Corotational derivative

If an orthonormal coordinate transformation tensor Q (e- = eo is used instead of the deformation gradient F, the concept of the convected derivative can be extended. That is, let ft = 2 2 an antisymmetric rotation tensor generated by Q as shown in (2.27), then the corotational derivative of a second-order tensor T due to Q is defined by... 

The material time derivative of a vector-valued or tensor-valued function is not always objective as described above even if the original function is objective. It can be said that the convected derivative and corotational derivative are introduced to ensure objectivity of the time-derivative. For example we have... 

Equation 2.179 gives the relationship between both corotational derivatives as... 

Again making use of the slippage tensor, this time we are led to an evolution equation involving the corotational 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. 

---