox{on}
\quad\Gamma_3\times (0,T),
\]
where $\varphi_F$ represents the
electric potential of the foundation and $k$ is the electric
conductivity coefficient. We use (\ref{2.9}) and the previous
equality to obtain
\begin{equation}\label{8}
e\partial_\nu
u-\beta\partial_\nu\varphi=k\,(\varphi-\varphi_F)\quad\mbox{on }
\Gamma_3\times (0,T).
\end{equation}
Finally, we prescribe the initial displacement,
\begin{equation}\label{9}
u(0)=u_0\quad\mbox{in }\Omega,
\end{equation}
where $u_0$ is a given function on $\Omega$.

We collect the above equations and conditions to obtain the
following mathematical model which describes  the antiplane shear
of an electro-viscoelastic cylinder in frictional contact with a
conductive foundation.


\subsection*{Problem $\mathcal{P}$}
Find the displacement
field $u:\Omega\times [0,T]\to\mathbb{R}$ and the electric
potential $\varphi:\Omega\times [0,T]\to\mathbb{R}$ such that
$(\ref{1})$--$(\ref{4})$, $(\ref{5})$, $(\ref{6})$,
$(\ref{7})$--$(\ref{9})$ hold. \smallskip


Note that once the displacement field $u$ and the electric
potential $\varphi$ which solve Problem $\mathcal{P}$ are known,
then the stress tensor $\boldsymbol{\sigma}$ and the electric
displacement field $\mathbf{D}$ can be obtained by using the
constitutive laws (\ref{2.5}) and (\ref{2.5n}), respectively.

\section{Variational formulation}\label{s:3}

We derive now the variational formulation of the Problem
$\mathcal{P}$. To this end we introduce the function spaces
\[
V=\{v\in H^1(\Omega):v =0  \mbox{ on } \Gamma_1 \},\quad
W=\{\,\psi\in H^1(\Omega): \psi=0  \mbox{ on } \Gamma_a\}
\]
where, here and below, we write $w$ for the trace $\gamma w$
of a function $w\in H^1(\Omega)$ on $\Gamma$. Since $meas\
\Gamma_1>0$ and $\mathop{\rm meas}\Gamma_a>0$, it is well known that $V$ and
$W$ are real Hilbert spaces with the inner products
\[
(u,v)_V=\int_{\Omega}\nabla u\cdot\nabla v\,dx\quad\forall u,v\in
V,\quad(\varphi,\psi)_W=\int_{\Omega}\nabla\varphi\cdot\nabla
\psi\,dx\quad\forall \varphi,\,\psi\in W.
\]
Moreover, the associated norms
\begin{equation}\label{norms}
\|v\|_V= \|\nabla v\|_{L^2(\Omega)^2}\quad\forall v\in V,\quad
\|\psi\|_V= \|\nabla\psi\|_{L^2(\Omega)^2}\quad\forall\psi\in W
\end{equation}
are equivalent on $V$ and $W$, respectively, with the usual norm
$\|\cdot\|_{H^1(\Omega)}$. By Sobolev's trace theorem we deduce
that there exist two positive constants $c_V>0$ and $c_W>0$  such
that
\begin{equation}\label{2.18}
 \|v\|_{L^{2}(\Gamma_3)}\leq c_V \|v\|_V \quad\forall
v\in V,\quad  \|\psi\|_{L^{2}(\Gamma_3)}\leq c_W\|\psi\|_W
\quad\forall \psi\in W.
\end{equation}

For a real Banach space $(X,\|\cdot\|_X)$ we use the usual
notation for the spaces $L^p(0,T;X)$ and $W^{k,p}(0,T;X$) where
$1\leq p \leq  \infty, \ k=1,2,\dots$; we also denote by $C([0,T];
X)$  and $C^1([0,T]; X)$ the spaces of continuous and continuously
differentiable functions on $[0,T]$ with values in $X$,  with the
respective norms
\begin{gather*}
\|x\|_{C([0,T]; X)}=\max_{t\in [0,T]} \|x(t)\|_X,
\\
\|x\|_{C^1([0,T]; X)}=\max_{t\in [0,T]} \|x(t)\|_X+\max_{t\in
[0,T]} \|\dot{x}(t)\|_X.
\end{gather*}

