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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 58, pp. 1--9.\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 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/58\hfil A quasistatic bilateral contact problem]
{A quasistatic bilateral contact problem with friction for 
nonlinear elastic materials}

\author[A. Touzaline \hfil EJDE-2006/58\hfilneg]
{Arezki Touzaline}

\address{Facult\'{e} de Math\'{e}matiques, USTHB \\
BP 32 EL ALIA, Bab-Ezzouar, 16111, Alg\'{e}rie}
\email{atouzaline@yahoo.fr}
\date{}

\thanks{Submitted December 26, 2005. Published May 1, 2006.}
\subjclass[2000]{35J85, 49J40, 47J20, 58E35}
\keywords{Nonlinear elasticity; bilateral contact; friction; 
\hfill\break\indent
 variational inequality; weak solution}

\begin{abstract}
   We consider a mathematical model describing the bilateral 
   contact between a deformable body and a foundation. We use 
   a nonlinear elastic constitutive law. The contact takes into 
   account the effects of friction, which are modelled with the 
   regularized friction law. We derive a variational formulation 
   of the problem and establish the existence of a weak solution
   under a smallness assumption of the friction coefficient. 
   The proof is based on arguments of compactness, lower 
   semicontinuity and time discretization.
\end{abstract}

\maketitle

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

\section{Introduction}

In this paper we study the existence of a solution for a quasistatic
bilateral contact problem with friction for nonlinear elastic materials. For
linear elastic materials the quasistatic contact problem using a normal
compliance law has been solved in \cite{a1} by considering incremental
problems and in \cite{k1} by an other method using a regularization relative
to time. The quasistatic contact problem with local or nonlocal friction has
been solved respectively in \cite{r1} and in \cite{c1} by using a
time-discretization. The same method was also used in \cite{t1} to solve the
quasistatic problem with unilateral contact involving nonlocal friction law
for nonlinear elastic materials. In \cite{a2} the quasistatic contact
problem with Coulomb friction was solved by the aid of an established
shifting technique used to obtain increased regularity at the contact
surface and by the aid of auxiliary problems involving regularized friction
terms and a so-called normal compliance penalization technique. Signorini 's
problem with friction for nonlinear elastic materials or viscoplastic
materials has been solved in \cite{d1} by using the fixed point's method. In
viscoelasticity, the quasistatic contact problem with normal compliance and
friction has been solved in \cite{r2} for nonlinear viscoelastic materials
by the same fixed point arguments. In the book \cite{h1} the authors resolve
the quasistatic contact problems in viscoelasticity and viscoplasticity.
Carrying out the variational analysis, the authors systymatically use
results on elliptic and evolutionary variational inequalities, convex
analysis, nonlinear equations with monotone operators, and fixed points of
operators.

Here we propose a variational formulation using a regularization of the
normal stress. We model the friction by Tresca's law, by nonlocal law as in 
\cite{t1} and by a modified version of Coulomb's law which has been derived
in \cite{s1} to take into account the wear of the contacting surface. The
variational formulation is written in the form of a single variational
inequality. By means of Euler's implicit scheme as in \cite{c1,r1,t1}, the
bilateral contact problem leads us to solve a well-posed variatonal
inequality at each time step. Finally by using lower semicontinuity and
compactness arguments we prove that the limit of the discrete solution is a
solution to the continuous problem.

\section{Problem statement and variational formulation}

We consider a nonlinear elastic body which is in frictional contact with a
rigid foundation. Time dependent volume forces and surface traction act on
it, and as result there is evolution of its mechanical state. Our interest
is in modelling this evolution. We assume that the forces and traction vary
slowly with time and therefore the accelerations in the system are
negligible. Also, we assume that there is no loss of contact between the
body and the foundation.

The physical setting is as follows. Let $\Omega \subset \mathbf{R}^{d}$; 
($d=2,3$), be the domain initially occupied by the nonlinear elastic body. 
$\Omega $ is supposed to be open, bounded, with a sufficiently regular
boundary $\Gamma $. $\Gamma $ is partitioned into three parts 
$\Gamma =\bar{\Gamma}_{1}\cup \bar{\Gamma}_{2}\cup \bar{\Gamma}_{3}$ 
where $\Gamma _{1},\Gamma _{2},\Gamma _{3}$ are disjoint open sets 
and $\mathop{\rm meas}\Gamma_{1}>0$. The body is acted upon by volume 
forces of density $\phi _{1}$ on $\Omega $ and surface traction of density
 $\phi _{2}$ on $\Gamma _{2}$. On $\Gamma _{3}$ the body is in bilateral 
contact with a rigid foundation.

