Lodge liquid constitutive equation

The strain is given by A. and the function h X) is a damping function which corrects for the too large strains which are predicted by the Lodge [21] elastic liquid constitutive equation which is identical to the equation above with h(X) = 1. 

Constitutive equations for the Rouse and Zimm models have been derived, and are found to be expressible in the form of Lodge s elastic liquid equation [Eq.(6.15)], with memory function given by (101) ... 

Equations 3.4-3 and 3.4-4 form the molecular theory origins of the Lodge rubberlike liquid constitutive Eq. 3.3-15 (23). For large strains, characteristic of processing flows, the nonlinear relaxation spectrum is used in the memory function, which is the product of the linear spectrum and the damping function h(y), obtained from the stress relaxation melt behavior after a series of strains applied in stepwise fashion (53)... 

The constitutive equations benefiting from the specific representations of reptation theory have the general form of the Lodge rubber-like liquid equation, since they are all... 

Just as there are various possible finite strain tensors, there are various time derivatives that can be used in place of the ordinary derivative of stress in Eq. 10.21 to satisfy the continuum mechanics requirements for a model to be able to describe large, rapid deformations in arbitrary coordinate systems. The derivative that yields a differential model equivalent to Lodge s Eq. 10.6 is the upper convected time derivative (defined in Eq. 11.19), and the resulting model is called the upper-convected Maxwell model. Other possibilities include the lower-convected derivative and the corotational derivative. Furthermore, a weighted-sum of two of these derivatives can be used to formulate a differential constitutive equation for polymeric liquids. In particular, the Gordon-Schowalter convected derivative [7] is defined in this manner. 

This Lodge network model result is a special case of the Lodge elastic liquid, in that the memory function is a sum of exponentials it is also of the same form as the constitutive equation for the Rouse and Zimm models, except that here the constants Aj and f]j are free parameters to be determined from the experimental data. If these quantities are both taken to be proportional to then the zero-shear-rate viscosity is proportional to and the first normal-stress... 

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