Budiansky 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 ... 

An especially interesting model that predicts dual-phase continuity is based on work by Budiansky. For two-phased systems, the Budiansky equation may be expressed... 

It is interesting to note that equation (6.55) is symmetrical, and the reversal of the geometric roles of components 1 and 2 does not change the composite property, provided its volume fraction remains the same. The Budiansky equation assumes cocontinuity of both polymers, in a manner... 

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. 

Note that like the Budiansky equation, the Bottcher equation is symmetrical in form. It has been suggested that both Budiansky and the Bottcher approaches predict the composite property especially well when the system undergoes a phase inversion or exhibits dual-phase continuity. 

Ramis and co-workers (48) made a similar study of SINs prepared from im-saturated polyester-styrene mixes with MDI-poly(propylene glycol). When the storage modulus, E, was plotted through a range of compositions, the Budiansky equation (47) was again found to fit better than several other relationships. 

In order to solve the system of the above-described equations, and which are derived by applying the self-consistent model, applied for composites by Budiansky 7), it is necessary to evaluate experimentally the moduli of elasticity (tension, shear, bulk) and Poisson s ratios of the constituent phases and the composite. Thus, the only unknown are the radius r of the mesophase layer and its mechanical properties and thermal expansion coefficient, which are then derived. 

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 order to account for the interaction between the phase domains, Budiansky (21) used the Kroner s "smearing out" approach (22) and assumed that an isolated spherical inclusion is embedded in an infinite isotropic matrix which itself is a composite material. For two-phase systems, his equation is expressed as ... 

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-...

Bottcher attempted to interpolate between the upper and lower bounds of the Maxwell equation. He assumed two co-continuous phases, in a manner similar to Budiansky " see Section 6.9, and especially (6.55). The Bottcher equation may be written... 

Budiansky derived the equation for shear modrdus which includes the values of mean deformations, yi, in each phase,(j)i volrune fraction of component i,... 

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