Isotropic elastic strain Green kernel (${\Gamma }_{ijkk}(x)=-(4\pi L{)}^{-1}{\partial }_i{\partial }_j|x{|}^{-1},L=\kappa +4\mu /3$)

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 $({\delta }_{ij}|x{|}^2-3x_i{x}_j)/(4\pi L|x{|}^5)$; the full distribution also includes ${\delta }_{ij}\delta /(3L)$. This contact term must not be discarded when computing hydrostatic strain.

When only the bulk modulus differs from an isotropic matrix, the elastic polarization is spherical: $\tau =(\kappa -{\kappa }^0)\mathrm{tr}(\epsilon )I$. The double trace of the isotropic elastic strain Green kernel makes the trace equation local, giving the displayed formula. Individual strain components still depend on a nonlocal potential of the spherical polarization, so the complete infinitesimal strain tensor is not generally a local multiple of its reference value.