Static elastic equilibrium (${\sigma }_{ij,j}+f_i=0$)

A symmetric stress tensor is in static elastic equilibrium when its divergence balances the prescribed body force. Its boundary traction is $t_i={\sigma }_{ij}{n}_j$. With zero body force, integration by parts gives ${\int }_{\Omega }\sigma :\mathrm{sym}\nabla v={\int }_{\partial \Omega }t\cdot v$. This is the weak principle of virtual work for a static elastic body.