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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 55, pp. 1--17.\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-2007/55\hfil Analysis of two contact problems]
{Analysis of two dynamic frictionless
contact problems for elastic-visco-plastic materials}

\author[Y. Ayyad, M. Sofonea\hfil EJDE-2007/55\hfilneg]
{Youssef Ayyad, Mircea Sofonea}  % in alphabetical order

\address{Youssef Ayyad \newline
Laboratoire de Math\'ematiques et de Physique pour les Syst\`emes,
Universit\'e de Perpignan, 52 Avenue Paul Alduy, 66860 Perpignan Cedex France}
\email{youssef.ayyad@univ-perp.fr}

\address{Mircea Sofonea \newline
Laboratoire de Math\'ematiques et Physique pour les Syst\`emes,
University of Perpignan, 52 avenue Paul Alduy,
66860 Perpignan, France}
\email{sofonea@univ-perp.fr}

\thanks{Submitted March 12, 2007. Published April 17, 2007.}
\subjclass[2000]{74M15, 74H20, 74H25, 49J40}
\keywords{Elastic-visco-plastic material; dynamic process;
 frictionless contact; \hfill\break\indent
normal compliance; normal damped response;
 variational formulation; weak solution}

\begin{abstract}
 We consider two mathematical models which describe the
 contact between an elastic-visco-plastic body and an obstacle,
 the so-called foundation. In both models the contact is
 frictionless and the process is assumed to be dynamic.
 In the first model the contact is described with a normal
 compliance condition  and, in the second one,  is described
 with a normal damped response condition.
 We derive a variational formulation of the models which is
 in the form of a system coupling an integro-differential
 equation with a second order variational equation for the
 displacement and the stress fields.
 Then we prove the unique weak solvability of the models.
 The proofs are based on arguments on nonlinear evolution
 equations with monotone operators and fixed point.
 Finally, we study the dependence of the solution with respect
 to a perturbation of the contact conditions and prove a
 convergence result.
\end{abstract}

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


\section{Introduction}

Phenomena of contact involving deformable bodies abound in
industry and everyday life. Contact of braking pads with wheels,
tires with roads, pistons with skirts are just three  simple
examples. Common industrial processes such as metal forming, metal
extrusion, involve contact evolutions. Owing to their inherent
complexity, contact phenomena are modelled by nonlinear
evolutionary problems.

The aim of this paper is to study two dynamic contact problems for
elastic-visco-plastic materials with a constitutive law of the
form
\begin{equation}\label{ce}
{\boldsymbol{\sigma}}(t) = \mathscr{A} \boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}(t)) + \mathscr{E}
\boldsymbol{\varepsilon}({\boldsymbol{u}(t)}) +\int_0^t \mathscr{G}
(\boldsymbol{\sigma}(s)-\mathscr{A}\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}(s)),\varepsilon({\boldsymbol{u}}(s)))\,ds,
\end{equation}
where $\boldsymbol{u}$ denotes the displacement field and $\boldsymbol{\sigma}$,
$\boldsymbol{\varepsilon}({\boldsymbol{u}})$ represent the stress and the linearized
strain tensor, respectively.  Here $\mathscr{A}$ and $\mathscr{E}
$ are nonlinear operators describing the purely viscous and the
elastic properties of the material, respectively, and $\mathscr{G}
$ is a nonlinear constitutive function which describes the
visco-plastic behaviour of the material. In (\ref{ce}) and
everywhere in this paper  the dot above a variable represents
derivative with respect to the time variable $t$.

Examples of constitutive laws of the form (\ref{ce}) can be
constructed by using rheological arguments, see e.g.  \cite{HS}.
It follows from (\ref{ce}) that, at each time moment $t$, the
stress tensor $\boldsymbol{\sigma}(t)$ is split into two parts:
$\boldsymbol{\sigma}(t)=\boldsymbol{\sigma}^V(t)+\boldsymbol{\sigma}^R(t)$, where $\boldsymbol{\sigma}^V(t)=\mathscr{A}
\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}(t))$ represents the purely viscous part of
the stress whereas $\boldsymbol{\sigma}^R(t)$ satisfies a rate-type
elastic-visco-plastic relation,
\begin{equation}\label{9}
\boldsymbol{\sigma}^R(t)=\mathscr{E} \boldsymbol{\varepsilon}({\boldsymbol{u}(t)}) +\int_0^t
\mathscr{G}(\boldsymbol{\sigma}^R(s),\boldsymbol{\varepsilon}({\boldsymbol{u}}(s)))\,ds.
\end{equation}
When $\mathscr{G} ={\bf 0}$ the constitutive law (\ref{ce}) reduces to the
Kelvin-Voigt viscoelastic constitutive relation,
\begin{equation}\label{KV}
{\boldsymbol{\sigma}} = \mathscr{A} \boldsymbol{\varepsilon}(\dot{\boldsymbol{u}})+ \mathscr{E}\boldsymbol{\varepsilon}({\boldsymbol{u}}).
\end{equation}

Examples and mechanical interpretation of elastic-visco-plastic
materials of the form (\ref{9}) can be found in \cite{CS,IS}.
Quasistatic contact problems for  materials of the form (\ref{9})
or (\ref{KV}) are the topic of numerous papers, e.g.\ \cite{AF,
AFS,FSV, RSS1}, and the comprehensive references \cite{HS, SST}.
Dynamic contact problems with Kelvin-Voigt materials of the form
(\ref{KV}) are studied in in \cite{J3,JE1,K,KS,MiO} and in the
monograph \cite{EJK}.

The two problems we consider in this paper are frictionless. In
the first one we assume that the normal stress on the contact
surface depends only on the normal displacement and therefore we
model the contact with normal compliance. 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 papers, see e.g.\
\cite{A1,A2,KO,KMS,RSS1} and the references therein. 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. In the
second problem we assume that the normal stress on the contact
surface depends only on the normal velocity and therefore we model
the contact with normal damped response. Such condition is
appropriate when the contact surfaces are lubricated; it was used
in a number of papers, see, e.g.  \cite{HS,AA,RSS2,SST} and the
references therein.

The paper is structured as follows. In Section \ref{Pre} we
introduce some notation and preliminaries. In Section \ref{Models}
we describe the two mathematical models for the frictionless
contact precess.  In Section \ref{Variational} we list the
assumption on the data and derive the variational formulation of
the problems, which is in the form of a nonlinear
integro-differential system for the displacement and the stress
fields. Then, we state our main existence and uniqueness results,
Theorems \ref{th1} and \ref{th2}. The proof of the theorems is
provided in Sections \ref{T} and are based on arguments of
abstract evolution equations with monotone operators and fixed
point. In Section \ref{cd} we study the dependence of the solution
with respect to a perturbation of the contact conditions and prove
a convergence result.

\section{Notation and preliminaries}\label{Pre}

In this section we present the notation we shall use and some
preliminary material. For further details, we refer the reader to
\cite{HS,SST}.

We denote by $\mathscr{S}^d$ the space of second order symmetric tensors
on $\mathbb{R}^d$ $(d=2,3)$, while $``\cdot"$ and $\|\cdot\|$ will
represent the inner product and the Euclidean norm on $\mathscr{S}^d$ and
$\mathbb{R}^d$. Let $\Omega\subset \mathbb{R}^d $ be a bounded domain with a
Lipschitz boundary $\Gamma$ and let $\boldsymbol{\nu}$ denote the unit outer
normal on $\Gamma$. Everywhere in the sequel the index $i$ and $j$
run from $1$ to $d$, summation over repeated indices is implied
and the index that follows a comma represents the partial
derivative with respect to the corresponding component of the
independent spatial variable.


We use the standard notation for Lebesgue  and Sobolev spaces
associated to $\Omega$ and $\Gamma$ and introduce the spaces
\begin{align*} \mathcal{H}&=\{\boldsymbol{\sigma}=(\sigma_{ij}):
\sigma_{ij}=\sigma_{ji} \in L^2(\Omega)\},\\
H_{1}&=\{\boldsymbol{u}=(u_{i}):\boldsymbol{\varepsilon}(\boldsymbol{u}) \in \mathcal{H} \},\\
\mathcal{H}_{1}& = \{\boldsymbol{\sigma} \in \mathcal{H} :\mathop{\rm Div} \boldsymbol{\sigma} \in
L^2(\Omega)^d\}.
\end{align*}
Here $ \boldsymbol{\varepsilon}$ and $\mathop{\rm Div}$ are the deformation and the
divergence operators, respectively, defined by
\[
\boldsymbol{\varepsilon}(\boldsymbol{u}) = (\varepsilon_{ij}(\boldsymbol{u})),\quad
\varepsilon_{ij}(\boldsymbol{u}) = \frac{1}{2}\,( u_{i,j} + u_{j,i} ),\quad
\mathop{\rm Div}\boldsymbol{\sigma} = (\sigma_{ij,j} ).
\]
The spaces $\mathcal{H}$, $H_{1}$ and $\mathcal{H}_1$ are real Hilbert
spaces endowed with the canonical inner products given by
\begin{align*}
( \boldsymbol{\sigma}, \boldsymbol{\tau} )_\mathcal{H} &= \int_{\Omega}{\sigma_{ij}\tau_{ij}dx},\\
(\boldsymbol{u},\boldsymbol{v})_{H_{1}} &= ( \boldsymbol{u},\boldsymbol{v} )_{L^2(\Omega)^d} + (
\boldsymbol{\varepsilon}(\boldsymbol{u}),\boldsymbol{\varepsilon}(\boldsymbol{v}) )_\mathcal{H},\\
(\boldsymbol{\sigma},\boldsymbol{\tau} )_{\mathcal{H}_1}&=(\boldsymbol{\sigma},\boldsymbol{\tau} )_\mathcal{H} + ( \mathop{\rm Div}
\boldsymbol{\sigma},\mathop{\rm Div} \boldsymbol{\tau} )_{L^2(\Omega)^d}.
\end{align*}
In general, we denote by $\|\cdot\|_{X}$ the norm on a Banach
space $X$ and note that this holds, in particular, for the
associated norms on the spaces $\mathcal{H}$, $H_{1}$ and
$\mathcal{H}_{1}$.

