Fiber volume fractions

Parts with fiber volume fractions up to 60% can be fabricated by filament winding. The procedure is often used to manufacture composite rocket motors, corrosion-resistant tanks and storage containers, and piping for below-ground appHcations. 

Eig. 10. The variation of the density of carbon-fiber reinforced epoxy resin with the fiber volume fraction, based on the rule of mixtures. 

Eor the case of high modulus fibers such as carbon fibers with = 240 GPa (3.5 x 10 psi), in a polymer matrix, such as epoxy resin with = 3.0 GPa (450,000 psi), the extensional modulus is approximately proportional to the fiber volume fraction and the modulus of the fibers ... 

Thus the addition of the stiff carbon fibers has a very great effect in stiffening the epoxy matrix. Eor the commonly used fiber volume fraction of 0.6 the unidirectional carbon—epoxy lamina has a predicted extensional stiffness E = 145 GPa (2.1 x 10 psi)-... 

The relationship between fiber and matrix moduH and fiber volume fraction for a unidirectional lamina loaded in the direction transverse to the fibers is not simple. A lower bound (1) is given by the expression of the series spring model. 

The variation of the ia-plane shear modulus normalised with respect to the matrix modulus as a function of the fiber volume fraction is shown ia Figure 11. As noted eadier, it is generally difficult to measure the shear modulus of the fibers, which may themselves be anisotropic. The equation should be used with caution. 

Fig. 11. The variation of the shear modulus G of carbon-fiber-reiaforced epoxy resia as a function of the fiber volume fraction for several values of the ratio of the fiber shear modulus to that of the matrix (G /G. Ratios are noted on the curves (100,10,2).

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... 

Fig. 8. Composite thermal conductivity as a function of fiber volume fraction.

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,... 

The results of the micromechanics studies of composite materials with unidirectional fibers will be presented as plots of an individual mechanical property versus the fiber-volume fraction. A schematic representation of several possible functional relationships between a property and the fiber-volume fraction is shown in Figure 3-4. In addition, both upper and lower bounds on those functional relationships will be obtained. 

Figure 3-6 Variation of with Fiber-Volume Fraction...

Rgure 3-14 Variation of Vi2 with Fiber-Volume Fraction 3.2.4 Determination of G 2... 

Halpin and Tsai [3-17] developed an interpolation procedure that is an approximate representation of more complicated micromechanics results. The beauty of the procedure is twofoldr-First. it is simple, so it can readily be used in the design process. Second, it enables the generalization of usually limited, although more exact, micromechanics results. Moreover, the procedure is apparently gnltp accurate if the fiber-volume fraction (Vf) does not approach one. ... 

The term r Vf in Equation (3.71) can be interpreted as a reduced fiber-volume fraction. The word reduced is used because q 1. Moreover, it is apparent from Equation (3.72) that r is affected by the constituent material properties as well as by the reinforcement geometry factor To further assist in gaining appreciation of the Halpin-Tsai equations, the basic equation. Equation (3.71), is plotted in Figure 3-39 as a function of qV,. Curves with intermediate values of can be quickly generated. Note that all curves approach infinity as qVf approaches one. Obviously, practical values of qV, are less than about. 6, but most curves are shown in Figure 3-39 for values up to about. 9. Such master curves for various vaiues of can be used in design of composite materiais. 

The preceding expressions, Equations (3.84) through (3.89), are more easily understood when they are plotted as in Figure 3-48. There, the composite material strength (i.e., the maximum composite material stress) is plotted as a function of the fiber-volume fraction. When V, is less than the composite material strength is controlled by the... 

Figure 3-50 Composite Tensile Strength versus Fiber-Volume Fraction (After Dow and Rosen [3 28])...

Dow and Rosen s results are plotted in another form, composite material strain at buckling versus fiber-volume fraction, in Figure 3-62. These results are Equation (3.137) for two values of the ratio of fiber Young s moduius to matrix shear modulus (Ef/Gm) at a matrix Poisson s ratio of. 25. As in the previous form of Dow and Rosen s results, the shear mode governs the composite material behavior for a wide range of fiber-volume fractions. Moreover, note that a factor of 2 change in the ratio Ef/G causes a factor of 2 change in the maximum composite material compressive strain. Thus, the importance of the matrix shear modulus reduction due to inelastic deformation is quite evident. 

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