 In the study of the Problem $\mathcal{P}$ we assume that the
viscosity coefficient and the electric permittivity coefficient
satisfy
\begin{gather}
\label{10}\theta\in L^\infty(\Omega)\mbox{ and there exists
}\theta^*>0 \mbox{ such that }\ \theta(\mathbf{x})\geq \theta^* \mbox{
a.e. }\mathbf{x}\in \Omega,
\\
\label{11}\beta\in L^\infty(\Omega)\mbox{ and there exists
}\beta^*>0 \mbox{ such that }\ \beta(\mathbf{x})\geq \beta^* \mbox{ a.e.
}\mathbf{x}\in \Omega.
\end{gather}
We also assume that the  Lam\'e coefficient and the piezoelectric
coefficient satisfy
\begin{gather}
\label{10n}\mu\in L^\infty(\Omega)\quad\mbox{and} \quad
\mu(\mathbf{x})>0\ \mbox{ a.e. }\mathbf{x}\in \Omega,
\\
\label{12} e\in L^\infty(\Omega).
\end{gather}

The forces, tractions, volume and surface free charge densities
have the regularity
\begin{gather}
\label{13}f_0\in W^{1,2}(0,T;L^2(\Omega)), \quad f_2\in
W^{1,2}(0,T; L^2(\Gamma_2)),\\
\label{14}q_0\in W^{1,2}(0,T;L^2(\Omega)), \quad q_2\in
W^{1,2}(0,T; L^2(\Gamma_b)).
\end{gather}
The friction bound and the electric conductivity coefficient
satisfy
\begin{gather}
\label{15}g\in L^2(\Gamma_3)\ \quad\mbox{and} \quad g(\mathbf{x})\geq
0\ \mbox{ a.e. }\mathbf{x}\in \Gamma_3,\\
\label{16}k\in L^\infty(\Gamma_3)\quad\mbox{and}\quad k(\mathbf{x})\geq
0\ \mbox{ a.e. }\mathbf{x}\in \Gamma_3.
\end{gather}
Finally, we assume that  the electric potential of the foundation
and the initial displacement are such that
\begin{gather}
\label{17}\varphi_F\in W^{1,2}(0,T;L^2(\Gamma_3)),\\
\label{18}u_0\in V.
\end{gather}

Next, we define the bilinear forms $a_\theta: V\times V\to\mathbb{R}$,
$a_\mu: V\times V\to\mathbb{R}$, $a_e: V\times W\to\mathbb{R}$,
$a_e^*: W\times V\to\mathbb{R}$, and $a_\beta: W\times W\to\mathbb{R}$,
 by equalities
\begin{gather}
\label{19n}a_\theta(u,v)=\int_\Omega\theta\,\nabla u\cdot\nabla v\,dx,\\
\label{19}a_\mu(u,v)=\int_\Omega\mu\,\nabla u\cdot\nabla v\,dx,\\
\label{20}a_e(u,\varphi)=\int_\Omega e\,\nabla
u\cdot\nabla\varphi\,dx=a_e^*(\varphi,u),\\
\label{21}a_\beta(\varphi,\psi)=\int_\Omega\beta\,\nabla
\varphi\cdot\nabla\psi\,dx+\int_{\Gamma_3}k\,\varphi\psi\,dx,
\end{gather}
for all $u,\, v\in V$,  $\varphi, \psi\in W$.
Assumptions (\ref{10})--(\ref{12}), (\ref{16}) imply that the
integrals above are well defined and, using (\ref{norms}) and
(\ref{2.18}), it follows that  the forms $a_\theta$, $a_\mu$,
$a_e$, $a_e^*$ and $a_\beta$ are continuous; moreover, the forms
$a_\theta$, $a_\mu$ and $a_\beta$ are symmetric and, in addition,
the form $a_\theta$ is $V$-elliptic and the form $a_\beta$ is
$W$-elliptic, since
\begin{gather}
\label{22}a_\theta(v,v)\ge \theta^*\|v\|^2_V\quad\forall v\in V,\\
\label{23}a_\beta(\psi,\psi)\ge \beta^*\|\psi\|^2_W\quad\forall
\psi\in W.
\end{gather}


We also define the mappings $f : [0,T]\to V$, $q : [0,T]\to W$ and
$j:V\to\mathbb{R}$, respectively, by
\begin{gather}
\label{24}(f(t),v)_V=\int_\Omega f_0(t)v\,dx+\int_{\Gamma_2}
f_2(t)v\,da,\\
\label{25}(q(t),\psi)_W=\int_\Omega
q_0(t)\psi\,dx-\int_{\Gamma_2}
q_b(t)\psi\,da+\int_{\Gamma_3}k\,\varphi_F(t)\psi\,da,\\
\label{26}j(v)=\int_{\Gamma_3}g|v|\,da,
\end{gather}
for all $v\in V$, $\psi\in W$ and $t\in[0,T]$. The
definition of $f$ and $q$ are based on Riesz's representation
theorem; moreover, it follows from assumptions by
(\ref{13})--(\ref{16}), that the integrals above are well-defined
and
\begin{gather}
\label{27}f\in W^{1,2}(0,T;V),\\
\label{28}q\in
W^{1,2}(0,T;W).
\end{gather}

Performing integration by parts and using notation
(\ref{19n})--(\ref{21}), (\ref{24})--(\ref{26}) it is
straightforward to derive the following variational formulation of
Problem $\mathcal{P}$.

