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     {\huge СИБИРСКИЕ \ ЭЛЕКТРОННЫЕ} \\[2mm]  %Headings in Russian
     {\huge МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ} \\[2mm]
     {\large Siberian Electronic Mathematical Reports} \\[1mm]
     {\LARGE\tt{http://semr.math.nsc.ru}}\\[0.5mm]
%     {\small Институт математики им. С.Л.Соболева СО РАН}\\[-1mm]
%     {\small Sobolev Institute of Mathematics SB RAS}
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     \noindent
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     \vspace{1mm}
     {\flushleft\it Том 16, стр. 144--144 (2019) \hspace{65mm}{\rm\small УДК \udcs}} %Volume, pages in Russian,
                                                                                    %filled by Editorial board
     \newline
         {\rm\small DOI~10.33048/semi.2019.16.xxx}\hphantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}{\rm\small MSC\ \ \mscs }
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\begin{document}
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\title[Asymptotic modeling of curvilinear narrow inclusions]{Asymptotic modeling of curvilinear narrow inclusions with    rough boundaries  in elastic bodies: case of a soft inclusion}
\author{{I.V. Fankina}}%
\address{Irina Vladimirovna Fankina
\newline\hphantom{iii} Sobolev Institute of Mathematics,
\newline\hphantom{iii} pr. Koptyuga, 4,
\newline\hphantom{iii} 630090, Novosibirsk, Russia
\newline\hphantom{iii} Lavrentyev Institute of Hydrodynamics,
\newline\hphantom{iii} pr. Lavrentyeva, 15,
\newline\hphantom{iii} 630090, Novosibirsk, Russia}%
\email{fankina.iv@gmail.com}%

\author{{A.I. Furtsev}}%
\address{Alexey Igorevich Furtsev
\newline\hphantom{iii} Sobolev Institute of Mathematics,
\newline\hphantom{iii} pr. Koptyuga, 4,
\newline\hphantom{iii} 630090, Novosibirsk, Russia
\newline\hphantom{iii} Lavrentyev Institute of Hydrodynamics,
\newline\hphantom{iii} pr. Lavrentyeva, 15,
\newline\hphantom{iii} 630090, Novosibirsk, Russia}%
\email{al.furtsev@mail.ru}%

\author{{E.M. Rudoy}}%
\address{Evgeny Mikhailovich Rudoy
\newline\hphantom{iii} Sobolev Institute of Mathematics,
\newline\hphantom{iii} pr. Koptyuga, 4,
\newline\hphantom{iii} 630090, Novosibirsk, Russia
\newline\hphantom{iii} Lavrentyev Institute of Hydrodynamics,
\newline\hphantom{iii} pr. Lavrentyeva, 15,
\newline\hphantom{iii} 630090, Novosibirsk, Russia}%
\email{rem@hydro.nsc.ru.ru}%

\author{{S.A. Sazhenkov}}%
\address{Sergey Alexandrovich Sazhenkov
\newline\hphantom{iii} Lavrentyev Institute of Hydrodynamics,
\newline\hphantom{iii} pr. Lavrentyeva, 15,
\newline\hphantom{iii} 630090, Novosibirsk, Russia}%
\email{sazhenkovs@yandex.ru}%

\thanks{\sc Fankina, I.V.,   Furtsev, A.I., Rudoy, E.M., Sazhenkov, S.A.
Asymptotic modeling of curvilinear narrow inclusions with    rough boundaries  in elastic bodies: case of a soft inclusion}
\thanks{\copyright \ 2022 Fankina  I.V.,   Furtsev  A.I., Rudoy  E.M., Sazhenkov  S.A.}
\thanks{\rm This work was supported by Mathematical Center in Akademgorodok under agreement No.
075-15-2019-1613 with the Ministry of Science and Higher Education of the Russian Federation.}
\thanks{\it Received  January, 1, 2015, published  March, 1,  2015.}%


\semrtop \vspace{1cm}
\maketitle {\small
\begin{quote}
\noindent{\sc Abstract. } Within the framework of  two-dimensional elasticity theory, a heterogeneous body with a narrow inclusion  lying strictly inside the body is considered. It is assumed  that the elastic properties of inclusion and its width depend on the small parameter $\delta>0$.  Moreover, we assume that the inclusion    has a curvilinear rough boundary.    We show  that there exist three type of limiting problem as $\delta\to0$: $p>1$ -- body with crack without interaction of its faces; $p=1$ -- body with crack with adhesive interaction of its faces;   $p\in[0,1)$ -- homogeneous body (no crack).



 \medskip

\noindent{\bf Keywords:} asymptotic analysis, inhomogeneous elastic body, narrow inclusion, curvilinear crack, interface conditions.
 \end{quote}
}


\section{Introduction} Modern aerospace,  automobile, and many other engineering   structures require new materials with low weight and high strength. One of the ways to obtain such materials is construction of composites containing various kinds of thin small inclusions. The numerical solution of models of the inclusions is fraught with large computational costs due to small size compared to the structure. To overcome this difficulty approximation models are derived. In these models inclusions are replaced by some submanifolds of codimension 1 on which the interface conditions inherited from the original full model are properly set.

In the present paper, within the framework of  two-dimensional elasticity theory, a heterogeneous body with a narrow inclusion  lying strictly inside the body is considered. It is assumed  that the elastic properties of inclusion and its width depend on the small parameter $\delta>0$: width is in direct ratio $\delta$, while Young’s modulus of the inclusion is  proportional to $\delta^p$, where $p$ is a   real number. We investigate the case $p\geq0$.  In this case such inclusion is called a soft one (see, e.g., \cite{Dumont,GeymonatKrasuckiSerpilli,SerpilliLenci}).  The inclusion has a curved rough border. The latter means that a perturbation of the inclusion boundary   depending  on the parameter of $ \delta $ is introduced.
Mixed boundary conditions
are prescribed  on the external  boundary of the   body. The equilibrium problem is formulated as the problem
of minimizing the energy functional in  Sobolev's space $H^1$.

We present some  methodology to derive  models of elastic bodies   for the crack problems. The methodology combines   asymptotic analysis with variational pro\-per\-ties of the equilibrium problem.  Its major advantage over existing asymptotic approaches consists of  the straightforward derivation of crack models  for any non-negative value of parameter $p $ simultaneously. It is shown that there exist three type of limiting problem as $\delta\to0$: $p>1$ -- body with crack without interaction of its faces; $p=1$ -- body with crack with adhesive interaction of its faces (so called spring type condition);   $p\in[0,1)$ -- homogeneous body (no crack).

