Elastic strain Green operator ($\epsilon =E-\Gamma \tau$)

For a homogeneous comparison elastic stiffness tensor $C^0$, the elastic strain Green operator maps an elastic polarization to minus the induced infinitesimal strain tensor. In a bounded domain the induced displacement field has zero boundary data; in an infinite medium it is the decaying perturbation. The defining static elastic equilibrium equation is $\nabla \cdot (C^0\mathrm{sym}\nabla u+\tau )=0$. The bounded-domain operator is self-adjoint, annihilates constant elastic polarizations, has zero mean output, and satisfies $\Gamma C^0\Gamma =\Gamma$.

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.

Let $B_{n}a=\mathrm{sym}(a\otimes n)$; its adjoint on symmetric stress tensors is $B_n^T\tau =\tau n$. Then $B_n^{T}C{B}_n$ is the acoustic tensor. The directional elastic strain Green operator satisfies the resolvent identity ${\tilde{\Gamma }}_r(C^r-C^s){\tilde{\Gamma }}_s={\tilde{\Gamma }}_s-{\tilde{\Gamma }}_r$, by the elementary inverse identity $K_r^{-1}(K_r-K_s)K_s^{-1}=K_s^{-1}-K_r^{-1}$.

The elastic polarization is the excess stress relative to a homogeneous comparison elastic stiffness tensor. In heterogeneous linear elasticity, $\tau =(C-C^0)\epsilon$. The comparison static elastic equilibrium equation gives the integral equation $\epsilon ={\epsilon }^0-\Gamma \tau$, where ${\epsilon }^0$ is the reference strain under the same external loading.