Closed-Form Critical-State Caputo Flow for a Teardrop Bounding Surface: Operator Semantics and Factorial Consequences

Abstract

Closed-form Caputo gradients in critical-state stress-fractional plasticity have relied on the polynomial Modified Cam-Clay surface. We extend them to the non-polynomial teardrop bounding surface by defining the operator as a Caputo derivative in the logarithm of normalized pressure, orientation-corrected for that decreasing map. On this axis, the surface becomes power–exponential, its critical-state terminal falls at exactly t* = 1/Ψ independently of Ω, and the fractional gradient reduces to incomplete-Beta–Kummer and Humbert-Φ1 closed forms, verified against singularity-aware quadrature over 1240 cases to relative errors below 10−11. The flow rule recovers associated flow as α → 1 and is exactly associated at the critical state. This is a computational study of operator semantics on inherited calibrations, not an experimental validation. Substituting it for the fixed-window Grünwald–Letnikov flow of a companion factorial collapses the dominant flow main effect from −60% to below 0.2% for both clays at every overconsolidation ratio tested, across drained and approximately undrained paths. The collapse is conditional: it is a property of the critical-state dwell, and at a laboratory-scale budget the operators still differ by about 15% of baseline. Window semantics, not fractionality alone, decide where flow influence resides in a factorial design.