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\begin{center}
A SOLID-FLUID MODEL WITH UNILATERAL CONTACT CONDITIONS

\vspace*{0.3cm}
{\it G. Chichua}

\vspace*{0.3cm}
{\it I.Vekua Institute of Applied Mathematics}\\
{\it I.Javakhishvili Tbilisi State University}
\end{center}

\vspace*{0.3cm}
\par
Let $ \Omega=\Omega^s\cup\Omega^f\cup\Sigma'' $ be a bounded 
domain in $ {\Bbb R}^3 $
with a Lipschitz boundary $\partial {\overline\Omega}$, 
where $ \Omega^s $ and $ \Omega^f $ are the subdomains occupied by the
elastic solid and viscous fluid respectively, with the 
Lipschitz interface $ \Sigma $
(see Fig.1).

\par
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%\vspace*{-1cm}
$$
\Sigma'=\{ x\in\Sigma\vert[u_iN_i]=0\},\;\;
u(x):=\{u_i\}^3_{i=1}=\left\{\begin{array}{l}
u_i^s\;\; {\rm if}\;\;x\in\Omega^s,\\ u_i^f\;\; {\rm if}\;\;x\in\Omega^f,
\end{array}\right.
$$
$u_i^s$ and $u_i^f$ denote components of the displacement vectors of the
solid and fluid respectively, by the symbol $[\varphi]$ we denote the jump of a
function $[\varphi]=$ $=\varphi\vert_f-\varphi\vert_s$ on the interface 
$\Sigma$,
$\Sigma''\subset\Sigma'$.
\par
In $ \Omega^s $ we have the equations of the linear theory of elasticity:
\begin{equation}
\rho^s\frac{\partial^2u_i^s}{\partial t^2}=
\frac{\partial\sigma^s_{ij}}{\partial x_j}+f_i,\;\;\;x\in\Omega^s
\;\;\;t\in[0,T],
\end{equation}
where\\
$
\displaystyle\sigma^s_{ij}=a^s_{ijkh}e_{kh}(u^s),\;i,j=1,2,3,\;
e_{kh}(u^s):=\frac{1}{2}
\left(\frac{\partial u^s_k}{\partial x_h}+\frac{\partial u^s_h}{\partial 
x_k}\right),\;k,h=1,2,3,
\;(2)$
\ \\
$ \sigma^s_{ij} $ is the stress tensor, 
%$ u^s=\{u_i^s\} $ is the displacement vector of elastic solid, 
$ f $ is a given force and the Einstein summation convention has been used (the 
Latin indices run the
values 1,2,3). The elastic coefficients $ a^s_{ijkh} $ are constants and they 
satisfy the conditions of symmetry and positive definiteness:
\setcounter{equation}{2}
\begin{equation}
a^s_{ijkh}=a^s_{jikh}=a^s_{jihk}=a^s_{khij},
\end{equation}
\begin{equation}
a^s_{ijkh}e_{ij}e_{kh}\ge\alpha e_{ij}e_{ij},\;\; \alpha>0,\;\;e_{ij}=e_{ji}.
\end{equation}
\par
In $\Omega ^f $ we have the following equations
\begin{equation}
\rho^f\frac{\partial^2 u_i^f}{\partial t^2}=
\frac{\partial\sigma^f_{ij}}{\partial x_j}+f_i,\;\;\;\;x\in\Omega^s,
\;\;t\in[0,T],
\end{equation}
\begin{equation}
\sigma^f_{ij}=-\delta_{ij}p+(\eta\delta_{ij}\delta_{kh}+
2\mu\delta_{ik}\delta_{jh})e_{kh}
\left(\frac{\partial u^f}{\partial t}\right),
\end{equation}
where $ \sigma^f_{ij} $ is the  component of the stress tensor, 
%$ u^f:=\{u_i^f\} $ is the displacement vector of viscous fluid, 
$ \eta $ and $ \mu $ are coefficients of viscosity satisfying the 
condition as follows: 
$$
\mu>0,\;\;\eta/\mu>-(2/3)\alpha;\;\;\;0<\alpha<1.
$$
$ \delta_{ij} $ is Kronecker delta.
\par
