Elastodynamic surface-jump representation ($u_i={\int }_0^t{\int }_{\Sigma }[-G_i^j[t_j]+M_{jk}{\partial }_{{\xi }_k}{G}_i^j]dSd\tau$)

Choose a surface normal $\mathbf{n}$ from its minus to its plus side and jumps $[u]=u^{+}-u^{-}$, $[t]=({\sigma }^{+}-{\sigma }^{-})\mathbf{n}$. The equivalent force density is $-[t]$ and the dipole density is $M_{jk}=c_{jk\ell m}[u_{\ell }]n_m$. This follows by applying the elastic wave operator to a piecewise smooth displacement field: its distributional source is $-[t_i]{\delta }_{\Sigma }-{\partial }_j(c_{ijkl}[u_k]n_l{\delta }_{\Sigma })$. Integration by parts against the elastodynamic Green tensor gives the displayed signs. At a material interface the dipole term is interpreted through the continuous reciprocal Green traction, rather than an undefined product of a discontinuous stiffness with a surface delta.