Surface displacements and stress field generated by a semi-ellipsoidal surface inclusion

This paper presents calculations of the displacement and stress fields generated by semi-ellipsoidal surface inclusions containing uniform transformation strains or eigenstrains. The inclusion is assumed to have the same elastic constants as the rest of the material. This is a reasonable assumption for modeling transformed zones in transformation toughened ceramics and localized plasticity in individual surface grains in alloys. Analytical results are obtained for special cases and numerical results for general cases. The approach is particularly useful for accurately calculating the anomalous fields at the intersection of the boundary of the inclusion and the free surface. It is shown that, in many physically important cases, all components of the stress tensor are Zero or constant on the free surface within the inclusion. For shallow inclusions or inclusions of general geometry suffering volume conserving transformation strains, the stress fields are also approximately uniform throughout the inclusion. This result greatly simplifies modeling of localized deformation in certain materials under complex external loads.