Abstract
A clay flow rule governs contraction or dilation during plastic shearing. A fractional replacement must preserve the zero-dilatancy critical state and remain compatible with elasticity and hardening. We examine these requirements for a terminal-directed Caputo operator on a teardrop surface. The operator integrates surface slopes over a logarithmic stress interval, not elapsed loading history. Analytical derivations and incremental tests separate terminal compatibility, operator composition, stress-path identifiability and admissibility. The operator zero matches the non-associated base-model critical state only if Ω = Ψ. For kaolin, the alternative zero raises the normally consolidated undrained terminal mean stress by 18.6%. An exact identity shows why the flow candidates share the same normally consolidated stress locus yet progress differently in a numerical coordinate. Mean-stress preservation permits exact scalar composition with the SMP transformation but supplies no physical strain mapping. Two direct overconsolidated substitutions expose distinct deficiencies: coupling the loading modulus to the fractional dilatancy (F2) reverses its sign above OCR=exp(1/Ψ), whereas retaining the inherited loading modulus (F1) causes hardening inconsistency, low-OCR surface drift or negative assigned work at high OCR. These results define requirements a fractional clay formulation must satisfy before stress and strain data can test its predictions.