Trace and determinant constraints in polymer rheology: analytical reformulations and benchmark solutions

Abstract We consider a certain class of widespread tensorial constitutive equations with structural constraints and introduce two lemmata that reformulate them in terms of unconstrained variables. For trace constraints, the transformation recasts a class of nonlinear evolution equations in terms of an auxiliary linear dynamics, revealing that tube models can reproduce nonlinear rheological behavior using only linear parameters and operations. This yields analytical benchmark solutions for the orientation dynamics of differential reptation models under uniaxial elongation. A determinant-constrained analogue is developed for strain formulations, including the Leonov and Zürich Rheology models. In contrast, this transformation does not eliminate nonlinearities but provides an alternative representation. For the classes considered here, the auxiliary variables exhibit markedly different behavior: trace-based formulations admit analytically tractable auxiliary dynamics, whereas determinant-based formulations remain nonlinear and may exhibit singular behavior.