Filler volume fraction modulus, function

The established concepts predict some features of the Payne effect, that are independent of the specific types of filler. These features are in good agreement with experimental studies. For example, the Kraus-exponent m of the G drop with increasing deformation is entirely determined by the structure of the cluster network [58, 59]. Another example is the scaling relation at Eq. (70) predicting a specific power law behavior of the elastic modulus as a function of the filler volume fraction. The exponent reflects the characteristic structure of the fractal heterogeneity of the CCA-cluster network. 

FIGURE 36.1. Filler particles with hard core relative increase of the elastic modulus as a function of filler volume fraction for different values of the ratio shell modulus to matrix modulus The ratio between shell (total) and core radius is taken as 4/3. 

Concerning the modulus evaluatirai of the fillers is always problematic. The modulus has been evaluated. Different composites had been processed with increasing LCFo i content. The fillers modulus has been estimated by fitting a semiempiri-cal Halpin-Tsai model on the evolution of the composites Young s modulus as a function of fillers volume fraction. By extrapolation at 100% of fillers, we obtain the filler modulus which is estimated at 6.7 GPa. This value is coherent with wheat straw data given in the literature (Hornsby et al. 1997 KrtMibergs 2000). 

As shown by Figs. 15 and 16 (carbon black) filled compounds exhibit a more complex dependency on strain and temperature essentially because strong interactions develop between the filler and a part of the rubber matrix. An approach of the complex modulus function G (y, T, [where stands for filler volume fraction] of carbon black filled mbber compounds was recently proposed by the author [28] but its detailed discussion is outside the scope of the present chapter. It will only be noted here that an appropriate equation for G (y, T, combines a member that represents the response of the polymer matrix to increasing strain (the so-called polymer component) and a member that describes the response of a rubber-filler network embedded in the matrix (the so-called filler component). 

The tensile modulus of a polymeric material has been shown to be remarkably improved when nanocomposites are formed with layered silicates. In the case of biodegradable polymer nanocomposites, most studies report the tensile properties as a function of clay content. In most conventionally filled polymer systems, the modulus increases linearly with the filler volume fraction, whereas for these nanoparticles much lower filler concentrations increase the modulus sharply and to a much larger extent. The dramatic enhancement of the modulus... 

By analyzing the compositional dependent relaxation time, the stress-strain relationships of polymer composites are determined as a function of the filler concentration and strain rate. As the volume fraction of filler increases, both the effective elastic modulus and yield stress increases. However, the system becomes more brittle at the same time. 

FIGURE 17.20 The effect of filler particles on gel properties, (a) Relative modulus (Gm/G0) as a function of particle volume fraction (broken lines are calculated for various values of the ratio Gp/Go, indicated near the curves. The drawn lines are average experimental values for acid casein gels (C) and polymer gels (polyvinyl alcohol, P), with emulsion droplets that are either bonded (B) or nonbonded (N) to the gel matrix, (b) Highly schematic pictures of the gel structure. Shaded area denotes primary gel. Particles are nonbonded (1) bonded (2) bonded but with intermediate layer (3) bonded and aggregated (4). (Adapted from T. van Vliet. Colloid Polymer Sci. 266 (1988) 518.)... 

Relative (a) tensile modulus and (b) yield stress of polypropylene nanocomposites plotted as a function of inorganic volume fraction of filler. The dotted lines serve as guides. (Reproduced from Mittal, V., Eur. Polym. 43, 3727, 2007. With permission from Elsevier.)... 

Figure 8.5 Average relative storage modulus at high frequencies as a function of the reduced volume fraction of filler. (Reprinted from Ref. 71 with kind permission from John Wiley Sons, Inc., New York, USA,)...

Figure 7.3 [6] shows the relative modulus values as a function of temperature indicating that the fillers with smaller particle size have a greater effect in increasing the modulus compared with the similar composites of the same volume fractions. 

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