For every element $\boldsymbol{v}\in H_1$ we also use the notation $\boldsymbol{v}$ for
the trace of $\boldsymbol{v}$ on $\Gamma$ and we denote by $v_\nu$ and
$\boldsymbol{v}_\tau$ the normal and the tangential components of $\boldsymbol{v}$ on
$\Gamma$ given by
\[
v_{\nu}=\boldsymbol{v}\cdot\boldsymbol{\nu},\quad \boldsymbol{v}_{\tau}=\boldsymbol{v}-v_{\nu}\boldsymbol{\nu}.
\]
We also denote by $\sigma_\nu$ and $\boldsymbol{\sigma}_\tau$
the normal and the tangential traces of a function $\boldsymbol{\sigma}\in
\mathcal{H}_1$, and we recall that when $\boldsymbol{\sigma}$ is a regular
function then
\[
\sigma_{\nu}=(\boldsymbol{\sigma}\boldsymbol{\nu})\cdot\boldsymbol{\nu}, \quad \boldsymbol{\sigma}_{\tau} =
\boldsymbol{\sigma}\boldsymbol{\nu} - \sigma_{\nu}\boldsymbol{\nu} ,
\]
 and the following Green's formula holds:
\begin{equation}\label{Green}
(\boldsymbol{\sigma},\boldsymbol{\varepsilon}(\boldsymbol{v}))_\mathcal{H}+
(\mathop{\rm Div}\boldsymbol{\sigma},\boldsymbol{v})_{L^2(\Omega)^d}= \int_{\Gamma}\boldsymbol{\sigma}\boldsymbol{\nu} \cdot
\boldsymbol{v}\,da  \quad \forall \boldsymbol{v}\in H_{1}.
\end{equation}


Let $\Gamma_1$ be a measurable part of $\Gamma$ such that
$meas\,\Gamma_1>0$ and let $V$ be the closed subspace of $H_1$
given by \[V=\{~\boldsymbol{v}\in H_1\ |\ \boldsymbol{v}={\bf 0}\ \ \mbox{on}\ \
\Gamma_1~\}.
\] Then,  the following Korn's inequality
holds:
\begin{equation}
\|\boldsymbol{\varepsilon}(\boldsymbol{v})\|_\mathcal{H}\ge
c_K\,\|\boldsymbol{v}\|_{H_1}\quad\forall\,\boldsymbol{v}\in V, \label{Korn}
\end{equation}
where  $c_K>0$ is a constant depending only on $\Omega$ and
$\Gamma_1$. A proof of Korn's inequality can be found in, for
instance, \cite[p.\ 16]{EJK}. Over the space $V$ we consider the
inner product given by
\[
(\boldsymbol{u},\boldsymbol{v})_V=(\boldsymbol{\varepsilon}(\boldsymbol{u}),\boldsymbol{\varepsilon}(\boldsymbol{v}))_\mathcal{H}
\]
 and let $\|\cdot\|_V$ be the associated norm. It follows from Korn's
inequality (\ref{Korn}) that $\|\cdot\|_{H_1}$ and $\|\cdot\|_V$
are equivalent norms on $V$. Therefore $(V,\|\cdot\|_V)$ is a real
Hilbert space. Moreover, by the Sobolev trace theorem, there
exists  a positive constant $c_0$ which depends on  $\Omega$,
$\Gamma_1$ and $\Gamma_3$ such that
\begin{equation}
\|\boldsymbol{v}\|_{L^2(\Gamma_3)^d}\le c_0\|\boldsymbol{v}\|_V\quad\forall \boldsymbol{v}\in V.
\label{2.2}
\end{equation}

Let $T>0$. For every real Banach space $X$ we use the notation
$C([0,T];X)$ and $C^1([0,T];X)$ for the space of continuous and
continuously differentiable functions from $[0,T]$ to $X$,
respectively; $C([0,T];X)$ is a real Banach space with the norm
\[
\|x\|_{C([0,T];X)}=\max_{t\in[0,T]}\|x(t)\|_X
\]
while $C^1([0,T];X)$ is a real Banach space  with the norm
\[
\|x\|_{C^1([0,T];X)}=\max_{t\in[0,T]}\|x(t)\|_X+\max_{t\in[0,T]}\|\dot
x(t)\|_X.
\]
Finally, for $k\in\mathbb{N}$ and $p\in [1,\infty]$, we use
the standard notation for the Lebesgue spaces $L^p(0,T;X)$ and for
the Sobolev spaces $W^{k,p}(0,T;X)$.

We complete this section with the following abstract result which may
be found in \cite[p.\ 140]{Ba} and which will be used in Section
\ref{T} of this paper.


\begin{theorem}\label{thB}
Let $V\subset H \subset V'$ be a Gelfand triple. Assume that
$A:V\to V'$ is a hemicontinuous and monotone operator which
satisfies
\begin{gather}
\langle Av,v\rangle_{V'\times V}\ge \omega\|v\|_{V}^{2}+\alpha\quad
\forall v \in V,
\label{2.5} \\
\|Av\|_{V'}\le C\,(\|v\|_{V}+1)\quad \forall v\in V, \label{2.6}
\end{gather}
for some constants $\omega>0,\, C>0$ and $\alpha\in \mathbb{R}$. Then,
given $u_{0}\in H$ and $f\in L^2(0,T;V')$, there exists a unique
function $u$ which satisfies
\begin{gather*}
 u\in L^2(0,T;V)\cap C([0,T];H),\
\dot{u}\in L^2(0,T;V'),\\
 \dot{u}(t) + Au(t) = f(t)
\quad \text{a.e. }t\in (0,T),\\
 u(0)=u_{0}.
\end{gather*}
\end{theorem}


\section{Statement of the Problems}\label{Models}

In this section we present the mathematical models  which describe
the frictionless contact process between an elastic-visco-plastic
body and the foundation.



The physical setting is as follows : an elastic-visco-plastic body
occupies a bounded domain $\Omega\subset\mathbb{R}^d$ ($d=2,3$) with a
regular boundary $\Gamma$ that is partitioned into three disjoint
measurable parts $\Gamma_1$, $\Gamma_2$ and $\Gamma_3$, such that
$meas\,\Gamma_1>0$. Let $T>0$ and let $[0,T]$ denote the time
interval of interest. The body is clamped on $\Gamma_1\times(0,T)$
and thus the displacement field vanishes there. A volume force of
density $\boldsymbol{f}_0$ acts in $\Omega\times(0,T)$ and a surface traction
of density $\boldsymbol{f}_2$ acts on $\Gamma_2\times(0,T)$. In the reference
configuration the body is in frictionless contact on $\Gamma_3$
with an obstacle, the so-called foundation. In the first problem
we assume that contact is modelled with normal compliance. Under
these assumptions, the classical formulation of the problem is the
following.


\subsection*{Problem $\mathscr{P}_1$} {\em Find a displacement field
$\boldsymbol{u}:\Omega\times[0,T]\to \mathbb{R}^d$ and a stress field
$\boldsymbol{\sigma}:\Omega\times[0,T]\to \mathscr{S}^d$ such that}
\begin{gather}
{\boldsymbol{\sigma}}(t) = \mathscr{A} \boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}(t))+
\mathscr{E} \boldsymbol{\varepsilon}({\boldsymbol{u}(t)}) +\int_0^t \mathscr{G}
(\boldsymbol{\sigma}(s)-\mathscr{A}\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}(s)),\boldsymbol{\varepsilon}({\boldsymbol{u}}(s)))\,ds
\quad \text{in } \Omega\times(0,T), \label{3.1}
\\
\rho\ddot{\boldsymbol{u}} =\mathop{\rm Div} \boldsymbol{\sigma}+{\boldsymbol{f}}_0 \quad \text{ in }
\Omega\times(0,T), \label{3.2}\\
\boldsymbol{u} ={\bf 0} \quad \text{ on }\Gamma_1\times(0,T), \label{3.3}\\
\boldsymbol{\sigma}\boldsymbol{\nu} ={\boldsymbol{f}}_2 \quad \text{on } \Gamma_2\times(0,T),
\label{3.4}\\
-\sigma_\nu =p(u_\nu) \quad \text{on }
\Gamma_3\times(0,T),
\label{3.5}\\
\boldsymbol{\sigma}_\tau ={\bf 0} \quad \text{on }\Gamma_3\times(0,T),
\label{3.6}\\
\boldsymbol{u}(0) =\boldsymbol{u}_0,\quad \dot{\boldsymbol{u}}(0) =\boldsymbol{v}_0\quad \text{in } \Omega.
\label{3.7}
\end{gather}

Here (\ref{3.1}) is the elastic-visco-plastic constitutive law
introduced in Section 1, (\ref{3.2}) represents the equation of
motion in which $\rho$ denotes the density of mass, (\ref{3.3})
and (\ref{3.4}) are the displacement and traction boundary
conditions, respectively. Condition (\ref{3.5}) represents the
normal compliance condition in which $\sigma_\nu$ denotes the
normal stress, $u_\nu$ is the normal displacement and $p$ is a
positive increasing function which vanishes for a negative
argument; this condition shows that when there is separation
between the body and the obstacle (i.e. when $u_\nu<0$), then the
reaction of the foundation vanishes (since $\sigma_\nu=0$); also,
when there is penetration (i.e. when $u_\nu\ge0$), then the
reaction of the foundation is towards the body (since
$\sigma_\nu\le0$) and it is increasing with the penetration (since
$p$ is an increasing function). Condition (\ref{3.6}) shows that
the tangential shear, denoted $\boldsymbol{\sigma}_\tau$, vanishes on the
contact surface, i.e. the process is frictionless. Finally, the
functions $\boldsymbol{u}_0$ and $\boldsymbol{v}_0$ in (\ref{3.7}) denote the initial
displacement and the initial velocity, respectively.

In the second problem we assume that contact is modelled with
normal damped response and, therefore, the classical formulation
of the problem is the following.


\subsection*{Problem $\mathscr{P}_2$} {\em Find a displacement field
$\boldsymbol{u}:\Omega\times[0,T]\to \mathbb{R}^d$ and a stress field
$\boldsymbol{\sigma}:\Omega\times[0,T]\to \mathscr{S}^d$ such that
$(\ref{3.1})$--$(\ref{3.4})$,$(\ref{3.6})$--$(\ref{3.7})$ hold
and, moreover,}
\begin{equation}\label{3.5n}
-\sigma_\nu =p(\dot u_\nu)\quad \text{on } \Gamma_3\times(0,T).
\end{equation}

The difference with respect problem $\mathscr{P}_1$  arise in the fact
that in problem $\mathscr{P}_2$ we replace the normal compliance condition
(\ref{3.5}) with the normal damped response condition
(\ref{3.5n}), which shows that now the normal stress depends on
the normal velocity on the contact surface.


\section{Variational formulation and main results}\label{Variational}

We now describe the assumptions on the data we consider in the
study of the mechanical problems $\mathscr{P}_1$ and $\mathscr{P}_2$. Then we
derive their variational foumulation and state our main existence
and uniqueness results.

