Reinforcement cluster-size distribution

Beside the characteristic stress softening up to large strains (Mullins effect) as shown in Fig. 36.13, the model also considers the hysteresis behavior of reinforced rubbers (Payne effect). Obviously, since the sum in Eq. (36.10) is taken over stretching directions with ds/dt > 0, only, up-and down cycles are described differently. An example considering a fit of the hysteresis cycles of silica filled EPDM rubber in the medium strain regime up to 50% is shown in Fig. 36.14. For these adaptations an alternative form of the cluster size distribution has been assumed, which allows for an analytical solution of the integrals in Eqs. (36.9) and (36.10) ... 

Figure 46a shows that a reasonable adaptation of the stress contribution of the clusters can be obtained, if the distribution function Eq. (55) with mean cluster size =25 and distribution width b= 0.8 is used. Obviously, the form of the distribution function is roughly the same as the one in Fig. 45b. The simulation curve of the first uniaxial stretching cycle now fits much better to the experimental data than in Fig. 45c. Furthermore, a fair simulation is also obtained for the equi-biaxial measurement data, implying that the plausibility criterion , discussed at the end of Sect. 5.2.2, is also fulfilled for the present model of filler reinforced rubbers. Note that for the simulation of the equi-biaxial stress-strain curve, Eq. (47) is used together with Eqs. (45) and (38). The strain amplification factor X(E) is evaluated by referring to Eqs. (53) and (54) with i= 2= and 3=(l+ ) 2-l.

The CCA-model considers the filler network as a result of kinetically cluster-cluster-aggregation, where the size of the fractal network heterogeneity is given by a space-filling condition for the filler clusters [60,63,64,92]. We will summarize the basic assumptions of this approach and extend it by adding additional considerations as well as experimental results. Thereby, we will apply the CCA-model to rubber composites filled with carbon black as well as polymeric filler particles (microgels) of spherical shape and almost mono-disperse size distribution that allow for a better understanding of the mechanisms of rubber reinforcement. 

Hence, the results stated above have shown that the integral structural parameters K and influence not only the elasticity modulus value of semi-crystalline polymers, but also their possible distribution in polymer structure. The decrease in cluster characteristic size and reduction in the number of segments in it, which is equal to F/2 (see Equation 5.29), result in an increase in the elasticity modulus for the indicated polymers. Let us note that the offered reinforcement mechanism principally differs from that considered earlier for polymer nanocomposites with inorganic filler, where reinforcement is realised at the expense of formation of interfacial regions [24]. 

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