Kraus model

Analogous to the spherical filler of radius R in the Kraus model, Bhattacharya and Bhowmick [31] consider an elliptical filler represented by R(1 + e cos 0), in the polar coordinate. The swelling is completely restricted at the surface and the restriction diminishes radially outwards (Fig. 40 where, qt and q, are the tangential and the radial components of the linear expansion coefficient, q0). This restriction is experienced till the hypothetical sphere of influence of the restraining filler is existent. One can designate rapp [> R( 1 + e cos 0)] as a certain distance away from the center of the particle where the restriction is still being felt. As the distance approaches infinity, the swelling assumes normality, as in a gum compound. This distance, rapp, however, is not a fixed or well-defined point in space and in fact is variable and is conceived to extend to the outer surface of the hypothetical sphere of influence. 

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 However, very recently Ulmer again evaluated the Kraus equations with data from several published sources and unpublished own data [66].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... 

Figure 14. The 8 cycle of strain sweep experiments at 140 °C and 1 Hz. (a) 30/70 DSF/SB composite (b) 20/80 DSF/SB composite (c) 30/70 SPI/SB composite (d) 20/80 SPI/SB composite. The samples were prepared by the casting method. Solid lines are the fit from the Kraus model. (Reproducedfrom reference 16.)...

Samples prepared by casting method and measured at 140 C Best fit of shear elastic modulus vs. strain with the Kraus Model Source Reproduced from reference 16. 

Starch nanocrystals were used to reinforce a non-vulcanised NR matrix. The NR was not vulcanised to enhance biodegradability of the total biocomposite. Non-linear dynamic mechanical experiments demonstrated a strong reinforcement by starch nanocrystals, with the presence of Mullins and Payne effects. The Payne effect was able to be predicted using a filler-filler model (Kraus model) and a matrix-filler model (Maier and Goritz model). The Maier and Goritz model showed that adsorption-desorption of NR onto the starch surface contributed the non-linear viscoelasticity. The Kraus model confirmed presence of a percolation network. ... 

Of the several mechanisms investigated, the most commonly adopted is based on the filler network breakage [48, 49]. Kraus [7, 50] proposed a phenomenological model of the Payne effect based on this interpretation. In this model, under dynamic deformation, filler-filler contacts are continuously broken and reformed. The Kraus model considers filler-filler interactions but the loss modulus and effect of temperature were not taken into account. In the model of Huber and Vilgis [9, 50, 51] the existence of dynamic processes of breakage and reformation of the filler network is explained. In this model, the Payne effect is related to the fractal nature of the filler surface. At sufficiently high volume fractions of filler, percolation occurs and a continuous filler network is formed, characterized by its fractal dimension and its... 

Fig. 16 Strain sweep measurements of the 40 phr filled nanocomposites at 20 °C (Lefi), the lines are fits according the Kraus model Payne effect measure at different temperatures Right) for SBS filled with 40 phr Aerosil 200 (Reprinted from [54])...

The amplitude of the Payne effect decreases dramatically with temperature. This is contrary to the theory of mbber elasticity, according to which the modulus should increase linearly with the temperature. In agreement with the former explanation given for the Payne effect, the temperature increases the rate of destmction of the network by weakening its cohesion. In the Kraus model, the temperature affects... 

Kraus model of deagglomeration-reagglomeration of filler aggregates, assimilated to "soft spheres."... 

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... 

To consider that deagglomeration and agglomeration are two symmetrical processes is surely a strong hypothesis and likely the reason for the deficiencies of the Kraus model. In order to somewhat circumvent it, recent theoretical developments were made with an explicit reference to the fractal description of CB aggregafes, and by considering that, above a percolation threshold, highly branched aggregates can flocculate and form a secondary... 

Kraus model as modified by Ulmer on SBR/carbon black (60 phr) compounds. 

In order to circumvent some deficiencies of the Kraus model, namely the fact that the whole set of model parameters has to be reconsidered if the frequency is changed. Lion et al. proposed a interesting phenomenological theory which leads also to a six-parameter model for the DSS effect. Both the frequency and the amplitude are taken into account with this model and by interpreting the observed history and recovery effects on the elastic and viscous moduli as manifestations of thixotropy, a so-called... 

From a theoretical point of view, the Lion et al. model has the merit to approach the DSS effect by applying constitutive laws formulated on the basis of fractional calculus, in other terms by formulating the behavior of materials with respect to fractional time derivatives of stress and strain an approach that in principle requires only a small number of material constants to express the material properties in the time or the frequency domain. However, deriving model parameters from experimental data is not straightforward and, for instance Lion et al. had to use a stochastic Monte Carlo method to estimate the model parameters for a comparison with experimental data on 60 phr CB filled rubber compound. Moreover, mathematical handling of the above equations (see Appendix 5.5) shows that, like the Kraus model, this one exhibits also horizontal symmetry for the G curve and vertical symmetry for the G" curve, and is therefore not expected to perfectly meet experimental data, at least in its present state of development. 

The Kraus model implies thus that, at the critical strain, the viscous character is maximum 2% of the elastic one and less than 0.1% at very high strain. Such limits somewhat depend on the value of fhe critical strain The variahons of G and G", as modeled by Kraus model, essentially reflect the mathematical forms of the Jo/Jc functions. 

The Kraus model fits reasonnably well the experimental G data but cannot meet the asymmetric shape of G" vs. strain data. 

A5.3 Ulmer Modification of the Kraus Model for Dynamic Strain Softening Fitting the Model... 

Comparison with Kraus model Heinrich and Kluppel... 

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