The classical formulation of the mechanical problem is written as follows.
\smallskip

\noindent \textbf{Problem P1}. Find a displacement field $u:\Omega \times
\lbrack 0,T]\rightarrow \mathbf{R}^{d}$ such that 
\begin{gather}
\mathop{\rm div}\sigma +\phi _{1}=0\quad \text{in }\Omega \times (0,T),
\label{e2.1} \\
\sigma =F(\varepsilon (u))\quad \text{in }\Omega \times (0,T),  \label{e2.2}
\\
u=0\quad \text{on }\Gamma _{1}\times (0,T),  \label{e2.3} \\
\sigma \nu =\phi _{2}\quad \text{on }\Gamma _{2}\times (0,T),  \label{e2.4}
\\
\left. \begin{gathered} u_{\nu }=0, \quad |\sigma _{\tau }|\leq \mu
p(|R\sigma _{\nu }(u)|)\\ |\sigma _{\tau }|<\mu p(|R\sigma _{\nu
}(u)|)\Longrightarrow \dot{u}_{\tau }=0 \\ |\sigma _{\tau }|=\mu p(|R\sigma
_{\nu }(u)|)\Longrightarrow \exists \lambda \geq 0: \sigma _{\tau }=-\lambda
\dot{u}_{\tau } \end{gathered}\right\} \text{on }\Gamma _{3}\times (0,T),
\label{e2.5} \\
u(0)=u_{0}\quad \text{in }\Omega .  \label{e2.6}
\end{gather}
Equality \eqref{e2.1} represents the equilibrium equation. Equality 
\eqref{e2.2} represents the elastic constitutive law of the material in
which $F$ is a given function and $\varepsilon (u)$ denotes the small strain
tensor; \eqref{e2.3} and \eqref{e2.4} are the displacement and traction
boundary conditions, respectively, in which $\nu $ denotes the unit outward
normal on $\Gamma $ and $\sigma \nu $ represents the Cauchy stress vector.
Conditions \eqref{e2.5} represents the bilateral contact boundary conditions
and the associate friction law in which $\sigma _{\tau }$ denotes the
tangential stress, $\dot{u}_{\tau }$ denotes the tangential velocity on the
boundary and $\mu $ is the coefficient of friction. Finally \eqref{e2.6}
represent the initial condition. In \eqref{e2.5} and below, a dot above a
variable represents its derivative with respect to time. We denote by $S_{d}$
the space of second order symmetric tensors on $\mathbf{R}^{d}$ $(d=2,3)$.
To proceed with the variational formulation, we need the following function
spaces: 
\begin{gather*}
H=L^{2}(\Omega )^{d},H_{1}=(H^{1}(\Omega ))^{d}, \\
Q=\{\tau =(\tau _{ij});\tau _{ij}=\tau _{ji}\in L^{2}(\Omega )\}, \\
H(\mathop{\rm div};\Omega )=\{\sigma \in Q;\mathop{\rm div}\sigma \in H\}.
\end{gather*}
Note that $H$ and $Q$ are real Hilbert spaces endowed with the respective
canonical inner products 
\begin{equation*}
(u,v)_{H}=\int_{\Omega }u_{i}v_{i}dx,\quad \langle \sigma ,\tau \rangle
_{Q}=\int_{\Omega }\sigma _{ij}\tau _{ij}dx.
\end{equation*}
The small strain tensor is 
\begin{equation*}
\varepsilon (u)=(\varepsilon _{ij}(u))=\big(\frac{1}{2}(u_{i,j}+u_{j,i})),
\quad i,j\in \{1,\dots ,d\};
\end{equation*}
$\mathop{\rm div}\sigma =(\sigma _{ij,j})$ is the divergence of $\sigma $.
Let $H_{\Gamma }=H^{1/2}(\Gamma )^{d}$ and let $\gamma :H_{1}\rightarrow
H_{\Gamma }$ be the trace map. For every element $v\in H_{1}$, we also use
the notation $v$ for the trace $\gamma v$ of $v$ on $\Gamma $ and we denote
by $v_{\nu }$ and $v_{\tau }$ the normal and tangential components of $v$ on 
$\Gamma $ given by 
\begin{equation*}
v_{\nu }=v.\nu ,\quad v_{\tau }=v-v_{\nu }\nu .
\end{equation*}
Let $H_{\Gamma }'$ be the dual of $H_{\Gamma }$, for every $\sigma
\in H(\mathop{\rm div}\Omega )$, $\sigma \nu $ can be defined as the element
in $H_{\Gamma }'$ which satisfies the Green's formula: 
\begin{equation*}
\langle \sigma ,\varepsilon (v)\rangle _{Q}+(\mathop{\rm div}\sigma
,v)_{H}=\langle \sigma \nu ,v\rangle _{H_{\Gamma }'\times H_{\Gamma
}}\quad \forall v\in H_{1}.
\end{equation*}
Denote by $\sigma _{\nu }$ and $\sigma _{\tau }$ the normal and tangential
traces of $\sigma $, respectively. If $\sigma $ is regular (say $C^{1}$),
then 
\begin{gather*}
\sigma _{\nu }=(\sigma \nu ).\nu ,\quad \sigma _{\tau }=\sigma -\sigma _{\nu
}\nu  \\
\langle \sigma \nu ,v\rangle _{H_{\Gamma }'\times H_{\Gamma
}}=\int_{\Gamma }\sigma \nu .v\,da
\end{gather*}
for all $v\in H_{1}$, where $da$ is the surface measure element. Let $V$ be
the closed subspace of $H_{1}$ defined by 
\begin{equation*}
V=\{v\in H_{1}:v=0\text{ on }\Gamma _{1},\;v_{\nu }=0\text{ on }\Gamma
_{3}\}.
\end{equation*}
Since $\mathop{\rm meas}\Gamma _{1}>0$, the following Korn's inequality
holds \cite{d3}, 
\begin{equation}
\Vert \varepsilon (v)\Vert _{Q}\geq c_{\Omega }\Vert v\Vert _{H_{1}}\quad
