Tensor logarithm

In the case of dynamic mechanical relaxation the Zimm model leads to a specific frequency ( ) dependence of the storage [G ( )] and loss [G"(cd)] part of the intrinsic shear modulus [G ( )] [1]. The smallest relaxation rate l/xz [see Eq. (80)], which determines the position of the log G (oi) and log G"(o>) curves on the logarithmic -scale relates to 2Z(Q), if R3/xz is compared with Q(Q)/Q3. The experimental results from dilute PDMS and PS solutions under -conditions [113,114] fit perfectly to the theoretically predicted line shape of the components of the modulus. In addition l/xz is in complete agreement with the theoretical prediction based on the pre-averaged Oseen tensor. 

Fattal, R. and Kupferman, R. (2004) Constitutive laws for the matrix-logarithm of the conformation tensor. 

It is seen that Eq. (3.35) becomes identical to Eq. (3.15) only when the symmetric part of tensor x (x - 2i Qd) disappears in other words, when x - 2i Qd = 0. The Giesekus model is very versatile in that it has an additional parameter, the so-called mobility parameter a, which allows for more accurate predictions of the material functions in steady-state shear flow as compared with the ZFD model. This will become clear later in this chapter when we present logarithmic plots of r versus y, and logarithmic plots of Alj versus y based on predictions from both the Giesekus and ZFD models. In Chapter 6 of Volume 2 we will use a modified version of the Giesekus model to simulate the high-speed melt spinning process. 

The first component of the angmented Hybrid Model relates to the elastic behavior of UHMWPE, which is captured by the elastic spring (E) in the rheological representation and the elastic deformation gradient F. The polar decomposition [26] of F involves a left stretch tensor, V , and a rotation tensor, R . Using V , the logarithmic tme strain (E )... 

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