Third-order bounds on the elastic moduli of metal-matrix composites
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Communications Third-Order Bounds on the Elastic Moduli of Metal-Matrix Composites
[
6~bl~b2(G1--G2) z] (G) 6((7) + ,~--1 J
L.C. DAVIS
[5]
6~blq~2(Gl - G2) 2]
Accurate prediction of the elastic moduli of a composite material consisting of particles in a continuous matrix is difficult for several reasons. Even for the simplest situation where all particles are spheres of the same size, the randomness of the spatial distribution of the particles is not easy to include in an exact calculation. Further difficulties arise when other realistic effects, such as clustering of particles, irregular particle shape, and nonuniform particle sizes, occur. It is the purpose of this communication to show that for the technologically important metal-matrix composites (A1/SiCp), sufficiently accurate results can be obtained over the useful composition range by considering third-order bounds on the bulk and shear moduli, rl-4] Typically, these bounds differ by only a few percent. Because they contain information on the phase geometry, they are nearly an order of magnitude better than the well-known Hashin-Shtrikman bounds, tSJ which depend only on volume fraction. Torquato and co-workers [~-4] have evaluated the thirdorder bounds and have tabulated results for the threepoint parameters characterizing the particle distribution, which are necessary for the calculation of the moduli. Fortunately, they simplified the procedure so that the bounds are easy to compute. In the present work, it is shown that by averaging the upper and lower bounds for the bulk modulus K, as well as for the shear modulus G, one can obtain an estimate of Young's modulus E that is in error by less than 4 pct for volume fractions up to 50 pct. Such accuracy is well within the current experimental variability. For completeness, the mathematical expressions for the bounds on K and G are given below in terms of the moduli of the matrix (phase 1) and the particles (phase 2) and their volume fractions, ~b~ and ~b2: (1/K) -
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