\subsection*{Problem $\mathcal{P}_V$}
Find a displacement field $u:[0,T]\to V$ and an electric potential field
$\varphi:[0,T]\to W$ such that
\begin{gather}
\label{29}
\begin{aligned}
&a_\theta(\dot{u}(t),v-\dot{u}(t))+a_\mu(u(t),v-\dot{u}(t))+
a_e^*(\varphi(t),v- \dot{u}(t))\\
&+j(v) - j(\dot{u}(t))\geq(f(t),v-\dot{u}(t))_V
\quad\forall v\in V,\ t\in[0,T],
\end{aligned}\\
\label{30}a_\beta(\varphi(t),\psi)-a_e(u(t),\psi) =(q(t),\psi)_W
\quad\forall \psi\in W,\ t\in[0,T],\\
\label{31}u(0)=u_0.
\end{gather}

Our main existence and uniqueness result, which we state now and
prove in the next section, is the following.

\begin{theorem}\label{t:1}
Assume that \eqref{10}--\eqref{18}  hold. Then there exists a
unique solution of problem $\mathcal{P}_V$. Moreover, the solution
satisfies
\begin{equation}\label{reg}
u\in W^{2,2}(0,T;V),\quad \varphi\in W^{1,2}(0,T;W).
\end{equation}
\end{theorem}


A couple of functions $(u,\varphi)$ which solves Problem $\mathcal{
P}_V$ is called a {\it weak solution} of the electro-mechanical
problem $\mathcal{P}$. We conclude by Theorem \ref{t:1} that the
antiplane contact problem $\mathcal{P}$ has a unique weak solution,
provided that \eqref{10}--\eqref{18}  hold.



\section{Proof of Theorem \ref{t:1}} \label{s:4}


The proof is based on  an abstract result for
evolutionary variational inequalities that we present in what
follows. 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 $u : [0,T]\to X$ such that
\begin{gather}\label{D1}
\begin{aligned}
& a(\dot u(t),v-\dot u(t))_X+b(u(t),v-\dot u(t))_X+ j(v)
-j(\dot u(t))\\
& \geq(h(t),v-\dot u(t))_X\quad\forall v\in X,\; t\in[0,T],
\end{aligned} \\
\label{D2} u(0)=u_0.
\end{gather}

In the study of the Cauchy problem (\ref{D1})--(\ref{D2}) we
assume that:
\begin{gather}\label{D3}
\parbox{10cm}{
$a:X\times X \to \mathbb{R}$ is a bilinear symmetric form and \\
(a) there exists $M>0$ such that
$|a(u,v)|\leq M\|u\|_X\|v\|_X$ for all $u,\,v\in X$.
\\
(b) there exists $m >0$ such that
 $a(v,v)\geq m\|v\|^2_X$ for all $v\in X$.
}
\\
\label{D4}
\parbox{10cm}{
$b:X\times X \to \mathbb{R}$ is a bilinear  form and
there exists $M'>0$  such that
$|b(u,v)|\leq M'\|u\|_X\|v\|_X$ for all $u,\,v\in X$.
}
\\
\label{D5}
j:X\to\mathbb{R} \mbox{ is a convex lower semicontinuous functional.} \\
\label{D6}
h \in C([0,T];X). \\
\label{D7}
u_0\in X.
\end{gather}

The following existence, uniqueness and regularity result
represent a particular case of a more general result  proved in
\cite[p.\,230--234]{HS}.

\begin{theorem}\label{th2}
Let \eqref{D3}--\eqref{D7} hold. Then
\\
$(1)$ There exists a unique solution $u\in C^1([0,T];X)$ of problem
\eqref{D1} and \eqref{D2}.
\\
$(2)$ If $u_1$ and $u_2$ are two solutions of \eqref{D1} and
\eqref{D2} corresponding to the data $h_1,\,h_2\in C([0,T];X)$,
then there exists $c>0$ such that
\begin{equation}
\|\dot u_1(t)-\dot u_2(t)\|_X\leq c\,(\|h_1(t)-h_2(t)\|_X +\|
u_1(t)-u_2(t)\|_X)\quad\forall\, t\in[0,T].
 \label{4.8}
\end{equation}
$(3)$ If, moreover, $h\in W^{1,p}(0,T;X)$, for some
$p\in[1,\infty]$, then the solution satisfies $u\in
W^{2,p}(0,T;X)$.
\end{theorem}


We turn now to the proof of Theorem \ref{t:1} which will be
carried out in several steps. We assume in what follows that
\eqref{10}--\eqref{18}  hold and, everywhere below, we denote
by $c$ various positive constants which are independent of time
and whose value may change from line to line. Let
$\eta\in C([0,T],V)$ be given and, in the first step, consider the
following intermediate variational problem.

