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

The critical-state branch of stress-fractional plasticity owes its analytical elegance to the Modified Cam-Clay surface, whose polynomial form admits closed-form Caputo gradients. We extend that elegance to the teardrop bounding surface of Chatwong et al., which is not polynomial in stress, by defining the operator as a Caputo derivative in the logarithm of normalized pressure, an orientation-corrected construction for that decreasing map (Section 2.3). On this axis the surface takes a power–exponential form, its critical-state terminal falls exactly at 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 1,240 cases to relative errors below 10⁻¹¹. The resulting flow rule recovers associated flow (in the log-stress conjugate representation) as α → 1, and its flow vector coincides with the associated vector exactly at the critical state. Under the present endpoint-dependent hardening law and critical-state-dominated loading budget, substituting this operator for the fixed-window Grünwald–Letnikov flow of a companion factorial study collapses the dominant flow main effect from −60% to below 1% for both clays of the factorial study at every overconsolidation ratio tested, robustly across drained and approximately undrained proportional strain paths and conjugacy conventions — below 1% under the calibrated α(OCR) law, and within 2.5% across a constant-α sensitivity grid; the residual is a critical-state dwell transient, and the window semantics decide where fractional flow influence resides in a factorial design.