Convected derivatives for differential constitutive equations.

To be useful, constitutive equations for polymer melts should describe the rheology of the material of interest in a broad range of deformation types and rates and should have a mathematical form suitable for numerical analysis. Differential constitutive equations are in many ways better suited to computation than integral equations, but to date none have been proposed that agree with each of four important experimental observation in relaxation after step shear strains: that the stress is factorable into time and strain dependent functions, that the strain dependent function is strongly shear thinning, that the ratio of first normal stress differences to shear stress equals the shear strain - that is, the Lodge-Meissner relationship holds, and that there is a negative second normal stress difference. The Johnson-Segalman model satisfies three of these, but fails to satisfy the Lodge-Meissner relationship, because in step strains the principal stress and strain axes do not rotate together. Using a mathematical technique for forcing co-rotation of stress and strain axes in an arbitrary deformation, we here present an explicit differential constitutive equation that satisfies all four of the above experimental observations.