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.