\subsection*{Problem} $\mathcal{P}^1_{\eta}$.
 Find a displacement field $u_{\eta} : [0,T]\to V$ such that
\begin{gather}\label{4.9}
\begin{aligned}
& a_\theta(\dot u_\eta(t),
v-\dot{u}_\eta(t))+a_\mu(u_\eta(t), v-\dot{u}_\eta(t))
+(\eta(t),v-\dot {u}_\eta(t))_V\\
&+j(v)-j(\dot{u}_\eta(t))\geq (f(t),
v-\dot{u}_\eta(t))_V\quad \forall \, v\in V,\ t\in
[0,T],
\end{aligned} \\
\label{4.10} {u}_\eta(0)={u}_0.
\end{gather}
We have  the following result for $\mathcal{P}^1_{\eta}$.

\begin{lemma}\label{l1}
$(1)$ There exists a unique solution $u_\eta \in C^1([0,T];V)$ to
the problem \eqref{4.9}--\eqref{4.10}.
\\
$(2)$ If $u_1$ and $u_2$ are two solutions of
\eqref{4.9}--\eqref{4.10} corresponding to the data $\eta_1$,
$\eta_2\in C([0,T];V)$, then there exists $c>0$ such that
\begin{equation}\label{4.11}
\|\dot u_1(t)-\dot u_2(t)\|_V\leq c\,(\|\eta_1(t)-\eta_2(t)\|_V+\|
u_1(t)-u_2(t)\|_V)\quad\forall\, t\in[0,T].
\end{equation}
\\
$(3)$ If, moreover, $\eta\in W^{1,2}(0,T;V)$,  then the solution
satisfies $u_\eta\in W^{2,2}(0,T;V)$.
\end{lemma}

\begin{proof}
 We apply Theorem \ref{th2}  on the space
$X=V$ with the inner product $(\cdot,\cdot)_V$\ and the associated
norm $\|\cdot\|_V$, with the choice $a=a_\theta$, $b=a_\mu$,
$h=f-\eta$. Clearly $a_\theta$ and $a_\mu$ satisfy conditions
(\ref{D3}) and (\ref{D4}), respectively, and using (\ref{15}) it
follows from   that the functional $j$ satisfies condition
(\ref{D5}). Moreover, using (\ref{27}) and the regularity $\eta\in
C([0,T],V)$ it is easy to see that $f-\eta\in C([0,T];V)$ i.e. $h$
satisfies (\ref{D6}). Finally, we note that (\ref{D7}) is
satisfied too and, if $\eta\in W^{1,2}(0,T;V)$ then $h=f-\eta\in
W^{1,2}(0,T;V)$. Lemma \ref{l1} is a direct consequence of Theorem
\ref{th2}.
\end{proof}

In the next step we use the solution $u_\eta \in C^1([0,T], V)$,
obtained in Lemma 4.2, to construct the following  variational
problem for the electrical potential.

\subsection*{Problem $\mathcal{P}^2_{\eta}$}
Find an electrical potential $\varphi_\eta : [0,T]\to W$ such that
\begin{equation}
\label{4.16} a_\beta(\varphi_\eta(t),\psi)-a_e(u_\eta(t),
\psi)=(q(t),\psi)_W\quad \forall\psi \in W,\, t\in [0,T].
\end{equation}

The well-posedness of problem $\mathcal{P}^2_\eta$ follows.


\begin{lemma}\label{l2}
There exists a unique solution $\varphi_\eta\in W^{1,2}(0,T;W)$
which satisfies $(\ref{4.16})$. Moreover, if $\varphi_{\eta_1}$
and $\varphi_{\eta_2}$ are the solutions of $(\ref{4.16})$
corresponding to $\eta_1$, $\eta_2\in C([0,T]; V)$ then, there
exists $c>0$, such that
\begin{equation}\label{4.17}
\|\varphi_{\eta_1}(t)-\varphi_{\eta_2}(t)\|_W\leq
c\,\|u_{\eta_1}(t)-u_{\eta_2}(t)\|_V\quad\forall\, t\in[0,T].
\end{equation}
\end{lemma}

