Rational Mechanics of Material Strength in Brittle Solids

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. Spatial covariance requires the physical strength criterion to be independent of the stress measure used to represent it. We show that a criterion depending only on one stress measure generally becomes a function of both stress and strain when expressed using another stress measure. This motivates taking the pair (stress,strain) ( stress , strain ) as the natural domain of a representation-independent formulation. Classical stress-based strength criteria remain admissible as a special class in which the strain dependence is absent in the chosen representation. 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. We also 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.