3.3.4 Mechanical Deformation

In mechanical theory, when atoms of a solid body move as a response to an applied force, their movement produces local strain which leads to changes in the body's shape and size [19]. This means that the body is subjected to deformation. In order to quantitatively study the deformation of a solid, the displacement field u at each point in the strained body, with respect to the initial position in the unstrained state, should be introduced. In linear elasticity, it is common to assume small displacement fields from the original configuration such that the total strain tensor is

\[\begin{equation} \epsilon_\text{ik}=\cfrac{1}{2}\left(\frac{\partial{u_\text{i}}}{\partial{x_\text{k}}}+\frac{\partial{u_\text{k}}}{\partial{x_\text{i}}}\right). \end{equation}\] (3.67)

The infinitesimal strain theory [67] describes the deformation of a solid body in which the displacements of the material are assumed to be much smaller than the body's dimensions. The evolution of the displacement vector is determined by the equation of motion derived from Newton's second law as follows

\[\begin{equation} \rho_\text{m}\frac{\partial^2{\vec u}}{\partial^2 t}=\nabla \cdot \overline{\overline\sigma}+\vec F_\text{b}, \end{equation}\] (3.68)

where F_b is the body force per unit volume which acts throughout the volume of a body, in contrast to the divergence of the stress tensor σ within the same volume. During electromigration, the accelerations in the structure are small and no external forces are acting on the line [133]. Equation (3.68) therefore reduces to the well known mechanical equilibrium equation in the form

\[\begin{equation} \nabla \cdot \overline{\overline\sigma}=0. \end{equation}\] (3.69)