Kraus-Ulmer equations

Figure 6.10 shows typical dynamic properties of vulcanized PDMS-silica systems, as investigated through strain sweep experiments at constant frequency and temperature. As can be seen, dynamic strain softening is observed in a qualitatively similar manner to other filled polymers. It follows that models, which successfully fit conventional filled rubbers (e.g., carbon black filled compounds), are expected to well suit such data. This is indeed the case, as shown by the curves in Figure 6.10, drawn by fitting the Kraus-Ulmer equations, i.e.. 

Modeling the Dynamic Strain Softening Effect on PDMS-Silica Systems with the Kraus-Ulmer Equations... 

The above data allows however to demonstrate how really strong are the PDMS-silica interactions. Indeed, using the fit parameters in Table 6.3 and the Kraus-Ulmer equations, one easily calculates low strain (let s say 0.001) values of G and G", in order to draw Figure 6.11. 

We note that the Kraus model provides a fairly good description of the experimental features of the Payne effect. However, very recently Ulmer again evaluated the Kraus equations with data from several published sources and unpublished own data [66]. He found that the description of G"(y0) according to Eq. (19) is not as good as the description of G (yo) according to Eq. (16). The basic deficiency of the Kraus-G"(y0) model is its inability to account for the G"-values at strains less than about 10 3. However, the G"(y0) description is improved considerably by the addition of a second, empirical term, for example an exponential term like... 

As previously commented, the postulate considered by Kraus (i.e.. Equation 5.34) imparts symmetries in both the G (Yo) and the G"(Yo) functions. If indeed, experimental data support an horizontal symmetry for the elastic modulus, with respect to the mid modulus value, no vertical symmetry (with respect to y<) is generally observed for the viscous modulus. The deficiencies of the Kraus model are therefore, embedded in the starting postulate. Various modifications have been proposed to account for the nonsymmetri-cal behavior of G" without changing the physical ideas leading to the model. Using different strain exponents for the deagglomeration and reagglomeration processes (Equation 5.34) was probed by Ulmer who concluded that it... 

The Ulmer s additional term to the Kraus equation for G" vs. strain gives a considerably improved fit of experimental data. 

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