Note   recent  studies \cite{RudoyIJSS,RudoySEMI,RudoyTech} where the analogous  technics  is applied to some models of elastic bodies and plates. Papers \cite{PhilTransA,FurtsevItouRudoy,Karnaev,Kh3,SazhenkovFankina} is devoted the in\-ves\-ti\-ga\-tion of thin inclusions problems including issues of numerical solutions, shape op\-ti\-mi\-za\-tion, homogenization. Papers \cite{RudoyIJSS,RaffaLebonRizzoni}, where     case of hard inclusions (with   $p<0$)  is studied  for some  models of Elasticity. At last, we mention several  works on investigations of problems with thin structure (not only inclusions) in elastic bodies (see \cite{Furtsev,Kh1,Kh2,Kh3,Pyatkin}).

Summarizing, in our consideration we take into account three key features, which, in our opinion, were not taken into account earlier
simultaneously; namely,

-- the inclusion is located strictly inside the body;

-- the inclusion has a curvilinear boundary;

-- the inclusion boundary is rough.





\section{Statement of the problem}  Let  $\Omega\subset\mathbb{R}^2$ be a convex bounded domain with   a Lipschitz boundary $\partial\Omega$; let $\Gamma_N$ and $\Gamma_D$ be parts of  $\partial\Omega$ such that  $\overline{\Gamma}_N\cup\overline{\Gamma}_D=\partial\Omega$, ${\rm meas}\,\Gamma_D>0$.
Let   $I_1$ and $I_2$ be two intervals lying on the abscissa axis $Oy_1$ such that  $\bar I_1\subset I_2$, $\bar I_2\subset I$, where  $I$ is an intersection of the domain   $\Omega$ with $Oy_1$-axis.

Let us consider   the Lipschitz functions  $\varphi$, $\psi_-$, $\psi_+$ defined on the interval  $I$ such that:

1. $\psi_+-\psi_->0$ on $I_1$;

2. $\psi_+-\psi_-=0$ on $  I\setminus\bar I_2$;

3. $\varphi=0$ on $ I\setminus\bar I_2$;

4. a graph of the function    $\varphi$ is located strictly inside the domain   $\Omega$.


Let us fix a small parameter $\delta>0$ and introduce the notations:
$$
\Omega_\pm=\{(x_1,x_2)\in\Omega\ \ |\ \ \pm x_2>\pm\varphi(x_1),\ x_1\in I\},$$
$$
\Omega_m^\delta=\{(y_1,y_2)\in \Omega\ \ | \ \ \varphi(y_1)-\delta\psi_-(y_1)<y_2<\varphi(y_1)+\delta\psi_+(y_1),\ y_1\in I_1\},
$$
$$
S_\pm^\delta=\{(y_1,y_2)\in \Omega\ \ | \ \ y_2=\varphi(y_1)\pm\delta\psi_\pm(y_1),\ y_1\in I_1\},
$$
%$$
%\Omega_e^\delta=\Omega\setminus\overline{\Omega_m^\delta},%\ \ \Omega_\pm^\delta=\Omega_e^\delta\cap\Omega_\pm.
%$$
Note that for all small enough $\delta>0$ the domain  $\Omega_m^\delta$ lies strictly
inside $\Omega$.

We assume that the domain   $\Omega$ is an elastic inhomogeneous body, containing a narrow inclusion  $\Omega_m^\delta$ with width of order $\delta$ and with    a rough boundary  (roughness is characterized by functions  $\psi_\pm$). By  $C^\delta$, $C^0$ we denote  tensors characterizing material constants of the inclusion $\Omega_m^\delta$ and an elastic matrix  $\Omega\setminus\overline{\Omega_m^\delta}$, respectively, with usual symmetrical and elliptical properties (see, e.g., \cite{KhKov}). We prescribe   homogeneous Dirichlet’s conditions  on a part  $\Gamma_D$ of the external boundary   $\partial\Omega$ and   Neumann's conditions on the rest part $\Gamma_N$ of $\partial\Omega$. It means that the body  $\Omega$ is clamped on $\Gamma_D$,   whereas the forces   $g\in L_2(\Gamma_N)$ is applied to $\Gamma_N$.



  An equilibrium problem of the body $\Omega$ with the inclusion  $\Omega_m^\delta $ is formulated as a variational one in the framework of two-dimensional Elasticity.
Let $u=(u_1,u_2)$ be a vector of displacements of the composite body, $\sigma(u)=(\sigma_{ij}(u))_{i,j=1,2}$ be the stress tensor, $\varepsilon(u)=(\varepsilon_{ij}(u))_{i,j=1,2}$ be the strain tensor, where
\[
\varepsilon_{ij}(u)=\frac{1}{2}\left( u_{i,j} +u_{j,i} \right),\ \ i,j={1,2},\]
$$
\sigma(u)=C\varepsilon(u), \ \mbox{where}\ C=\left\{\begin{array}{ll}C^0 \ \mbox{in}\ \Omega\setminus\overline{\Omega_m^\delta} \\
C^\delta \ \mbox{in}\ \Omega_m^\delta.\end{array}\right.$$
Lower indices after comma denote the operation of differentiation with respect
 to corresponding coordinate; the summation over repeated indices is performed.
Additionally, we assume that  the inclusion $\Omega_m^\delta $ is an isotropic and homogeneous body such that    $C^\delta=\delta^p C^1$ with  $p\geq0$, where
$$
C^1\varepsilon(u)=
\lambda_m I{\rm tr}\,\varepsilon(u)+2\mu_m \varepsilon(u),$$
with the associated Lam\'{e} coefficients given by $\lambda_m$ and  $\mu_m$. We assume  $3\lambda_m+2\mu_m>0$ and $\mu_m>0$, in which case   the tensor $C^1$ is positive definite.  In general, the elastic matrix   is anisotropic and homogeneous with elastic coefficients from $L^\infty_{loc}(\mathbb{R}^2)$.


Thus, the equilibrium problem is as follows:
for a given $g\in L_2(\Gamma_{N})$,   find $u_\delta\in H^1_{\Gamma_D}(\Omega)$ satisfying the following variational equality:
\beq\label{eq2}
\int\limits_\Omega\sigma(u_\delta):\varepsilon(u)\,dy=\int\limits_{\Gamma_N}gu\,ds\eeq  for all kinematically admissible displacement functions $u\in H^1_{\Gamma_D}(\Omega)$,
where
$$
H^1_{\Gamma_D}(\Omega)=\{u\in H^1(\Omega)^2\ \ | \ \ u=0\ \mbox{a.e. on}\ \Gamma_D\}.$$

