Constitutive relations for random polycrystals

Many ceramics are used in a random polycrystalline form and thus, it is useful to be able to predict the elastic constants from those of the single crystals. The approaches outlined in the last two sections are used for this procedure by considering the random polycrystal as an infinite number of phases with all possible orientations. For example, Voigt and Reuss used a technique based on averaging the stiffness or compliance constants and obtained upper and lower bounds. The Voigt upper bounds for the bulk (B) and shear (/i) moduli of the composite can be written as 

Expressions are available for the HS bounds for the other crystal classes but the mathematics increases in complexity and, thus, the VRH averages are often used. The data in Table 2.1 were calculated using this procedure and Table 3.4 compares some experimental values with the theoretical values. It appears from the data that these constitutive equations are extremely successful in estimating the elastic constants of the polycrystalline material. 

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