On the Eshelby's inclusion problem for ellipsoids with nonuniform dilatational Gaussian and exponential eigenstrains

This work investigates the three-dimensional elastic state of inclusions in which the prescribed stress-free transformation strains or eigenstrains are spatially nonuniform and distributed either in a Gaussian, or an exponential manner. The prescribed eigenstrain distributions are taken to be dilatational. Typical research in the micromechanics of inclusions and inhomogeneities has dealt, by and large, with spatially uniform eigenstrains and, to some limited degree, with polynomial distributions. Solutions to Eshelby's inclusion problem, where eigenstrains are Gaussian and exponential in nature, do not exist. Such eigenstrain distributions arise naturally due to highly localized point-source type heating (typical in electronic chips), due to compositional differences, and those due to diffusion related mechanisms among others. The current paper provides such a solution for ellipsoidal shaped inclusions located in an infinite isotropic elastic matrix. It is shown, similar to the much-discussed uniform eigenstrain problem, that the elastic state is completely, determined in closed form save for some simple one-dimensional integrals that are evaluated trivially using numerical quadrature. For the specialized case of a spherical shape, solutions in terms of known functions are derived and numerical results are presented. The elastic state both within and outside the inclusion is investigated. For the specific case of a sphere, the elastic strain energies are given in terms of simple formulas. Some applications of the current work in various areas such as electronics, micromechanics of composites, and material science are also discussed.