 Let us rewrite problem \eqref{eq2} in equivalent form. First of all, we introduce an extended inclusion
 $$
\Omega_M^\delta=\{(y_1,y_2)\in \Omega\ \ | \ \ \varphi(y_1)-\delta\psi(y_1)<y_2<\varphi(y_1)+\delta\psi_+(y_1),\ y_1\in I_2\},
$$
containing the inclusion $ \Omega_m^\delta$, and     put
$$
T_\pm^\delta=\{(y_1,y_2)\in \Omega\ \ | \ \ y_2=\varphi(y_1)\pm\delta\psi_\pm(y_1),\ y_1\in I_2\},
$$ containing curves   $ S_\pm^\delta$. Moreover, we define   domains
$$
\omega_m^\delta=\Omega_M^\delta\setminus\overline{\Omega_m^\delta},\ \ \Omega_e^\delta=\Omega\setminus\overline{\Omega_M^\delta},\ \ \Omega_\pm^\delta=\Omega_e^\delta\cap\Omega_\pm$$
We introduce  a set
\begin{multline*}
 K^\delta=\{(v^+,v^-,v^m)\in H^1(\Omega^\delta_+)\times H^1(\Omega^\delta_-)\times H^1(\Omega^\delta_M)\ \ | \\
 v^\pm=0\ \mbox{a.e. on}\ \Gamma_D\cap\partial\Omega_\pm^\delta,\ \ v^\pm=v^m\ \mbox{a.e. on}\ T^\delta_\pm,\ v^+=v^-\ \mbox{a.e. on}\ \partial\Omega^\delta_-\cap\partial\Omega^\delta_+\}.
\end{multline*}
Then problem  \eqref{eq2} is rewritten as follows: find a triplet   $(u^+_\delta,u^-_\delta,u^m_\delta)\in K^\delta$, satisfying a variational equality
\begin{multline}\label{eq3}
\int\limits_{\Omega_+^\delta}C^0\varepsilon(u^+_\delta):\varepsilon(u^+)\,dy+
\int\limits_{\Omega_-^\delta}C^0\varepsilon(u^-_\delta):\varepsilon(u^-)\,dy+
\int\limits_{\omega_m^\delta}C^0\varepsilon(u^m_\delta):\varepsilon(u^m)\,dy+\\+
\delta^p\int\limits_{\Omega_M^\delta}C^1\varepsilon(u^m_\delta):\varepsilon(u^m)\,dy=\int\limits_{\Gamma_N\cap\partial\Omega_+^\delta}gu^+\,ds+
\int\limits_{\Gamma_N\cap\partial\Omega_-^\delta}gu^-\,ds
\end{multline}
for all  $(u^+,u^-,u^m)\in K^\delta.$

By the Lions-Stampacchia theorem, problem  \eqref{eq3} has a unique solution $(u^+_\delta,u^-_\delta,u^m_\delta)$ for all  $\delta>0$, and the following relations
$$
u_\delta|_{\Omega_\pm^\delta}=u^\pm_\delta,\ \ u_\delta|_{\Omega_M^\delta}=u^m_\delta$$ hold.



\section{Coordinate transformations}
We put
$$
\Omega_M=\{(z_1,z_2)\in \mathbb{R}^2\ |\ \psi_-(z_1)<z_2<\psi_+(z_1),\ z_1\in I_2\},$$
$$
\Omega_m=\{(z_1,z_2)\in \mathbb{R}^2\ |\ \psi_-(z_1)<z_2<\psi_+(z_1),\ z_1\in I_1\},$$
$$
T_\pm=\{(z_1,z_2)\in\mathbb{R}^2\ |\ z_2=\psi_\pm(z_1),\ z_1\in I_2\},$$
$$
S_\pm=\{(z_1,z_2)\in\mathbb{R}^2\ |\ z_2=\psi_\pm(z_1),\ z_1\in I_1\},$$
$$
T=\{(x_1,x_2)\in\mathbb{R}^2\   |\ x_2=\varphi(x_1), \ x_1\in I_2\},$$
$$
S=\{(x_1,x_2)\in\mathbb{R}^2\   |\ x_2=\varphi(x_1), \ x_1\in I_1\},$$
and introduce coordinate transformation mapping domains $\Omega_\pm^\delta$ and $\Omega_M^\delta$ into  domains $\Omega_\pm$ and $\Omega_M$, respectively. Let us  take two   convex open domains $D_1$ and $D_2$   such that  $\overline I_2\subset D_1$, $\overline D_1\subset D_2$, $\overline D_2\subset \Omega$, and consider the following domains:
$$
D^\varphi_i=\{(z_1,z_2)\ \ | \ \ z_1=y_1,\ z_2=\varphi(y_1)+y_2,\ (y_1,y_2)\in D_i\}, \ i=1,2.$$  As a result, we have the following  inclusions: $\overline T\subset D_1^\varphi$, $\overline D_1^\varphi\subset D_2^\varphi$. Additionally, we also suppose that the domain $D_2^\varphi$ lies strictly inside $\Omega$.

 Now we consider a cut-off  function $\theta\in C^1(\overline\Omega)$ such that
$$
\theta=1\ \mbox{in}\ D_1^\varphi;\ \ 0<\theta<1 \ \mbox{in}\ D_2^\varphi;\ \theta =0\ \mbox{in}\ \Omega\setminus\overline{D}_2^\varphi.$$
In the domain $\Omega_\pm$ and $\Omega_M$ we define coordinate transformations as follows
\beq\label{transpm}
y_1=x_1,\  y_2=x_2\pm\delta\psi_\pm(x_1)\theta(x_1,x_2),\ \ (x_1,x_2)\in\Omega_\pm,\   (y_1,y_2)\in\Omega_\pm^\delta,
\eeq
\beq\label{transm}
y_1=z_1,\ y_2=\delta z_2+\varphi(z_1), \ \ (z_1,z_2)\in\Omega_M,\ (y_1,y_2)\in\Omega_M^\delta.
\eeq
It is easy to see that transformations  \eqref{transpm} and \eqref{transm} are one-to-one. Note that trans\-for\-ma\-tion  of the type \eqref{transpm}are widely used in  shape optimization  and crack theory (see, e.g., \cite{kh2002,rudoi,Kovper,KovpLaz,rush}).

Due to smoothness properties of functions   $\varphi$ and $\psi_\pm$, coordinate transformations \eqref{transpm} and \eqref{transm} generate one-to-one mappings between spaces  $H^1(\Omega_\pm)$, $H^1(\Omega_M)$ and  $H^1(\Omega_\pm^\delta)$, $H^1(\Omega_M^\delta)$, respectively (see \cite{Necas}, Chapter 2, Lemma 3.4).
Preimage of     $K^\delta$ in  $H^1(\Omega_-)\times H^1(\Omega_+)\times H^1(\Omega_M)$ is a set
\begin{multline*}
 K=\{(v^+,v^-,v^m)\in H^1(\Omega_+)\times H^1(\Omega_-)\times H^1(\Omega_M)\ \ | \\ v^\pm|_{T}=v^m|_{T_\pm},\ \ \
 v^\pm=0\ \mbox{a.e. on}\ \Gamma_D\cap\partial\Omega_\pm, \\ v^+=v^-\ \mbox{a.e. on}\ (\partial\Omega_-\cap\partial\Omega_+)\setminus\overline{T}\}.\end{multline*}
 Hereinafter, $v^\pm|_{T}=v^m|_{T_\pm}$ means that
$$
v^\pm(x_1,\varphi(x_1))=v^m(x_1,\psi_\pm(x_1)),\ x_1\in I_2.$$

