Ellipsoidal inclusions lose their yes-or-no answer: shape deviation and internal strain heterogeneity now come with two-sided bounds
Under a uniform dilatational eigenstrain, an ellipsoidal inclusion has a spatially uniform internal strain; a two-sided inequality now links how far a shape sits from the ellipsoid family to how heterogeneous its interior strain becomes, with a half-order Sobolev scale that brings morphological wavelength into the mechanics. For protruded, indented, and necked geometries, ellipsoidal subtraction yields an exact residual-field budget whose bounds tighten monotonically and agree with independent displacement-based finite-element results.