\forall v\in V  \label{e2.7}
\end{equation}
where the constant $c_{\Omega }$ depends only on $\Omega $ and $\Gamma _{1}$. 
We equip $V$ with the inner product 
\begin{equation*}
(u,v)_{V}=\langle \varepsilon (u),\varepsilon (v)\rangle _{Q}
\end{equation*}
and $\Vert \cdot \Vert _{V}$ is the associated norm. It follows from Korn's
inequality \eqref{e2.7} that the norms $\Vert \cdot \Vert _{H_{1}}$ and 
$\Vert \cdot \Vert _{V}$ are equivalent on $V$. Then $(V,\Vert \cdot \Vert
_{V})$ is a real Hilbert space. Moreover by the Sobolev's trace theorem,
there exists $d_{\Omega }>0$ which only depends on the domain $\Omega $, 
$\Gamma _{1}$ and $\Gamma _{3}$ such that 
\begin{equation}
\Vert v\Vert _{L^{2}(\Gamma _{3})^{d}}\leq d_{\Omega }\Vert v\Vert _{V}\quad
\forall v\in V  \label{e2.8}
\end{equation}
For $p\in \lbrack 1,\infty ]$ , we use the standard norm of $L^{p}(0,T;V)$.
We also use the Sobolev space $W^{1,\infty }(0,T;V)$ equipped with the norm 
\begin{equation*}
\Vert v\Vert _{W^{1,\infty }(0,T;V)}=\Vert v\Vert _{L^{\infty
}(0,T;V)}+\Vert \dot{v}\Vert _{L^{\infty }(0,T;V)}.
\end{equation*}
For every real Banach space $(X,\Vert \cdot \Vert _{X})$ and $T>0$ we use
the notation $C([0,T];X)$ for the space of continuous functions from $[0,T]$
to $X$; recall that $C([0,T];X)$ is a real Banach space with the norm 
\begin{equation*}
\Vert x\Vert _{C([0,T];X)}=\max_{t\in \lbrack 0,T]}\Vert x(t)\Vert _{X}.
\end{equation*}
We suppose 
\begin{equation}
\phi _{1}\in W^{1,\infty }(0,T;H),\quad \phi _{2}\in W^{1,\infty
}(0,T;L^{2}(\Gamma _{2})^{d})  \label{e2.9}
\end{equation}
and we denote by $f(t)$ the element of $V$ defined by 
\begin{equation}
(f(t),v)_{V}=\int_{\Omega }\phi _{1}(t).v\,dx+\int_{\Gamma _{2}}\phi
_{2}(t).v\,da\text{ \ \ }\forall v\in V,\text{ for }t\in \lbrack 0,T]
\label{e2.10}
\end{equation}
Using \eqref{e2.9} and \eqref{e2.10} yields $f\in W^{1,\infty }(0,T;V)$. 
$\langle .,.\rangle $ shall denote the duality pairing on 
$H^{1/2}(\Gamma_{3})\times H^{-1/2}(\Gamma _{3})$, where 
\begin{equation*}
H^{1/2}(\Gamma _{3})=\{w\big|_{\Gamma _{3}}:w\in H^{1/2}(\Gamma ),w=0\text{
on }\Gamma _{1}\}.
\end{equation*}
The normal stress $\sigma _{\nu }(u(t))\in H^{-1/2}(\Gamma _{3})$ associated
with a function $u(t)\in V$ is defined by 
\begin{equation}
\begin{gathered} \forall \text{ }w\in H^{1/2}(\Gamma _{3}): \\ \langle
\sigma _{\nu },w\rangle =\langle F ( \varepsilon (u(t))),\varepsilon (
v)\rangle _{Q}-(f(t),v)_{V} \\ \forall \text{ }v\in H_{1}: v=0\text{ on
}\Gamma _{1}\text{ and } v_{\nu }=w,\quad v_{\tau }=0\quad \text{on }\Gamma
_{3}. \end{gathered}  \label{e2.11}
\end{equation}
$R:H^{-1/2}(\Gamma _{3})\rightarrow L^{2}(\Gamma _{3})$ is a continuous
regularizing operator representing the averaging of the normal stress over a
small neighborhood of the contact point. In the case where $p$ is a known
function which is independent of $\sigma _{\nu }$, i.e., $p(r)=g$, the
friction law involved in \eqref{e2.5} becomes the Tresca friction law, and 
$H=\mu g$ is the friction bound. By choosing $p(r)=r$ in $(2.5)$, we recover
the usual regularized Coulomb friction law used in the literature. The
choice $p(r)=r_{+}(1-\delta r)_{+}$, where $\delta $ is a small positive
coefficient related to the wear and hardness of the surface, was employed in 
\cite{s1}. We assume that $R:H^{-1/2}(\Gamma _{3})\rightarrow L^{2}(\Gamma
_{3})$ is a linear compact mapping. The assumptions on the friction function 
$p$ are: 
\begin{equation}
\begin{aligned} (a)& p:\Gamma _{3}\times \mathbf{R}\to \mathbf{R}_{+} \\
(b)& \text{There exists }L_{p}>0\text{ such that }
|p(x,u_{1})-p(x,u_{2})|\leq L_{p}|u_{1}-u_{2}|\\ & \text{for all }
u_{1},u_{2}\in \mathbf{R}, \text{ a.e. } x\in \Gamma _{3}. \\ (c)& \text{For
each }u\in \mathbf{R}, x\to p(x,u) \text{ is measurable on }\Gamma _{3}. \\
(d)& \text{The mapping }x\to p(x,0)\in L^{2}(\Gamma _{3}). \end{aligned}
\label{e2.12}
\end{equation}
We observe that the above assumptions on $p$ are quite general. Clearly, the
functions $p(r)=g$, $p(r)=r$, $p(r)=r_{+}(1-\delta r)_{+}$ satisfy these
conditions, when $g$ is a known function and $\delta $ is a given positive
constant. So, the results presented below hold true for the boundary value
problems with each one of these tangential functions.