After applying coordinate transformations \eqref{transpm} and \eqref{transm} to variational equality \eqref{eq3},   we conclude that   transformed displacements
\beq\label{tilde1}
\tilde u^\pm_\delta(x_1,x_2)=u^\pm_\delta(x_1,x_2\pm\delta\psi_\pm(x_1)\theta(x_1,x_2)),
\eeq\beq\label{tilde2}
\tilde u^m_\delta(z_1,z_2)=u^m_\delta(z_1,\delta z_1+\varphi(z_1))
\eeq
satisfy
the following variational equality:
\begin{multline}\label{3'}
A^\delta_+(\tilde u^+_\delta,v^+)+A^\delta_-(\tilde u^-_\delta,v^-)+\delta\int\limits_{\omega_m}C^0e^\delta(\tilde u^m_\delta): e^\delta(v^m)\,dz+\\+
\delta^{p+1}\int\limits_{\Omega_m}C^1e^\delta(\tilde u^m_\delta): e^\delta(v^m)\,dz=
\int\limits_{\partial\Omega^+\cap\Gamma_N}gv^+\,ds+
\int\limits_{\partial\Omega^-\cap\Gamma_N}gv^-\,ds
\end{multline}
for any triplet $(v^+,v^-,v^m)\in K$, where $\omega_m=\Omega_M\setminus\overline{\Omega_m}$,
$$
A^\delta_\pm(u^\pm,v^\pm)=\int\limits_{\Omega_\pm}J^\pm_\delta C^0 E(\Psi^\pm_\delta; u^\pm): E(\Psi^\pm_\delta;v^\pm)\,dx,
$$
$$
E_{ij}(\Psi_\delta^\pm;v)={1}/{2}
\left({ v_{i,k}}(\Psi_\delta^\pm)_{kj}+{ v_{j,k}}(\Psi_\delta^\pm)_{ki}\right),\ i,j=1,2;$$
$$
e^\delta(v^m)=\left(\begin{array}{cc}v_{1,1}^m-\frac{\varphi^\prime}{\delta}v_{1,2}^m& \frac{1}{2}\left(\frac{1}{\delta}v_{1,2}^m+v_{2,1}^m-\frac{\varphi^\prime}{\delta}v_{2,2}^m\right)\\
\frac{1}{2}\left(\frac{1}{\delta}v_{1,2}^m+v_{2,1}^m-\frac{\varphi^\prime}{\delta}v_{2,2}^m\right)& \frac{1}{\delta}v_{2,2}^m\end{array}\right),
$$
 matrices  $\Psi_\delta^\pm$ are Jacobi matrices of  transformations \eqref{transpm}, and    $J^\pm_\delta$ is Jacobian of transformations \eqref{transpm}.

Since the main goal of the our investigation is to find a limit problem as   $\delta\to0$, in the sequel it is convenient to use the following asymptotic expansion:
 \beq\label{Aas1}
 A_\pm^\delta(u,v)=\int\limits_{\Omega_\pm}C^0 \varepsilon(u):\varepsilon(v)\,dx+\delta r_\pm(\delta,u,v),\eeq
 where
\beq\label{Aas2}
 |r_\pm(\delta,u,v)|\leq c(\|u\|_{H^1(\Omega_\pm)}^2+ \|v\|_{H^1(\Omega_\pm)}^2)\eeq  with a constant  $c$ independent of  $\delta$, $u$, and $v$.

 \section{Asymptotic analysis}

 We need to recall   auxiliary lemmas from \cite{RudoySEMI,RudoyIJSS}.


\begin{lemma} For any function  $\tilde v\in H^1(\Omega_M)$ the following holds:
$$%\beq\label{lemma1}
\|\tilde v\|_{L_2(\Omega_M)}^2\leq c\left(\|\tilde v_{,2}\|_{L_2(\Omega_M)}^2+\|\tilde v\|_{L_2(T_s)}^2\right),\ \ s\in\{-,+\},
$$ with a constant  $c$ independent of $\tilde v$.
\end{lemma}

\begin{lemma}  For any small enough  $\delta>0$ and any triplet  $(\tilde v^+,\tilde v^-,\tilde v^m)\in K$ the following inequality holds:
\begin{multline}\label{estim5}
c
\left(
\|\tilde v^+\|^2_{H^1(\Omega_+)}+\|\tilde v^-\|^2_{H^1(\Omega_-)}+
\right.\\
\left.+
\delta\|\tilde v_{1,1}^m-\frac{\varphi^\prime}{\delta}\tilde v_{1,2}^m\|^2_{L_2(\Omega_M)}+
\frac{1}{\delta}\|\tilde v_{1,2}^m\|^2_{L_2(\Omega_M)}+\delta\|\tilde v_{2,1}^m-\frac{\varphi^\prime}{\delta}\tilde v_{2,2}^m\|^2_{L_2(\Omega_M)}+
\frac{1}{\delta}\|\tilde v_{2,2}^m\|^2_{L_2(\Omega_M)}\right)\leq\\\leq
\int\limits_{\Omega_+}C^0\varepsilon(\tilde v^+):\varepsilon(\tilde v^+)\,dx+
\int\limits_{\Omega_-}C^0\varepsilon(\tilde v^-):\varepsilon(\tilde v^-)\,dx+\\+
\delta \int\limits_{\omega_m}C^0e^\delta(\tilde v^m): e^\delta(\tilde v^m)\,dz+
\delta \int\limits_{\Omega_m}C^1e^\delta(\tilde v^m): e^\delta(\tilde v^m)\,dz
\end{multline}
 with a constant $c$ independent of $\delta$ and $(\tilde v^+,\tilde v^-,\tilde v^m)$.
\end{lemma}

\begin{proof} Due to the fact that the domain   $\Omega$ and the set  $K$ do not depend on  $\delta$, from the Korn inequality it follows existence a constant  $c$ independent of $v\in H^1_{\Gamma_D}(\Omega)$ and  $\delta>0$ such that
\begin{multline}\label{nerlemma2}
c(\|v^+\|^2_{H^1(\Omega_+^\delta)}+
\|v^-\|^2_{H^1(\Omega_-^\delta)}+
\|v^m\|^2_{H^1(\Omega_m^\delta)}\leq  \int\limits_{\Omega}C \varepsilon(v ):\varepsilon(v )\,dy= \\
=\int\limits_{\Omega_+^\delta}C^0\varepsilon(v^+):\varepsilon(v^+)\,dy+
\int\limits_{\Omega_-^\delta}C^0\varepsilon(v^-):\varepsilon(v^-)\,dy+\\+
\int\limits_{\omega_m^\delta}C^0\varepsilon(v^m):\varepsilon(v^m)\,dy+
\int\limits_{\Omega_m^\delta}C^1\varepsilon(v^m):\varepsilon(v^m)\,dy.\end{multline} Here  $v^+$, $v^-$, and  $v^m$ are the restrictions of the function   $v$ onto  domains  $\Omega_+^\delta$, $\Omega_-^\delta$, and   $\Omega_M^\delta=\Omega_m^\delta\cup\omega_m^\delta$, respectively.

