Rational Mechanics of Material Strength in Brittle Solids

TL;DR

This paper develops a comprehensive geometric and constitutive framework for material strength in brittle solids, emphasizing stress-strain dependence, covariance, and extensions to anisotropic and residual stress effects.

Contribution

It introduces a unified, symmetry-theoretic formulation of material strength that generalizes classical criteria to finite elasticity and complex material behaviors.

Findings

Strength surface is a smooth, compact hypersurface under regularity assumptions.

Safe domain is star-shaped for isotropic solids under proportional-reduction hypothesis.

The theory reduces to classical stress-based criteria in the small-strain limit.

Abstract

Material strength is a classical concept with renewed importance in fracture mechanics, particularly in crack nucleation in brittle solids. We formulate material strength in finite elasticity and examine its geometric, constitutive, and symmetry-theoretic foundations. Spatial covariance requires a strength function to depend on both stress and the corresponding strain measure, so that strength is governed by the pair (stress,strain), not stress alone, and only then can representations based on different stress measures be consistently related, with classical stress-based criteria recovered as a special case. We analyze covariance under spatial diffeomorphisms and relate formulations based on the first Piola--Kirchhoff, second Piola--Kirchhoff, and Cauchy stresses. For stress-based criteria, we define the strength hypersurface as a subset of the constitutively admissible stress manifold and study the associated safe domain. Under standard regularity assumptions and the requirement that sufficiently large stresses are inadmissible, the strength surface is a smooth compact hypersurface of this manifold. For isotropic solids, we show that the safe domain is star-shaped under a proportional-reduction hypothesis. We extend the formulation to anelastic brittle solids, showing that residual stresses and eigenstrains modify the strength surface through the material metric, and discuss anisotropic strength via material symmetry. Finally, in the small-strain limit, the theory reduces to classical stress-based criteria.