Right stretch tensor

For some purposes, it is convenient to express the constitutive equations for an inelastic material relative to the unrotated spatial configuration, i.e., one which has been stretched by the right stretch tensor U from the reference configuration, but not rotated by the rotation tensor R. The referential constitutive equations of Section 5.4.2 may be translated into unrotated terms, using the relationships given in the Appendix. 

Equations (4.11) and (4.12) are called polar decomposition of the deformation gradient. Particularly, U is called the right stretch tensor and V is called the left stretch tensor. 

R is referred to as the rotation tensor, U the right stretch tensor, V the left stretch tensor. R is orthonormal R R = /, RR = i), which gives the rotation of C and B to their principal axes. Under the polar decomposition we have... 

The deformation gradient F contains not only information about the deformation, but also about rigid-body rotations of the material. These, however, do not contribute to the deformation itself, and the two contributions thus have to be separated. This can be done by considering the deformation gradient as a composition of a deformation U, called the right stretch tensor (or, sometimes, material stretch tensor) and a subsequent rotation R. These two are multiplied using the tensor product ... 

The last term on the right hand side in (1.66) expresses the elastic stress through the components of the orientatiOTi-deformation tensor L. The evolution of this tensor in the jet flow is described by the following equations accounting for macromolecular stretching and relaxation... 

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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