Evolution equation

Fundamental principle of dynamics in the entire volume \(\Omega\)
Change of momentum = External forces (in volume) + Traction (stress applied on exterior surface normals)

\(\Rightarrow \displaystyle \underbrace{\int_{\Omega} \rho\,\,p''(t)\,\mathrm{d}\Omega}_{\scriptsize{Change \; of \; momentum}} = \underbrace{\int_{\Omega} F(t)\,\mathrm{d}\Omega}_{\scriptsize{External\; forces}}+\underbrace{\int_{\partial \Omega}\,\sigma\,n\,\mathrm{d}S}_{\scriptsize{Traction\; force\; on\; the\; boundary}}\)

Using divergence theorem \(\displaystyle \int_{\partial \Omega} \sigma\,n\,\mathrm{d}S=\int_{\Omega}\,div(\sigma)\,\mathrm{d}\Omega\)

Equation in volume satisfied at each position \(p\in\Omega\)

\(\rho\;p''(t) = F(t)+div(\sigma(t))\)

\(\sigma=\left(\begin{array}{c}\sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{zx} & \sigma_{zy} & \sigma_{zz} \end{array} \right)\)

\(div(\sigma)=\left(\begin{array}{c}\frac{\partial \sigma_{xx}}{\partial x}+\frac{\partial \sigma_{yx}}{\partial y}+\frac{\partial \sigma_{zx}}{\partial z} \\ \frac{\partial \sigma_{xy}}{\partial x}+\frac{\partial \sigma_{yy}}{\partial y}+\frac{\partial \sigma_{zy}}{\partial z} \\ \frac{\partial \sigma_{xz}}{\partial x}+\frac{\partial \sigma_{yz}}{\partial y}+\frac{\partial \sigma_{zz}}{\partial z} \end{array} \right)\)