K-BKZ equation

For example, Figs. 2.43 and 2.44 present the measured [55] viscosity and first normal stress difference data, respectively, for three blow molding grade high density polyethylenes along with a fit obtained from the Papanastasiou-Scriven-Macosko [59] form of the K-BKZ equation. A memory function with a relaxation spectrum of 8 relaxation times was used. 

The coefficients used to fit the data are summarized in Table 2.11 [43], The viscosity and first normal stress coefficient data presented in Figs. 2.30 and 2.31 where fitted with the Wagner form of the K-BKZ equation [41],... 

Since second normal stresses are generally difficult to obtain from the experimental point of view, it may seem attractive to cancel the Cauchy term of the K-BKZ equation setting h2di, I2) = 0 and to find a suitable material function hidi, I2). Wagner [26] wrote such an equation in the form ... 

Using a factorizable K-BKZ equation (21), Wagner and Demarmels [32, 33] showed that an equation of the damping function such as ... 

The Wagner equation finds its theoretical basis in the derivation of the more general K-BKZ equation. Unfortunately, it loses part of its original thermod3mamic consistency since, for simplification purposes, only the Finger strain measure is taken into account. Doing so, it is no more derivable from any potential function and additionally it does not predict second normal stress differences any more. 

The Doi—Edwards equation is a special case of a separable K—BKZ equation, for which... 

The development of molecular constitutive equations for commercial melts is still a challenging unsolved problem in polymer rheology. Nevertheless, it has been found that for many melts, especially those without long-chain branching, the rheological behavior can be described by empirical or semiempirical constitutive equations, such as the separable K-BKZ equation, Eq. (3-72), discussed in Section 3.7.4.4 (Larson 1988). To use the separable K-BKZ equation, the memory function m(t) and the strain-energy function U, or its strain derivatives dU/dli and W jdh, must be obtained empirically from rheological data. 

As seen in the next paragraphs, the DE-NIA equations sometimes do better and sometimes worse than the DE-IA or K-BKZ equations. Also note that in all instances the measured single-step relaxation fimction is used rather than the theoretical DE fimction, which as already discussed is too narrow to capture the full breadth of the relaxation response of entangled polymers. 

Step-Up. Figure 49 shows the response for a two-step history in which the second step is approximately the same magnitude as the first step, ie, K2 = 2y i. The results are from Osaki s work (138) on polystyrene solutions and illustrate that both the shear stress response and the normal stress response are well represented with the DE independent alignment approximation (ie, the K-BKZ equations). This result is similar to what was found previously for the K-BKZ model (Fig. 34) (138). 

Fig. 65. Half-step torque (a) and normal force (b) responses for a polycarbonate glass comparing the experimental data (open circles) with the K-BKZ equation (closed circles) and the Bernstein-Shokooh stress-clock model (155) modified to an energy clock (inverted triangles) predictions. After Pesce and McKenna (162).

Steady state shear viscosity and primary normal stress coefficient for low density polyethylene melt T and from the Kaye-Bernstein, Kearsley, Zapas (K-BKZ) equation wim the double exponential damping function, eq4.4.13 (solid lines) and with the single exponential, eq4.4.12 (dotted Une). Data at different temperatures have been shifted to one master curve by ar T). Replotted from Laun (1978). 

Several theoreticians have worked with an integral equation that is somewhat more general than the factored K-BKZ equation (Birdetal. 1987) ... 

The DE Constitutive Equations. The DE model (52-56) made a major breakthrough in polymer viscoelasticity in that it provided an important new molecular physics based constitutive relation (between the stress and the applied deformation history). This section outlines the DE approach that built on the reptation-tube model developed above and gave a nonlinear constitutive equation, which in one simplified form gives the K-BKZ equation (70,71). The model also inspired a significant amount of experimental work. One should begin by... 

A constitutive equation that includes the features in both (b) and (c) above is the factorized K-BKZ equation - ... 

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