Rule of mixture

Substituting the net Young s modulus times the area of the composite for kAx and the Young s modulus times the areas of each component for kiAx and k2Ax, the net modulus becomes 

Upper and lower bounds of the performance of a particle-reinforced composite set by the rule of mixtures. In this illustration the strength of the fiber is five times the matrix. The dashed line in the middle is the estimated performance of a randomly oriented discontinuous fibers (1/3 aligned with the applied stress, 2/3 normal to the applied stress). 

The other extreme is represented by placing the springs in series. Now the Fa is the same on each spring, but the displacements will be given by Axi = Fa/ i and A%2 = Fa/ z- Now the net displacement is given by Ax = Axi + Axz and the net spring constant becomes 

Now by assuming that A /Ac = Vi and A /Aq = Vz, we obtain ElEz FMatrixF Particle 

These derivations, while far from rigorous, do set upper and lower limits on the combined effect of the two components. The net thermal conductivity and CTE can also be estimated in this manner. 

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The density p of the composite material is then related to the density of the fiber and matrix p by the rule of mixtures ... 

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

Poisson s ratio, is the negative of the ratio of the strain transverse to the fiber direction, 8, and the strain ia the fiber direction, S, when the lamina is loaded ia the fiber direction and can also be expressed ia terms of the properties of the constituents through the rule of mixtures. 

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... 

In any structural application it is the peak stress which matters. At the peak, the fibres are just on the point of breaking and the matrix has yielded, so the stress is given by the yield strength of the matrix, d and the fracture strength of the fibres, (t, combined using a rule of mixtures... 

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,... 

This is an important relationship. It states that the modulus of a unidirectional fibre composite is proportional to the volume fractions of the materials in the composite. This is known as the Rule of Mixtures. It may also be used to determine the density of a composite as well as other properties such as the Poisson s Ratio, strength, thermal conductivity and electrical conductivity in the fibre direction. 

From the rule of mixtures, the stresses are related as follows. 

As shown in Fig. 3.4 stress-strain tests on uniaxially aligned fibre composites show that their behaviour lies somewhere between that of the fibres and that of the matrix. In regard to the strength of the composite, Ocu, the rule of mixtures has to be modified to relate to the matrix stress, o at the fracture strain of the fibres rather than the ultimate tensile strength, o u for the matrix. 

Thus, as shown earlier, the ultimate strength of the composite may be predicted by the rule of mixtures. 

The ultimate tensile strength of a uniaxially aligned fiber-reinforced composite is given to reasonable accuracy by the rule of mixtures relation ... 

The foregoing are but examples of the types of mechanics of materials approaches that can be used. Other assumptions of physical behavior lead to different expressions for the four elastic moduli for a unidirectionally reinforced lamina. For example, Ekvall [3-2] obtained a modification of the rule-of-mixtures expression for and of the expression for E2 in which the triaxial stress state in the matrix due to fiber restraint is accounted for ... 

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... 

For the modulus in the direction of the fibers. Tsai modified the rule of mixtures to cqunt for imperfectiorisanjioer ajignment ... 

Equation (3.68) does not represent a very significant departurerom the rule of mixtures. Of course, k is an experimentally determined constant and is highly dependent on the manufacturing process. 

Note that the expressions for E., and v.,2 are the generally accepted rule-of-mixtures results. The Halpin-Tsai equations are equally applicable to fiber, ribbon, or particulate composites. For example, Halpin and... 

Blending of ionomers with other homopolymers is also one means of enhancing mechanical performance. Frequently, in ionomer/polymer blends, synergistic effects are realized and properties may be significantly increased over anticipated values based on the rule of mixtures. This area of study has not been extensively explored and the probability clearly exists that new materials and new blends, having even a greater degree of property enhancement, will become available in the near future. 

Kohli et al. [27], for instance, showed that the tensile modulus of a highly drawn PC-TLCP composite could be modeled effectively by the simple additivity rule of mixtures, while the compression molded composite samples with a spherical TLCP morphology had moduli according to the inverse rule. In both cases, the tensile modulus of the TLCP (Ei,c) itself was assumed to be a constant value determined from a tensile test of the pure TLCP samples. But whether or not the dispersed TLCP fibers and deformed droplets have the same modulus as the bulk TLCP samples remains a question. 

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