Skip to main content
U.S. flag

An official website of the United States government

Official websites use .gov
A .gov website belongs to an official government organization in the United States.

Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.

Elastic Moduli of a Material Containing Composite Inclusions: Effective Medium Theory and Finite Element Computations



Edward J. Garboczi, J G. Berryman


Concrete is a good example of a composite material in which the inclusions (rocks and sand) are surrounded by a thin shell of altered matrix material and embedded in a normal matrix. Concrete therefore may be viewed as consisting of a matrix material containing composite inclusions. Assigning each of these phases different linear elastic moduli results in a complicated effective elastic moduli problem.A new kind of differential effective medium theory (D-EMT) is presented in this paper that is intended to address this problem. The key new idea is that each inclusion particle, surrounded by a shell of another phase, is mapped onto an effective particle of uniform elastic moduli. The resulting simpler composite, with a normal matrix, is then treated in usual D-EMT. Before use, however, the accuracy of this method must be determined, as effective medium theory of any kind is an uncertain approximation. One good way to assess the accuracy of effective medium theory is to compare to exact results for known microstructures and phase moduli. Exact results, however, only exist for certain microstructures (e.g., dilute limit of inclusions) or special choices of the moduli (e.g., equal shear moduli). Recently, a special finite element method has been developed that can compute the linear elastic moduli of an arbitrary digital image in 2-D or 3-D. If a random microstructure can be represented with enough resolution by a digital image, then its elastic moduli can be readily computed. This method is used, after proper error analysis, to provide stringent tests of the new D-EMT equations, which are found to compare favorably to numerically exact finite element simulations, in both 2-D and 3-D, with varying composite inclusion particle size distributions.
Mechanics of Materials
No. 8


composite, concrete, differential effective medium, elastic moduli, finite element, microstructure, random material


Garboczi, E. and Berryman, J. (2001), Elastic Moduli of a Material Containing Composite Inclusions: Effective Medium Theory and Finite Element Computations, Mechanics of Materials, [online], (Accessed February 26, 2024)
Created August 1, 2001, Updated June 2, 2021