Strength criteria stop being hand-picked: a sparse differentiable search bakes convexity and uniqueness into the fit, cutting concrete's cross-validated error fourfold
Synopsis
Multiaxial strength tests give only a few dozen failure stress states, and choosing which limit surface to fit has long relied on manual comparison and visual checks. Researchers recast that choice as sparse model discovery with differentiable penalties, define the surface by the first root along rays so the equivalent stress is explicitly solvable, and add PBP atoms with learnable shape parameters plus competition-gated mixtures of experts. On concrete data the PBP model's cross-validated error is about a quarter of the monomial library's, and measured strength ratios are reproduced within a few per cent.
Interpretation
Choosing a limit surface becomes sparse model discovery under physical constraints: convexity, absence of spurious outer contours and bounds on inelastic Poisson ratios are written as differentiable penalties or hold by construction. Kolupaev's systematised catalogue of criteria required checking these conditions largely by hand, held that spurious branches of high-power criteria cannot be removed by new constraints, and required visual control at each step. In the synthetic benchmark all 24 constrained fits pass the discrete audit, against 45.8% of unconstrained fits and 75.0% of the purely algebraic baseline.
The surface is defined by the first root along rays from the origin, so the equivalent stress is the gauge of that star-shaped set and explicit solvability and first-order homogeneity hold for every library member. Kolupaev treated ray-based physical objectives such as distance along the ray as too expensive for parameter identification and used them only to compare fits. Features at fixed directions are evaluated once, root finding reduces to evaluating a degree-six polynomial, and exact derivatives follow from one implicit Newton correction, all by automatic differentiation in JAX.
Adding PBP atoms beyond monomials resolves data the monomial library cannot, and for concrete the selected surface is a cone with a nearly Rankine-type triangular section. The monomial library ties Lode-angle resolution to stress degree, so sharp deviatoric shapes need many terms or cannot be represented at all. On concrete the PBP library reaches a cross-validated error of 0.044 against 0.194 for the monomial library, and the measured ratios 10.54 and 11.19 are reproduced within a few per cent.
Perspective
The demonstrated scope is isotropic limit surfaces, with data from two PMI foams, polyamide PA6 and concrete in plane stress, and the structural silicone TSSA in Hencky-strain space; convexity and uniqueness are imposed and audited on finite direction sets, not certified on the continuum. The structural identification is validated on synthetic notched-bar failure loads with a linear-elastic pre-failure model, which the authors present as a proof of concept.
The competition-gate exponent does not quantitatively match the Weibull moduli estimated from replicate scatter, so the authors treat the statistical reading as a hypothesis these data do not support. Values the tests do not constrain, above all the hydrostatic tensile strength, often differ between admissible model classes by several bootstrap widths. The tabulated equibiaxial tensile strengths of 110IG are about twice the values implied by the book's force–stress relation; the authors use one half and note the correction should be confirmed with the original test records.