Material strength is a classical concept that has recently found renewed applications in fracture mechanics, especially in models for crack nucleation in brittle solids. In this paper, we formulate material strength in the setting of finite elasticity and examine its geometric, constitutive, and symmetry-theoretic foundations. We show that spatial covariance requires a strength function to depend on both stress and the corresponding strain measure, so that strength is not controlled by stress alone, but by the pair (stress,strain). Only in this case can a strength function written in terms of one stress measure be consistently rewritten in terms of another, while classical stress-based strength criteria are recovered as a special case in which the strain dependence is suppressed. We discuss the covariance of strength functions under arbitrary spatial diffeomorphisms and use this to relate representations in terms of the first Piola-Kirchhoff, second Piola-Kirchhoff, and Cauchy stresses. Restricting attention to the stress-based criteria that appear in the existing literature, we define the associated strength hypersurface as a subset of the constitutively admissible stress manifold, distinguish constitutive admissibility from fracture, and analyze the geometric and topological properties of the corresponding safe domain. We show that, for stress-based strength functions satisfying standard regularity conditions and the requirement that sufficiently large stresses are inadmissible, the strength surface is a smooth compact hypersurface of the constitutively admissible stress manifold. For isotropic solids, we study the symmetry of strength surfaces in principal stress space and show that the safe domain is star-shaped under a natural proportional-reduction hypothesis. We then extend the formulation to anelastic brittle solids and examine the effects of residual stresses and anelastic distortions on material strength. We also discuss the action of material symmetry on strength functions for anisotropic solids. Finally, we discuss material strength in the setting of linear elasticity and show how the general theory reduces to the classical stress-based criteria in this limit.
https://arxiv.org/abs/2605.01156