Field discontinuities

If a volume contains an arbitrary surface $\Sigma$ across which A is discontinuous, then application of Greens Theorem to the subvolumes, Figure turns a volume integral in a sum of surface integrals.

$$\oint_V {\nabla}_{} \cdot {A}_{} ~dV = \oint_S {A}_{} \cdot ... ...t_{\Sigma} ( {A}_{1} \cdot {n}_{1} + {A}_{2} \cdot {n}_{2} ) dS$$ (7)

n₁ and n₂ are the unit normals when approaching the discontinuity surface either from side 1 or the side 2. A₁ and A₂ are forces on opposite sides of the surface. The first term is the integral over the exterior surface, while the second term accounts for effects created by interior discontinuity surfaces. Applying the Greens Theorem (6) to the energy integral 5, which we want to calculate, we get:

$$\oint_V {\sigma}_{} : {\epsilon}_{} ~dV = - \oint_{S_{tot}... ... ~dS + \oint_V {u}_{} \cdot({\bf \nabla}\cdot {\sigma}_{} )~dV$$ (8)

where $S_{\rm tot} = S + \Sigma$ is the sum of exterior and interior surfaces. In the static case the divergence of the stress ${\bf \nabla}\cdot {\sigma}_{}$ at any point in the medium is zero. If it were not zero we would have a net force acting on some part of the medium; then the medium would not be in equilibrium and would accelerate according to Newton's law (${\bf \nabla}\cdot {\sigma}_{} = {\partial^2 {u}_{} \over {\partial t^2}} - {F}_{\rm source}$). This static argument is equivalent to saying: after an external force is applied to the medium, we wait for a long time until the medium is in equilibrium (or there is no energy any more in form of elastic waves traveling though the medium).