We assume that the operators $\mathscr{A}$, $\mathscr{E} $ and  $\mathscr{G} $ satisfy the
following conditions.
\begin{equation}
\left\{ \begin{array}{l}
\text{(a)} \mathscr{A}:\Omega\times \mathscr{S}^d\to\mathscr{S}^d.\\
\text{(b) There exists $L_{\mathscr{A}}>0$ such that}\\
  \quad \|{\mathscr{A}}(\boldsymbol{x},\boldsymbol{\varepsilon}_1)-{\mathscr{A}}(\boldsymbol{x},\boldsymbol{\varepsilon}_2)\|
      \le L_{\mathscr{A}} \|\boldsymbol{\varepsilon}_1-\boldsymbol{\varepsilon}_2\|\\
   \quad \forall\,\boldsymbol{\varepsilon}_1,\boldsymbol{\varepsilon}_2
    \in\mathscr{S}^d,\text{ a.e. }\boldsymbol{x}\in \Omega.\\
\text{(c)  There exists $m_{\mathscr{A}}>0$  such that}\\
\quad ({\mathscr{A}}(\boldsymbol{x},\boldsymbol{\varepsilon}_1)-{\mathscr{A}}(\boldsymbol{x},\boldsymbol{\varepsilon}_2))
       \cdot(\boldsymbol{\varepsilon}_1-\boldsymbol{\varepsilon}_2)\ge m_{\mathscr{A}}\,
      \|\boldsymbol{\varepsilon}_1-\boldsymbol{\varepsilon}_2\|^2 \\
 \quad \forall\,\boldsymbol{\varepsilon}_1, \boldsymbol{\varepsilon}_2 \in\mathscr{S}^d,
  \text{ a.e. $\boldsymbol{x}\in \Omega$}.\\
\text{(d) For each $\boldsymbol{\varepsilon}\in\mathscr{S}^d$,
   $\boldsymbol{x}\mapsto {\mathscr{A}}(\boldsymbol{x},\boldsymbol{\varepsilon})$ is measurable on $\Omega$}.\\
\text{(e) The mapping $\boldsymbol{x}\mapsto {\mathscr{A}}(\boldsymbol{x},{\bf 0})$
belongs to $\mathcal{H}$}.
\end{array} \right.
\label{3.9}
\end{equation}

\begin{equation}
\left\{\begin{array}{l}
\text{(a) }\mathscr{E} : \Omega\times\mathscr{S}^d\to\mathscr{S}^d.\\
\text{(b) There exists $L_\mathscr{E} >0$  such that}\\
 \quad \|\mathscr{E} (\boldsymbol{x},\boldsymbol{\sigma}_1,\boldsymbol{\varepsilon}_1) -\mathscr{E}
(\boldsymbol{x},\boldsymbol{\sigma}_2,\boldsymbol{\varepsilon}_2)\|\le L_\mathscr{E}
\,(\|\boldsymbol{\sigma}_1-\boldsymbol{\sigma}_2\|+
\|\boldsymbol{\varepsilon}_1-\boldsymbol{\varepsilon}_2\|)\\
\quad \forall\, \boldsymbol{\sigma}_1,\boldsymbol{\sigma}_2,\boldsymbol{\varepsilon}_1,\boldsymbol{\varepsilon}_2\in
\mathscr{S}^d,\text{ a.e.  }\boldsymbol{x}\in\Omega.\\
\text{(c) For each }\boldsymbol{\sigma},\boldsymbol{\varepsilon}\in\mathscr{S}^d,\, \boldsymbol{x}\mapsto
\mathscr{E} (\boldsymbol{x},\boldsymbol{\sigma},\boldsymbol{\varepsilon})
{\rm\ is\ measurable\ on\ }\Omega.\\
\text{(d) The mapping $\boldsymbol{x}\mapsto \mathscr{E} (\boldsymbol{x},{\bf 0},{\bf 0})$
belongs to $\mathcal{H}$}.
\end{array} \right.
\label{3.10}
\end{equation}

\begin{equation}
\left\{ \begin{array}{l}
\text{(a) }\mathscr{G} : \Omega \times \mathscr{S}^d\times\mathscr{S}^d\to \mathscr{S}^d.\\
\text{(b) There exists $L_\mathscr{G} >0$  such that }\\
\quad \|\mathscr{G} (\boldsymbol{x},\boldsymbol{\sigma}_1,\boldsymbol{\varepsilon}_1) -\mathscr{G}
(\boldsymbol{x},\boldsymbol{\sigma}_2,\boldsymbol{\varepsilon}_2)\|\le L_\mathscr{G}
\,(\|\boldsymbol{\sigma}_1-\boldsymbol{\sigma}_2\|+
\|\boldsymbol{\varepsilon}_1-\boldsymbol{\varepsilon}_2\|)\\
\quad \forall\, \boldsymbol{\sigma}_1,\boldsymbol{\sigma}_2,\boldsymbol{\varepsilon}_1,\boldsymbol{\varepsilon}_2\in
\mathscr{S}^d,\text{ a.e. $\boldsymbol{x}\in\Omega$}.\\
\text{(c) For each $\boldsymbol{\sigma},\boldsymbol{\varepsilon}\in\mathscr{S}^d,\, \boldsymbol{x}\mapsto
\mathscr{G} (\boldsymbol{x},\boldsymbol{\sigma},\boldsymbol{\varepsilon})$
 is measurable on $\Omega$}.\\
\text{(d)  The mapping $\boldsymbol{x}\mapsto \mathscr{G} (\boldsymbol{x},{\bf 0},{\bf 0})$
belongs to $\mathcal{H}$}.
\end{array} \right.
\label{3.11}
\end{equation}

The contact function $p$ satisfies
\begin{equation}\label{p}
\left\{ \begin{array}{l}
\text{(a) } p:\Gamma_3\times\mathbb{R}\to\mathbb{R}.\\
\text{(b) There exists $ L_p>0$ such that}\\
\quad |p(\boldsymbol{x},r_1)-p(\boldsymbol{x},r_2)|\le L_p|r_1-r_2|\quad\forall r_1,\,
 r_2\in \mathbb{R}, \text{ a.e. $\boldsymbol{x}\in\Gamma_3$}.\\
\text{(c) }(p(\boldsymbol{x},r_1)-p(\boldsymbol{x},r_2))(r_1-r_2)\geq0\quad\forall
r_1,\, r_2\in \mathbb{R} ,\text{ a.e. $\boldsymbol{x}\in\Gamma_3$}.
\\
\text{(d)  For each $r\in \mathbb{R},\, \boldsymbol{x}\mapsto p(\boldsymbol{x},r)$ is
measurable on $\Gamma_3$}.
\\
\text{(e) $p(\boldsymbol{x},r)=0$ for all $r<0$  a.e. $\boldsymbol{x}\in\Gamma_3$}.
\end{array} \right.
\end{equation}


We also suppose that the mass density satisfies
\begin{equation} \label{3.13}
\rho \in L^{\infty}(\Omega),
\quad\text{there exists $\rho^{*}>0$  such that $\rho(\boldsymbol{x})\geq\rho^{*}$
a.e. $\boldsymbol{x} \in \Omega$,}
\end{equation}
the body forces and surface tractions have the regularity
\begin{equation}
\label{3.15} \boldsymbol{f}_0\in L^2(0,T;L^2(\Omega)^d),\quad
\boldsymbol{f}_2\in L^2(0,T;L^2(\Gamma_2)^d),
\end{equation}
and the initial data satisfy
\begin{equation}
\label{3.16} \boldsymbol{u}_0\in V, \quad \boldsymbol{v}_0\in L^2(\Omega)^d.
\end{equation}

We turn now to the variational formulations of Problems $\mathscr{P}_1$
and $\mathscr{P}_2$. To this end we use a modified inner product on the
Hilbert space $H=L^2(\Omega)^d$, given by
\begin{equation}
\label{3.17} ( \boldsymbol{u},\boldsymbol{v} )_{H} = (\rho\, \boldsymbol{u},\boldsymbol{v})_{L^2(\Omega)^d}
\quad \forall\, \boldsymbol{u},\, \boldsymbol{v} \in H,
\end{equation}
that is, it is weighed with $\rho$, and we let $\|\cdot\|_H$ be
the associated norm, i.e.,
\begin{equation}
\label{3.18} \|\boldsymbol{v}\|_{H} = (\rho\, \boldsymbol{v},\boldsymbol{v})_{L^2(\Omega)^d}^{1/2}
\quad \forall\, \boldsymbol{v} \in H.
\end{equation}
It follows from assumption (\ref{3.13}) that  $\|\cdot\|_H$ and
$\|\cdot\|_{L^2(\Omega)^d}$ are equivalent norms on $H$, and also
the inclusion mapping of $(V,\|\cdot\|_V)$ into $(H, \|\cdot\|_H)$
is continuous and dense. We denote by $V'$ the dual space of $V$.
Identifying $H$ with its own dual, we can write the Gelfand triple
\[
V\subset H \subset V'.
\]
We use the notation $\langle\cdot,\cdot\rangle_{V'\times V}$ to represent
the duality pairing between $V'$ and $V$ and recall that
\begin{equation}
\label{3.19} \langle\boldsymbol{u},\boldsymbol{v}\rangle_{V'\times V}=( \boldsymbol{u}, \boldsymbol{v})_H \quad
\forall \boldsymbol{u} \in H,\, \boldsymbol{v} \in V
\end{equation}

Assumptions (\ref{3.15}) allow us, for almost any $t\in (0,T)$, to
define $\boldsymbol{f}(t) \in V'$ by
\begin{equation}
\label{3.21} \langle\boldsymbol{f}(t),\boldsymbol{v}\rangle_{V'\times V} =
\int_\Omega\boldsymbol{f}_0(t)\cdot\boldsymbol{v}\,dx +
\int_{\Gamma_2}\boldsymbol{f}_2(t)\cdot\boldsymbol{v}\,da\quad \forall \boldsymbol{v}\in V,
\end{equation}
and note that
\begin{equation}
\label{3.22} \boldsymbol{f}\in L^{2}(0,T;V').
\end{equation}
Finally, we consider the functional $j:V\times V\to\mathbb{R}$ defined by
\begin{equation}\label{j}
j(\boldsymbol{u},\boldsymbol{v})= \int_{\Gamma_3} p(u_\nu)v_\nu\, da,\quad \forall\boldsymbol{u},\,
\boldsymbol{v} \in V
\end{equation}
and we note that, by assumption (\ref{p}), the integral in
(\ref{j}) is well defined.


We assume in what follows that $(\boldsymbol{u},\boldsymbol{\sigma})$ are smooth
functions satisfying (\ref{3.2})--(\ref{3.6}) and let $t\in
[0,T]$. We take the dot product of equation (\ref{3.2}) with $\boldsymbol{w}$
where $\boldsymbol{w}$ is an arbitrary element of $V$, integrate the result
over $\Omega$, and use Green's formula (\ref{Green}) to obtain
\begin{equation}\label{3.23}
(\rho\,\ddot{\boldsymbol{u}}(t),\boldsymbol{w})_{L^2(\Omega)^d} +
(\boldsymbol{\sigma}(t),\boldsymbol{\varepsilon}(\boldsymbol{w}))_\mathcal{H}
=\int_{\Omega}\,\boldsymbol{f}_0(t)\cdot\boldsymbol{w}\,dx + \int_{\Gamma}
\boldsymbol{\sigma}(t)\boldsymbol{\nu} \cdot\boldsymbol{w}\, da.
\end{equation}
Applying the boundary conditions (\ref{3.4}) and (\ref{3.6}) and
noting that $\boldsymbol{w}={\bf 0}$ on $\Gamma_1$, we have
\begin{equation}\label{3X}
\int_{\Gamma} \boldsymbol{\sigma}(t)\boldsymbol{\nu} \cdot\boldsymbol{w}\, da = \int_{\Gamma_2}
\boldsymbol{f}_2(t) \cdot\boldsymbol{w}\, da+\int_{\Gamma_3} \sigma_\nu(t)\,w_\nu\,da.
\end{equation}
Moreover, (\ref{3.5}) combined with (\ref{j}) lead to
\begin{equation}\label{3XX}
\int_{\Gamma_3} \sigma_\nu(t)\,w_\nu\, da= -j(\boldsymbol{u}(t),\boldsymbol{w}).
\end{equation}
We now use (\ref{3.23})--(\ref{3XX}) and the equalities
(\ref{3.17}), (\ref{3.19}) and (\ref{3.21}) to find
\begin{equation}
\label{3.24} \langle\ddot{\boldsymbol{u}}(t),\boldsymbol{w}\rangle_{V'\times V} +
(\boldsymbol{\sigma}(t),\boldsymbol{\varepsilon}(\boldsymbol{w}))_Q+j(\boldsymbol{u}(t),\boldsymbol{w})=
\langle\boldsymbol{f}(t),\boldsymbol{w}\rangle_{V'\times V}.
\end{equation}
Finally, we combine (\ref{3.1}), (\ref{3.24}),  and (\ref{3.7}) to
derive the following variational formulation of Problem $\mathscr{P}_1$.



