Isotropic elastic strain Green kernel (source code)

= Isotropic elastic strain Green kernel
{title2=$\Gamma_{ijkk}(x)=-(4\pi L)^{-1}\partial_i\partial_j|x|^{-1},\quad L=\kappa+4\mu/3$}

For a three-dimensional isotropic comparison material, the contraction of its <elastic strain Green operator> is the displayed <distributional derivative>. Its double trace is $\Gamma_{iikk}=\delta/L$, because $\Delta|x|^{-1}=-4\pi\delta$. Away from zero the contraction is $(\delta_{ij}|x|^2-3x_ix_j)/(4\pi L|x|^5)$; the full <distribution> also includes $\delta_{ij}\delta/(3L)$. This contact term must not be discarded when computing hydrostatic <strain>.