\subsection*{Hypotheses on the nonlinear elasticity operator}

As in \cite{t1} we assume $F : \Omega \times S_{d} \to S_{d}$ satisfies the
following conditions: 
\begin{equation}
\begin{aligned} (a)&\text{There exists }L_{1}>0\text{ such that } |F
(x,\varepsilon _{1})-F (x,\varepsilon _{2})|\leq L_{1}|\varepsilon
_{1}-\varepsilon _{2}|, \\ &\text{for all }\varepsilon _{1},\varepsilon
_{2}\text{ in } S_{d}, \text{ a.e. }x\text{ in }\Omega . \\ (b) &
\text{There exists }L_{2}>0\text{ such that } (F (x,\varepsilon _{1})
-F(x,\varepsilon _{2})). ( \varepsilon _{1}-\varepsilon _{2}) \geq
L_{2}|\varepsilon _{1}-\varepsilon _{2}|^{2}, \\ & \text{for all
}\varepsilon _{1},\varepsilon _{2}\text{ in } S_{d},\text{ a.e. }x\text{ in
}\Omega . \\ (c) &\text{The mapping }x\to F (x,\varepsilon ) \text{ is
Lebesgue measurable on }\Omega , \\ &\text{for any }\varepsilon \text{ in
}S_{d}. \\ (d) & F (x,0)=0\text{ for all }x\text{ in }\Omega . \end{aligned}
\label{e2.13}
\end{equation}

\begin{remark} \label{rmk2.1} \rm
$F  (x,\tau ( x))\in Q$, for all $\tau \in Q$ and thus it is possible
to consider $F$ as an operator defined from $Q$ to $Q$.
\end{remark}

