Kerner model

Figure 26.12 Relative flex modulus of ATH- and MgOH2-filled ES30. The solid line represents the Kerner model [Equation (1)] prediction...

The objective here is not to pick the correct model, but rather to pick one which is capable of reproducing all the observed phenomena while introducing as few extra parameters as possible. All candidate models must be designed to compute the overall modulus of a composite from the individual moduli of the components, which are usually represented as a matrix phase and a dispersed phase. For example, the Kerner model (24) describes the elastic behavior of a suspension of spheres, while the Halpin... 

An important variation of the self-consistent model is the three-phase model, introduced by Kerner 20), according to which the inclusion is enveloped by a matrix annulus, which in turn is embedded in an infinite medium with the unknown macroscopic properties of the composite. 

Figure 5. Modulus-composition curves for crass-polybutadiene-inier-cross-polystyrene semi-I and full IPNs (16). (a) Kerner equation (upper bound) (b) Budiansky model (c) Davies equation and (d) Kerner equation (lower bound). (Reproduced from ref. 23. Copyright 1981 American Chemical Society.)...

It is instructive to consider the predictions of the mean field model for the above type of composite. A frequently used mean field model is that of Kerner [32],... 

When the inclusion is an air bubble in a viscoelastic solid the shear modulus of the air is zero, and also it is reasonable to neglect the bulk modulus of the air since it is several orders of magnitude less than the bulk modulus of the solid. The Kerner mean field model then predicts the following expression for the effective bulk modulus of the air-polymer composite,... 

One of the most successful theories is that of Kerner (8). Kerner employs a three phase model consisting of an average size spherical inclusion surrounded by a shell of the host material and imbedded in the equivalent homogeneous medium. The inclusions are distributed randomly and there is no interaction between them. 

Other investigators (1 ) have obtained results similar to those of Kerner. The results of Dewey (ll) which are valid for a dilute solution agree with the Kerner equation in the dilute solution limit. Christensen ( 12) reviews and rederives the effective modulus calculations for spherical inclusions. The three models which are... 

The Kerner equation, a three phase model, is applicable to more than one type of inclusion, Honig (14,15) has extended the Hashin composite spheres model to include more than one inclusion type. Starting with a dynamic theory and going to the quasi-static limit, Chaban ( 6) obtains for elastic inclusions in an elastic material... 

Theory. Basic theories for the prediction of the modulus of a composite from those of the components were derived by, for example, Hashin in 1955 (I), Kerner in 1956 (2), and van der Poel in 1958 (3). Takayanagi (4, 5), and Fujino et al. (6) developed a very promising and instructive model theory which includes calculation of the loss spectra of composites, and it may easily be extended to anisotropic morphologies. Furthermore, Nielsen and coworkers may be cited for fundamental theoretical and experimental contributions (7, 8, 9,10). 

Kerner, Z, and T, Pajkossy, Impedance of rough capacitive electrodes the role of surface disorder. Journal of Electroanalytical Chemistry, 1998. 448 pp. 139-142 Ldng, G, and K.E, Heusler, Remarks of the energetics of interfaces exhibiting constant phase element behaviour. Journal of Electroanalytical Chemistry, 1998. 457 pp, 257-260 Liu, S.H, Eractal model for the ac response of a rough interface. Physical Review Letters, 1985, 55 pp, 529-532... 

Various composite models such as parallel model, series model, Halpin-Tsai equation, and Kerner s model can be used to predict and compare the mechanical properties of polymer blends [43-45]. For the theoretical prediction of the tensile behavior of PMMA/EMA blends, some of these models... 

While the Takayanagi models have proved useful because of their simplicity, the effects of changes in mechanical behavior with composition and phase structure may also be profitably explored using several analytical relations, which include equations derived by Kerner (1956b), Hashin and Shtrikman (1963), and Halpin and Tsai (Ashton et al, 1969, Chapter 5). The most widely applied of these is the Kerner equation, which presents the... 

Kerner [104] made the first sophisticated analysis of thermoelastic properties of composite media using a model which had been considered earlier by van der Poel for calculation of the mechanical properties of composite materials. Here the dispersed phase has been assumed for spherical particles. Kemer s model accounts for both the shear and isotactic stresses developed in the component phases and gives for the composite ... 

Starting with Kerner s model Bousmina [289] derived a simplifled expression for the complex shear modulus of emulsions containing two viscoelastic liquids within the linear region ... 

Using the Kerner-Nielsen model for expressing the effect of filler concentration on Young s modulus (85) ... 

More quantitative information depends on the use of models. The Takayanagi models were already mentioned in connection with Figure 6.29. More analytical models have been evolved by Kerner/ Hashin and Shtrikman/ and Davies.Briefly, the first two theories assume spherical particles dispersed in an isotropic matrix. From the modulus of each phase, the composite modulus is calculated. An upper or lower bound modulus is arrived at by assuming the higher or lower modulus phase to be the matrix, respectively. The theory is reviewed elsewhere. 

Figure 6.35. Young s modulus vs. polyurethane concentration for the polyurethane-poly(methyl methacrylate) SINs at 23 C. Solid lines are based on the theoretical models with K, the Kerner equation, assuming the polyurethane as the continuous phase Z>i and D2 the respective Dickie equations, M the Mooney equation and B, the Budiansky equa-...

Lewis-Nielsen or modified Kerner equation [17] is one of the most successful models ... 

Nielsen developed a model for thermal conductivity, based on the Kerner equations describing the modulus of two phase systems, that provides a different response that of the SOLBM [20]. Nielsen s model added to the consideration of the relative c nductivity of the two phases the effect of the maximum packing... 

In the case of micro-composites, specific filler smface area of common fillers is usually less than the value 10 m /g. In these systems, a very small portion of polymei- molecules is in direct interaction with the filler surface. Moreover, dimensions of polymer chains and volumes chai acteristic of microscopic relaxar tion modes are orders of magnitude smaller compared to the dimensions of filler particles (see Table 6.1). Thus, continuum mechanics approaches, such as micromechauical models can be successfully used for description of their mechanical behavior. The continuum mechanics model, well usable for prediction of the mechanical contribution (i) to the modulus of elasticity of polymer composites, is the simple Kerner-Nielsen (K-N) model [66] ... 

Figure 6.4 Comparison of experimental modulus data for an emulsion blend of poly(vinyl acetate)/ethy-lene-vinyi acetate copolymer with the parallel, series and Kerner s models (reproduced from Robeson, L. M. and Berner, R. A., J. Polym. Scl Part B Polym. Phys. (2001) 39, p. 1093, with permission of John Wiley Sons, Inc.)...

Foamed materials can be considered to be the opposite of fiber reinforcement. In a foam, the gas inclusions make the matrix softer and weaker. The introduction of even small amounts of air into a polymer can have a dramatic effect on acoustic properties. One approach to modeling the effect of a small air content 4> on the acoustic properties of a rubber is to use Kerner s theory of composites (30) ... 

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