\begin{proof}
 Let $t\in[0,T]$. We use the properties of
the bilinear form $a_\beta$ and the Lax-Milgram lemma to see that
there exists a unique element $\varphi_\eta(t)\in W$ which solves
(\ref{4.16}) at any moment $t\in[0,T]$. Consider now $t_1,\,
t_2\in[0,T]$; using (\ref{4.16}) and (\ref{23}) we find that
\[
\beta^*\,\|\varphi(t_1)-\varphi(t_2)\|_W^2 \leq
\|e\|_{L^\infty(\Omega)}\|u(t_1)-u(t_2)\|_V+ \|q(t_1)-q(t_2)\|_W
\|\varphi(t_1)-\varphi(t_2)\|_W
\]
which implies
\begin{equation}
\label{46}
\|\varphi(t_1)-\varphi(t_2)\|_W\leq
c\,(\|u(t_1)-u(t_2)\|_V+ \|q(t_1)-q(t_2)\|_W).
\end{equation}
We note that regularity $u_\eta \in C^1( [0,T]; V)$ combined with
(\ref{28}) and (\ref{46}) imply that $\varphi_\eta\in W^{1,2}(0,T;
W)$. Also, arguments similar to those used in the proof of
(\ref{46}) lead to (\ref{4.17}), which concludes the
proof.
\end{proof}

We now  use Riesz's representation theorem to define the element
$\Lambda \eta(t)\in V$ by equality
\begin{equation}
\label{4.20} (\Lambda \eta(t),v)_V=a_e^*(\varphi_\eta(t),v) \quad
\forall v\in V,\ t\in [0,T].
\end{equation}
Clearly, for a given
$\eta\in C([0,T];V)$ the function $t\mapsto\Lambda \eta(t)$
belongs to $C([0,T];V)$. In the newt step we show that the
operator $\Lambda : C([0,T];V) \to C([0,T];V)$ a unique
fixed point.

\begin{lemma}\label{l3}
There exists a unique  $\widetilde\eta \in W^{1,2}(0,T;V)$ such
that $\Lambda\widetilde\eta =\widetilde\eta$.
\end{lemma}

\begin{proof}
 Let $\eta_1$, $\eta_2\in C([0,T];V)$ and
denote by $u_i$ and $\varphi_i$ the functions $u_{\eta_i}$ and
$\varphi_{\eta_i}$ obtained in Lemmas \ref{l1} and \ref{l2}, for
$i=1,2$. Let $t\in [0,T]$. Using (\ref{4.20})  and (\ref{20}) we
obtain
\[
\|\Lambda\eta_1(t)-\Lambda\eta_2(t)\|_V\leq c\,
\|\varphi_1(t)-\varphi_2(t)\|_W,
\]
and, keeping in mind (\ref{4.17}), we find
\begin{equation}\label{4.21}
\|\Lambda\eta_1(t)-\Lambda\eta_2(t)\|_V\leq
c\,\|u_1(t)-u_2(t)\|_V.
\end{equation}
On the other hand, since  $\displaystyle u_i(t)=u_0+\int_0^t \dot
u_i(s)\,ds$, we have
\begin{equation}
\label{4.22} \|u_1(t)-u_2(t)\|_V\leq \int_{0}^t \|\dot u_1(s)-\dot
u_2(s)\|_V\,ds,
\end{equation}
and using this inequality in (\ref{4.11}) yields
\[
\|\dot u_1(t)-\dot u_2(t)\|_V\leq
c\Big(\|\eta_1(t)-\eta_2(t)\|_{V}+\int_{0}^t \|\dot u_1(s)-\dot
u_2(s)\|_V\,ds\Big).
\]
It follows now from a Gronwall-type argument that
\begin{equation}\label{4.23}
\int_{0}^t \|\dot u_1(s)-\dot u_2(s)\|_V\,ds\leq
c\int_0^t\|\eta_1(t)-\eta_2(t)\|_{V}\,ds.
\end{equation}
Combining (\ref{4.21})--(\ref{4.23}) leads to
\[
\|\Lambda\eta_1(t)-\Lambda\eta_2(t)\|_{V}\leq
c\int_0^t\|\eta_1(t)-\eta_2(t)\|_{V}\,ds
\]
and, reiterating this inequality $n$ times results in
\[
\|\Lambda^n\eta_1(t)-\Lambda^n\eta_2(t)\|_{V}\leq\frac{c^n}{n!}
\|\eta_1(t)-\eta_2(t)\|_{C([0,T];V)}.
\]
This last inequality shows that for a sufficiently large $n$ the
operator $\Lambda^n$ is a contraction on the Banach space
$C([0,T];V)$ and, therefore, there exists a unique element
$\widetilde\eta \in C([0,T];V)$ such that
$\Lambda\widetilde\eta=\widetilde\eta$. It follows from Lemma
\ref{l2} that $\varphi_\eta\in W^{1,2}(0,T;W)$ and, therefore, the
definition (\ref{4.20}) of the operator $\Lambda$ combined with
the properties of the bilinear form $a_e^*$ implies that
$\Lambda\widetilde\eta\in W^{1,2}(0,T;V)$; this regularity
combined with equality $\Lambda\widetilde\eta=\widetilde\eta$
shows that $\widetilde\eta\in W^{1,2}(0,T;V)$ which concludes the
proof.
\end{proof}

We have now all the ingredients to provide the proof of the
Theorem \ref{t:1}.