Let us apply coordinate transformations   \eqref{transpm} and  \eqref{transm} in integrals in \eqref{nerlemma2}. After that we use expansions  \eqref{Aas1} and the fact that after coordinate transformation \eqref{transpm} the following expansion hold:
$$
\|v^\pm\|^2_{H^1(\Omega_\pm^\delta)}=\|\tilde v^\pm\|^2_{H^1(\Omega_\pm)}+\tilde r_\pm(\delta;\tilde v^\pm)$$  with an estimate
$$
|\tilde r_\pm(\delta;v^\pm)|\leq c\delta\|\tilde v^\pm\|^2_{H^1(\Omega_\pm)},$$ where  $\tilde v^+$, $\tilde v^-$, and $\tilde v^m$ are defined akin to   \eqref{tilde1} and \eqref{tilde2}.
Thus, inequality  \eqref{estim5} is valid for all small enough   $\delta>0$.
\end{proof}

\begin{theorem} The solution   $(\tilde u^+_\delta,\tilde u^-_\delta,\tilde u^m_\delta)$  to problem  \eqref{3'} satisfies the uniform  estimates:
\beq\label{estimm1}
\|\tilde u^+_\delta\|_{H^1(\Omega_+)^2}\leq c,\ \ \|\tilde u^-_\delta\|_{H^1(\Omega_-)^2}\leq c,
\eeq
\beq\label{estimm2}
\delta^{\frac{p+1}{2}}\|\tilde u^m_{\delta 1,1}-\frac{\varphi^\prime}{\delta}\tilde u^m_{\delta 1,2}\|_{L_2(\Omega_m)}\leq c,
\  \delta^{\frac{p-1}{2}}\|\tilde u^m_{\delta 2,2}\|_{L_2(\Omega_m)}\leq c,
\eeq
%\beq\label{estimm3}
%\delta^{\frac{p+1}{2}}\| \frac{1}{\delta}\tilde u^m_{\delta 1,2}+\tilde u^m_{\delta 2,1}-\frac{\varphi^\prime}{\delta}\tilde u^m_{\delta %2,2}\|_{L_2(\Omega_m)}\leq c,
%\eeq
\beq\label{estimm4}
\delta^{\frac{p-1}{2}}\|\tilde u^m_{\delta 1,2}\|_{L_2(\Omega_m)}\leq c,\
\delta^{\frac{p+1}{2}}\|\tilde u^m_{\delta 2,1}-\frac{\varphi^\prime}{\delta}\tilde u^m_{\delta 2,2}\|_{L_2(\Omega_m)}\leq c,
\eeq
\beq\label{estimm12}
%\delta^{\frac{1}{2}}\|\tilde u^m_{\delta 1,1}-\frac{\varphi^\prime}{\delta}\tilde u^m_{\delta 1,2}\|_{L_2(\omega_m)}\leq c,\
   \delta^{-\frac{1}{2}}\|\tilde u^m_{\delta 2,2}\|_{L_2(\omega_m)}\leq c,
\eeq
\beq\label{estimm14}
\delta^{-\frac{1}{2}}\|\tilde u^m_{\delta 1,2}\|_{L_2(\omega_m)}\leq c
%\delta^{\frac{1}{2}}\|\tilde u^m_{\delta 2,1}-\frac{\varphi^\prime}{\delta}\tilde u^m_{\delta 2,2}\|_{L_2(\omega_m)}\leq c
.
\eeq
\end{theorem}

\begin{proof} First of all, let us substitute $(\tilde u^+_\delta, \tilde u^-_\delta, \tilde u^m_\delta)$ in \eqref{3'}. Taking into account \eqref{Aas1} and \eqref{Aas2} and the Korn inequality, we easily get estimates \eqref{estimm1}, \eqref{estimm2}, and \eqref{estimm12}.

To prove \eqref{estimm4} we substitute $(\tilde u^+_\delta, \tilde u^-_\delta, \tilde u^m_\delta)$ in \eqref{estim5} and multiply the obtained inequality  by $\delta^p$. Due to estimates \eqref{estimm1} and since   $p\geq0$,  we have the following chain of inequalities:
\begin{multline*}
c
\left(
\delta^{p-1}\|\tilde u_{\delta 1,2}^m\|^2_{L_2(\Omega_m)}+\delta^{p+1}\|\tilde u_{\delta 2,1}^m-\frac{\varphi^\prime}{\delta}\tilde u_{\delta 2,2}^m\|^2_{L_2(\Omega_m)}\right)\leq\\
\leq
\delta^p\int\limits_{\Omega_+}C^0\varepsilon(\tilde u^+_\delta):\varepsilon(\tilde u^+_\delta)\,dx+
\delta^p\int\limits_{\Omega_-}C^0\varepsilon(\tilde u^-_\delta):\varepsilon(\tilde u^-_\delta)\,dx+\\+
\delta^{p+1} \int\limits_{\omega_m}C^0e^\delta(\tilde u^m_\delta): e^\delta(\tilde u^m_\delta)\,dz+
\delta^{p+1} \int\limits_{\Omega_m}C^1e^\delta(\tilde u^m_\delta): e^\delta(\tilde u^m_\delta)\,dz\leq\\
\leq\int\limits_{\Omega_+}C^0\varepsilon(\tilde u^+_\delta):\varepsilon(\tilde u^+_\delta)\,dx+
\int\limits_{\Omega_-}C^0\varepsilon(\tilde u^-_\delta):\varepsilon(\tilde u^-_\delta)\,dx+\\+
\delta \int\limits_{\omega_m}C^0e^\delta(\tilde u^m_\delta): e^\delta(\tilde u^m_\delta)\,dz+
\delta^{p+1} \int\limits_{\Omega_m}C^1e^\delta(\tilde u^m_\delta): e^\delta(\tilde u^m_\delta)\,dz=\\=
\int\limits_{\partial\Omega^+\cap\Gamma_N}g\tilde u_\delta^+\,ds+
\int\limits_{\partial\Omega^-\cap\Gamma_N}g\tilde u_\delta^-\,ds-\delta r_\pm(\delta,\tilde u_\delta^\pm,\tilde u_\delta^\pm)\leq c.
\end{multline*}

Estimate   \eqref{estimm14} is followed from \eqref{estim5} and \eqref{estimm1}-\eqref{estimm4}.
\end{proof}

Below we consider the limiting cases separately depending on the positive parameter $p$.

