Since the elastic properties of both 2-D and 3-D models will be computed in this paper, it is important to write out both sets of elastic properties and indicate the relationship between them. The following discussion is taken from Ref. [8]. Hooke's law for an isotropic 3-D body, which defines the elastic moduli, is:
along with cyclic permutations of x, y, and z. The Young's modulus is E
and the Poisson's ratio is 
. Simply writing Hooke's law for an isotropic 2-D body results in:
along with cyclic permutation of x and y. The subscripts on the moduli indicate the dimension in which they are defined. The relationships between the various isotropic moduli for 3-D are
and for 2-D are
Note that for K (bulk modulus) and G (shear modulus) to be non-negative, and so give a stable elastic system, the bounds on Poisson's ratio are
Torquato has written equivalent relations between the various isotropic elastic moduli for general dimensions in his recent book on the theory of composite materials [9]. We can relate 3-D to 2-D moduli if we assume that the 2-D equations have been derived from the 3-D equations via the assumption of plane strain (all out of the plane strains are zero) or plane stress (all out of the plane stresses are zero). If we make the plane strain assumption, then the 3-D equations transform to the equivalent of the 2-D equations, with the mapping
If we make the assumption of plane stress, then the same thing happens again, but with the new mapping
which, interestingly enough, makes the 2-D and the 3-D moduli numerically equal to each other but with different units.
The finite element technique (in FORTRAN 77) used to solve the elastic equations has been thoroughly described [10], and a manual covering details of theory and usage has been written [4]. Briefly, each pixel is treated as a finite element, so that the finite element mesh is just the lattice of the digital image itself. The continuum elastic equations can be solved exactly via a variational principle, where the correct solution is found by minimizing the elastic energy stored in the material, given the constraint of the applied strain. The finite element technique minimizes the digital elastic energy, so that the best solution, given the discretization of the problem, is found. The elastic displacements are found in every pixel, and the average strain and stress in each pixel is computed and averaged over the entire microstructure to give the effective elastic properties of the porous material. The memory required to run these models is about 150 bytes per pixel, in 2-D, and about 230 bytes per pixel in 3-D. The total memory required for two typical systems used, 2002 in 2-D and 1003 in 3-D, then required, respectively, 6 Mbytes and 230 Mbytes. Obviously, much larger systems can be studied in 2-D. In 3-D, somewhat larger systems can be studied, although the memory required quickly becomes in the multi-gigabyte range. Run times will vary with the computer used.
The materials we are trying to simulate are elastically isotropic random collections of overlapping, elongated crystals. In analyzing the elastic properties of the models, we want to achieve isotropic moduli, as this is what is measured. But when bars are placed only in the principal directions, then the effective elastic moduli are not isotropic. In addition, they do not even have square (2-D) or cubic (3-D) properties, because of their randomness. There are then two kinds of averaging that must be done. The first is averaging over random configurations, because of statistical fluctuation, which is described in the next section. The second is angular averaging. In the following, we describe an averaging technique that results in isotropic elastic moduli.
Consider the 3-D case first. The elastic moduli tensor obtained for one configuration of a given model cannot be assumed to follow any particular crystal pattern. In Voigt notation, a general symmetric 6 x 6 matrix, with 21 independent elements, is of the following form:
where 1=xx, 2=yy, 3=zz, 4=xz, 5=yz, and 6=xy. The tensor in 2-D, with 6 independent elements, has the form:
where 1=xx, 2=yy, and 6=xy (we use "6" instead of the more natural "3," in order to link more clearly to 3-D).
The computed elastic tensor of one of the 3-D models will, in general, have
the symmetry of eq. (8). One can analytically
average this tensor, using the
full Cijkl form of the elastic moduli
tensor, over the three Euler angles (
,
, 
) in
3-D and the single polar angle (
) in 2-D.
The 3-D Euler angle matrix is [11]:
The 3-D equation for angularly averaging the elastic moduli tensor is then [11]:
where Cmnpq is a constant, evaluated in some set of Cartesian axes. This will result in an isotropic tensor, with values of K and G given by:
Note that the ij terms where i=1,2,3 and j=4,5,6, and vice versa, do not appear in the averages. This leads one to formulate a simpler averaging process, where the angular averaging process need not be used explicitly. Define an average cubic material with cubic symmetry in 3-D, so that there are three independent elastic moduli: (C11 )avg, (C12 )avg, and (C44 )avg , defined in the following straightforward way:
Equations (14) look like the equations obtained when angularly averaging a
cubic elastic moduli tensor [12], with (C11)avg
C11 , (C44)avg
C44 , and (C12)avg
C12 , the three independent elements of a cubic elastic moduli tensor.
The relevant rotation matrix in 2-D is:
The same averaging procedure can be done in 2-D [ 11], according to
with the results
If we define (C11)avg = (1/2) (C11 + C22), then eqs. (17) become the equations for an effective square symmetry elastic moduli tensor [12]:
If the bars were oriented randomly in all directions, then the effective moduli would not need to be rotationally averaged, but just averaged over different configurations. However, for a nominally isotropic random system, a rotational average should be very similar to a configurational average. Note that since eqs. (11) and (16) are linear in Cmnpq, the configurational average and the rotational average can be taken in any order.
In summary, the "experimental" technique is as follows: build a microstructure using one of the above rules and a random number generator, assign elastic properties to the solid phase(s), and use the programs to compute the effective elastic moduli of the model. To get all 21 of the elements in eq.
(8), six runs must be made for each system
considered, with only one of the six independent strains
( 
1,
2,
3,
4,
5,
6 )
non-zero each time. Using the average stress tensor, <
ij>, one can then define the effective elastic
tensor by <s
ij> º
<Cijkl>
ij , where
ij is the applied strain. Using this technique, all 21 elements can be evaluated. The symmetry of the tensor is then a check of the numerical technique, as elasticity theory requires that the elastic moduli be symmetric for infinite systems. Using periodic boundary conditions in this case gives an effectively infinite system, as there is no surface. This general symmetry elastic moduli tensor should then be angularly averaged to produce isotropic elastic moduli. In the next section, we discuss how solid moduli were picked, and how to analyze the sources of error in this process.