Displacements and stress associated with localized and distributed inelastic deformation with piecewise-constant elastic variations

We present a semi-analytical method and expressions for computing the displacements, strains and stress due to localized (e.g. faulting) and distributed (volumetric) inelastic deformation in heterogeneous elastic full- and half-spaces. Variations in elastic properties are treated as piecewise-constant homogeneous subregions as in orthodox multiregion approaches. The deformation in the subregions is solved by matching the interface traction and displacement conditions for contrasting elastic parameters. We show equivalence between the integral equation convolving boundary traction and those convolving displacement discontinuities and volumetric inelastic strain in the representation theorem for a bounded volume. This equivalence allows us to express the deformation fields in the half-/full-space which comprises those subregions by using virtual fault displacement elements or volumetric eigenstrain elements, the integral kernels of which have known analytic forms for finite sources in homogeneous volumes. We include computer programs that implement our method with known analytic solutions of homogeneous volumes free of major singular points. We provide an extension to the existing toolkit available for the observational and theoretical analyses of deformation fields allowing users to model heterogeneous geological structures, with a number of primary geophysical applications, including earthquake and volcano deformation, where variations in elastic parameters may present a substantial contribution to the observed deformation.