\subsection{Case $0<p<1$ (homogeneous body)} From  \eqref{estimm1}, \eqref{estimm12}, and \eqref{estimm14}, we conclude   that there exists sequence still denoting by  $\delta$, functions $u^\pm\in H^1(\Omega_\pm)$, and $\sigma^1\in L_2(\Omega_M)^4$, such that the following convergences
$$
\tilde u^\pm_\delta \to   u^\pm \ \ \mbox{weakly in}\ \ H^1(\Omega_\pm),$$
$$
\delta^\frac{p+1}{2} C^1e^\delta(\tilde u^m_\delta)\to \sigma^1\ \ \mbox{weakly in}\ \ L_2(\Omega_m)^4,$$
$$
\delta^\frac{1}{2} C^0e^\delta(\tilde u^m_\delta)\to \sigma^1\ \ \mbox{weakly in}\ \ L_2(\omega_m)^4,$$
\beq\label{um,2}
\tilde u^m_{\delta ,2}\to   u^m_{,2}=0\ \ \mbox{strongly in}\ \ L_2(\Omega_M)^2
\eeq take place.

From  \eqref{um,2} it follows that  $u$  is independent of  $z_2$. Moreover, the traces of the functions  $u^+$ and  $u^+$ on the curve  $T$ coincide with each other  (see \cite{Itou}), i.e.,
\beq\label{u-=u+}
u^-|_T=u^+|_T.
\eeq

Let us take a smooth function   $v\in C^1(\overline\Omega)$ and define functions
$$
v^\pm=v|_{\Omega_\pm},\ \ v^m(z_1,z_2)=v(z_1,\varphi(z_1)).$$ Since $v^m$ does not depend on $z_2$,
$$
\delta^{\frac{p+1}{2}}  e^\delta(v^m)\to 0\ \ \mbox{strongly in}\ \ L_2(\Omega_m)^4,
$$
$$
\delta^{\frac{ 1}{2}}  e^\delta(v^m)\to 0\ \ \mbox{strongly in}\ \ L_2(\omega_m)^4.
$$

After substituting   $(v^-,v^+,v^m)$  in  \eqref{3'} and passing to the limit as $\delta\to0$ we have
\begin{multline}\label{finalupm3}
\int\limits_{\Omega_+}C^0\varepsilon(  u^+):\varepsilon(v^+)\,dx+
\int\limits_{\Omega_-}C^0\varepsilon(  u^-):\varepsilon(v^-)\,dx
=\\=
\int\limits_{\partial\Omega^+\cup\Gamma_N}gv^+\,ds+
\int\limits_{\partial\Omega^-\cup\Gamma_N}gv^-\,ds,
\end{multline}

Due to  \eqref{u-=u+}, variational equality   \eqref{finalupm3} can be  rewritten in an equivalent form
\beq\label{finalupm33}
\int\limits_{\Omega}C^0\varepsilon(  u):\varepsilon(v)\,dx
=
\int\limits_{\Gamma_N}gv\,ds \ \ \forall v\in H^1_{\Gamma_D}(\Omega),
\eeq where  $u\in H^1_{\Gamma_D}(\Omega)$ is defined as follows:
\beq\label{defu}
  u(x_1,x_2)=\left\{\begin{array}{ll}  u^-(x_1,x_2),\ \ \ (x_1,x_2)\in \Omega_-,\\
  u^+(x_1,x_2),\ \ \ (x_1,x_2)\in \Omega_+\end{array}\right.\eeq
Problem  \eqref{finalupm33} describes equilibria of an homogeneous body without any inclusions.

\subsection{Case $p=1$ (crack with  adhesion of its faces)}  From Lemma 1 and estimates \eqref{estimm1}-\eqref{estimm14} it follows
$$
\|\tilde u^m_{\delta}\|_{L_2(\Omega_M)^2}\leq c,$$
uniformly with respect to $\delta$.

Theorem 1 allows us to find a sequence still denoting by  $\delta$ and   functions $u^\pm\in H^1_{\Gamma_D}(\Omega_\pm)$, $u^m\in L_2(\Omega_M)^2$, $\sigma^1\in L_2(\Omega_M)^4$ such that the following convergences
$$
\tilde u^\pm_\delta \to u^\pm \ \ \mbox{weakly in}\ \ H^1_{\Gamma_D}(\Omega_\pm),$$
$$
\tilde u^m_{\delta   } \to u^m \ \ \mbox{weakly in}\ \ L_2(\Omega_m)^2,\ $$
$$
\tilde u^m_{\delta  ,2} \to u^m_{ ,2}\ \ \mbox{weakly in}\ \ L_2(\Omega_m)^2, $$
$$
\delta \tilde u^m_{\delta  ,1} \to 0\ \ \mbox{weakly in}\ \ L_2(\Omega_m)^2, $$
$$
\delta  \tilde u^m_{\delta  ,1}-\varphi^\prime\tilde  u^m_{\delta  ,2} \to -\varphi^\prime  u^m_{  ,2}\ \ \mbox{weakly in}\ \ L_2(\Omega_m)^2,$$
$$
\delta^\frac{1}{2} C^1e^\delta(\tilde u^m_\delta)\to \sigma^1\ \ \mbox{weakly in}\ \ L_2(\omega_m)^4$$
hold as $\delta\to0$.
Then we have
\beq\label{deltaedelta}
\delta C^1e^\delta(\tilde u^m_\delta)\to \sigma^0(u^m)\ \ \mbox{weakly in}\ \ L_2(\Omega_m)^4,\eeq
where
$$
\sigma^0( u^m)=\left(\begin{array}{cc}-(\lambda_m+2\mu_m)\varphi^\prime   u^m_{ 1,2}+\lambda_mu_{2,2}^m&\mu_m  (u^m_{1,2}-\varphi^\prime   u^m_{ 2,2})\\
\mu_m (u^m_{1,2}-\varphi^\prime   u^m_{ 2,2})& (\lambda_m+2\mu_m)\tilde u^m_{2,2}-\lambda_m \varphi^\prime   u^m_{ 1,2}\end{array}\right).$$
Moreover, for any $v^m\in H^1(\Omega_m)^2$ we have
\beq\label{deltae1v}
\delta e^\delta(v^m)\to e^0(v)\ \ \mbox{strongly in}\ \ L_2(\Omega_m)^4,
\eeq
where
$$
e^0(v^m)=\left(\begin{array}{cc}-\varphi^\prime v^m_{1,2}&\frac{1}{2}( v^m_{1,2}-\varphi^\prime v^m_{2,2})\\
\frac{1}{2}( v^m_{1,2}-\varphi^\prime v^m_{2,2})& v^m_{2,2}\end{array}\right).$$

Let us investigate properties limit functions. Firstly, by  the same reasons as the previous case, we have equality \eqref{u-=u+} on part $ T\setminus\overline{S}$ of the curve $T$.

