On configuration-dependent generalized oldroyd derivatives

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After a short derivation of the upper and lower Oldroyd derivatives, linear combinations of them are examined with regard to their suitability for describing the non-affine motion of polymer structures occuring because of hampered flow. This leads to the introduction of a new type of convected derivative called a “configuration-dependent” derivative. While the combined Oldroyd derivatives can be considered to be upper convected derivatives supplemented by the addition of terms containing the rate of deformation tensor, in the new derivatives the irreversible rate of deformation tensor is substituted instead. This tensor, first introduced by Leonov, is strongly related to the configuration tensor, which is a measure of recoverable strain, at least in so-called one-mode models. The force-free motion of particles in steady simple shear flow as predicted by the respective generalized derivatives is analysed and it is found that the configuration-dependent derivative supplies a more realistic result for hampered flow than the combined Oldroyd derivative, except in the case of permanently isotropic structures for which both predictions are identical. For the special case of application of the configuration-dependent derivative to the configuration tensor, the expression is equivalent to that obtained by the application of the simple upper Oldroyd derivative itself. Thus it is concluded that the effects predicted by simple constitutive equations, when combined Oldroyd derivatives are used but not when only the upper Oldroyd derivative is used, may be more appropriately described by taking into account the non-isotropic character of the relative motion of structural elements occuring because of mutual interactions.