\subsection*{Problem $\mathscr{P}_1^V$} {\em Find a displacement field
$\boldsymbol{u}:[0,T]\to V$ and a stress field $\boldsymbol{\sigma}:
[0,T]\to \mathcal{H}$ such that, for a.e. $t\in(0,T)$,}
\begin{gather}
 {\boldsymbol{\sigma}}(t) = \mathscr{A} \boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}(t))+\mathscr{E}
\boldsymbol{\varepsilon}({\boldsymbol{u}}(t)) +\int_0^t \mathscr{G}
(\boldsymbol{\sigma}(s)-\mathscr{A}\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}(s)),\boldsymbol{\varepsilon}({\boldsymbol{u}}(s)))\,ds,
\label{3.26}
\\
\label{3.27} \langle\ddot{\boldsymbol{u}}(t),\boldsymbol{w}\rangle_{V'\times V} +
(\boldsymbol{\sigma}(t),\boldsymbol{\varepsilon}(\boldsymbol{w}))_\mathcal{H} +
j(\boldsymbol{u}(t),\boldsymbol{w})=\langle\boldsymbol{f}(t),\boldsymbol{w}\rangle_{V'\times V}\quad\forall\boldsymbol{w}\in V,
\\
\boldsymbol{u}(0)=\boldsymbol{u}_0,\quad\dot{\boldsymbol{u}}(0)=\boldsymbol{v}_0. \label{3.28}
\end{gather}

Using similar arguments we derive the following variational
formulation of Problem $\mathscr{P}_2$.

\subsection*{Problem $\mathscr{P}_2^V$}
{\em Find a displacement field
$\boldsymbol{u}:[0,T]\to V$ and a stress field $\boldsymbol{\sigma}:
[0,T]\to \mathcal{H}$ such that $(\ref{3.26}$),
$(\ref{3.28})$ hold and, moreover,
\begin{equation}
\label{3.27n} \langle\ddot{\boldsymbol{u}}(t),\boldsymbol{w}
\rangle_{V'\times V} +
(\boldsymbol{\sigma}(t),\boldsymbol{\varepsilon}(\boldsymbol{w}))_\mathcal{H} 
+j(\dot{\boldsymbol{u}}(t),\boldsymbol{w})=\langle\boldsymbol{f}(t),\boldsymbol{w}\rangle_{V'\times V},
\end{equation}
for all $\boldsymbol{w}\in V$, a.e. $t\in(0,T)$.}



Our main results that we state here and prove in the next section
are the following.

\begin{theorem}\label{th1}
{Assume that conditions \eqref{3.9}--\eqref{3.16}
hold. Then, Problem $\mathscr{P}_1^V$ has a unique solution. Moreover, the
solution satisfies}
\begin{gather}
\label{4.29} \boldsymbol{u}\in W^{1,2}(0,T;V)\cap C^1([0,T];H), \quad
\ddot{\boldsymbol{u}}\in L^2(0,T;V'),\\
\label{4.30} \boldsymbol{\sigma}\in L^2(0,T;\mathcal{H}), \quad {\rm
Div}\,\boldsymbol{\sigma}\in L^2(0,T;V').
\end{gather}
\end{theorem}

\begin{theorem}\label{th2} {Assume that conditions
\eqref{3.9}--\eqref{3.16} hold. Then, Problem $\mathscr{P}_2^V$ has at
least a solution. Moreover, the solution has the regularity
expressed in \eqref{4.29}--\eqref{4.30}.}
\end{theorem}

We conclude by Theorems \ref{th1} and \ref{th2} that, under the
assumptions \eqref{3.9}--\eqref{3.16}, both the dynamic contact
problem $\mathscr{P}_1$ and the dynamic contact problem $\mathscr{P}_2$ have a
unique {\it weak solution} with regularity
(\ref{4.29})--(\ref{4.30}).


\section{Proof of Theorems \ref{th1} and \ref{th2}}\label{T}

We start with the proof of Theorem \ref{th1} which will be carried
out in several steps. We assume in the rest of this section that
\eqref{3.9}--\eqref{3.16} hold  and $c$ will denote a generic
positive constant which may depend on $\Omega$, $\Gamma_1$,
$\Gamma_2$, $\Gamma_3$, $\mathscr{A} $, $\mathscr{E}$, $\mathscr{G}$, $p$  and $T$, but
does not depend on $t$, nor on the rest of the input data, and
whose value may change from place to place. Let $\boldsymbol{\eta}\in
L^2(0,T;V')$ be given.  In the first step we consider the
following variational problem :

\subsection*{Problem $\mathscr{P}_1^{\eta-disp}$}
{\it Find a displacement field
$\boldsymbol{u}_{\eta}: [0,T]\to V$ such that}
\begin{align}
&\langle\ddot{\boldsymbol{u}}_{\eta}(t),\boldsymbol{w}\rangle_{V'\times V} + (\mathscr{A}
\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}_{\eta}(t)),\boldsymbol{\varepsilon}(\boldsymbol{w}))_\mathcal{H}
+ \langle{\boldsymbol{\eta}}(t),\boldsymbol{w}\rangle_{V'\times V}\label{4.1}\\
&= \langle\boldsymbol{f}(t),\boldsymbol{w}\rangle_{V'\times V} \quad \forall\,
\boldsymbol{w}\in V,\ \text{a.e. }t\in (0,T), \nonumber\\
&\boldsymbol{u}_{\eta}(0)=\boldsymbol{u}_0, \quad \dot{\boldsymbol{u}}_{\eta}(0)=\boldsymbol{v}_0. \label{4.2}
\end{align}


In the study of Problem $\mathscr{P}_1^{\eta-disp}$ we have the following
result.

\begin{lemma}\label{l1}
There exists a unique solution to Problem $\mathscr{P}_1^{\eta-disp}$ and
it has the regularity expressed in $ (\ref{4.29})$. Moreover, if
$\boldsymbol{u}_i$ represents the solution of Problem $\mathscr{P}_1^{\eta_i-disp}$
for $\boldsymbol{\eta}_i\in L^2(0,T;V')$, $i=1,2$, then there exists $c>0$
such that
\begin{equation}\label{4.3}
\int_0^t\|\dot{\boldsymbol{u}}_1(s)-\dot{\boldsymbol{u}}_2(s)\|^2_{V}\,ds\leq
c\,\int_0^t\|\boldsymbol{\eta}_1(t)-\boldsymbol{\eta}_2(t)\|^2_{V'}\,ds\quad\forall
t\in[0,T].
\end{equation}
\end{lemma}

\begin{proof}
We define the operator $A : V\to V'$ by
\begin{equation}
\label{4.4} \langle A\boldsymbol{v},\boldsymbol{w}\rangle_{V'\times V}= (\mathscr{A}
\boldsymbol{\varepsilon}(\boldsymbol{v}),\boldsymbol{\varepsilon}(\boldsymbol{w}))_\mathcal{H} \quad
\forall\,\boldsymbol{v},\,\boldsymbol{w}\in V.
\end{equation}
It follows from  (\ref{4.4}) and (\ref{3.9})(b) that
\begin{equation}
\label{DA.Lip} \|A\boldsymbol{v}-A\boldsymbol{w}\|_{V'} \leq
L_{\mathscr{A}}\,\|\boldsymbol{v}-\boldsymbol{w}\|_V\quad \forall\,\boldsymbol{v},\,\boldsymbol{w}\in V,
\end{equation}
which shows that $A : V\to V'$ is continuous and so is
hemicontinuous. Now, by (\ref{4.4}) and (\ref{3.9})(c), we find
\begin{equation}
\label{DA.4.9} \langle A\boldsymbol{v}-A\boldsymbol{w},\boldsymbol{v}-\boldsymbol{w}\rangle_{V'\times V} \ge
m_{\mathscr{A}}\,\|\boldsymbol{v}-\boldsymbol{w}\|_{V}^2 \quad \forall\,\boldsymbol{v},\,\boldsymbol{w}\in V,
\end{equation}
i.e., that  $A:V\to V'$ is a monotone operator. Choosing
$\boldsymbol{w}={\bf 0}_V$ in (\ref{DA.4.9}) we obtain
\begin{align*}
\langle A\boldsymbol{v},\boldsymbol{v}\rangle_{V'\times V} & \ge m_{\mathscr{A}}\|\boldsymbol{v}\|_{V}^2-\|A{\bf 0}_V\|_{V'}\| \boldsymbol{v}\|_{V}\\
& \geq \frac{1}{2}\,m_{\mathscr{A}}\,\|\boldsymbol{v}\|_{V}^2-\frac {1}{2 m_{\mathscr{A}}}\,
\|A{\bf 0}_V\|_{V'}^2 \quad\forall\,\boldsymbol{v}\in V.
\end{align*}
Thus, $A$ satisfies condition (\ref{2.5}) with $\omega =
m_{\mathscr{A}}/2$ and $\alpha = -\|A{\bf 0}_V\|_{V'}^2/(2m_{\mathscr{A}})$. Next,
by (\ref{DA.Lip}) we deduce that
\[
\|A\boldsymbol{v}\|_{V'}\leq L_{\mathscr{A}}\,\|\boldsymbol{v}\|_V+\|A{\bf 0}_{V}\|_{V'} \quad
\forall\, \boldsymbol{v}\in V.
\]
This inequality {implies that} $A$ satisfies condition
(\ref{2.6}). Finally, we recall that by (\ref{3.22}), (\ref{3.16})
we have $\boldsymbol{f}-\boldsymbol{\eta}\in L^2(0,T;V')$ and $\boldsymbol{v}_0\in H$. It follows
now from Theorem \ref{thB} that there exists a unique function
$\boldsymbol{v}_{\eta}$ which satisfies
\begin{gather}
\label{4.5} \boldsymbol{v}_{\eta}\in L^2(0,T;V)\cap C([0,T];H),\
\dot{\boldsymbol{v}}_{\eta}\in L^2(0,T;V'),\\
\label{4.6} \dot{\boldsymbol{v}}_{\eta} (t)+ A\boldsymbol{v}_{\eta}(t) +
{\boldsymbol{\eta}}(t)= \boldsymbol{f}(t)\quad \text{a.e. }t\in (0,T),\\
\label{4.7} \boldsymbol{v}_{\eta}(0)=\boldsymbol{v}_0.
\end{gather}

Let $\boldsymbol{u}_{\eta} : [0,T] \to V$ be the function defined by
\begin{equation}
\label{4.8} \boldsymbol{u}_{\eta}(t)=\int_{0}^t\,\boldsymbol{v}_{\eta}(s)\,ds + \boldsymbol{u}_0
\quad \forall\, t\in [0,T].
\end{equation}
It follows from (\ref{4.4}), (\ref{4.5})--(\ref{4.8}) that
$\boldsymbol{u}_{\eta}$ is a solution of the variational problem
$\mathscr{P}_1^{\eta-disp}$ and it satisfies  the regularity expressed in
(\ref{4.29}). This concludes the existence part of Lemma \ref{l1}.
The uniqueness part follows from the uniqueness of the solution of
problem (\ref{4.5})--(\ref{4.7}), guaranteed by Theorem \ref{thB}.

