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.
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.
(a) corresponds to the maximum possible overal property from a composite (arithmetic average)
E=faEa+fbEb
(b) corresponds to the minimum possible overal property from a composite
f
f
1
 a  b
E E a Eb
An improvement is to assume the coated-sphere geometry, according to Hashin and Shtrikman 2.
It's applicable to isotropic materials.
If we assume Ea<Eb, then the lower value corresponds to Eb-material is in the core and Eamateiral is the coating; this equation is also called the Maxwell-Garnett formula.
E min  E a 
fb
fa
1

Eb  E a d  E a
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.
E max  Eb 
fa
f
1
 b
E a  Eb d  Eb
where d is the dimensionality.
Another famous formula is the Bruggeman's symmetric mixing rule, which is based on the
structure of dilute spherical inclusions.
fa 
Ea  E
E E
 fb  b
0
Ea  2E
Eb  2 E
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.
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.
1.
2.
3.
4.
5.
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).
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).
Milton, G. W. The Theory of Composites (eds. Ciarlet, P. G., Iserles, A., Kohn, R. V. &
Wright, M. H.) (Cambridge University Press, 2002).
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).
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).