\subsection*{Existence}
Let $\widetilde\eta\in W^{1,2}(0,T;V) $
be the fixed point of the operator $\Lambda$, and let
$u_{\widetilde\eta}$, $\varphi_{\widetilde\eta}$ be the solutions
of problems $\mathcal{P}^1_\eta$ and $\mathcal{P}^2_\eta$, respectively, for
$\eta=\widetilde\eta$. It follows from (\ref{4.20}) that
\[(\widetilde\eta(t),v)_V=a^*_e(\varphi_{\widetilde\eta}(t),v)
\quad\forall v\in V,\ t\in[0,T]\] and, therefore, (\ref{4.9}),
(\ref{4.10}) and (\ref{4.16}) imply that $(u_{\widetilde\eta},
\varphi_{\widetilde\eta})$ is a solution of problem $\mathcal{P}_{V}$.
Regularity (\ref{reg}) of the solution follows from Lemmas
\ref{l1} (3) and \ref{l2}.

\subsection*{Uniqueness}
The uniqueness of the solution follows
from the uniqueness of the fixed point of the operator $\Lambda$.
It can also be obtained by using arguments similar as those used
in \cite{HS,SST}.


\section{A convergence result} \label{s:5}

In this section we investigate the behavior of the weak solution
of the antiplane frictional problem as the viscosity converges to
zero.  In order to outline the dependence on the viscosity
coefficient $\theta$, we reformulate Problem $\mathcal{P}_V$ as
follows.

\subsection*{Problem $\mathcal{P}_V^\theta$}
Find a displacement field
$u_\theta:[0,T]\to V$ and an electric potential field
$\varphi_\theta:[0,T]\to W$ such that
\begin{gather}
\label{29m}
\begin{aligned}
&a_\theta(\dot{u}_\theta(t),v-\dot{u}_\theta(t))+
a_\mu(u_\theta(t),v-\dot{u}_\theta(t))+a_e^*(\varphi_\theta(t),v-
\dot{u}_\theta(t))\\
&+j(v) -j(\dot{u}_\theta(t))\geq(f(t),v-\dot{u}_\theta(t))_V
\quad\forall v\in V,\  t\in[0,T],
\end{aligned}\\
\label{30m}a_\beta(\varphi_\theta(t),\psi)-a_e(u_\theta(t),\psi)
=(q(t),\psi)_W
\quad\forall \psi\in W,\ t\in[0,T],
\\
\label{31m} u_\theta(0)=u_0.
\end{gather}

 We also consider the inviscid problem associated to
(\ref{29m})--(\ref{31m}); i.e., the problem obtained for
$\theta=0$, which is formulated as follows.

\subsection*{Problem $\mathcal{P}_V^0$}
Find a displacement field $u:[0,T]\to V$ and an electric potential
field $\varphi:[0,T]\to W$ such that
\begin{gather}\label{29n}
\begin{aligned}
& a_\mu(u(t),v-\dot{u}(t))+a_e^*(\varphi(t),v-
\dot{u}(t))+j(v) - j(\dot{u}(t))\\
&\geq(f(t),v-\dot{u}(t))_V
\quad\forall v\in V, \mbox{ a.e. } t\in(0,T),
\end{aligned}\\
\label{30n}a_\beta(\varphi(t),\psi)-a_e(u(t),\psi)
=(q(t),\psi)_W \quad\forall \psi\in W,\ t\in[0,T],\\
\label{31n}u(0)=u_0.
\end{gather}

Clearly, Problem $\mathcal{P}_V^0$ represents the
variational formulation of the model  in Section \ref{s:2}, in the
case when the piezoelectric cylinder is assumed to be
electro-elastic.


