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\begin{center}
ON A CLASS OF BOUNDARY VALUE PROBLEMS OF THERMOELASTICITY FOR AN
 ORTHOTROPIC BODY IN THE
RECTANQULAR CARTESIAN COORDINATE SYSTEM

\vspace*{0.2cm}
{\it I. Khomasuridze}

\vspace*{0.2cm}
{\it I.Vekua Institute of Applied Mathematics}\\
{\it I.Javakhishvili Tbilisi State University}
\end{center}

\vspace*{0.2cm}
The basic equations system describing the thermoelastic equilibrium of a 
plane-stressed, orthotropic homogeneous body has the form [1]

\begin{equation}
\begin{array}{l}
\displaystyle
\Delta _{21}\varphi _{1}=\frac{g}{\gamma _{4}}\frac{\partial ^{2}\tilde{T}}{%
\partial z^{2}}, \\ 
\ \\\displaystyle
\Delta _{22}\Phi _{1}=-\frac{\gamma _{2}}{\gamma _{4}}\frac{\partial
^{2}\varphi _{1}}{\partial z^{2}}+\frac{\gamma _{5}\gamma _{2}}{\gamma _{4}}%
\frac{\partial ^{2}\tilde{T}}{\partial z^{2}}
\end{array}
\label{1}
\end{equation}

Further we use the following notation 
$$a=\sqrt{\gamma _{1}\gamma
_{4}-\gamma _{2}^{2}}i,\;{\rm if}\;
\gamma _{2}^{2}-\gamma _{1}\gamma _{4}<0,\;\;i=%
\sqrt{-1}; \;\; a=\sqrt{\gamma _{2}^{2}-\gamma _{1}\gamma _{4}},\;{\rm if}\;\gamma
_{2}^{2}-\gamma _{1}\gamma _{4}\ge 0,$$
$$a_{1}=a-\frac{1}{2\mu },\;\;a_{2}=a+%
\frac{1}{2\mu },\;\;g=\gamma _{2}\gamma _{5}-a\gamma _{5}-\gamma _{3}\gamma
_{4},$$
$$\Delta _{21}=\frac{\gamma _{2}-a}{\gamma _{4}}\frac{\partial ^{2}}{%
\partial z^{2}}+\frac{\partial ^{2}}{\partial x^{2}},\;\;  
\Delta _{22}=\frac{\gamma _{2}+a}{\gamma _{4}}\frac{\partial ^{2}}{\partial z^{2}}+\frac{%
\partial ^{2}}{\partial x^{2}},\;\;  
\Delta _{23}=\frac{\lambda _{3}}{\lambda
_{1}}\frac{\partial ^{2}}{\partial z^{2}}+\frac{\partial ^{2}}{\partial x^{2}%
},$$ 
$$
\gamma _{1}=\frac{E_{1}-E_{2}\nu _{12}^{2}}{E_{1}^{2}},\;\; \gamma _{2}=
\frac{E_{1}-2\mu \left( \nu _{13}+\nu _{23}\nu _{12}\right) }{2\mu E_{1}}, 
\gamma _{3}=-\left( \frac{E_{2}\nu _{12}k_{2}}{E_{1}}+k_{1}\right),
$$
$$ 
\gamma _{4}=\frac{E_{2}-E_{3}\nu _{23}^{2}}{E_{3}E_{2}},\;\;\gamma
_{5}=-\left( \nu _{23}k_{2}+k_{3}\right),
$$
where $E_{1},$ $E_{2},$ $E_{3}$ are Young's modulus along $Ox,Oy,Oz$; 
$\nu _{12}$ is Poisson's ratio in the plane $Oxy$; 
$\nu _{13}$ is Poisson's ratio in the plane $Oxz$; 
$\nu _{23}$ is Poisson's ratio in the plane $Oyz$;  
$k_{1},$ $k_{2},$ $k_{3}$ are temperature linear expansion coefficients
along $Ox,$ $Oy,$ $Oz$ respectively; 
$\lambda _{1},$ $\lambda _{2},$ $\lambda _{3}$ are heat
conductivity coefficients along $Ox,$ $Oy,$ $Oz$ respectively and  
$\displaystyle\frac{\partial ^{2}%
\tilde{T}}{\partial z^{2}}=T$ is a temperature in the elastic body $\left( \Delta
_{23}T=0\text{, }\Delta _{23}\tilde{T}=0\right) $. 