We assume that the friction coefficient satisfies 
\begin{equation}
\mu \in L^{\infty }(\Gamma _{3})\text{ and } \mu \geq 0\text{ a.e. on }
\Gamma _{3}.  \label{e2.14}
\end{equation}
Also we assume that the initial data $u_{0}\in V$ satisfies 
\begin{equation}
\langle F (\varepsilon (u_{0})),\varepsilon (v)\rangle _{Q}+j(u_{0},v)\geq
(f(0),v)_{V}\quad \forall v\in V.  \label{e2.15}
\end{equation}
Now by assuming the solution to be sufficiently regular, we obtain by using
Green's formula and techniques similar to those exposed in \cite{d3} that
the problem $(P_{1})$ has the following variational formulation. \smallskip

\noindent \textbf{Problem P2.} Find a displacement field $u\in W^{1,\infty
}(0,T;V)$ such that $u( 0)=u_{0}$ in $\Omega $ and for almost all $t\in
[0,T] $: 
\begin{equation}
\begin{aligned} &\langle F (\varepsilon (u(t))),\varepsilon (v)-\varepsilon
(\dot{u}(t))\rangle _{Q}+j(u(t) ,v) -j(u(t),\dot{u}(t))\\ &\geq
(f(t),v-\dot{u}(t))_{V}\quad \forall v\in V. \end{aligned}  \label{e2.16}
\end{equation}
where 
\begin{equation*}
j(u,v)=\int_{\Gamma _{3}}\mu |R\sigma _{\nu }(u) ||v_{\tau }|da.
\end{equation*}

Our main result of this section, which will be established in the next is
the following theorem.

\begin{theorem} \label{thm2.2}
Let $T>0$ and assume that
\eqref{e2.9}, \eqref{e2.12}, \eqref{e2.13}, \eqref{e2.14},
and \eqref{e2.15} hold. Then problem ($P_{2}$) has at least a
solution  for a small enough friction coefficient $\mu $.
 Moreover, there exists a constant $C>0$ such that
\begin{equation*}
\| u\| _{W^{1,\infty }(0,T;V)}\leq C\| f\| _{W^{1,\infty }(0,T;V)}.
\end{equation*}
\end{theorem}

\section{Existence of solutions}

This evolution problem can be integrated in time by an implicit scheme as in 
\cite{t1}. Let $0=t_{0}<t_{1}<\dots <t_{n}=T$ be a uniform partition of the
time interval $[0,T] $, i.e., $t_{i}=i\Delta t$, $i=0,1,\dots ,n $, $\Delta
t=\frac{T}{n}$ the step-size. We denote by $u^{i}$ the approximation of $u$
at the time $t_{i}$ and by the symbol $\Delta u^{i}$ the backward difference 
$u^{i+1}-u^{i}$. We obtain a sequence of incremental problems, for $u^{0}\in
V$, defined as

\noindent\textbf{Problem $(P_{n}^{_{i}})$.} Find $u^{i+1}\in V$ such that 
\begin{equation}
\begin{aligned} &\langle F (\varepsilon (u^{i+1}) ),\varepsilon
(w)-\varepsilon (u^{i+1})\rangle _{Q}+j(u^{i+1},w-u^{i})
-j(u^{i+1},u^{i+1}-u^{i})\\ &\geq ( f^{i+1},w-u^{i+1}) _{V}\quad \forall
w\in V \end{aligned}  \label{e3.1}
\end{equation}
where $\ u^{0}=u_{0}$, $f^{i+1}=f(t_{i+1})$.

\begin{lemma} \label{lem3.1}
 There exists a positive constant $\mu _{1}>0$ such that for
$\| \mu \|_{L^{\infty }(\Gamma _{3})}<\mu _{1}$,
 problem $(P_{n}^{_{i}})$  admits a unique solution.
\end{lemma}

For the proof of the above lemma, see \cite{t1} in the case of unilateral
contact.

\begin{lemma} \label{lem3.2}
We have the following estimates: For
a positive constant $\mu _{2}>0$, when
$\| \mu \|_{L^{\infty }(\Gamma _{3})}<\mu _{2}$,
there exists $d_{i}>0$,  $i=1, 2$, such that
\begin{gather}
\| u^{i+1}\| _{V}\leq d_{1}\| f^{_{i+1}}\| _{V}  \label{e3.2}
\\
\| \Delta u^{i}\| _{V}\leq d_{2}\| \Delta
f^{i}\| _{V}  \label{e3.3}
\end{gather}
\end{lemma}

\begin{proof}
By setting $w=0$ in the inequality \eqref{e3.1} and by using hypothesis 
\eqref{e2.13}(b), hypothesis \eqref{e2.12}(b), \eqref{e2.11} and the
properties of $j$, there exists $c_{1}>0$ such that for $\| \mu \|
_{L^{\infty}(\Gamma _{3})}<c_{1}$, we deduce that there exists $d_{1}>0$
such that \eqref{e3.2} is satisfied.