The symbols $ \rho ^s $ and $ \rho^f $ in (1) and (5)
are respectively the densities of solid and fluid in the position 
of a rest. Let us assume that they are constants.
\par
$ p $ is a perturbation of pressure with respect to the 
position of a rest. It is connected with the perturbation of the density
$ \rho $ by the following expression:
\begin{equation}
p=c^2\rho,
\end{equation}
where $ c>0 $ is the velocity of sound spreading in the position of 
a rest of fluid.
\par
Besides, $ \rho $ and $ v^f:=\displaystyle\frac{\partial u^f}{\partial t} $
are connected with
each other by the continuity equation, which in the linear case  has the
following form:
$$
\frac{\partial\rho}{\partial t}+\rho^f{\rm div} v^f=0,
$$
Hence taking into account, that $\rho=0 $ when $ u^f=0 $, after integration
we obtain:
$$
\rho+\rho^f{\rm div}u^f=0.
$$
\par
Finally, the expresion (7) takes the following form:
\begin{equation}
p=-c^2\rho^f{\rm div}u^f.
\end{equation}
\par
From expression (6) using (8) we get
\begin{equation}
\sigma^f_{ij}=\delta_{ij}c^2\rho^f{\rm div}u^f+
(\eta\delta_{ij}\delta_{kh}+2\mu\delta_{ik}\delta_{jh})
e_{kh}\left(\frac{\partial u^f}{\partial t}\right),
\end{equation}
\par Therefore $ \sigma _{ij}^f $ is a function of the vector $ u^f $ as well
as of the velocity vector $ \displaystyle\frac{\partial u^f}{\partial t}. $
\par Let us consider the following boundary and initial conditions:
\begin{equation}
u=0\;\;\;\;\;\;{\rm on}\;\;\partial\Omega
\end{equation}
\begin{equation}
u=\frac{\partial u}{\partial t}=0\;\;\;\;{\rm in}\;\;\Omega\;\;\;\;\;{\rm 
when}\;\;t=0.
\end{equation}
%where $u(x):=\cases{u^s(x)\;\;\;\;if\;\;\;\;x\in\Omega^s,\cr
%u^f(x)\;\;\;\;if\;\;\;\;x\in\Omega^f}$
\par
On the interface $ \Sigma $ we set conditions as follows:
\begin{equation}
[u_iN_i]\ge 0\;\;\;\;\;\;\;\;on\;\;\Sigma;
\end{equation}
\begin{equation}
\sigma^s_{ij}n^s_j=\sigma^s_{NN}N_i;\;\;\;\;
\sigma^f_{ij}n^f_j=-\sigma^s_{NN}N_i,\;\;\;\;i=1,2,3;
\end{equation}
\begin{equation}
\sigma^s_{NN}\le 0
\end{equation}
\begin{equation}
{\rm and}\;\;\;\;\sigma^s_{NN}=0\;\;\;\;if\;\;\;\;[u_iN_i]>0,
\end{equation}
where (see Fig.1)  $ N $ is a unit normal vector  to the interface $ \Sigma $ 
with an external 
direction with respect to $ \Omega ^s $, $ n $ is a unit outer normal vector to
$ \partial(\bar\Omega\backslash\Sigma), $ 
$\varphi $, 
$ \sigma_{NN} $ is the projection on the direction $N$ of the
stress vector acting on the surface $\Sigma$.
\par 
The relation (12) means that on the interface $ \Sigma $ the quantity of
the projection $u^f_N$ of the vector
$ u^f $ on the direction $N$ can not be less than $ u^s_N $; the relations (13) 
denote
that the inner force acting on $ \Sigma $ is normal to
$ \Sigma $ and the action is opposite to the reaction, (14)
means that there is compression (but not traction!) on $ \Omega^s $ 