Then,   take a function $v^m\in C_0^\infty(\Omega_m)^2$ and substitute a triplet  $(0,0,v^m)\in K$ in variational equality  \eqref{3'} as a test function. After passing to the limit as $\delta\to0$, due to \eqref{deltaedelta} and \eqref{deltae1v}, we get
$$
\int\limits_{\Omega_m}\sigma^0(  u^m): e^1(v^m)\,dz=0,$$
which is valid for all $v^m\in C_0^\infty(\Omega_m)^2$. Then there exist functions $\alpha_1(z_1)$ and $\alpha_2(z_1)$ such that
\beq\label{13_1}
-\varphi^\prime(z_1)(\mu_m+\lambda_m)  u^m_{1,2}+(\varphi'^2(z_1)\mu_m+(\lambda_m+2\mu_m))  u^m_{2,2}=\alpha_1(z_1)\ \ \mbox{a.e. in}\ \ \Omega_m,
\eeq
\beq\label{13_2}
((\lambda_m+2\mu_m)\varphi'^2(z_1)+\mu_m)  u^m_{1,2}-\varphi'(z_1)(\mu_m+\lambda_m)  u^m_{2,2}=\alpha_2(z_1)\ \ \mbox{a.e. in}\ \ \Omega_m,
\eeq
Integrating  \eqref{13_1} and \eqref{13_2} with respect to  $z_2$ from $\psi_-(z_1)$ to  $\psi_+(z_1)$, we find
$$
\alpha_1(z_1)=\frac{-\varphi^\prime(z_1)(\mu_m+\lambda_m)[  u_{1} ]+(\varphi'^2(z_1)\mu_m+(\lambda_m+2\mu_m))[  u_{2} ]}{\psi_+(z_1)-\psi_-(z_1)},
$$
$$
\alpha_2(z_1)=\frac{((\lambda_m+2\mu_m)\varphi'^2(z_1)+\mu_m)[  u_{1} ]-\varphi'(z_1)(\mu_m+\lambda_m)[  u_{2} ]}{\psi_+(z_1)-\psi_-(z_1)},
$$
where $[  u_i  ]=u_i^+(z_1,\varphi(z_1))-  u_i^-(z_1,\varphi(z_1))$, $z_1\in I_1$, $i=1,2$.

Note that    system of algebraic equations \eqref{13_1}, \eqref{13_2} for  $u^m_{1,2}$ and $u^m_{2,2}$ is non\-de\-ge\-ne\-rate, because its determinant is equal to     $-\mu_m(\lambda_m+2\mu_m)((\varphi^\prime)^2+1)^2$ and, therefore, is non-zero. After obvious calculation,      we find
$$
  u^m_{i,2}(z_1,z_2)=\frac{u_i^+(z_1,\varphi(z_1))-  u_i^-(z_1,\varphi(z_1))}{\psi_+(z_1)-\psi_-(z_1)}, \ \ i=1,2.
$$
Integrating this representation with respect to $z_2$ from $\psi_+(z_1)$ to $z_2$, we get
\begin{multline}\label{tildeuim}
  u_i^m(z_1,z_2)=\frac{u_i^+(z_1,\varphi(z_1))-  u_i^-(z_1,\varphi(z_1))}{\psi_+(z_1)-\psi_-(z_1)}z_2+\\+\frac{  u_i^-(z_1,\varphi(z_1))\psi_+(z_1)-  u_i^+(z_1,\varphi(z_1))\psi_-(z_1)}{\psi_+(z_1)-\psi_-(z_1)},
\end{multline}
$i=1,2$.

Let us find a variational problem for functions $  u^\pm$, $  u^m$.  For this, we take an arbitrary  function   $v \in C^1(\Omega)^2 \cap H^{1,0}(\Omega\setminus\overline{S})^2$,
where
$$
H^{1,0}(\Omega\setminus\overline{S})=\{v\in H^1(\Omega\setminus\overline{S})\ \ |\ \ v=0\ \mbox{a.e. on}\ \Gamma_D\},
$$
and  define a function   $v^m=(v^m_1,v^m_2)$ as follows:
\begin{multline}\label{formulavm}
v^m_i(z_1,z_2)=\frac{v^+_i(z_1,\varphi(z_1))-v^-_i(z_1,\varphi(z_1))}{\psi_+(z_1)-\psi_-(z_1)}z_2+\\+\frac{-v^+_i(z_1,\varphi(z_1))
\psi^-(z_1)+v^-_i(z_1,\varphi(z_1))\psi_+(z_1)}{\psi_+(z_1)-\psi_-(z_1)}
\end{multline} in the domain  $\Omega_m$;
\beq\label{formulavmw}
v^m_i(z_1,z_2)=v^-_i(z_1,\varphi(z_1)),
\eeq
in the domain  $\omega_m$, where  $v^\pm$ is a restriction of   $v$ onto  $\Omega_\pm$, $i=1,2$. Since $v \in C^1(\Omega)^2 \cap H^{1,0}(\Omega\setminus\overline{S})^2$,   we have
$$
v^-(z_1,\varphi(z_1))=v^+(z_1,\varphi(z_1)),\ \ z_1\in T\setminus\overline{S}$$ and the function $v^m$ is defined correctly. Moreover, $$
\delta^{\frac{1}{2}}  e^\delta(v^m)\to 0\ \ \mbox{strongly in}\ \ L_2(\omega_m)^4
$$
as $\delta$ goes to $0$.


Now substitute a triplet $(v^+,v^-,v^m)$ in   \eqref{3'} as a test function. After passing to the limit as   $\delta\to0$, we get
\begin{multline}\label{varum}
\int\limits_{\Omega_+}C^0\varepsilon(  u^+):\varepsilon(v^+)\,dx+
\int\limits_{\Omega_-}C^0\varepsilon(  u^-):\varepsilon(v^-)\,dx+
\int\limits_{\Omega_m}\sigma^0(  u^m): e^1(v^m) \,dz=\\=
\int\limits_{\partial\Omega^+\cup\Gamma_N}gv^+\,ds+\int\limits_{\partial\Omega^-\cup\Gamma_N}gv^-\,ds.
\end{multline}

Due to \eqref{tildeuim},   \eqref{formulavm}, and \eqref{formulavmw} we can integrate   the third term in \eqref{varum} with respect to $z_2$. Finally,  the function $u^+$ and $u^-$ satisfy the following variational equality:
\begin{multline}\label{finalupm}
\int\limits_{\Omega_+}C^0\varepsilon(\tilde u^+):\varepsilon(v^+)\,dx+
\int\limits_{\Omega_-}C^0\varepsilon(\tilde u^-):\varepsilon(v^-)\,dx+
\int\limits_{S}\frac{A_\varphi [  u]\cdot [v]}{\psi_+-\psi_-} \,ds=\\=
\int\limits_{\partial\Omega^+\cup\Gamma_N}gv^+\,ds+
\int\limits_{\partial\Omega^-\cup\Gamma_N}gv^-\,ds\quad \forall v\in H^{1,0}(\Omega\setminus\overline{S}),
\end{multline}
where $v^-$ and $v^+$ are restrictions of $v$ onto domains $\Omega_-$ and $\Omega_+$, respectively;
$$
A_\varphi=\frac{1}{(1+(\varphi^\prime)^2)^\frac{1}{2}}\left(\begin{array}{cc} (\lambda_m+2\mu_m)\varphi'^2+\mu_m&- (\lambda_m+\mu_m)\varphi'\\
- (\lambda_m+\mu_m)\varphi'&\lambda_m+2\mu_m+ \mu_m\varphi'^2\end{array}\right).
$$

Due to the Korn inequality and the fact that matrix $A_\varphi$ is positive definite, the problem \eqref{finalupm} has the unique solution $  u^+,   u^-$.