Assume in what follows that \eqref{10}--\eqref{18} hold. Then,
it follows from Theorem \ref{t:1} that Problem $\mathcal{P}_V^\theta$
has a unique solution $(u_\theta,\varphi_\theta)$ which satisfies
$u_\theta\in W^{2,2}(0,T;V)$, $\varphi_\theta\in W^{1,2}(0,T;W)$.
In order to state an existence and uniqueness result in the study
of Problem $\mathcal{P}_V^0$ we need additional assumptions. First,
we reinforce (\ref{10n}) with assumption
\begin{equation}\label{mux}
\mu\in L^\infty(\Omega)\mbox{ and there exists }\mu^*>0 \mbox{
such that }\ \mu(\mathbf{x})\geq \mu^* \mbox{ a.e. }\mathbf{x}\in \Omega
\end{equation}
and note that in this case the bilinear form $a_\mu$ is
$V$-elliptic, since it safisfies
\begin{equation}
\label{22n}a_\mu(v,v)\ge \mu^*\|v\|^2_V\quad\forall v\in V.
\end{equation}
Next, we employ the $W$-ellipticity of the form $a_\beta$,
(\ref{23}), and the Lax-Milgram lemma to see that there exists a
unique element $\varphi_0\in W$ such that
\begin{equation}\label{p0}
a_\beta(\varphi_0,\psi)-a_e(u_0,\psi) =(q(0),\psi)_W \quad\forall
\psi\in W.
\end{equation}
We use the element $\varphi_0$ defined above to introduce
the condition
\begin{equation}\label{32}
a_\mu(u_0,v)_V+a_e^*(\varphi_0,v)+j(v)\geq (f(0),v)_V
\quad\forall v\in V.
\end{equation}
This inequality  represents a compatibility condition on the
initial data that is necessary in many quasistatic problems, see
for instance \cite{SST}. Physically, it is needed so as to
guarantee that initially the state is in equilibrium, since
otherwise the inertial terms cannot be neglected and the problems
become dynamic. It follows from Theorem 4.1 in \cite{SDA} that,
under assumptions (\ref{10})--(\ref{18}), (\ref{mux}) and
(\ref{32}), the electro-elastic Problem $\mathcal{P}_V^0$ has a
unique solution $(u,\varphi)$ with regularity $u\in
W^{1,2}(0,T;V)$, $\varphi\in W^{1,2}(0,T;W)$.


Consider now the assumption
\begin{equation}\label{co}
\frac{1}{\theta^*}\,\|\theta\|^2_{L^\infty(\Omega)}\to 0.
\end{equation}
We have the following convergence result.


\begin{theorem}\label{t:2}
Assume that \eqref{10}--\eqref{18}, \eqref{mux},
\eqref{32} and \eqref{co} hold. Then the solution
$(u_\theta,\varphi_\theta)$ of Problem $\mathcal{P}_V^\theta$
converges to the solution $u$ of Problem $\mathcal{P}_V^0$, i.e.
\begin{equation}\label{con}
\|u_\theta-u\|_{C([0,T];V)}\to
0,\quad\|\varphi_\theta-\varphi\|_{C([0,T];W)}\to 0.
\end{equation}
\end{theorem}

\begin{proof} The equalities and inequalities below hold
for almost any $t\in(0,T)$. We take $v=\dot{u}(t)$ in (\ref{29m}),
$v=\dot{u}_\theta(t)$ in (\ref{29n}) and add the resulting
inequalities to obtain
\begin{align*}
&a_\theta(\dot{u}_\theta(t),\dot{u}(t)-\dot{u}_\theta(t))+
a_\mu(u_\theta(t)-u(t),\dot{u}(t)-\dot{u}_\theta(t))\\
&+a_e^*(\varphi_\theta(t)-\varphi(t),\dot{u}(t)-\dot{u}_\theta(t))\ge0.
\end{align*}
This implies that
\begin{equation}\label{P}
\begin{aligned}
&a_\theta(\dot{u}_\theta(t)-\dot{u}(t),\dot{u}_\theta(t)-\dot{u}(t))+
a_\mu(u_\theta(t)-u(t),\dot{u}_\theta(t)-\dot{u}(t))
\\
&\leq a_\theta(\dot{u}(t),\dot{u}(t)-\dot{u}_\theta(t))+
a_e^*(\varphi_\theta(t)-\varphi(t),\dot{u}(t)-\dot{u}_\theta(t)).
\end{aligned}
\end{equation}
We use now assumption (\ref{10})  to see that
\begin{align*}
&\theta^*\|\dot{u}_\theta(t)-\dot{u}(t)\|^2_V+
a_\mu(u_\theta(t)-u(t),\dot{u}_\theta(t)-\dot{u}(t))\label{0}\\
&\leq \|\theta\|_{L^\infty(\Omega)}\,
\|\dot{u}(t)\|_V\|\dot{u}_\theta(t)-\dot{u}(t)\|_V+
a_e^*(\varphi_\theta(t)-\varphi(t),\dot{u}(t)-\dot{u}_\theta(t))
\end{align*}
and combine this inequality with the elementary inequality
\[
\|\theta\|_{L^\infty(\Omega)}\,\|\dot{u}(t)\|\,\|\dot{u}_\theta(t)-\dot{u}(t)\|_V\leq
\frac{\|\theta\|_{L^\infty(\Omega)}^2}{4\theta^*}\,\|\dot{u}(t)\|_V^2+
\theta^*\,\|\dot{u}_\theta(t)-\dot{u}(t)\|_V^2.
\]
As a result we obtain
\begin{equation}
\label{PP}
a_\mu(u_\theta(t)-u(t),\dot{u}_\theta(t)-\dot{u}(t))
 \le
\frac{\|\theta\|_{L^\infty(\Omega)}^2}{4\theta^*}\,
\|\dot{u}(t)\|_V^2+a_e^*(\varphi_\theta(t)-\varphi(t),
\dot{u}(t)-\dot{u}_\theta(t)).
\end{equation}

