Interrelations Between Shear Functions

The introduction emphasized that in general one cannot use linear viscoelastic data to predict a nonlinear viscoelastic material function, nor as a rule can one use one nonlinear material function to predict another. Nevertheless, in shearing flows a few usefiil in-terrelaficms between material functions have often been observed to hold, at least approximately, for polymer melts and solutions. 

Steady state shear viscosity versus shear rate for a low density polyethylene melt (solid line) compared to predictions of the Cox-Merz rule, eq 4.2.6 (open points), and the Gleissle mirror rule, eq 4.2.7 (solid points). Replotted from Retting and Laun (1991). 

The first of these relationships, eq. 4.2.4, follows from little more than the definitions of rj and t], while eq. 4.2.5 is less obvious (Coleman and Markovitz, 1964). Both relationships are of limited usefulness because they are relevant only for low shear rate properties. However, an empirical relationship, called the Cox-Merz rule, often holds fairly well at high shear rates. This rule states that the shear rate dependence of the steady state viscosity ij is equal to the frequency dependence of the linear viscoelastic viscosity ri that is. 

The quantities t and rf were defined in eqs. 3.3.28-3.3.30. An analogous relationship, the mirror rule (Gleissle, 1980) between rfiy) and ri (t), has been proposed  

A rule that can be derived theoretically from general premises likely to be valid for polymer melts and solutions is the Lodge-AfeicSfrier relationship (L ge and Meissner, 1972) between the shear stress and the first nomml stress difference after a step shear strain  

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LDPE, and with polypropylene, PP, was studied In steady state shear, dynamic shear and uniaxial extenslonal fields. Interrelations between diverse rheological functions are discussed In terms of the linear viscoelastic behavior and Its modification by phase separation Into complex morphology. One of the more Important observations Is the difference In elongational flow behavior of LLDPE/PP blends from that of the other blends the strain hardening (Important for e.g. fllm blowing and wire coating) occurs In the latter ones but not In the former. 

It is possible to define other small strain material functions, such as stress growth under constant rate of straining (Example 3.2.1) or recoverable strain after constant strain rate. However, these deformation histories are better suited for large strain studies and are discussed in Section 3.2. The small strain material functions will be seen as limits of the large strain ones. Table 3.3.2 lists some of the interrelations between the various experiments for linear viscoelastic behavior. Note that the limiting low shear rate viscosity rio can be calculated from... 

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