Consider now $\boldsymbol{\eta}_1,\boldsymbol{\eta}_2\in L^2(0,T;V')$ and denote
$\boldsymbol{u}_i=\boldsymbol{u}_{\eta_i}$, $\boldsymbol{v}_i=\boldsymbol{v}_{\eta_i}=\dot{\boldsymbol{u}}_{\eta_i}$ for
$i=1,2$.  We use (\ref{4.1}) to obtain
\begin{align*}
&\langle\dot{\boldsymbol{v}}_1-\dot{\boldsymbol{v}}_2,\boldsymbol{v}_{1} - \boldsymbol{v}_{2}\rangle_{V'\times V} +
(\mathscr{A} \,\boldsymbol{\varepsilon}(\boldsymbol{v}_{1})-\mathscr{A} \,\boldsymbol{\varepsilon}(\boldsymbol{v}_{2}),
\boldsymbol{\varepsilon}(\boldsymbol{v}_{1})-\boldsymbol{\varepsilon}(\boldsymbol{v}_{2}))_\mathcal{H} \\
&+ \langle {\boldsymbol{\eta}_1} - {\boldsymbol{\eta}_2},\boldsymbol{v}_{1} -
\boldsymbol{v}_{2}\rangle_{V'\times V} =0\quad{\rm a.e.\ on}\ \ (0,T).
\end{align*}
Let $t\in[0,T]$. We integrate the previous equality with respect
to time and use the initial conditions
$\boldsymbol{v}_{1}(0)=\boldsymbol{v}_{2}(0)=\boldsymbol{v}_0$ and the properties of the operator
$\mathscr{A} $ to find
\[
m_\mathscr{A}\, \int_0^t\|\boldsymbol{v}_{1}(s) - \boldsymbol{v}_{2}(s)\|_{V}^2\,ds \le
-\int_0^t\langle {\boldsymbol{\eta}_1}(s) - {\boldsymbol{\eta}_2}(s),\boldsymbol{v}_{1}(s) -
\boldsymbol{v}_{2}(s)\rangle_{V'\times V}\,ds.
\]
Now,
\begin{align*}
& -\int_0^t\langle {\boldsymbol{\eta}_1}(s) - {\boldsymbol{\eta}_2}(s),\boldsymbol{v}_{1}(s) -
\boldsymbol{v}_{2}(s)\rangle_{V'\times V}\,ds \\
& \leq\int_0^t\|{\boldsymbol{\eta}_1}(s) -
{\boldsymbol{\eta}_2}(s)\|_{V'}\|\boldsymbol{v}_{1}(s) -
\boldsymbol{v}_{2}(s)\|_{V}\,ds\\
& \leq\frac{1}{m_\mathscr{A}}\int_0^t\|{\boldsymbol{\eta}_1}(s) -
{\boldsymbol{\eta}_2}(s)\|_{V'}^2 ds+\frac{m_\mathscr{A}}{4} \int_0^t\|\boldsymbol{v}_{1}(s) -
\boldsymbol{v}_{2}(s)\|_{V}^2\,ds.
\end{align*}
The previous two inequalities lead to
\[\int_0^t\|\boldsymbol{v}_{1}(s) -
\boldsymbol{v}_{2}(s)\|_{V}^2\,ds \leq c\,\int_0^t\|{\boldsymbol{\eta}_1}(s) -
{\boldsymbol{\eta}_2}(s)\|_{V'}^2 ds,
\]
which implies (\ref{4.3}). \end{proof}


We use the displacement field $\boldsymbol{u}_\eta$ obtained in Lemma
\ref{l1} to construct the following Cauchy problem for the stress
field.



\subsection*{Problem $\mathscr{P}_1^{\eta-st}$.}
{\it Find a stress field
$\boldsymbol{\sigma}_\eta:[0,T]\to\mathcal{H}$ such that
\begin{equation}\label{4.9}
{\boldsymbol{\sigma}}_\eta(t) = \mathscr{E} \boldsymbol{\varepsilon}(\boldsymbol{u}_\eta(t))+\int_0^t
\mathscr{G}(\boldsymbol{\sigma}_\eta(s),\boldsymbol{\varepsilon}({\boldsymbol{u}}_\eta(s)))\,ds
\end{equation}
for all  $t\in[0,T]$}.

In the study of Problem $ \mathscr{P}_V^{\eta-st}$ we have the following
result.

\begin{lemma}\label{l2}
There exists a unique solution of Problem $\mathscr{P}_1^{\eta-st}$ and it
satisfies $\boldsymbol{\sigma}_\eta\in W^{1,2}(0,T;\mathcal{H})$. Moreover, if
$\boldsymbol{\sigma}_i$ and $\boldsymbol{u}_i$ represent the solutions of problem
$\mathscr{P}_1^{\eta_i-st}$ and $\mathscr{P}_1^{\eta_i-disp}$, respectively, for
$\boldsymbol{\eta}_i\in L^2(0,T;V')$, $i=1,2$, then there exists $c>0$ such
that
\begin{equation}\label{4.10}
\|\boldsymbol{\sigma}_1(t)-\boldsymbol{\sigma}_2(t)\|_\mathcal{H}
\leq c\,\Big(\|{\boldsymbol{u}}_1(t)-{\boldsymbol{u}}_2(t)\|_V
+\int_0^t\|{\boldsymbol{u}}_1(s)-{\boldsymbol{u}}_2(s)\|_V\,ds\Big)
\end{equation}
for all $t\in[0,T]$.
\end{lemma}

\begin{proof}
  Let $\Lambda_\eta:L^2(0,T;\mathcal{H})\to L^2(0,T;\mathcal{
H})$ be the operator given by
\begin{equation}\label{4.11}
\Lambda_\eta\boldsymbol{\sigma}(t)= \mathscr{E} \boldsymbol{\varepsilon}({\boldsymbol{u}}_\eta(t))+
\int_0^t\,\mathscr{G} (\boldsymbol{\sigma}(s),
  \boldsymbol{\varepsilon}(\boldsymbol{u}_\eta(s)))\,ds
\end{equation}
for all $\boldsymbol{\sigma}\in L^2(0,T;\mathcal{H})$ and  $t\in[0,T]$. For
$\boldsymbol{\sigma}_1,$ $\boldsymbol{\sigma}_2\in L^2(0,T;\mathcal{H})$ we use (\ref{4.11})
and (\ref{3.11}) to obtain
\[
\|\Lambda_\eta\,\boldsymbol{\sigma}_1(t)-\Lambda_\eta\,\boldsymbol{\sigma}_2(t)\|_\mathcal{
H}\leq L_\mathscr{G} \,\int_0^t\|\boldsymbol{\sigma}_1(s)-\boldsymbol{\sigma}_2(s)\|_\mathcal{H}\,ds
\]
for all $t\in [0,T]$. It follows from this inequality that for $p$
large enough, a power $\Lambda_\eta^p$ of the operator
$\Lambda_\eta$ is a contraction on the Banach space $L^2(0,T;V)$
and, therefore, there exists  a unique element $\boldsymbol{\sigma}_\eta\in
L^2(0,T;\mathcal{H})$ such that
$\Lambda_\eta\boldsymbol{\sigma}_\eta=\boldsymbol{\sigma}_\eta$. Moreover, $\boldsymbol{\sigma}_\eta$
is the unique solution of Problem $\mathscr{P}_1^{\eta-st}$ and, using
(\ref{4.9}), the regularity of $\boldsymbol{u}_\eta$ and the properties of
the operators $\mathscr{A}$, $\mathscr{E}$ and $\mathscr{G}$, it follows that
$\boldsymbol{\sigma}_\eta\in W^{1,2}(0,T;\mathcal{H})$.

Consider now $\boldsymbol{\eta}_1,$ $\boldsymbol{\eta}_2\in L^2(0,T;V')$ and, for
$i=1,2$, denote  $\boldsymbol{u}_{\eta_i}=\boldsymbol{u}_i$,
$\boldsymbol{\sigma}_{\eta_i}=\boldsymbol{\sigma}_i$. We have
\[
{\boldsymbol{\sigma}}_i(t) = \mathscr{E} \boldsymbol{\varepsilon}({\boldsymbol{u}}_i(t)) +\int_0^t \mathscr{G}
(\boldsymbol{\sigma}_i(s),\boldsymbol{\varepsilon}({\boldsymbol{u}}_i(s)))\,ds \quad\forall
t\in[0,T],\nonumber
\]
and, using the properties (\ref{3.10}) and (\ref{3.11}) of $\mathscr{E} $
and $\mathscr{G} $, we find
\begin{align*}
\|\boldsymbol{\sigma}_1(t)-\boldsymbol{\sigma}_2(t)\|_\mathcal{H}
&\leq c\,\Big(\|{\boldsymbol{u}}_1(t)-{\boldsymbol{u}}_2(t)\|_V
 + \int_0^t\|{\boldsymbol{\sigma}}_1(s)-{\boldsymbol{\sigma}}_2(s)\|_\mathcal{H}\,ds\\
&\quad +\int_0^t\|{\boldsymbol{u}}_1(s)-{\boldsymbol{u}}_2(s)\|_V\,ds \Big)\quad\forall t\in
[0,T].
\end{align*}
Using now a Gronwall argument in the previous inequality we deduce
(\ref{4.10}), which concludes the proof.
\end{proof}

We now introduce the  operator $\Theta:L^2(0,T;V')\to L^2(0,T;V')$
which maps every element $\boldsymbol{\eta}\in L^2(0,T;V')$ to the element
$\Theta\boldsymbol{\eta}\in L^2(0,T;V')$ defined by
\begin{equation}\label{4.14}
\begin{split}
\langle\Theta\boldsymbol{\eta}(t),\boldsymbol{w}\rangle_{V'\times V}
&=(\mathscr{E}
\boldsymbol{\varepsilon}({\boldsymbol{u}}_\eta(t)),\boldsymbol{\varepsilon}(\boldsymbol{w}))_\mathcal{H}+(\int_0^t
\mathscr{G} (\boldsymbol{\sigma}_\eta(s),
\boldsymbol{\varepsilon}(\boldsymbol{u}_\eta(s)))\,ds,\boldsymbol{\varepsilon}(\boldsymbol{w}))_\mathcal{H}\\
&\quad +j(\boldsymbol{u}_\eta(t),\boldsymbol{w}) \quad\forall\,\boldsymbol{w}\in
V,\quad\forall\ t\in[0,T].\end{split}
\end{equation}
Here, for every $\boldsymbol{\eta}\in L^2(0,T;V')$, $\boldsymbol{u}_\eta$ and
$\boldsymbol{\sigma}_\eta$ represent the displacement field and the stress
field obtained in Lemmas \ref{l1} and \ref{l2}, respectively. We
have the following result.

