Models filler volume fraction

In Figure 5.23 the finite element model predictions based on with constraint and unconstrained boundary conditions for the modulus of a glass/epoxy resin composite for various filler volume fractions are shown. 

Fig. 9. The variation of the adhesion exponentsT andT)2, the adhesion coefficient A for the three-term modes, as well as the adhesion coefficient 2r for the two-term model, versus the filler volume fraction of...

Fig. II. The variation of the elastic modulus for the composite Ec, versus the filler-volume fraction, uf, and the mode of variation of the average mesophase modulus, E", as derived from the model...

Langley, K.R., Martin, A. and Ogin, S.L. (1994). The effect of filler volume fraction on the fracture toughness of a model food composite. Composites Sci. Tech. 50, 259-264. 

Figure 12.11. Dependence of the stress at 100% strain of NR / MWNTs composites on the filler volume fraction and comparison with predictions of theoretical models.

On surface it is very simple model but effective concentration of filler includes observation that some layer of polymer is bound to the surface of filler and the mechanisms of this bonding is mathematically expressed by effectiveness factor. The recent model assumes that filler particles are spheres which might be connected to form chain-like agglomerates. Each particle is surface coated with matrix polymer. The elastomeric layer is considered immobilized. The effective filler volume is higher than filler volume fraction by the amount of adsorbed polymer. The effectiveness factors is given by equation ... 

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). 

Fig. 17.7 Fittings on the evolution of the modulus of LCFo i-based biocomposites versus filler volume fraction content (Takayanagi, Voigt and Reuss models). Reproduced with permission (Averous and Le Digabel 2006). Copyright of Elsevier...

The resulting empirical expression for particulate filler reinforcement for the Lipatov model can be used to calculate the effective interphase thickness Ar, using given values of filler volume fraction, filler particle radius, and measured calorimetric evaluation of X ... 

The Guth-Gold model (29) is an extension of the Einstein model, whereby interactions of neighboring particles are incorporated. Normalized initial moduli, normalized against the polymer modulus, of filled polymers are related to the filler volume fraction by a polynomial series as follows ... 

Here, Es is the modulus of the polymer and cp is the filler volume fraction. Further modification of the Guth-Gold model leads to the Guth model (30) in which a geometric factor, g, was introduced. The Guth model is shown below. 

Here, P is the permeabiUly, Pg is the permeability of the polymer without fillers, (p is the filler volume fraction, and a is the aspect ratio. Cussler and coworkers (50) and Fredrickson and Bicerano (51) provided fiirther refinements of the model by removing the position order but with perfect orientation. Recently, Gusev and Lusti (52) conducted direct 3D finite-element permeability calculations with a multiinclusion computer model comprising of randomly dispersed, perfectly oriented, and nonoverlapping platelets. They found all their numerical simulated permeability values could be well represented with a stretched exponential function as... 

In summary, the crack pinning model, as described so far, explains why fracture energy increases with filler volume fraction and it accounts for the particle size effect. In the next section a rationalisation of the maxima in the plots of G]c (c) against Vf is presented. 

Figure 8 shows the comparison of theoretical predictions of equation (9) with experimental values for Young s modulns in starch/PVOH/Na-MMT and starch/PVOH/ LRD nanocomposites. Similar to WVP models, some obvious expectations can be quantified higher aspect ratio filler provide substantial improvement of Young s modulus for given filler volume fraction. In Figure 8, it is observed that modulus of starch/ PVOH/Na-MMT does not follow well with model predictions with aspect ratio a = 100. The reason for this is because most mechanical model predictions are based on complete layer exfoliation and perfect orientation. However, a state of delamination... 

The first question is which model we should use The lower bounds of the previous models can be used to predict the ETC of composites but only for low filler volume fraction. Indeed, as shown by Dupuis [6], ETC models agree with experimental values (PBT and aluminum fiber) for filler volume fraction lower than 35% or for an ETC 3 times less than the matrix one. 

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