Filler strained, stress contribution

For filler reinforced rubbers, both contributions of the free energy density Eq. (35) have to be considered and the strain amplification factor X, given by Eq. (39) differs from one. The nominal stress contributions of the cluster deformation are determined by oAtfJ=dWA/dzA, where the sum over all stretching directions, that differ for the up- and down cycle, have to be considered. For uniaxial deformations E =e, E2=Ej= +E) m- one obtains a positive contribution to the total nominal stress in stretching direction for the up-cycle if Eqs. (29)-(36) are used ... 

Figure 45a-c shows an adaptation of the developed model to uniaxial stress-strain data of a pre-conditioned S-SBR-sample filled with 40 phr N220. The fits are obtained for the third stretching cycles at various prestrains by referring to Eqs. (38), (44), and (47) with different but constant strain amplification factors X=Xmax for every pre-strain. For illustrating the fitting procedure, the adaptation is performed in three steps. Since the evaluation of the nominal stress contribution of the strained filler clusters by the integral in Eq. (47) requires the nominal stress aR>1 of the rubber matrix, this quantity is developed in the first step shown in Fig. 45a. It is obtained by demanding an intersection of the simulated curves according to Eqs. (38) and (44) with the measured ones at maximum strain of each strain cycle, where all fragile filler clusters are broken and hence the stress contribution of the strained filler clusters vanishes. The adapted polymer parameters are Gc=0.176 MPa and neITe= 100, independent of pre-strain. According to the considerations at the end of Sect. 5.2.2, the tube constraint modulus is kept fixed at the value Ge=0.2 MPa, which is determined by the plateau modulus Gn° 0.4 MPa [174, 175] of the uncross-linked S-SBR-melt (Ge=l/2GN°). The adapted amplification factors Xmax for the different pre-strains ( max=l> 1-5, 2, 2.5, 3) are listed in the insert of Fig. 45a.

Figure 45b (upper part) shows the residual stress contribution of the strained filler clusters for the different pre-strains, obtained by subtracting the polymer contributions (solid lines) from the experimental stress-strain data (symbols) of Fig. 45a. The resulting data (symbols) are fitted to the second addend of Eq. (47) (solid lines), whereby the size distribution of filler clusters Eq. (37), shown in the lower part of Fig. 45b, has been used. The size distribution (x ) is determined by the adapted mean cluster size =< ifd>=26 and the pre-chosen distribution width Q=-0.5, which allows for an analytical solution of the integral in Eq. (47). The tensile strength of filler-filler bonds is found as Q b/d3=24 MPa. The different fit lines result from the different stress-strain curves ctR1( ) that enter the upper boundary of the integral in Eq. (47). Note that this integral, representing the contribution of the strained filler clusters to the total stress, becomes zero at = max for every pre-strain.

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.

Fig. 46 a Stress contributions of the strained filler clusters for the different pre-strains (upper part), obtained as in Fig. 45b. The solid lines are adapted with the integral term of Eq. (47) and the log-normal cluster size distribution Eq. (55), shown in die lower part. The obtained parameters of the filler clusters are Qe /d 3=26 MPa, =25, and b=0.8. b Uniaxial stress-strain data (symbols) as in Fig. 45c. The insert shows a magnification for the smaller strains, which also includes equi-biaxial data for the first stretching cycle. The lines are simulation curves with the log-normal cluster size distribution Eq. (55) and material parameters as specified in the insert of Fig. 45a and Table 4, sample type C40...

The stress-strain curves of the vulcanizates with 40 phr filler loading are shown in Fig. 28. SBR reinforced with plasma-coated carbon black shows a slight improvement in tensile strength relative to SBR with uncoated carbon black. Polyacetylene-coated carbon black can better interact chemically and physically with the elastomer and thus contributes extra to the reinforcement of the elastomer. 

In view of an illustration of the viscoelastic characteristics of the developed model, simulations of uniaxial stress-strain cycles in the small strain regime have been performed for various pre-strains, as depicted in Fig. 47b. Thereby, the material parameters obtained from the adaptation in Fig. 47a (Table 4, sample type C60) have been used. The dashed lines represent the polymer contributions, which include the pre-strain dependent hydrodynamic amplification of the polymer matrix. It becomes clear that in the small and medium strain regime a pronounced filler-induced hysteresis is predicted, due to the cyclic breakdown and re-aggregation of filler clusters. It can considered to be the main mechanism of energy dissipation of filler reinforced rubbers that appears even in the quasi-static limit. In addition, stress softening is present, also at small strains. It leads to the characteristic decline of the polymer contributions with rising pre-strain (dashed lines in... 

Of the experimental obstacles, attainment of a reliable equilibrium stress (the statistical theory is an equilibrium theory) is perhaps the most serious one. Pre-straining to a higher extension is sometimes resorted to as a means for hastening the attainment of a level stress value, but stress-softening is an undesired complication. At low extensions secondary filler aggregation makes a large contribution to the modulus which is... 

The response of unvulcanized black-filled polymers (in the rubbery zone) to oscillating shear strains (151) is characterized by a strong dependence of the dynamic storage modulus, G, on the strain amplitude or the strain work (product of stress and strain amplitudes). The same behavior is observed in cross-linked rubbers and will be discussed in more detail in connection with the dynamic response of filled networks. It is clearly established that the manyfold drop of G, which occurs between double strain amplitudes of ca. 0.001 and 0.5, is due to the breakdown of secondary (Van der Waals) filler aggregation. In fact, as Payne (102) has shown, in the limit of low strain amplitudes a storage modulus of the order of 10 dynes/cm2 is obtained with concentrated (30 parts by volume and higher) carbon black dispersions made up from low molecular liquids or polymers alike. Carbon black pastes from low molecular liquids also show a very similar functional relationship between G and the strain amplitude. At lower black concentrations the contribution due to secondary aggregation becomes much smaller and, in polymers, it is always sensitive to the state of filler dispersion. 

In attempting to predict the direction that future research in carbon black technology will follow, a review of the literature suggests that carbon black-elastomer interactions will provide the most potential to enhance compound performance. Le Bras demonstrated that carboxyl, phenolic, quinone, and other functional groups on the carbon black surface react with the polymer and provided evidence that chemical crosslinks exist between these materials in vul-canizates (LeBras and Papirer, 1979). Ayala et al. (1990, 1990) determined a rubber-filler interaction parameter directly from vulcanizatemeasurements. The authors identified the ratio a jn, where a = slope of the stress-strain curve that relates to the black-polymer interaction, and n = the ratio of dynamic modulus E at 1 and 25% strain amplitude and is a measure of filler-filler interaction. This interaction parameter emphasizes the contribution of carbon black-polymer interactions and reduces the influence of physical phenomena associated with networking. Use of this defined parameter enabled a number of conclusions to be made ... 

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