Rules of Mixtures for Composites

Rule of mixtures equations (often modified according to the type, shape, and orientation of the reinforcement/filler) are commonly used to describe certain properties of the composites. For example 

1) Concentrations are usually expressed by volume, as volume fractions of filler Vf 

2) Volume fractions are also used to predict a theoretical density of the composite q, based on the respective densities of the components and assuming a total absence of voids  

3) The total cost per unit weight of the composite C can also be calculated from the volume fractions and the costs of the individual components and the cost of compounding per unit weight of the composite Q [7]  

After introducing incorporation costs, the cost of the composite may be higher or lower than that of the unfilled polymer. For low-cost commodity plastics, the term filler (implying cost reduction) may be a misnomer since manufacturing costs may offset the lower cost of most mineral fillers. For higher cost specialty high-temperature thermoplastics, the final cost of, for example, glass fiber-reinforced polyetherimide is usually less than that of the unmodified polymer. 

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Fig. 5 Young s modulus (E) as a function of volume fraction PPO (Vppo) for the compatible Copolymer B/PPO blends. The curve was drawn by use of the modified rule of mixtures for composites (Eq. 4).

Both compatible and semi-compatible PPO blends that show blend densification exhibit a small synergistic maximum in their modulus-composition plots which can be modeled by the classical rule of mixtures for composites with an additional interaction term. The incompatible PPO blends exhibit no blend densification and can be modeled adequately by the series model for composites. 

For a crystalline polymer, its viscoelastic properties are relevant to the crystallinity besides temperature. At skin region of injection molded crystalline polymers, their material parameters can be expressed as the following equations according to the rule of mixtures for composites. 

Another simple relationship between the constituent moduli results from the observation that the compliance of the composite material, 1/E, must agree with the compliance of the matrix, l/En, vvheD V. = 1 and with the compliance of the dispersed material when = 1 The resulting rule of mixtures for compliances is... 

The rule of mixtures for discontinuous fiber composites may be expressed as ... 

It is well known that the Young s modulus of a composite can be calculated by the rule of mixtures for long-fibre reinforced material. In the case of whiskers, the rule of mixture is also applied to estimate the change of modulus (conventionally, reinforcements are added to improve the stiffness of a material, though for ceramic matrix composites this is not always the primary concern). 

If the composite strain rate is known, the composite stress during steady-state isothermal creep can be computed from the rule of mixtures for the stress, Eqn. (9). This gives... 

A rule of mixture for one phase systems with an adjustable compatibility parameter gives adequate fit to the observed composition dependence of the modulus. 

Considering the orientation of the fiber, the rule of mixture for the strength and modulus of the continuous fiber-reinforced composites can be represented by... 

The data of Figs. 11.15 and 11.16 can be manipulated to produce results suitable for comparison with the predictions of the rule of mixtures. The procedure is to fit smooth curves to the data and differentiate to find the densification rate at any time. Figure 11.17 shows the results of the comparison. Drastic deviations are observed for the ZnO matrix composite for v,- as low as 3-6 vol%. The glass matrix composite shows good agreement with the predictions of the rule of mixtures for v,- less than 20 vol% but the deviations become increasingly severe at higher v, values. 

Figure 11.17 Comparison of the sintering data for the composites described in Figs. 11.15 and 11.16 with the predictions of the rule of mixtures. The densification rate of the composite relative to that for the free matrix is plotted versus the volume fraction of inclusions. Note the strong deviation from the predictions of the rule of mixtures for the ZnO matrix even at fairly small inclusion content.

Data were considered in terms of the material constituents, specification and fabrication method and whether these were typical of the resulting composite. It has been tacitly assumed that authors have used valid test methods even though these have not always been cited. Reliability, scatter and reproducibility were difficult to assess unless the authors had commented on their data. Comparison with other data has generally been undertaken with the aid of the rule of mixtures for calculating the modulus (section 1.3). 

Alkali treatment of jute fibers also improves the mechanical properties of thermoplastics. The vinyl ester resin reinforced with alkali-treated fibers shows improved mechanical properties. The maximum improvement was noted for the composites prepared with 4 h alkali-treated fibers at 35% fiber loading. The flexural strength improved by 20% and the modulus by 23%. The strength and modulus of the composites were found to be lower than the values estimated from the general rule of mixtures. For the jute/vinyl ester composites with 35% fiber content, the strength was decreased by 29% and 16% for the untreated and 4h alkali-treated fibers and the modulus was lower by 51% and 37% for the untreated and 4h alkali-treated fibers, respectively [154]. 

Equation 8.10 is referred to as the rule of mixtures for the apparent Young s modulus of the composite in the direction of the fibers. The rule of mixtures predicts a simple linear variation of ranging in value from 7 , to , as the fiber volume fraction goes from 0 to 1 (Figure 8.6). 

Sisal was chosen for this work, because the fibres are relatively stiff [7], meaning the buckling and wrinkling in the composite is lower, and their shape is also much cylinder like, therefore getting the possibility to use micromechanical models, i.e. rule of mixtures, for glass fibre reinforced polymers to consider the reinforcement properties of sisal fibres. 

The rule of mixture for calculating composite strength is given in Equation 1 ... 

The difference between the bounds defined by the simple models can be large, so that more advanced theories are needed to predict the transverse modulus of unidirectional composites from the constituent properties and fiber volume fractions (1). The Halpia-Tsai equations (50) provide one example of these advanced theories ia which the rule of mixtures expressions for the extensional modulus and Poisson s ratio are complemented by the equation... 

The premise that discontinuous short fibers such as floating catalyst VGCF can provide structural reinforcements can be supported by theoretical models developed for the structural properties of paper Cox [36]. This work was recently extended by Baxter to include general fiber architecture [37]. This work predicts that modulus of a composite, E can be determined from the fiber and matrix moduli, Ef and E, respectively, and the fiber volume fraction, Vf, by a variation of the rule of mixtures,... 

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