A microstructure made up of a digital image is already naturally discretized and so lends itself to numerical computation of many quantities. For computing elastic moduli, there are two methods available: a finite element method [18], and a finite difference method [29]. The finite element method uses a variational formulation of the linear elastic equations, and finds the solution by minimizing the elastic energy via a fast conjugate gradient method. The finite difference method formulates the linear elastic equations directly in a finite difference approach, and solves the resulting set of linear equations with a similar conjugate gradient method.
For a porous material, with one solid phase and one pore phase, either method can be used, as the zero normal force boundary condition at a solid-pore boundary is easy to handle in either method. When there are solid-solid boundaries between two different phases, the boundary conditions become continuity of displacement and continuity of normal force. This is harder to implement in the finite difference method, while it is just as easy to do as in the solid-pore case for the finite element method. We have used the finite element method exclusively in this paper.
The finite element method is one that has been especially adapted for digital images. It is for linear elasticity only. Each pixel, in 3-D, is taken to be a tri-linear finite element [14]. There can be any number of phases, whether isotropic or anisotropic, as long as each phase can be adequately depicted within the resolution of the digital image used, and can be described with a single elastic moduli tensor. Thermal strains can also be easily handled [17]. The digital image is assumed to have periodic boundary conditions. A strain is applied, with the average stress or the average elastic energy giving the effective elastic moduli [41,19]. Details of the theory and programs used are reported in the papers of Garboczi & Day [18] and Garboczi [17]. The actual programs are available at http://ciks.cbt.nist.gov/garboczi/, Chapter 2.