For the overwhelming majority of
orthotropic bodies 
$$\displaystyle\mu =\frac{E_{1}E_{2}}{E_{1}\left( 1+2\nu _{12}\right)
+E_{2}},$$
where $\mu $ is a shift modulus in the plane $Ozx$; 

The following dependences
$$
E_{1}\nu _{21}=E_{2}\nu _{12},\;E_{2}\nu _{32}=E_{3}\nu _{23},\;E_{3}\nu
_{13}=E_{1}\nu _{31}.
$$

If $a\ne 0$, $\displaystyle\gamma _{3}=\frac{\lambda _{3}}{\lambda _{1}}\gamma _{5}$, then
solution of the system (1) will take the form
\begin{equation}
\varphi _{1}=\tilde{\varphi}+\gamma _{5}\tilde{T},\;\;\Phi _{1}=\tilde{\Phi}%
+G_{2}\tilde{\varphi}  \label{2}
\end{equation}
where $G_{2}=-\displaystyle\frac{\gamma _{2}}{2a}$, $\Delta _{21}\tilde{\varphi}=0$,  $%
\Delta _{22}\tilde{\Phi}=0$.

Taking into consideration (1) , (2), 
components of the displacement vector $\vec{U}$ and components of the stress
tensor could be presented by means of the functions $\tilde{\Phi}$, $\tilde{%
\varphi}$, $\tilde{T}$ in the following form

\begin{equation}
\begin{array}{l}
\displaystyle
u=\left( 1+\frac{a_{1}G_{2}}{\gamma _{2}}\right) \frac{\partial \tilde{%
\varphi}}{\partial x}+\frac{a_{1}}{\gamma _{2}}\frac{\partial \tilde{\Phi}}{%
\partial x}+\gamma _{5}\frac{\partial \tilde{T}}{\partial x}, \\ 
\ \\\displaystyle
w=-\left( 1+\frac{a_{2}G_{2}}{\gamma _{2}}\right) \frac{\partial \tilde{%
\varphi}}{\partial z}-\frac{a_{2}}{\gamma _{2}}\frac{\partial \tilde{\Phi}}{%
\partial z}-\gamma _{5}\frac{\partial \tilde{T}}{\partial z}, 
\end{array}
\end{equation}
%\newpage
$$
Y_{y}=k_rE_r\frac{\partial^2 \tilde{T}}{\partial z^2},
$$

$$
Z_{z}=\frac{1}{\gamma _{2}}\frac{\partial ^{2}\tilde{\Phi}}{\partial x^{2}}+%
\frac{G_{2}}{\gamma _{2}}\frac{\partial ^{2}\tilde{\varphi}}{\partial x^{2}},
$$

$$
X_{x}=\frac{1}{\gamma _{2}}\frac{\partial ^{2}\tilde{\Phi}}{\partial z^{2}}+%
\frac{G_{2}}{\gamma _{2}}\frac{\partial ^{2}\tilde{\varphi}}{\partial z^{2}},
$$
 
$$
Z_{x}=-\frac{1}{\gamma _{2}}\frac{\partial ^{2}\tilde{\Phi}}{\partial
x\partial z}-\frac{G_{2}}{\gamma _{2}}\frac{\partial ^{2}\tilde{\varphi}}{%
\partial x\partial z}.
$$

We consider thermoelastic equilibrium of a plane stressed body which occupies
the domain $\Omega _{p}=\left\{ 0<x<x_{1},\,0<z<z_{1}\right\} $.

Let us consider the following boundary value conditions
\begin{equation}
\begin{array}{l}
\text{If }x=x_{0}=0\text{ then }X_{x}=0\text{, }Z_{x}=0\text{, }%
T=f_{1}\left( z\right) \text{,} \\ 
\text{If }x=x_{1}\text{ then }X_{x}=0\text{, }Z_{x}=0\text{, }T=f_{2}\left(
z\right) \text{,} \\ 
\text{If }z=z_{0}=0\text{ then }Z_{z}=0\text{, }Z_{x}=0\text{, }%
T=f_{3}\left( x\right) \text{,} \\ 
\text{If }z=z_{1}\text{ then }Z_{z}=0\text{, }Z_{x}=0\text{, }T=f_{4}\left(
x\right) .
\end{array}
\label{4}
\end{equation}

The solution of problem (1), (4)
(solution for temperature problem in the domain $\Omega _{p}$ is trivial and
can be easily constructed using the method of separation of variables) could
be presented assuming in (3) $\tilde{\varphi}=0$, $%
\tilde{\Phi}=0$ . But in this case everywhere in the domain the components of
the stress tensor

\[
\begin{array}{l}
Z_{z}=0, \\ 
X_{x}=0, \\ 
Z_{x}=0,
\end{array}
\]
regardless of temperature field, but $Y_y$, $u$, $w$ depend on the function $T$,
and this is the generalization for orthotropic bodies of temperature effect obtained
by N.Muskhelishvili [2].

\vspace*{0.2cm}
\footnotesize
\begin{center}
{\bf R e f e r e n c e s}

\vspace*{0.1cm}
\end{center}
\par
[1]
Khomasuridze N., Khomasuridze I., Two-dimensional Problems of Thermoelasticity 
in Cylindricall and Rectangular Coordinates. 
Proceedings of I.Vekua Institute of Applied Mathematics, 48, 1998.
\par
[2]
Muskhelishvili N., Some Basic Problems of the Mathematicals Theory of Elasticity.
Nauka, Moscow, 1966, (Russian).

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