Ellipsoidal inclusions lose their yes-or-no answer: shape deviation and internal strain heterogeneity now come with two-sided bounds
Synopsis
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.
Interpretation
An ellipsoidal inclusion has a spatially uniform internal strain, and how much of that uniformity is lost when the shape departs from an ellipsoid previously had no quantitative answer; a two-sided relation now connects the shape deviation to the internal strain heterogeneity. Earlier rigidity results answered only the exact question of which shapes admit a uniform field, and perturbation analyses characterized infinitesimal directions and leading-order changes without controlling a finite morphology. Near a fixed nondegenerate ellipsoid, removing the nine ellipsoidal tangent directions leaves every remaining mode with a positive lower gain and a finite upper gain, with constants independent of perturbation frequency and eigenstrain amplitude.
Shape error cannot be read from boundary height alone, because morphological wavelength matters: a fine corrugation and a broad protrusion of equal height generate different strain fields. A height RMS weights all wavelengths equally and a slope RMS weights them by frequency, so neither yields wavelength-independent two-sided control. On the sphere at fixed height RMS, degree 12 produces about 2.48 times the bulk response of degree 3; with the half-order distance, geometry and mechanical response decay at the same order at high degree and both bounds hold.
For any prescribed shape, ellipsoidal subtraction gives an exact residual-field budget from which upper and lower bounds tighten monotonically as interior and exterior field information is resolved. The local theorem covers only an infinite-dimensional neighborhood of the ellipsoid family and does not directly treat a prescribed finite geometry outside it. Across five refinement levels, interval widths for the protruded, indented, and necked shapes shrink by roughly 4, 4, and 8 times, and all 15 intervals contain the independent displacement-based finite-element reference values.
Perspective
The results hold for an infinite homogeneous isotropic solid with identical inclusion and matrix moduli, a perfectly bonded interface, no remote load, and a uniform dilatational eigenstrain. The local theorem holds in shape space and permits asymmetric, localized, and arbitrary combinations of admissible perturbations, but does not cover shapes far from the ellipsoid family. The finite-shape bounds apply to one prescribed geometry and tighten monotonically as interior and exterior field information is resolved. A reader can watch for extensions to general eigenstrains, elastic mismatch, anisotropy, interfaces, and multiple inclusions.
The finite-shape numerical endpoints are working evaluations with quadrature-convergence checks, not directed-rounding-certified enclosures, so numerical certification is distinct from agreement with FEM. The local theorem's constants depend on fixed norm conventions and are not uniform over degenerating ellipsoids. The FEM results are numerical convergence checks rather than rigorous continuum error intervals. Whether near-uniform interior fields support a global geometric conclusion fails in related two-dimensional settings.