To show the inequality \eqref{e3.3} we consider the translated inequality of 
\eqref{e3.1} at the time $t_{i}$\ that is 
\begin{equation}
\begin{aligned} &\langle F (\varepsilon (u^{i})),\varepsilon (w)-\varepsilon
(u^{i})\rangle _{Q}+j(u^{i},v-u^{i-1}) -j(u^{i},u^{i}-u^{i-1})\\ &\geq
(f^{i},w-u^{i})_{V} ,\quad \forall w\in V \end{aligned}  \label{e3.4}
\end{equation}
By setting $w=u^{i}$ in \eqref{e3.1} and $w=u^{i+1}$ in \eqref{e3.4} and add
them up, we obtain the inequality 
\begin{align*}
&-\langle F (\varepsilon (u^{i+1})) -F (\varepsilon (u^{i})),\varepsilon
(\Delta u^{i}) \rangle _{Q}-j(u^{i+1},\Delta u^{i}) \\
&+j(u^{i},u^{i+1}-u^{i-1})-j( u^{i},u^{i}-u^{i-1}) \\
&\geq (-\Delta f^{i},\Delta u^{i})_{V}\,.
\end{align*}
Furthermore, using the inequality 
\begin{equation*}
||u_{\tau }^{i+1}-u_{\tau }^{i-1}|-| u_{\tau }^{i}-u_{\tau }^{i-1}||\leq
|u_{\tau }^{i+1}-u_{\tau }^{i}|
\end{equation*}
we have 
\begin{equation*}
j(u^{i},u^{i+1}-u^{i-1})-j(u^{i},u^{i}-u^{i-1})\leq j(u^{i},\Delta u^{i}).
\end{equation*}
Therefore, 
\begin{equation*}
-\langle F (\varepsilon (u^{i+1})) -F (\varepsilon (u^{i})),\varepsilon
(\Delta u^{i}) \rangle _{Q}+j(u^{i},\Delta u^{i}) -j(u^{i+1},\Delta
u^{i})\geq (-\Delta f^{i},\Delta u^{i})_{V}
\end{equation*}
Using hypothesis \eqref{e2.12}(b), \eqref{e2.11}, hypothesis \eqref{e2.13}
(b) and the properties of $j$, we deduce that there exists two positive
constants $d_{3}$ and $d_{4}$ such that 
\begin{equation*}
L_{2}\| \Delta u^{i}\| _{V}^{2}\leq d_{3}\| \mu \| _{L^{\infty }(\Gamma
_{3}) }\| \Delta u^{i}\| _{V}^{2}+d_{4}\| \Delta f^{i}\| _{V}\| \Delta
u^{i}\|_{V}.
\end{equation*}
Then we deduce that there exists a constant $c_{2}>0$ such that if $\| \mu
\| _{L^{\infty }(\Gamma _{3}) }<c_{2}$, there exists $d_{2}>0$ such that 
\begin{equation*}
\| \Delta u^{i}\| _{V}\leq d_{2}\| \Delta f^{i}\| _{V}
\end{equation*}
It suffices to take $\mu _{2}=\min (c_{1},c_{2})$ and the lemma is proved.
\end{proof}

The proof of Theorem 2.2 is done as in \cite{c1}, but in $L^{\infty }$. For
the next lemma, we define the continuous function $u^{n}$ on $[0,T] \to V$
by 
\begin{equation*}
u^{n}(t)=u^{i}+\frac{(t-t_{i})}{\Delta t} \Delta u^{i} \quad \text{on }
[t_{i},t_{i+1}] ,\quad i=0,\dots ,n-1.
\end{equation*}

\begin{lemma} \label{lem3.3}
There exists a function $u\in W^{1,\infty }(0,T;V)$
such that passing to a subsequence still denoted
$(u^{n})$ we have
\begin{equation*}
u^{n}\to u\textit{ weak $\ast$ in} W^{1,\infty }(0,T;V).
\end{equation*}
\end{lemma}

