On the problem of simple shear of an incompressible viscoelastic solid under finite deformations

TL;DR

The paper investigates simple shear of an incompressible viscoelastic solid under finite deformations using a one-parameter Gordon–Schowalter derivative. It combines a Maxwell-type constitutive framework with an objective derivative $D_a[\mathbf{S}]$, yielding $\bm{\sigma}=\mathbf{S}-p\mathbf{I}$ and equations that depend on the derivative parameter $a$, which recovers Oldroyd, Cotter–Rivlin, and Jaumann cases. Under accelerated shear, nonzero normal stresses appear (Poynting effect), and an analytical reduction describes the coupled evolution of $S_{23}$ and $S_{33}$, while numerical integration confirms the full stress evolution. For a shear rate linear in a normally distributed random variable, the Lomakin approach provides a mean-stress solution and a variance estimate via linearization, revealing pronounced dependence of stress variance on $a$ with no universally minimizing derivative. These results have practical implications for boundary-condition specification and predictability in finite-deformation viscoelastic analyses of simple shear.

Abstract

In the framework of a viscoelastic material model, whose constitutive relation is given by a one-parameter family of Gordon-Schowalter derivatives, the problem of simple shear under acceleration and random velocity motion is considered. For motion with acceleration, the presence of non-zero normal stresses is discovered, which corresponds to the Poynting effect previously discovered for this material. A problem in which the shear rate was determined as a linear function of a random variable given from a normal distribution was studied. Within the framework of the methodology proposed by V.A. Lomakin, an analytical solution of the problem is constructed. A significant dependence of the dispersion of the stress tensor components on the choice of the objective derivative was found.