3.3.5 Model Equations

Substituiting the stress-strain relation (equation (3.64)), and the standard relationship between the total strain tensor and the displacement field (equation (3.67)) into equation (3.69), the equation for the displacement field in the metal line is obtained [132]

\[\begin{equation} \mu\nabla^2 u_\text{i}+(\lambda+\mu)\cfrac{\partial}{\partial x_\text{i}} (\nabla \cdot \vec u)= B \frac{\partial{\mathrm{Tr}(\epsilon^\text{th}+\epsilon^\text{v})}}{\partial{x_\text{i}}}. \end{equation}\] (3.70)

This is the Navier-Cauchy equation which enables the determination of the total line displacement in terms of the trace of the non-elastic stress tensors for given boundary conditions.

The stresses in the line can also be obtained by solving the mechanical equilibrium equation (3.69). In this way, 3D equations for the mechanical stress distribution inside the metal line, subjected to specific boundary conditions, can be derived [133]. In particular, the three normal components of the stress tensor determine the hydrostatic pressure p acting on the line as follows

\[\begin{equation} p=\pm\cfrac{\sigma_\text{xx}+\sigma_\text{yy}+\sigma_\text{zz}}{3}, \end{equation}\] (3.71)

where the positive sign results in tension, and the negative one in compression.