through $ \Sigma $; (15) means that if the crack is open at a point, 
the acting force is zero at this point. 
\par
Let us introduce the following spaces: 
\begin{equation}
V:=\{u\vert u_i\in 
H^1(\Omega^s\cup\Omega^f\cup\Sigma'');u_i\vert_{\partial\overline\Omega}=0\}
\end{equation}
and
\begin{equation}
K:=\{u\vert u_i\in V;\;[u_iN_i]\ge 0\;a.e.\;on\;\Sigma\},
\end{equation}
where $ H^1(\Omega^s\cup\Omega^f\cup\Sigma'') $  is the usual Sobolev space. 
\par{\bf Theorem 1.} {\it $ K $ is the closed convex set.}
\par
{\bf Proof.} Let us first prove that $ K $ is closed. Let $ v^m\in K 
,\;m=1,2,..., $
and $ v^m\to v_,$ if $m\to\infty, $ by the norm of the space $ V $. Let us
prove that $ v\in K $. Obviously,
\begin{equation}
v_N^m\vert_f-v_N^m\vert_s\ge 0
\end{equation}
According to the trace theorem, we have
$$
v_N^m\vert_f\to v_N\vert_f,\;\;\;\;{\rm and}\;\;\;\;
v_N^m\vert_s\to v_N\vert_s,
$$
so $ 0\le v_N^m\vert_f-v_N^m\vert_s\to v_N\vert_f-v_N\vert_s,$ that means, that 
$ v_N\vert_f-v_N\vert_s\ge 0,$ hence $ v\in K $.
\par
Now let us prove the convexity of the set $ K $.
Let $ u,v\in K $. Now, we shall prove that
$$
\alpha u+\beta v\in K,\;\forall\alpha,\beta=const>0,\;\;\alpha+\beta=1. 
$$
\par
We have
$$
u_N\vert_f-u_N\vert_s\ge 0,\;\;\;\;{\rm and}\;\;\;\;
v_N\vert_f-v_N\vert_s\ge 0
$$
i.e.
$$
(\alpha u_N)\vert_f-(\alpha u_N)\vert_s\ge 0\;\;\;{\rm and}\;\;\;
(\beta v_N)\vert_f-(\beta v_N)\vert_s\ge 0,
$$
from which we obtain that
$$
(\alpha u_N+\beta v_N)\vert_f-(\alpha u_N+\beta v_N)\vert_s\ge 0.
$$
Hence 
$$
\alpha u+\beta v\in K.
$$
\par Let  $ H:=L^2(\Omega) $.
\par{\bf Definition.} A vector-function $u $ is called a generalized solution of
the problem (1)-(6), (10)-(15) , 
if $ u $ satisfies the following conditions
$$
\begin{array}{l}
u,u'\in L^\infty(0,T;V),\;\;\;\;u''\in L^\infty(0,T;H),\;\;\;\;u'\in K,\\
(\rho u''(t),v- u'(t))+a(u(t),v-u'(t))+b(u'(t),v-u'(t))\ge\left(f(t),v-
u'(t)\right),
\\
\hspace{9.64cm}\forall v\in K,\;\forall t\in[0,T],
\\
u=0\;\;\;\;{\rm and}\;\;\;\;u'=0\;\;\;\;{\rm when}\;\;\;\;  t=0,
\end{array}
$$
where $ L^\infty(0,T;V):=\{f(t)\vert f(t):[0,T]\to V,$ is measurable
and ${\mathop{\rm ess\;sup}\limits_{t\in[0,T]}} \Vert f(t)\Vert_v=\Vert f
\Vert_{L^\infty(0,T,V)}<\infty\},$
by the symbol (.,.) we denote the scalar product in $ H $,
$$
\rho=\rho(x)=\left\{
\begin{array}{l}
\rho^s\;\;\;\; if\;\; x\in\Omega^s,\\
\rho^f\;\;\;\; if\;\; x\in\Omega^f;
\end{array} \right.
$$
$$
\begin{array}{l}
a(u,w)=\int\limits_{\Omega}a_{ijkh}(x) e_{kh}(u) e_{ij}(w)dx,\\
b(v,w)=\int\limits_{\Omega}b_{ijkh}(x) e_{kh}(v) e_{ij}(w)dx;
\end{array}
$$
$$
a_{ijkh}(x)=\left\{
\begin{array}{l}
a_{ijkh}^s\;\;\;\; if\;\; x\in\Omega^s,\\
a_{ijkh}^f\equiv\rho^fc^2\delta_{ij}\delta_{kh}\;\;\;\; if\;\;
 x\in\Omega^f; 
\end{array}
\right.
$$
$$
b_{ijkh}(x)=\left\{
\begin{array}{l}
b_{ijkh}^s= 0\;\;\;\; if\;\; x\in\Omega^s,\\
