Codeformational time derivative

With the homogeneous flow assumption, whereby V(RR) = 0, the definition of (RR)(i) as given by Eq. (6.25) is the same as the codeformational time derivative or convected time derivative shown in Appendix 6.A. 

Both yjo] and X[oj are codeformational rheological tensors. They can describe the deformation and the stress response to a deformation without interference from the rigid-body rotation (see Chapter 5). In other words, these tensors and their time derivatives, integrations, and combinations... 

The above equation is generalized to three dimensions by replacing the stress component T by the stress matrix and the strain component 7 by the strain matrix. This procedure works well as long as one confines oneself to small strains. For large strains, the time derivative of the stress requires special treatment to ensure that the principle of material objectivity(78) is not violated. This principle requires that the response of a material not depend on the position or motion of the observer. It turns out that one can construct several different time derivatives all of which satisfy this requirement and also reduce to the ordinary time derivative for infinitesimal strains. By experience over many years, it has been found (see Chapter 3 of Reference 79) that the Oldroyd contravariant derivative also called the codeformational derivative or the upper convected derivative, gives the most realistic results. This derivative can be written in Cartesian coordinates as(79)... 

By replacing the partial time derivative with a nonlinear time derivative, which is based on a codeforming and translating coordinate system, the constitutive equation given in Eq. 3.1 now becomes (Bird et al., 1987a) ... 

Here, b/br is called the convected derivative due to Oldroyd (1950), and it is the fixed coordinate equivalent of the material derivative of a second-order tensor referred to in convected coordinates. The physical interpretation of the right-hand side of Eq. (2.104) may be given as follows. The first two terms represent the derivative of tensor a j with time, with the fixed coordinate held constant (i.e., Da /Dr), which may be considered as the time rate of change as seen by an observer in a fixed coordinate system. The third and fourth terms represent the stretching and rotational motions of a material element referred to in a fixed coordinate system. This is because the velocity gradient dv fdx (or the velocity gradient tensor L defined by Eq. (2.59)) may be considered as a sum of the rate of pure stretching and the material derivative of the finite rotation. For this reason, the convected derivative is sometimes referred to as the codeformational derivative (Bird et al. 1987). 

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