<p>For simplicity, let's start with binary composites: Component A, with Young's modulus Ea and volume fraction fa; Component B, with Young's modulus Eb and volume fraction fb.</p><p>The simplest mixing rule was developed by Wiener (e.g. 1) that calculated the effective property of parallel slabs, and the arrow indicates stress direction.</p><p>(a) corresponds to the maximum possible overal property from a composite (arithmetic average) E=faEa+fbEb</p><p>(b) corresponds to the minimum possible overal property from a composite 1 f f  a  b E Ea Eb</p><p>An improvement is to assume the coated-sphere geometry, according to Hashin and Shtrikman 2. It's applicable to isotropic materials.</p><p>If we assume Ea<Eb, then the lower value corresponds to Eb-material is in the core and Ea- mateiral is the coating; this equation is also called the Maxwell-Garnett formula. f E  E  b min a 1 f  a Eb  Ea d  Ea The upper value corresponds to Ea-material is in the core and Eb-material is the coating. The actual modulus stays between these two limiting cases.</p><p> f E  E  a max b 1 f  b Ea  Eb d  Eb where d is the dimensionality.</p><p>Another famous formula is the Bruggeman's symmetric mixing rule, which is based on the structure of dilute spherical inclusions. </p><p>Ea  E Eb  E f a   fb   0 Ea  2E Eb  2E Here 2 is related to the depolarization factor of sphere (1/3). If the inclusion shape is ellipsoid or others, we should replace 2 with appropriate values.</p><p>There are also some other mixing rules, but there're much more complex. If you are interested, please take a look at: 3-5 Also you might find other references yourself.</p><p>1. Karkkainen, K. K., Sihvola, A. H. & Nikoskinen, K. I. Effective permittivity of mixtures: numerical validation by the FDTD method. Geoscience and Remote Sensing, IEEE Transactions on 38, 1303-1308 (2000). 2. Hashin, Z. & Shtrikman, S. A Variational Approach to the Theory of the Effective Magnetic Permeability of Multiphase Materials. Journal of Applied Physics 33, 3125-3131 (1962). 3. Milton, G. W. The Theory of Composites (eds. Ciarlet, P. G., Iserles, A., Kohn, R. V. & Wright, M. H.) (Cambridge University Press, 2002). 4. Walpole, L. J. On bounds for the overall elastic moduli of inhomogeneous systems--I. Journal of the Mechanics and Physics of Solids 14, 151-162 (1966). 5. Walpole, L. J. On bounds for the overall elastic moduli of inhomogeneous systems--II. Journal of the Mechanics and Physics of Solids 14, 289-301 (1966).</p>