b_{ijkh}^f\equiv 2\mu\delta_{ik}\delta_{jh}+\eta\delta_{ij}\delta_{kh}\;\;\;\;
 if\;\;x\in\Omega^f.
\end{array}
\right.$$
\par {\bf Theorem 2.} {\it The classical solution is also the generalized
solution of the problem} (1)-(6), (10)-(15).
\par {\bf Proof.} Let $ u $ be the classical solution of problem (1)-(6),
(10)-(15) and let $ v:=\{v_i\}\in K $. If we multiply
both sides of equations (1) and (5) by
$$
v_i=\cases{v_i^s\;\;\;\;if\;\;\;\;x\in\Omega^s,\;\;\;i=1,2,3,\cr
v_i^f\;\;\;\;if\;\;\;\;x\in\Omega^f,}
$$
and use the formula of integration by parts respectively on $ \Omega^s $
and $ \Omega ^f $, we get the relations as follows
\begin{equation}
\int\limits_{\Omega^s}
\rho^s\frac{\partial u_i^s}{\partial t^2}v_i^sdx-
\int\limits_{\Sigma}
n_j\sigma_{ij}^sv_i^sd{\Sigma}=
\int\limits_{\Omega^s}f_iv_i^sdx-
\int\limits_{\Omega^s}
\sigma_{ij}^s\frac{\partial v_i^s}{\partial x_j}dx
\end{equation}
and
\begin{equation}
\int\limits_{\Omega^f}
\rho^f\frac{\partial u_i^f}{\partial t^2}v_i^fdx-
\int\limits_{\Sigma}
n_j\sigma_{ij}^fv_i^fd{\Sigma}=
\int\limits_{\Omega^f}f_iv_i^fdx-
\int\limits_{\Omega^f}
\sigma_{ij}^f\frac{\partial v_i^f}{\partial x_j}dx
\end{equation}
Let us sum the relations (19) and (20), and take 
into account the interface conditions (13), then we obtain:
\begin{equation}
\int\limits_{\Omega}
\rho\frac{\partial^2u_i}{\partial t^2}v_idx+
\int\limits_{\Sigma}
\sigma_{NN}^s[v_iN_i]d{\Sigma}=
\int\limits_{\Omega}f_iv_idx
-\int\limits_{\Omega}\sigma_{ij}
\frac{\partial v_i}{\partial x_j}dx,
\end{equation}
where
$
\rho=\cases{\rho^s\;\;{\rm if}\;x\in\Omega^s,
\cr \rho^f\;\;{\rm if}\;x\in\Omega^f,}
\;\;
\sigma_{ij}(x)=\cases{\sigma^f_{ij}\;\;{\rm if}\;x\in\Omega^s,
\cr \sigma^s_{ij}\;\;{\rm if}\;x\in\Omega^f,}
$
$
u_i=\cases{u^s_i\;\;{\rm if}\;x\in\Omega^s,\cr
u^f_i\;\;{\rm if}\;x\in\Omega^f,}$ $i=1,2.3.
$\par
From (21) and (17) we get
$$
\int\limits_{\Omega}
\rho\frac{\partial^2u_i}{\partial t^2}v_idx+
\int\limits_{\Omega}
\sigma_{ij}\frac{\partial v_i}{\partial x_j}dx\ge
\int\limits_{\Omega}f_iv_idx.
$$
Substituting here $v-u'$ instead of $v$ the proof is fulfilled.
\par
The following theorems can also be proved:
\par {\bf Theorem 3.} {\it  If the generalized solution is smooth enough then it
will be also a classical solution of the problem } (1)-(6), (10)-(15).
\par {\bf Theorem 4.} {\it There exists the unuque generalized solution of the
problem} (1)-(6), (10)-(15).


\vspace*{0.4cm}
\footnotesize
\begin{center}
{\bf R e f e r e n c e s}

\vspace*{0.2cm}
\end{center}
\par
[1] Sanches-Palencia E., Non-homogeneous Media and Vibration Theory. 
Lecture Notes in Physics 127. Springer-Verlag, New-York, 1980, (Russian).
\par
[2] Lions J.L., Quelqes Methodes de Resolution de Problems aux Limites Non-
lineares.
Dunod, Gauthier-Villars, Paris, 1969, (Russian).


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