Fig. 3 Traction of an volume element
To determine the deformations corresponding to the stresses applied to
this volume element, we use the principle of superposition [29]. The
elastic deformation of a material is only given with constraints, the
relationship between stress and strain can be written in the following
matrix form:
\(\par
\begin{bmatrix}\end{bmatrix}=\frac{1}{E}\par
\begin{bmatrix}\par
\begin{matrix}\ 1&-\nu&-\nu\\
-\nu&\ 1&-\nu\\
-\nu&-\nu&1\\
\end{matrix}&\par
\begin{matrix}0&\ \ \ \ \ \ \ 0&\ \ \ \ \ \ \ 0\\
0&\ \ \ \ \ \ 0&\ \ \ \ \ \ \ 0\\
0&\ \ \ \ \ \ 0&\ \ \ \ \ \ \ 0\\
\end{matrix}\\
\par
\begin{matrix}\ 0&\ \ \ 0&\ \ \ \ 0\\
\ 0&\ \ \ 0&\ \ \ \ 0\\
\ 0&\ \ \ 0&\ \ \ \ 0\\
\end{matrix}&\par
\begin{matrix}1+\nu&0&0\\
0&1+\nu&0\\
0&0&1+\nu\\
\end{matrix}\\
\end{bmatrix}\)*\(\par
\begin{bmatrix}\end{bmatrix}\) (5)
where \(\varepsilon_{\text{ij}}\) ; axial strain,ν ; poisson ratio, E ; elasticity modulus of material.