Kinematic Inconsistencies and Initial-Value Boundary Paradoxes in Rate-Dependent Viscoelastic Yield Stress Models

Abstract

We present a rigorous analysis of the mathematical boundaries and kinematic consistency of a recently proposed rate-dependent relaxation time framework intended to unify pre- and post-yield dynamics in yield-stress fluids. By evaluating the governing constitutive equations under an idealized transient creep protocol from a state of physical rest, we show that the model encounters an unavoidable boundary paradox. To avoid predicting perfectly rigid solid behavior or falling into a division-by-zero mathematical singularity under a constant applied stress below the yield threshold ($σ\le σ_y$), the framework requires an unphysical, instantaneous velocity or strain-rate step at $t = 0^+$. We show that assuming a non-zero initial strain rate explicitly violates momentum conservation and fluid inertia. Consequently, the framework preserves the piecewise, discontinuous drawbacks of classic viscoplastic models.