Gordon-Schowalter convected derivative

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. 

Differential models obtained by replacing the ordinary time derivative in Eq. 10.21 by one that can describe large, rapid deformations are able to describe some nonlinear viscoelastic phenomena, but only qualitatively. To improve on such models, it is necessary to introduce additional nonlinearity into the equation. In the popular Phan-Thien/Tanner model, the Gordon-Schowalter convected derivative is used, and nonlinearity is introduced by multiplying the stress term by a function of the trace of the stress tensor. The Giesekus and Leonov models are other examples of nonlinear differential models. All of the models mentioned above are described in the monograph by Larson [7j. 

It requires that the principal stress axes should coincide with the principal strain axes. This rrile has been experimentally checked hy many authors [24, 56] Actually, the use of the Gordon-Schowalter derivative involves the violation of the Lodge - Meissner rule, indeed when a equals 0 or 2, either the upper or the lower convected derivatives implies that the relationship is respected. In the general case, the double value of the slip parameter is a natural way to accommodate this rule. 

In its general form, the Phan Thien Tanner equation includes two different contributions of strain to the loss of network junctions, through the use of a particular convected derivative which materializes some slip of the junctions and through the use of stress-dependent rates of creation and destruction of junctions. The use of the Gordon-Schowalter derivative brings some improvement in shear and a second normal stress is predicted, whereas the... 

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