\begin{lemma}\label{l3}
The  operator $\Theta$ has a unique fixed point $\boldsymbol{\eta}^*\in
L^2(0,T,V')$.
\end{lemma}

\begin{proof}
 Let $\boldsymbol{\eta}_1,\,\boldsymbol{\eta}_2\in L^2(0,T;V')$,
let $t\in[0,T]$ and denote $\boldsymbol{u}_{\eta_i}=\boldsymbol{u}_i$,
$\boldsymbol{\sigma}_{\eta_i}=\boldsymbol{\sigma}_i$, $i=1,2$. We use (\ref{4.14}),
(\ref{3.10}), (\ref{3.11}) and elementary algebraic manipulations
to obtain
\begin{equation}\label{4.15}
\begin{split}
&|\langle\Theta\boldsymbol{\eta}_1(t)-\Theta\boldsymbol{\eta}_2(t),\boldsymbol{w}\rangle_{V'\times V}|\\
&\le c\,\Big(\|{\boldsymbol{u}}_1(t)-{\boldsymbol{u}}_2(t)\|_V+\int_0^t
\|{\boldsymbol{\sigma}}_1(s)-{\boldsymbol{\sigma}}_2(s)\|_\mathcal{H}\,ds\\
&\quad+ \int_0^t\|{\boldsymbol{u}}_1(s)-{\boldsymbol{u}}_2(s)\|_V\,ds\Big)\|\boldsymbol{w}\|_V
+|j(\boldsymbol{u}_1(t),\boldsymbol{w})-j(\boldsymbol{u}_2(t),\boldsymbol{w})|.
\end{split}
\end{equation}
Now, it follows from (\ref{j}) and (\ref{p}) that
\begin{align*}
|j(\boldsymbol{u}_1(t),\boldsymbol{w})-j(\boldsymbol{u}_2(t),\boldsymbol{w})|
&\le c\,\int_{\Gamma_3}\|{\boldsymbol{u}}_1(t)-{\boldsymbol{u}}_2(t)\|\,\|\boldsymbol{w}\|\,dx\\
&\leq c\,\|{\boldsymbol{u}}_1(t)-{\boldsymbol{u}}_2(t)\|_{L^2(\Gamma_3)^d}
\|\boldsymbol{w}\|_{L^2(\Gamma_3)^d};
\end{align*}
Using (\ref{2.2}), we find
\begin{equation}\label{4.16}
|j(\boldsymbol{u}_1(t),\boldsymbol{w})-j(\boldsymbol{u}_2(t),\boldsymbol{w})|\le
c\,\|{\boldsymbol{u}}_1(t)-{\boldsymbol{u}}_2(t)\|_{V}\|\boldsymbol{w}\|_{V}.
\end{equation}
We substitute (\ref{4.16}) in (\ref{4.15}) and deduce that
\begin{equation}\label{4.17}
\begin{split}
\|\Theta\boldsymbol{\eta}_1(t)-\Theta\boldsymbol{\eta}_1(t)\|_{V'}
&\leq c\,\Big(\|{\boldsymbol{u}}_1(t)-{\boldsymbol{u}}_2(t)\|_{V}+\int_0^t
\|{\boldsymbol{u}}_1(s)-{\boldsymbol{u}}_2(s)\|_V\,ds\\
&\quad+\int_0^t \|{\boldsymbol{\sigma}}_1(s)-{\boldsymbol{\sigma}}_2(s)\|_\mathcal{
H}\,ds\Big).\end{split}
\end{equation}
We use now (\ref{4.10}) in (\ref{4.17}) to obtain
\begin{equation}\label{y}
\|\Theta\boldsymbol{\eta}_1(t)-\Theta\boldsymbol{\eta}_1(t)\|_{V'}\leq
c\,\Big(\|{\boldsymbol{u}}_1(t)-{\boldsymbol{u}}_2(t)\|_{V}+\int_0^t
\|{\boldsymbol{u}}_1(s)-{\boldsymbol{u}}_2(s)\|_V\,ds\Big)
\end{equation}
and, since $\boldsymbol{u}_1(0)=\boldsymbol{u}_2(0)=\boldsymbol{u}_0$, we have
\begin{gather}\label{yy}
\|{\boldsymbol{u}}_1(t)-{\boldsymbol{u}}_2(t)\|_{V}\leq \int_0^t
\|\dot{\boldsymbol{u}}_1(s)-\dot{\boldsymbol{u}}_2(s)\|_V\,ds,
\\ \label{yyy}
\int_0^t \|{\boldsymbol{u}}_1(s)-{\boldsymbol{u}}_2(s)\|_V\,ds\leq c\,\int_0^t
\|\dot{\boldsymbol{u}}_1(s)-\dot{\boldsymbol{u}}_2(s)\|_V\,ds.
\end{gather}
It follows from (\ref{y})--(\ref{yyy}) that
\[
\|\Theta\boldsymbol{\eta}_1(t)-\Theta\boldsymbol{\eta}_1(t)\|_{V'}\leq
c\,\int_0^t\|\dot{\boldsymbol{u}}_1(s)-\dot{\boldsymbol{u}}_2(s)\|_{V}\,ds,
\]
which implies
\begin{equation}\label{4.18}
\|\Theta\boldsymbol{\eta}_1(t)-\Theta\boldsymbol{\eta}_1(t)\|_{V'}^2\leq
c\,\int_0^t\|\dot{\boldsymbol{u}}_1(s)-\dot{\boldsymbol{u}}_2(s)\|_{V}^2\,ds.
\end{equation}
Lemma \ref{l3} is now a direct consequence of inequalities
(\ref{4.18}), (\ref{4.3}) and Banach's fixed point
theorem.
\end{proof}


We now have all the ingredients to prove  Theorem \ref{th1}.


\begin{proof}[Proof of Theorem \ref{th1}]
 Let $\boldsymbol{\eta}^*\in L^2(0,T;V')$ be
the fixed point of the operator $\Theta$ defined by (\ref{4.14})
and denote
\begin{equation}\label{100}
\boldsymbol{u}^*=\boldsymbol{u}_{\eta^*},\quad
\boldsymbol{\sigma}^*=\mathscr{A}\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}^*)+\boldsymbol{\sigma}_{\eta^*}.
\end{equation}
We prove that the couple $(\boldsymbol{u}^*,\boldsymbol{\sigma}^*)$ satisfies
(\ref{3.26})--(\ref{3.28}), (\ref{4.29}) and (\ref{4.30}). Indeed,
we write (\ref{4.9}) for $\boldsymbol{\eta}=\boldsymbol{\eta}^*$ and use (\ref{100}) to
obtain that (\ref{3.26}) is satisfied. Then we use (\ref{4.1}) for
$\boldsymbol{\eta}=\boldsymbol{\eta}^*$ to find
\begin{equation} \label{4.20}
\begin{split}
&\langle\ddot{\boldsymbol{u}}^*(t),\boldsymbol{w}\rangle_{V'\times V} + (\mathscr{A}
\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}^*(t)),\boldsymbol{\varepsilon}(\boldsymbol{w}))_\mathcal{H}
+ \langle{\boldsymbol{\eta}}^*(t),\boldsymbol{w}\rangle_{V'\times V}\\
&= \langle\boldsymbol{f}(t),\boldsymbol{w}\rangle_{V'\times V} \quad \forall\, \boldsymbol{w}\in
V,\ \text{a.e. }t\in (0,T). \end{split}
\end{equation}
Equality $\Theta\boldsymbol{\eta}^*=\boldsymbol{\eta}^*$ combined with (\ref{4.14}) and
(\ref{100}) shows that
\begin{equation}\label{4.21} \begin{split}
\langle\boldsymbol{\eta}^*(t),\boldsymbol{w}\rangle_{V'\times V}
&=(\mathscr{E}\boldsymbol{\varepsilon}(\boldsymbol{u}^*(t)),\boldsymbol{\varepsilon}(\boldsymbol{w}))_\mathcal{H}\\
&\quad+(\int_0^t \mathscr{G}
(\boldsymbol{\sigma}^*(s)-\mathscr{A}\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}^*(s)),
\boldsymbol{\varepsilon}(\boldsymbol{u}^*(s)))\,ds,\boldsymbol{\varepsilon}(\boldsymbol{w}))_\mathcal{H}\\
&\quad+j(\boldsymbol{u}^*(t),\boldsymbol{w}) \quad\forall\,\boldsymbol{w}\in V,\ t\in[0,T].
\end{split}
\end{equation}
We now substitute (\ref{4.21}) in (\ref{4.20}) and use
(\ref{3.26}), to see that $(\boldsymbol{u}^*,\boldsymbol{\sigma}^*)$ satisfies
(\ref{3.27}). Next, (\ref{3.28}) and (\ref{4.29}) follow from
Lemma \ref{l1} and the regularity $\boldsymbol{\sigma}^*\in L^2(0,T;\mathcal{H})$
follows from Lemmas \ref{l1}, \ref{l2} and the second equality in
(\ref{100}). Finally (\ref{3.27}) implies that
\[
\rho\ddot{\boldsymbol{u}}^*(t)={\rm Div}\, \boldsymbol{\sigma}^*(t)+{\boldsymbol{f}}_0(t)\quad{\rm in}\
\ V',\quad \text{a.e. }\ t\in(0,T),
\]
and therefore by
(\ref{3.15}) we find that ${\rm Div}\,\boldsymbol{\sigma}^*\in L^2(0,T;V')$.
We deduce that (\ref{4.30}) holds which concludes the existence
part of the theorem. The uniqueness part is a consequence of the
uniqueness of the fixed point of the operator $\Theta$ defined by
(\ref{4.14}).
\end{proof}

The proof of Theorem \ref{th2} is similar to that of Theorem
\ref{th1} and is carried out in several steps. Since the
modifications are straightforward, we omit the details.

\begin{proof}[Proof of Theorem \ref{th2}]
 The steps of the proof are the following.

(i) For every $\boldsymbol{\eta}\in L^2(0,T;V')$ we prove that there exists a
unique function $\boldsymbol{u}_\eta$ with regularity (\ref{4.29}) such that
\begin{gather}
\begin{aligned}
&\langle\ddot{\boldsymbol{u}}_{\eta}(t),\boldsymbol{w}\rangle_{V'\times V} + (\mathscr{A}
\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}_{\eta}(t)),\boldsymbol{\varepsilon}(\boldsymbol{w}))_\mathcal{H}+
j(\dot{\boldsymbol{u}}_{\eta}(t)),\boldsymbol{w})
+ \langle{\boldsymbol{\eta}}(t),\boldsymbol{w}\rangle_{V'\times V}\\
&= \langle\boldsymbol{f}(t),\boldsymbol{w}\rangle_{V'\times V} \quad \forall\,
\boldsymbol{w}\in V,\ \text{a.e. }t\in (0,T),
\end{aligned} \label{4.1n}\\
\boldsymbol{u}_{\eta}(0)=\boldsymbol{u}_0, \quad \dot{\boldsymbol{u}}_{\eta}(0)=\boldsymbol{v}_0.
\label{4.2n}
\end{gather}
To prove that this holds, we  define the operator $A :
V\to V'$ by
\begin{equation}
\label{4.4n} \langle A\boldsymbol{v},\boldsymbol{w}\rangle_{V'\times V}= (\mathscr{A}
\boldsymbol{\varepsilon}(\boldsymbol{v}),\boldsymbol{\varepsilon}(\boldsymbol{w}))_\mathcal{H} + j(\boldsymbol{v},\boldsymbol{w}) \quad
\forall\,\boldsymbol{v},\,\boldsymbol{w}\in V.
\end{equation}
We prove that $A$ satisfies conditions (\ref{2.5}) and
(\ref{2.6}), then we use again Theorem \ref{thB} and proceed like
in the proof of Lemma \ref{l1}.

Moreover, we use estimates similar to those in the prof of Lemma
\ref{l1} to see that, if $\boldsymbol{u}_i$ represents the solution of
problem (\ref{4.1n})--(\ref{4.2n}) for $\boldsymbol{\eta}_i\in L^2(0,T;V')$,
$i=1,2$, then there exists $c>0$ such that (\ref{4.3}) holds.


