Constant-shear bulk-modulus inclusion ($\mathrm{tr}\epsilon =\frac{3{\kappa }^0+4\mu }{3\kappa (x)+4\mu }\mathrm{tr}{\epsilon }^0$)

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.