Kerner equation

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.)...

In the blend of the previous question we blend glass fibres. Now = 2400 MPa and Ei= 75,000 MPa, so a = 31.25. E should be increased up to 3000 MPa so E/Eq = 1.25. When we substitute this into the Kerner equation, again with A = 1.14 (not wholly correct since n has become somewhat higher because of the blending with rubber), Kerner s formula then becomes ... 

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... 

There is a relatively sharp drop in storage modulus corresponding to the a relaxation of the rubber phase. At temperature above the Tg of the rubber, the storage modulus of the rubber becomes negligible compared with that of the rigid matrix, and the modulus of the composite is due solely to the matrix. Under these conditions, the system can be described by the modified Kerner equation [12,53] ... 

For composite PHE-Gr, it was shown [23] that the dependence of the elasticity modulus (E) on (pf is well described by the Kerner equation which assumes strong interactions at the interface. Because of it we can suppose that by fracture of a composite there exists the possibility of transfer of the applied stress through the interfacial border. The tensile strength of the composite in the given case should be the function of the shear strength of the... 

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... 

Additional evidence for the validity (to a good approximation) of a Kerner-type equation—in this case, the unmodified Kerner equation— as applied to rigid polymeric matrices filled with rigid particles (up to a filler volume fraction of 0.5) has been given by Kenyon and Duffey (1967), Ishai and Cohen (1967), Moehlenpah et al. (1970, 1971), Manson and Chiu (1972) [based on Chiu (1973)], and Brassell and Wischmann (1974). 

Figure 12.2. Dependence of relative modulus (composite/polymer) on concentration for (A) the original Kerner equation (B) the modified Kerner equation (C) the Mooney equation. Circles correspond to experimental results. Solid curves are calculated with v = 0.35, Gf/Gp (relative modulus filler/polymer) = 25, and = 0.64.

Figure 12.30. Comparison of experimental and predicted thermal conductivities for glass-sphere-filled polymers. The upper curves are for polyethylene, the lower curves for polystyrene. Except for Kerner equation plot ( ), curves and data are from Sundstrom and Chen (1970). (--) Maxwell (—) Cheng-Vachon (- -) Behrens and Peterson-Hermans. (From Sundstrom, D. W., and Chen, S. Y., 1970, J. Compos. Mater. 4, 113 courtesy Technomic Publishing Co.)...

Subscripts 1 and 2 denote component materials and)/ is the Poisson s ratio of the composite. It is interesting to note that eq 10 is symmetrical and the reversal of geometrical roles of the materials 1 and 2 does not change the composite property in a given concentration. In other words, one does not differentiate materials 1 and 2 as the matrix or the inclusion. The Mooney equation (eq 1 and 2), the Kerner equation (eq 3 and 5), the Dickie equation (eq 8) (IT =1 for shear modulus, G, in eq 8) and the Budiansky equation (eq 10) were compared with the experimental results at 23 C. The Poisson s ratio of the polyurethane phase was assumed to be 0.5 and that of the acrylic to be 0.35 at 23°C. The Poisson s ratio of the composite was assumed to be the volume additive of its components ... 

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-...

The Kerner Equation Perhaps the most widely used relationship is the Kerner equation (10) ... 

However, in using the Kerner equation, it is necessary to take into account that interaction occurs at the phase border and a part of pol3rmer is bound by filler, as a result of which, the effective proportion of filler has to be increased to... 

Taking this into account, the Kerner equation may be transformed by incorporating the value (pe = [Pg.206]

The mechanical losses diminish when filler is introduced. The equation presented above takes into account the real structru e of a fdled system and the existence of an intermediate layer between the particle and matrix, despite the fact that their contribution to the change of properties of pol3rmer matrix has not been estimated. Introduction of parameter B (Eq 5.10) has a very formal character. This parameter may be estimated from the experimental dependence of E/Ep on (p (Eq 5.10). Experimental data show that B diminishes with increase of (p. The thickness of the bound layer decreases with decreasing thickness of the intermediate layer between two filler particles which has no meaning. Dependence of B on (p means that we cannot use the modified Kerner equation (5.10), which does not take such a dependence into account. The values of B are changed in the range of 6-1, i.e., in the limit of the case when the surface layer is absent. Therefore, this approach is only interesting because it takes into account the existence of the bound surface layer, but the dependence of B on (p makes its justification doubtful, especially because the thickness of the bound layer, as determined from mechanical measurements, depends on the frequency. ... 

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... 

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