(ii) We use the displacement field $\boldsymbol{u}_\eta$ obtained in step i)
and Lemma \ref{l2} to prove that there exists a unique function
$\boldsymbol{\sigma}_\eta\in W^{1,2}(0,T;\mathcal{H})$ which satisfies (\ref{4.9})
for all $t\in[0,T]$. Moreover, if $\boldsymbol{\sigma}_i$ and $\boldsymbol{u}_i$
represent the solutions obtained above for $\boldsymbol{\eta}_i\in
L^2([0,T];V')$, $i=1,2$, then there exists $c>0$ such that
(\ref{4.10}) holds.


(iii)  We now introduce the  operator $\Theta:L^2(0,T;V')\to
L^2(0,T;V')$ which maps every element $\boldsymbol{\eta}\in L^2(0,T;V')$ to
the element $\Theta\boldsymbol{\eta}\in L^2(0,T;V')$ defined by
\begin{equation}\label{4.14n}
\langle\Theta\boldsymbol{\eta}(t),\boldsymbol{w}\rangle_{V'\times V}
=(\mathscr{E} \boldsymbol{\varepsilon}({\boldsymbol{u}}_\eta(t)),\boldsymbol{\varepsilon}(\boldsymbol{w}))_\mathcal{H} \\
+ (\int_0^t \mathscr{G} (\boldsymbol{\sigma}_\eta(s),
\boldsymbol{\varepsilon}(\boldsymbol{u}_\eta(s)))\,ds,\boldsymbol{\varepsilon}(\boldsymbol{w}))_\mathcal{H}
\end{equation}
for all $\boldsymbol{w}\in V$, $t\in[0,T]$.
Here, for every $\boldsymbol{\eta}\in L^2(0,T;V')$, $\boldsymbol{u}_\eta$ and
$\boldsymbol{\sigma}_\eta$ represent the displacement field and the stress
field obtained in steps i) and ii) respectively. We use
(\ref{4.10}) and estimates similar to those used in Lemma \ref{l3}
to prove that the operator $\Theta$ satisfies (\ref{4.18}). It
follows now from (\ref{4.3}) and Banach's fixed point theorem and
that the operator $\Theta$ has a unique fixed point $\boldsymbol{\eta}^*\in
L^2(0,T,V')$.

(iv) Let $\boldsymbol{\eta}^*\in L^2(0,T;V')$ be the fixed point of the
operator $\Theta$ defined by (\ref{4.14n}) and denote
\begin{equation}\label{100n}
\boldsymbol{u}^*=\boldsymbol{u}_{\eta^*},\quad
\boldsymbol{\sigma}^*=\mathscr{A}\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}^*)+\boldsymbol{\sigma}_{\eta^*}.
\end{equation}
We use equality $\Theta\boldsymbol{\eta}^*=\boldsymbol{\eta}^*$, (\ref{100n}) and the
definition (\ref{4.14n}) of the operator $\Theta$ to prove that
the couple $(\boldsymbol{u}^*,\boldsymbol{\sigma}^*)$  is a solution of Problem $\mathscr{P}_2^V$
and it satisfies \eqref{4.29}--\eqref{4.30}. This concludes
the existence part of the theorem. The uniqueness follows from the
uniqueness of the fixed point of the operator $\Theta$ defined by
(\ref{4.14n}), obtained in step iii).\end{proof}


\section{Continuous dependence results}\label{cd}

In this section we study the dependence of the solution of the
problem $\mathscr{P}_1^V$ and $\mathscr{P}_2^V$ with respect to a perturbation of
the contact conditions. To avoid repetitions we restrict ourselves
to the study of Problem $\mathscr{P}_1^V$ and we note that a result
similar to  that in Theorem \ref{th3} below can be obtained in the
study of Problem $\mathscr{P}_2^V$. We suppose in what follows that
(\ref{3.9})--(\ref{3.16})  hold and denote by $(\boldsymbol{u},\boldsymbol{\sigma})$ the
solution of Problem $\mathscr{P}_1^V$ obtained in Theorem \ref{th1}. Also,
for all $\alpha>0$ we denote by $p^\alpha$ a perturbation of $p$,
which satisfies (\ref{p}) with $L_p$ replaced by $L_p^\alpha$. We
introduce the functional $j^\alpha$ defined by (\ref{j}) replacing
$p$ with $p^\alpha$ and we consider the following variational
problem.


\subsection*{Problem $\mathscr{P}_1^{V^\alpha}$}
{\em Find a displacement field
$\boldsymbol{u}^\alpha:[0,T]\to V$ and a stress field
$\boldsymbol{\sigma}^\alpha:[0,T]\to \mathcal{H}$ such that, for a.e.
$t\in(0,T)$,}
\begin{gather}
{\boldsymbol{\sigma}^\alpha}(t) = \mathscr{A} \boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}^\alpha(t))+\mathscr{E}
\boldsymbol{\varepsilon}({\boldsymbol{u}^\alpha}(t)) +\int_0^t \mathscr{G}
(\boldsymbol{\sigma}^\alpha(s)-\mathscr{A}\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}^\alpha(s)),
\boldsymbol{\varepsilon}({\boldsymbol{u}^\alpha}(s)))\,ds, \label{5.1}
\\
\label{5.2} \langle\ddot{\boldsymbol{u}}^\alpha(t),\boldsymbol{w}\rangle_{V'\times V} +
(\boldsymbol{\sigma}^\alpha(t),\boldsymbol{\varepsilon}(\boldsymbol{w}))_\mathcal{H} +
j^\alpha(\boldsymbol{u}^\alpha(t),\boldsymbol{w})=\langle\boldsymbol{f}(t),\boldsymbol{w}\rangle_{V'\times
V}\quad\forall\boldsymbol{w}\in V,
\\
\boldsymbol{u}^\alpha(0)=\boldsymbol{u}_0,\quad\dot{\boldsymbol{u}}^\alpha(0)=\boldsymbol{v}_0. \label{5.3}
\end{gather}

We deduce from the Theorem \ref{th1} that, for every $\alpha>0$,
problem $\mathscr{P}_1^{V^\alpha}$ has a unique solution
$(\boldsymbol{u}^\alpha,\boldsymbol{\sigma}^\alpha)$ which satisfies
(\ref{4.29})--(\ref{4.30}). Assume that the contact function
satisfies the following hypothesis :
\begin{gather}\label{palpha}
\left\{
\begin{array}{l}
\text{There exist }\beta \in\mathbb{R}_{+}\text{ and }\theta
:\,]0,+\infty[\to [0,+\infty[\,\text{ such that} \\
\mathrm{(a)}\left| p^{\alpha }(\boldsymbol{x},r)-p(\boldsymbol{x},r)\right| \leq \theta
(\alpha )(\left| r\right| +\beta )  \quad\forall \alpha>0,\ \
r\in\mathbb{R},\text{ p.p. }\boldsymbol{x}\in
\Gamma _{3}. \\
\mathrm{(b)}\underset{\alpha \to 0}{\lim }\theta (\alpha
)=0.
\end{array} \right.
\\
\label{palph}
{\rm There\ exists}\ \ L_{0}>0\ \ {\rm such\ that}\ \ L^{\alpha
}\leq L_{0}\  \ {\rm for\ all}\ \ \alpha>0.
\end{gather}


Under these hypotheses, we have the following convergence result.


\begin{theorem}\label{th3}
{The solution $(\boldsymbol{u}^\alpha,\boldsymbol{\sigma}^\alpha)$ of the
problem $\mathscr{P}_1^{V^\alpha}$ converges to the solution
$(\boldsymbol{u},\boldsymbol{\sigma})$ of the problem $\mathscr{P}_1^V$, i.e.
\begin{equation}\label{5.4}
\boldsymbol{u}^\alpha\to\boldsymbol{u}\quad{\rm in}\quad W^{1,2}(0,T;V),\quad
\boldsymbol{\sigma}^\alpha\to\boldsymbol{\sigma}\quad{\rm in }\quad L^2(0,T;\mathcal{
H})\quad{\rm as }\quad \alpha\to 0.
\end{equation}}
\end{theorem}


In addition to the interest in this convergence result from the
asymptotic analysis point of view, it is important from mechanical
point of view since it shows that small perturbations on the
contact conditions lead to small perturbations of the weak
solution of the dynamic contact problem $\mathscr{P}_1$.