\begin{proof}
 From \eqref{e3.2}, the sequence $(u^{n})$ is bounded in $C([0,T] ;V)$ and
there exists a constant $c_{3}>0$ such that 
\begin{equation*}
\max_{0\leq t\leq T} \| u^{n}(t)\| _{V}\leq c_{3}\| f\| _{C([ 0,T];V)}\,.
\end{equation*}
 From \eqref{e3.3}, the sequence $(\dot{u}^{n})$ is bounded in $L^{\infty
}(0,T;V)$ and there exists $c_{4}>0$ such that 
\begin{equation*}
\| \dot{u}^{n}\| _{L^{\infty }(0,T;V)}= \max_{0\leq i\leq n-1} \| \frac{
\Delta u^{t_{i}}}{\Delta t} \| _{V}\leq c_{4}\| \dot{f}\|_{L^{\infty
}(0,T;V)}\,.
\end{equation*}
Consequently, the sequence $(u^{n})$ is bounded in $W^{1,\infty}(0,T;V)$,
and thus we can extract a subsequence still denoted $(u^{n})$ such that 
$u^{n}\to u$ in $W^{1,\infty }(0,T;V)$ weak $\ast $ as $n\to \infty $, and
satisfying 
\begin{equation*}
\| u\| _{W^{1,\infty }(0,T;V)}\leq C\| f\| _{W^{1,\infty }(0,T.V)}.
\end{equation*}
\end{proof}

As in \cite{r2} let's introduce the functions $\widetilde{u}^{n}:[0,T] \to V$,
 $\widetilde{f}^{n}:[0,T] \to V$ defined by 
\begin{equation*}
\widetilde{u}^{n}(t)=u^{i+1}=u(t_{i+1}),\quad \widetilde{f}
^{n}(t)=f(t_{i+1})\quad \forall t\in (t_{i},t_{i+1}],i=0,\dots ,n-1
\end{equation*}
As in \cite{t1} we have the following result.

\begin{lemma} \label{lem3.4}
There exists a subsequence still denoted by $(\tilde{u}^{n})$ such that
\begin{itemize}
\item[(i)] $\widetilde{u}^{n}\to u$ weak $\ast$
 in $L^{\infty }(0,T;V)$
\item[(ii)] $\widetilde{u}^{n}(t)\to u(t)$ weakly in
$V$ a.e. $t\in [0,T]$
\end{itemize}
\end{lemma}

\begin{remark} \label{rmk3.5} \rm
Since $f\in W^{1,\infty }(0,T;V)$, $u\in W^{1,\infty }(0,T;V)$,
 we have
\begin{gather}
\widetilde{f}^{n}\to f \quad \text{strongly in }L^{2}(0,T;V)
\label{e3.5}\\
\widetilde{u}^{n}\to u\quad \text{strongly in }L^{2}(0,T;V)\label{e3.6}
\end{gather}
To prove that $u$ is a solution of the problem, in the inequality
of problem $(P_{n}^{i})$, for $v\in V$, we set $w=u^{i}+v\Delta t$ and
divide by $\Delta t$.  We obtain
\begin{align*}
&\langle F  (\varepsilon (u^{i+1})) ,\varepsilon
(v)-\varepsilon (\frac{\Delta u^{i}}{\Delta t})\rangle _{Q}
+j(u^{i+1},v) -j(u^{i+1},\frac{\Delta u^{i}}{\Delta t})\\
&\geq \Big(
f(t_{i+1}),v-\frac{\Delta u^{i}}{\Delta t}\Big)_{V}
\quad \forall v\in V
\end{align*}
Whence for any $v\in L^{2}(0,T;V)$, we have
\begin{align*}
&\langle F  (\varepsilon ( \widetilde{u}^{n}(
t))),\varepsilon (v(t))-\varepsilon (\dot{u}^{n}(t)) \rangle
_{Q}+j(\widetilde{u}^{n}(t),v(t))
-j(\widetilde{u}^{n}(t),\dot{u}^{n}(t))\\
&\geq \Big(\widetilde{f}^{n}(t),v(t)-\dot{
u}^{n}(t)\Big)_{V}\,.
\end{align*}
Integrating both sides of the previous inequality on $(0,T) $, we
obtain
\begin{equation}
\begin{aligned}
&\int_{0}^{T}\langle F  (\varepsilon (\widetilde{u}
^{n}(t))),\varepsilon (v(t))-\varepsilon (\dot{u}^{n}(t) )
\rangle _{Q}dt \\
&+\int_{0}^{T}j(\widetilde{u}^{n}(t),v(t))dt
-\int_{0}^{T}j(\widetilde{u}^{n}(t),\dot{u}^{n}( t))dt\\
&\geq \int_{0}^{T}(\widetilde{f}^{n}(t) ,v-\dot{u}^{n}(t))_{V}dt
\end{aligned}  \label{e3.7}
\end{equation}
\end{remark}