Let us rewrite problem  \eqref{finalupm} in  an equivalent form.   Since    $  u^+=  u^-$ a.e. on $(\partial\Omega_-\cap\partial \Omega_+)\setminus \overline{S}$,  the  function $u$ defined by \eqref{defu}
 belongs to the space  $H^{1,0}(\Omega)$, and it is a unique solution of the following variational problem:
\beq\label{FINAL1}
\int\limits_\Omega C^0\varepsilon(  u ):\varepsilon(v )\,dx+\int\limits_{S}\frac{A_\varphi [  u]\cdot [v]}{\psi_+-\psi_-} \,ds=
\int\limits_{ \Gamma_N}gv \,ds\ \ \forall v\in H^{1,0}(\Omega\setminus\overline{S}).\eeq

In the sense of distributions, problem  \eqref{FINAL1} implies  a solution of the following boundary value problem:
$$
-{\rm div}\,\sigma(u)=0,\ \ \sigma(u) =C^0\varepsilon(  u) \ \ \mbox{in}\ \ \Omega\setminus\overline{S},
$$
$$
[\sigma(u) \nu]=0\ \mbox{on}\      S,
$$
$$
\sigma(u) \nu=\frac{A_\varphi [  u]}{\psi_+-\psi_-}\ \mbox{on}\      S,
$$
$$
  u=0\ \mbox{on}\ \Gamma_D,\ \ \sigma(u)\nu=g\ \mbox{on}\ \Gamma_N.
$$


Note that the interfacial condition on $S$ is generalized known ones because it takes into account simultaneously both curvature and roughness of the inclusion (see, e.g., \cite{Baranova,Benveniste,Dumont,Rizzoni,Kushch}).

Let us analyze the interface condition in the differential formulation of the equilibrium problem. Define by $u_\nu$ and $u_\tau$ normal and tangential components of the displacement vector on  the crack $S$; and denote by $\sigma_\nu$ and $\sigma_\tau$ the shear and normal components  of the traction $\sigma\nu$ on $S$, i.e.,
$$
u_\nu=u\nu=u_i\nu_i,\ \ u_\tau=u\tau=u_i\tau_i,
$$
$$
\sigma_\nu=(\sigma\nu)\nu=\sigma_{ij}\nu_i\nu_j,\ \
\sigma_\tau=(\sigma\nu)\tau=\sigma_{ij}\nu_i\tau_j,
$$
where $\nu$ and $\tau$ are unit normal and tangent vectors to $S$, respectively,
$$
\nu=\frac{(-\varphi^\prime,1)}{(1+(\varphi^\prime)^2)^\frac{1}{2}},\ \ \tau=\frac{(1,\varphi^\prime)}{(1+(\varphi^\prime)^2)^\frac{1}{2}}.
$$
After simple calculations we get
$$
A_\varphi[u]=(\lambda_m+2\mu_m)[u_\nu]\nu+\mu_m[u_\tau]\tau.
$$
Then the interface condition on the crack $S$ can be rewritten  as follows:
$$
\sigma_\nu =(\lambda_m+2\mu_m)\frac{[u_\nu]}{\psi_+-\psi_-},\ \ \sigma_\tau =\mu_m\frac{[u_\tau]}{\psi_+-\psi_-}.
$$
It is seen the shear and normal components  of the traction are proportional to the jump of normal and tangential components of the displacements, respectively.
 These conditions, known in interface models of  Elasticity as  spring type conditions, is derived  for  crack case.





\subsection{Case $p>1$ (crack with stress-free faces)}   Again from Lemma 1,   estimates \eqref{estimm2} and \eqref{estimm4} we can conclude that there exists sequence still  denoting by $\delta$, and functions $u^\pm\in H^1(\Omega_\pm)$, $\sigma^0\in L_2(\Omega_m)^4$, such that the following convergences
$$
\tilde u^\pm \to   u^\pm \ \ \mbox{weakly in}\ \ H^1_{\Gamma_D}(\Omega_\pm),$$
$$
\delta^\frac{p+1}{2} C^1e^\delta(\tilde u^m_\delta)\to \sigma^0\ \ \mbox{weakly in}\ \ L_2(\Omega_m)^4$$ hold.
Moreover, for any $v^m\in H^1(\Omega_m)$ we have
$$
\delta^{\frac{p+1}{2}}  e^\delta(v^m)\to 0\ \ \mbox{strongly in}\ \ L_2(\Omega_m)^4.
$$

To find a problem for   $  u^\pm$, take     $v \in C^1(\Omega) \cap H^{1,0}(\Omega\setminus\overline{S})$
and define in the domain   $\Omega_M$ a function  $v^m$ by    \eqref{formulavm} and \eqref{formulavmw}. Then we substitute  a triplet $(v^+,v^-,v^m)$    in  \eqref{3'} as a test function. After passing to the limit as $\delta\to0$, we have the following variational equality:
\begin{multline}\label{finalcrack}
\int\limits_{\Omega_+}C^0\varepsilon(  u^+):\varepsilon(v^+)\,dx+
\int\limits_{\Omega_-}C^0\varepsilon(  u^-):\varepsilon(v^-)\,dx=\\=
\int\limits_{\partial\Omega^+\cup\Gamma_N}gv^+\,ds+
\int\limits_{\partial\Omega^-\cup\Gamma_N}gv^-\,ds.
\end{multline}

As before, we define a function   $u$ by  \eqref{defu}. Then variational equality  \eqref{finalcrack} can be rewritten in the following for:
\beq\label{FINALcrack}
\int\limits_{\Omega\setminus\overline{S}}C^0\varepsilon(  u):\varepsilon(v)\,dx+
\int\limits_{\Gamma_N}gv\,ds \ \
\forall v\in H^{1,0}(\Omega\setminus\overline{S}).
\eeq


 It is known that  \eqref{FINALcrack} describes equilibria of an elastic body with the crack  $S$. In the sense of distributions, the function  $u$ satisfies the following boundary value problem:
 $$
-{\rm div}\,\sigma(u)=0,\ \ \sigma(u) =C^0\varepsilon(  u) \ \ \mbox{in}\ \ \Omega\setminus\overline{S},
$$
$$
\sigma(u) \nu=0\ \mbox{on}\     S.
$$
$$
  u=0\ \mbox{on}\ \Gamma_D,\ \ \sigma(u)\nu=g\ \mbox{on}\ \Gamma_N.
$$






\bigskip
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