Ellipsoids are no longer the only answer: how far a shape drifts from an ellipsoid sets how uneven its interior strain gets, checked by independent finite elements

The classical statement that an ellipsoidal inclusion has a uniform interior strain now has a quantitative robustness version: the farther a shape departs from the ellipsoid family, the more heterogeneous the interior strain, and conversely a nearly uniform field constrains how far the shape can sit from that family. Earlier results on the Eshelby conjecture were rigidity statements that only answered which shapes are possible when the field is exactly uniform, and perturbation analyses characterized only infinitesimal directions and leading-order changes, leaving finite non-ellipsoidal shapes uncontrolled. For a uniform dilatational eigenstrain in an infinite homogeneous isotropic solid with identical inclusion and matrix moduli, a perfectly bonded interface, and no remote load, a two-sided inequality is proved for every fixed nondegenerate ellipsoid between a geometry-only distance to the ellipsoid family and the normalized RMS strain fluctuation, with constants independent of perturbation frequency, eigenstrain amplitude, and the admissible isotropic elastic constants.

Shape error cannot be read off boundary height alone, because the spatial wavelength of the imperfection matters, so the measure is a half-order Sobolev norm that folds morphological length scale into the mechanics. A height RMS weights all wavelengths equally and a slope RMS weights them by wavenumber, and neither alone yields two-sided control. On a locally flat interface the first-order field decays into the inclusion with wavenumber, so the squared bulk response scales as amplitude squared times one power of wavenumber; on a sphere, at fixed height RMS degree 12 produces about 2.48 times the RMS response of degree 3, while at fixed RMS slope the ranking reverses.

For any prescribed finite geometry, ellipsoidal subtraction yields an exact residual-field budget from which partial field information gives upper and lower bounds that tighten monotonically to the same response as interior and exterior regions are resolved. The local theorem covers only an infinite-dimensional neighborhood of the ellipsoid family and cannot directly answer how heterogeneous a specified geometry outside that neighborhood can be. Protruded, indented, and equal-volume necked two-lobe shapes are built on an aspect-ratio-1.5 reference spheroid, and across five refinement levels all 15 intervals contain their displacement-based finite-element reference values, with interval widths from first to final level reduced by roughly factors of four, three, and two.

Protrusion, indentation, and necking produce different spatial distributions of interior heterogeneity, and the equal-volume necked shape produces the largest. Volume change, maximum height, or fitted aspect ratio alone cannot quantify the loss of Eshelby uniformity. The finite-element fields show variation around the polar feature and attachment for the protrusion, concentration around the rim for the indentation, and a broader heterogeneous region through the constriction for the necked particle, with finite-element responses of 0.0524, 0.0763, and 0.1547.

The results apply to a single bounded inclusion carrying a uniform dilatational eigenstrain in an infinite homogeneous isotropic solid, with identical inclusion and matrix moduli, a perfectly bonded interface, and no remote load. The local theorem is local in shape amplitude and permits asymmetric, localized, and arbitrary combinations of admissible spatial variations, but does not cover shapes far from the ellipsoid family. The finite-shape theory gives systematically improvable bounds for one prescribed geometry, and the numerical intervals for the protruded, indented, and necked shapes are working evaluations with fixed-region quadrature checks rather than directed-rounding-certified enclosures. Full computer-assisted certification additionally requires outward numerical enclosure of the field integrals.

The authors state that the reported numerical endpoints are empirically quadrature-stable working evaluations, not directed-rounding numerical certificates, and that numerical certification is distinct from agreement with the finite-element results. The finite-shape theory is exact at the continuum level, but the convergence proof supplies no algebraic rate, and the observed contraction orders are a feature of the computed range. The local theorem's constants depend on norm conventions, are not uniform over degenerating ellipsoids, and are not numerically evaluated. Partial Lean formalization covers selected mathematical components rather than the full local theorem, and the staged AI review found no substantive gap within its scope, though separate AI review contexts do not ensure independent model errors.