\begin{lemma} \label{lem3.6}
For every $v\in L^{2}(0,T;V)$ we have following properties
\begin{gather}
\begin{aligned}
&\lim_{n\to \infty }\int_{0}^{T}\langle
F  (\varepsilon (\widetilde{u}^{n}(t)))
,\varepsilon (v(t))-\varepsilon (\dot{u}
^{n}(t))\rangle _{Q}dt \\
&=\int_{0}^{T}\langle F  (\varepsilon (u( t) )
),\varepsilon (v(t))-\varepsilon (\dot {u}(t)) \rangle_{Q}dt,
\end{aligned}\label{e3.8}
\\
\lim_{n\to \infty } \int_{0}^{T}j(\widetilde{u}
^{n}(t),v(t))dt=\int_{0}^{T}j(u( t),v(t))dt, \label{e3.9}
\\
\liminf_{n\to \infty } \int_{0}^{T}j(\widetilde{u}
^{n}(t),\dot{u}^{n}(t))dt\geq \int_{0}^{T}j(u(t)
,\dot {u}(t))dt,\label{e3.10}
\\
\lim_{n\to \infty } \int_{0}^{T}(\widetilde{f}^{n}(t),v(t)
-\dot{u}^{n}(t))_{V}dt=\int_{0}^{T}(f(t),v(t)-\dot {u}
(t))_{V}dt \,. \label{e3.11}
\end{gather}
\end{lemma}

\begin{proof}
For \eqref{e3.8}, see \cite{t1}. For \eqref{e3.9} it suffices to use 
\eqref{e3.6}. To prove \eqref{e3.10}, we write 
\begin{equation*}
j(\widetilde{u}^{n}(t),\dot{u}^{n}(t))=( j(\widetilde{u}^{n}(t),\dot{u}
^{n}(t))-j(u(t) ,\dot{u}^{n}(t)))+j( u( t),\dot{u}^{n}(t))
\end{equation*}
then by \eqref{e2.12}(b), we have 
\begin{align*}
&\big|\int_{0}^{T}(j(\widetilde{u}^{n}(t),\dot{ u}^{n}(t)-j(u(t),\dot{u}
^{n}(t) )))dt\big| \\
&\leq L_{p}\| \mu \| _{L^{\infty }(\Gamma _{3}) }\| R(\sigma _{\nu }(
\widetilde{u}^{n})-\sigma _{\nu }(u))\| _{L^{2}(0,T;L^{2}( \Gamma _{3})) }\| 
\dot{u}_{\tau }^{n}\| _{L^{2}(0,T;L^{2}(\Gamma _{3})^{d})}\,.
\end{align*}
Since the mapping $R$ is compact, we have 
\begin{gather*}
\lim_{n\to \infty } \| R(\sigma _{\nu }(\widetilde{u}^{n})-\sigma _{\nu
}(u)) \| _{L^{2}(0,T:L^{2}(\Gamma _{3}))}=0\,, \\
\liminf_{n\to \infty } \int_{0}^{T}j(u( t) ,\dot{u}^{n}(t))dt\geq
\int_{0}^{T}j(u( t) ,\dot {u}(t))dt\ 
\end{gather*}
by Mazur's lemma. For proving \eqref{e3.11} it suffices to use \eqref{e3.5}.
Passing to the limit in inequality \eqref{e3.7}, we obtain 
\begin{align*}
&\int_{0}^{T}\langle F (\varepsilon (u( t) ) ) ,\varepsilon
(v(t))-\varepsilon (\dot {u}(t)) \rangle _{Q}dt+\int_{0}^{T}j(u(t),v(t))dt
-\int_{0}^{T}(j(u(t),\dot {u}(t)))dt \\
&\geq \int_{0}^{T}(f(t),v(t) -\dot {u}(t))_{V}dt\,.
\end{align*}
In this inequality we set 
\begin{equation*}
v(s)=
\begin{cases}
z & \text{for }s\in (t,t+\lambda ) \\ 
\dot{u}(s) & \text{elsewhere}.
\end{cases}
\end{equation*}
Then we obtain 
\begin{align*}
&\frac{1}{\lambda }\int_{t}^{t+\lambda }(F \langle (\varepsilon
(u(s))),\varepsilon ( z) -\varepsilon (\dot {u}(s))\rangle
_{Q}+j(u(s),z)-j(u( s),\dot {u}(s)))ds \\
&\geq \frac{1}{\lambda }\int_{t}^{t+\lambda }(f(s),z- \dot {u}(s))_{V}ds\,.
\end{align*}
Passing to the limit, one obtains that $u$ satisfies \eqref{e2.16}.
\end{proof}

\subsection*{Conclusion}

In this article we have obtained the existence of a weak solution of the
quasistatic bilateral contact problem for nonlinear elastic materials under
a smallness assumption of the friction coefficient. The uniqueness of
solution represents, as far as we know, an open question.

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\end{document}