Dickie equation

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-...

In a recent study of the viscodlastic behavior of polyurethane/poly (methyl methacrylate) SINs, Kim et al. (97) evaluated the several equations treated above. They found that the Budiansky equation fit their data best (see Figure 6.35). The S-shape character of the Budiansky curve suggests a phase inversion, which was experimentally established by these workers (see Figure 6.4). For sake of comparison, the Kerner, Mooney, and Dickie equations are also illustrated. 

Differing from the previous studies (5-7) where the parameters Goo, m, loay have been treated as constants, we find that they depend on cross-link density which is consistent with the measurements of Dickie and Ferry (4). Figure 5 shows the dependence of the viscoelastic relaxation on cross-link density. The solid curves are calculated from Equations 17, 19 and 20 by using a value of xq = 2.5 x 102 hrs at T = 25°C. Figure 5 resembles the corresponding figure in ref. 5. 

Vinberg, 1956,1962,1986 Paloheimo and Dickie, 1966 MacKinnon, 1973 Greze, 1979). The last two authors supposed that because the coefficient was stable, the metabolic expenditure of an individual or a population could be evaluated from estimates of the food consumed and the resulting production, which can be calculated by the following equation ... 

To convert into dry or fresh weights or calorie content, the same coefficients are used in the equation, while conversion into protein requires different coefficients. The inverse relationship between AT, and K2 on die one hand and the ration on the other has been pointed out by several workers (Paloheimo and Dickie, 1966 Kitchell et al., 1977 Paloheimo and Plowright, 1979). 

One of the eharacteristics of viscoelastic foods is that when a shear rate is suddenly imposed on them, the shear stress displays an overshoot and eventually reaches a steady state value. Figure 3-43 illustrates stress overshoot data as a function of shear rate (Kokini and Dickie, 1981 Dickie and Kokini, 1982). The data can be modeled by means of equations which contain rheological parameters related to the stresses (normal and shear) and shear rate. One such equation is that of Leider and Bird (1974) ... 

Several studies were conducted on the stress overshoot and/or decay at a constant shear rate. Kokini and Dickie (1981) obtained stress growth and decay data on mayonnaise and other foods at 0.1, 1.0, lO.Oand 100 s . As expected from studies on polymers, shear stresses for mayonnaise and other food materials displayed increasing degrees of overshoot with increasing shear rates. The Bird-Leider empirical equation was used to model the transient shear stresses. 

Kokini and Dickie (1981) found that the Bird-Leider equation provided moderately good predictions for peak shear stresses (trmax) and the corresponding times (tmax), but the prediction of shear stress decay was poor. They suggested that a series of relaxation times, as opposed to a single exponential term, would be needed for mayonnaise and other materials. 

Donatelli et also applied the Kerner and Davies theories to their sequential IPNs and found that the Davies equation fits best (see Figure 6.33). Two of these compositions, shown in the lower part of Figure 2.3, suggest dual-phase continuity in a qualitative way. Semi-II IPNs, illustrated in the center of Figure 2.3, fell below the Davies line, suggesting one continuous and one discontinuous phase. While the Dickie et al latex materials would not be expected to exhibit dual-phase continuity, it is interesting that both the Allen et al and the Donatelli et al. materials do. 

FIG. 10-6. Relaxation spectram H in dynes/cm, plotted logarithmically, for natural rubber lightly cross-linked by dicumyl peroxide with v= 1.86 X 10 moles/cm and tangent with slopeto calculate fo from equation 39. Points top black, from C bottom black, from C". (Data of Dickie and Ferry. )... 

FIG. 14-19. Plots of log E and log tan 5 against temperature at a frequency of 110 Hz for a two-phase blend of equal weight fractions of polyfmethyl methacrylate) and slightly cross-linked polyfbutyl acrylate). Curves calculated from modified Kemer equation for a polyfmethyl methacrylate) matrix with inclusions of polyfbutyl acrylate) containing 14.4% poly(methyl methacrylate) by volume. (Dickie and Cheung. ) Reproduced, by permission, from the Journal of Polymer Science. 

Alternative formulations, which may be equivalent to the Takayanagi model in certain special cases, have been reviewed in detail by Nielsen and Dickie. Dickie 58,159 used with considerable success a modification of an equation... 

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