On the other hand, we recall that  $a_\beta$ and $a_e$ are
bilinear continuous forms and the functions $u_\theta$, $u$,
$\varphi_\theta$, $\varphi$ and $q$  have the regularity
$W^{1,2}$. Therefore, the two sides of equalities (\ref{30m}) and
(\ref{30n}) are derivable with respect to the time variable. We
derive (\ref{30m}) and (\ref{30n}), subtract the resulting
equalities and use the definition of the form $a_e^*$ to obtain
\[
a_\beta(\dot{\varphi}_\theta(t)-\dot{\varphi}(t),\psi)=
a_e(\dot{u}_\theta(t)-\dot{u}(t),\psi)=
a_e^*(\psi,\dot{u}_\theta(t)-\dot{u}(t))\quad \forall \psi\in W.
\]
We take now $\psi=\varphi(t)-\varphi_\theta(t)$ in the previous
equality to find that
\begin{equation}\label{Z}
a_e^*(\varphi_\theta(t)-\varphi(t),\dot{u}(t)-\dot{u}_\theta(t))=
a_\beta(\dot{\varphi}_\theta(t)-\dot{\varphi}(t),\varphi(t)-\varphi_\theta(t)).
\end{equation}
Next, we write (\ref{30m}) and (\ref{30n}) at $t=0$, use the
initial condition $u_\theta(0)=u(0)=u_0$ and the unique
solvability of the variational equation (\ref{p0}) to see that
\begin{equation}
\label{ZZ} \varphi_\theta(0)=\varphi(0)=\varphi_0.
\end{equation}

We now combine (\ref{PP}) and (\ref{Z}) to obtain
\[
a_\mu(u_\theta(t)-u(t),\dot{u}_\theta(t)-\dot{u}(t))
\le \frac{\|\theta\|_{L^\infty(\Omega)}^2}{4\theta^*}\,
\|\dot{u}(t)\|_V^2+a_\beta(\dot{\varphi}_\theta(t)-
\dot{\varphi}(t),\varphi(t)-\varphi_\theta(t)).
\]
Let $s\in[0,T]$. We integrate the previous inequality on $[0,s]$
with the initial conditions (\ref{31m}), (\ref{31n}) and
(\ref{ZZ}) and use (\ref{23}), (\ref{22n}) to obtain
\begin{equation}\label{xy}
\frac{\mu^*}{2}\,\|u_\theta(s)-u(s)\|_V^2\le
\frac{\|\theta\|_{L^\infty(\Omega)}^2}{2\theta^*}\int_0^s\|\dot{u}(t)\|_V^2\,dt.
\end{equation}
Next, we write (\ref{30m}) and (\ref{30n}) with $t=s$,
$\psi=\varphi_\theta(s)-\varphi(s)$ and subtract the resulting
equalities to obtain
\[
a_\beta(\varphi_\theta(s)-\varphi(s),\varphi_\theta(s)-\varphi(s))=
a_e(u_\theta(s)-u(s),\varphi_\theta(s)-\varphi(s)).
\]
Then, we use the coercivity of the form $a_\beta$, (\ref{23}), and the
continuity of the form $a_e$; as a result we find that
\begin{equation}\label{yx}
\|\varphi_\theta(s)-\varphi(s)\|_W \le
\frac{\|e\|_{L^\infty(\Omega)}}{\beta^*}\,\|u_\theta(s)-u(s)\|_V.
\end{equation}
Assume now that (\ref{co}) hold. Then (\ref{xy}) and (\ref{yx})
yield the convergence result (\ref{con}) which concludes the
proof.
\end{proof}

Consider now the case of homogeneous viscosity, i.e. the case when
assumption (\ref{co}) is replaced by the assumption
\[
\theta(x)=\theta\quad{\rm a.e.}\quad \mathbf{x}\in \Omega,
\]
where $\theta$ is given
positive constant. In this case
$\|\theta\|_{L^\infty(\Omega)}=\theta$, $\theta^*=\theta$ and the
convergence (\ref{co}) is equivalent to $\theta\to 0$. Therefore,
by (\ref{con}) we conclude that the weak solution to the antiplane
electro-viscoelastic problem with Tresca's friction law may be
approached by the weak solution to the antiplane electro-elastic
problem with Tresca's friction law, as the viscosity is small
enough. From mechanical point of view this convergence result
shows that the electro-elasticity with friction may be considered
as a limit case of electro-viscoelasticity with friction as the
viscosity decreases.


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\end{document}