\begin{proof}
 Let $\alpha>0$.
Everywhere below $c$ denotes a positive constant which may depend
on the data and on the solution $(\boldsymbol{u},\boldsymbol{\sigma})$ but does not
depend on $\alpha$, nor on the time variable, and whose value may
change from line to line. Using (\ref{3.27}) and (\ref{5.2}) we
obtain
\begin{equation}\label{5.5}
\begin{aligned}
&(\ddot{\boldsymbol{u}}^\alpha(t)-\ddot{\boldsymbol{u}}(t),\dot{\boldsymbol{u}}^\alpha(t)
-\dot{\boldsymbol{u}}(t))_{V'\times V} +
(\boldsymbol{\sigma}^\alpha(t)-\boldsymbol{\sigma}(t),\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}^\alpha(t))-
\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}(t)))_\mathcal{H}\\
&+j^\alpha(\boldsymbol{u}^\alpha(t),\dot{\boldsymbol{u}}^\alpha(t)-\dot{\boldsymbol{u}}(t))-j(\boldsymbol{u}(t),
\dot{\boldsymbol{u}}^\alpha(t)-\dot{\boldsymbol{u}}(t))\\
&=0 \quad\text{a.e. }t\in(0,T).
\end{aligned}
\end{equation}
We define
\begin{equation}\label{sigmaR}
\boldsymbol{\sigma}^{\alpha R}(t)=\boldsymbol{\sigma}^\alpha (t)- \mathscr{A}
\boldsymbol{\varepsilon}({\dot{\boldsymbol{u}}}^\alpha(t)), \quad
\boldsymbol{\sigma}^{R}(t)=\boldsymbol{\sigma}(t)-\mathscr{A} \boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}(t))
\end{equation}
and note that (\ref{5.1}) and (\ref{3.26}) yield
\begin{equation}\label{5.22}
\begin{gathered}
\boldsymbol{\sigma}^{\alpha R}(t)=\mathscr{E}\boldsymbol{\varepsilon}(\boldsymbol{u}^{\alpha}(t))+
\int_0^t\mathscr{G}(\boldsymbol{\sigma}^{\alpha
R}(s),\boldsymbol{\varepsilon}(\boldsymbol{u}^\alpha (s)))ds,
\\
\boldsymbol{\sigma}^{R}(t)=\mathscr{E}\boldsymbol{\varepsilon}(\boldsymbol{u}(t))+\int_0^t
\mathscr{G}(\boldsymbol{\sigma}^{R}(s),\boldsymbol{\varepsilon}(\boldsymbol{u}(s)))ds
\end{gathered}
\end{equation}
for all $t\in[0,T]$. We combine   (\ref{5.5}) and (\ref{sigmaR})
to obtain
\begin{equation}\label{x}
\begin{aligned}
&(\ddot{\boldsymbol{u}}^\alpha(t)-\ddot{\boldsymbol{u}}(t),\dot{\boldsymbol{u}}^\alpha(t)-\dot{\boldsymbol{u}}(t))_{V'\times
V}\\
&+(\mathscr{A}\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}^\alpha(t))-\mathscr{A}\boldsymbol{\varepsilon}
(\dot{\boldsymbol{u}}(t)),\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}^\alpha(t))-\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}(t)))_\mathcal{
H}\\
&=-(\boldsymbol{\sigma}^{\alpha
R}(t)-\boldsymbol{\sigma}^R(t),\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}^\alpha(t))-
\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}(t)))_\mathcal{
H}\\
&\quad+j(\boldsymbol{u}(t),\dot{\boldsymbol{u}}^\alpha(t)-\dot{\boldsymbol{u}}(t))-j^\alpha(\boldsymbol{u}^\alpha(t),
\dot{\boldsymbol{u}}^\alpha(t)-\dot{\boldsymbol{u}}(t))\quad\text{a.e. }t\in(0,T).
\end{aligned}
\end{equation}
It follows from (\ref{3.9}) that
\begin{equation}\label{5.6}
(\mathscr{A}\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}^\alpha(t))-\mathscr{A}\boldsymbol{\varepsilon}
(\dot{\boldsymbol{u}}(t)),\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}^\alpha(t))-\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}(t)))_\mathcal{
H}\geq m_{\mathscr{A}}\,\|\dot{\boldsymbol{u}}^\alpha(t)-\dot{\boldsymbol{u}}(t)\|_V^2
\end{equation}
a.e. $t\in(0,T)$. Using (\ref{5.22}),
\begin{equation}\label{5.23}
\begin{aligned}
\boldsymbol{\sigma}^{\alpha R}(t)-\boldsymbol{\sigma}^{R}(t)
&=\mathscr{E}\boldsymbol{\varepsilon}
(\boldsymbol{u}^{\alpha}(t))-\mathscr{E}\boldsymbol{\varepsilon}(\boldsymbol{u}(t))
+\int_0^t\mathscr{G}
(\boldsymbol{\sigma}^{\alpha R}(s),\boldsymbol{\varepsilon}(\boldsymbol{u}^\alpha(s)))ds\\
&\quad - \int_0^t\mathscr{G}(\boldsymbol{\sigma}^{R}(s),\boldsymbol{\varepsilon}(\boldsymbol{u}(s)))ds\quad\forall
t\in[0,T].
\end{aligned}
\end{equation}
We use now   (\ref{5.23}), (\ref{3.10}) and (\ref{3.11}) to obtain
\begin{align*}
\|\boldsymbol{\sigma}^{\alpha R}(t)-\boldsymbol{\sigma}^{ R}(t)\|_\mathcal{H}
&\leq c\,\Big(\| {\boldsymbol{u}}^{\alpha}(t) \ -\boldsymbol{u}(t)\|_V + \int_0^t\|\boldsymbol{\sigma}^
{ \alpha R}(s)-\boldsymbol{\sigma}^{R}(s)\|_\mathcal{H}ds\\
&\quad + \int_0^t \|\boldsymbol{u}^{\alpha}(s)-\boldsymbol{u}(s)\|_V\,ds\Big)\quad\forall t\in[0,T].
\end{align*}
After a Gronwall argument we deduce
\begin{equation}\label{ineq}
 \|\boldsymbol{\sigma}^{ \alpha R}(t)-\boldsymbol{\sigma}^R(t)\|_\mathcal{H}\leq
c\,\Big(\|\boldsymbol{u}^{\alpha}(t)-\boldsymbol{u}(t)\|_V+ \int_0^t
\|\boldsymbol{u}^{\alpha}(s)-\boldsymbol{u}(s)\|_V\,ds\Big)
\end{equation}
for all $t\in[0,T]$.
The above inequality shows that
\begin{equation}\label{5.28}
\begin{aligned}
&-(\boldsymbol{\sigma}^{\alpha R}(t)-\boldsymbol{\sigma}^{R}(t),\boldsymbol{\varepsilon}
(\dot{\boldsymbol{u}}^\alpha(t))-\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}(t)))_\mathcal{H}\\
&\leq c\,\Big(\|\boldsymbol{u}^{\alpha}(t)-\boldsymbol{u}(t)\|_V
+\int_0^t \|\boldsymbol{u}^{\alpha}(s)-\boldsymbol{u}(s)\|_V\,ds\Big)
\|\dot{\boldsymbol{u}}^\alpha(t)-\dot{\boldsymbol{u}}(t)\|_V
\quad \text{a.e. }t\in(0,T).
\end{aligned}
\end{equation}
Note that from the definition of the functionals $j$ and
$j^\alpha$ it follows that
\begin{align*}
& j(\boldsymbol{u}(t),\dot{\boldsymbol{u}}^{\alpha}(t)-\dot{\boldsymbol{u}}(t))-j^{\alpha
}(\boldsymbol{u}^{\alpha}(t),\dot{\boldsymbol{u}}^{\alpha }(t)-\dot{\boldsymbol{u}}(t))\\
&=\int_{\Gamma _{3}}(p(u_{\nu}(t))-p^{\alpha
}(u_\nu(t)))(\dot{u}_{\nu}^{\alpha}(t))-\dot{u}_{\nu}(t))\,da
\\
&\quad+\int_{\Gamma_{3}}(p^{\alpha}(u_{\nu}(t))-
p^{\alpha}(u^\alpha_\nu(t)))(\dot{u}_{\nu
}^{\alpha}(t))-\dot{u}_{\nu}(t))\,da\quad\text{a.e. }t\in(0,T).
\end{align*}
Using (\ref{palpha}), (\ref{palph}) and (\ref{2.2}) we deduce
that
\begin{equation}\label{5.35}
\begin{aligned}
&j(\boldsymbol{u}(t),\dot{\boldsymbol{u}}^{\alpha}(t)-\dot{\boldsymbol{u}}(t))-j^{\alpha
}(\boldsymbol{u}^{\alpha}(t),\dot{\boldsymbol{u}}^{\alpha }(t)-\dot{\boldsymbol{u}}(t))\\
&\leq c\,\big(\theta(\alpha) +
\|{\boldsymbol{u}}^\alpha(t)-\boldsymbol{u}(t)\|_V\big)
\|{\dot{\boldsymbol{u}}}^\alpha(t)-\dot{\boldsymbol{u}}(t)\|_V\quad\text{a.e. }t\in(0,T).
\end{aligned}
\end{equation}
We use now  (\ref{x}), (\ref{5.6}), (\ref{5.28}) and (\ref{5.35})
to obtain
\begin{equation}\label{5.11}
\begin{aligned}
& (\ddot{\boldsymbol{u}}^{\alpha}(t)-\ddot{\boldsymbol{u}}(t),\dot{\boldsymbol{u}}^{\alpha
}(t)-\dot{\boldsymbol{u}}(t))_{V^{\prime ^{\prime }}\times
V}+m_{\mathscr{A}}\,\|\dot{\boldsymbol{u}}^{\alpha
}(t)-\dot{\boldsymbol{u}}(t)\|_{V}^{2}\\
&\leq c\,\Big(\theta(\alpha) + \|{\boldsymbol{u}}^\alpha(t)-\boldsymbol{u}(t)\|_V+
\int_0^t \|\boldsymbol{u}^{\alpha}(s)-\boldsymbol{u}(s)\|_V ds\Big)
\|{\dot{\boldsymbol{u}}}^\alpha(t)-\dot{\boldsymbol{u}}(t)\|_V
\end{aligned}
\end{equation}
a.e. $t\in(0,T)$.
Using the inequality
\[
ab\leq\frac{1}{2m_{\mathscr{A}}}\,a^2+ \frac{m_{\mathscr{A}}}{2}\,b^2,
\]
 after some algebra we find that
\begin{align*}
& (\ddot{\boldsymbol{u}}^{\alpha}(t)-\ddot{\boldsymbol{u}}(t),\dot{\boldsymbol{u}}^{\alpha
}(t)-\dot{\boldsymbol{u}}(t))_{V^{\prime ^{\prime }}\times
V}+\frac{m_{\mathscr{A}}}{2}\,\|\dot{\boldsymbol{u}}^{\alpha
}(t)-\dot{\boldsymbol{u}}(t)\|_{V}^{2}\\
&\leq c\,\Big(\theta^2(\alpha) +
\|{\boldsymbol{u}}^\alpha(t)-\boldsymbol{u}(t)\|^2_V+ \int_0^t
\|\boldsymbol{u}^{\alpha}(s)-\boldsymbol{u}(s)\|^2_V ds\Big) \quad\text{a.e. }t\in(0,T).
\end{align*}

We integrate the previous inequality on $[0,s]$ and use the
initial conditions $\dot{\boldsymbol{u}}^\alpha(0)=\dot{\boldsymbol{u}}(0)=\boldsymbol{v}_0$ to
find that
\begin{equation}\label{xx}
\frac{m_{\mathscr{A}}}{2}\int_0^s\|\dot{\boldsymbol{u}}^{\alpha
}(t)-\dot{\boldsymbol{u}}(t)\|_{V}^{2}\leq c\,\Big(\theta^2(\alpha)+\int_0^s
\|\boldsymbol{u}^{\alpha}(t)-\boldsymbol{u}(t)\|^2_V ds\Big) \quad\forall s\in[0,T].
\end{equation}
We use now (\ref{3.28}) and (\ref{5.3}) to see that
\begin{equation}\label{5.13}
\|\boldsymbol{u}^\alpha(s)-\boldsymbol{u}(s)\|_V^2\leq
c\,\int_0^s\|\dot{\boldsymbol{u}}^\alpha(t)-\dot{\boldsymbol{u}}(t)\|_V^2 dt
\hspace{5mm}\forall s\in [0,T].
\end{equation}
We substitute  (\ref{xx}) in (\ref{5.13}) then we use again the
Gronwall inequality to find that
\begin{equation}\label{5.14}
\|\boldsymbol{u}^\alpha(s)-\boldsymbol{u}(s)\|_V^2\leq c\,\theta^2(\alpha)\quad\forall
s\in[0,T].
\end{equation}
Using  now (\ref{xx}) and (\ref{5.14}) it follows that
\begin{equation}\label{5.15}
\int_0^s\|\dot{\boldsymbol{u}}^\alpha(t)-\dot{\boldsymbol{u}}(t)\|_V^2 ds\leq
c\,\theta^2(\alpha)\quad\forall s\in[0,T].
\end{equation}
We combine now (\ref{5.14}), (\ref{5.15}) and use assumption
(\ref{palpha})(b) to see that
\begin{equation}\label{5.16}
\boldsymbol{u}^\alpha\to \boldsymbol{u}\quad{\rm in}\quad
W^{1,2}(0,T;V)\quad{\rm as}\quad\alpha\to 0.
\end{equation}
It follows from (\ref{sigmaR}) that
\[
\boldsymbol{\sigma}^\alpha (t)-\boldsymbol{\sigma}(t)= \boldsymbol{\sigma}^{\alpha
R}(t)-\boldsymbol{\sigma}^{R}(t)+ \mathscr{A} \boldsymbol{\varepsilon}({\dot{\boldsymbol{u}}}^\alpha(t))-\mathscr{A}
\boldsymbol{\varepsilon}(\dot{\boldsymbol{u}}(t))\quad\text{a.e. }t\in(0,T).
\]
Using this
inequality, (\ref{ineq}), the properties (\ref{3.9}) of the
operator $\mathscr{A}$ and (\ref{5.16}) in obtain
\begin{equation}\label{5.17}
\boldsymbol{\sigma}^\alpha\to\boldsymbol{\sigma}\quad\text{in }
L^{2}(0,T;\mathcal{H})\quad\text{as }\alpha\to 0.
\end{equation}
Theorem \ref{th3} is now a consequence of (\ref{5.16}),
(\ref{5.17}).
 \end{proof}

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