AI and the World's Hard Problems · Physics · arXiv · 2026-10-22
Quantitative Eshelby stability: local and finite-shape theory through AI-assisted mechanics
The study explores how human researchers can use AI on difficult mechanics problems while retaining control of physical significance, mathematical rigor, and validation, through a quantitative stability problem rooted in the Eshelby conjecture. For a uniform dilatational eigenstrain in a homogeneous isotropic solid, it establishes a nonlinear two-sided relation between whole-interior strain heterogeneity and a geometry-only distance from the ellipsoid family with half-order Sobolev scaling, and develops a complementary finite-shape theory that yields systematically convergent upper and lower bounds for a prescribed geometry. Applied to protruded, indented, and necked inclusions, the bounds agree with independent displacement-based finite-element calculations, and the paper documents a human-led, AI-assisted research process as well as Open Problems in Mechanics, a versioned community-facing map of selected mechanics questions.