New D-EMT

The resulting effective particle is now treated as the inclusion phase in the usual D-EMT, as described above. When an inclusion particle size distribution is used, the functions k and g are averages over this size distribution, as was stated above. The differential equations can be easily solved numerically by a 4th order Runge-Kutta method [35,36].

There are a few differences, however, involving the effective inclusion volume fraction. Each particle is now of diameter a_j = b_j + h_j, where h_j is the shell thickness for the j-th kind of inclusion, so that the volume fraction of "effective inclusions" now goes to the renormalized value $c^{\prime}$, not c. The value of $c^{\prime}$ must be known in order to know where to terminate integration of the D-EMT differential equations, which start at $c^{\prime} = 0$. Possible overlaps are ignored in this calculation. The total volume of these effective particles is equal to the original volume of inclusions, plus the volume of a complete shell around each particle. The effect of overlaps, which is fairly minor, will be discussed in a later section.

The new volume fraction of effective particles can be determined simply by considering the number of particles of a certain type. If V_j is the total volume of the j-th kind of particle, and N_j is the total number of this kind of particle, then

$$N_j {\pi \over 6} ( b_j )^3 = V_j,$$ (17)

and, therefore,

$${N_j \over V} {\pi \over 6} ( b_j )^3 = {V_j \over V} = f_j \: c$$

$$n_j {\pi \over 6} ( b_j )^3 = f_j\: c,$$ (18)

where V is the total volume of the system and n_j is the number of particles of type j per unit total material volume.

The values of $f_j^{\prime}$ and $c^{\prime}$ are defined directly by writing

$$c^{\prime} = \sum_{j=1}^M n_j {\pi \over 6} a_j^3,$$ (19)

$$f_j^{\prime} = \frac{n_j a_j^3}{ \sum_{i=1}^M n_i a_i^3}.$$ (20)

Combining the above equations, we can then derive forms for $f_j^{\prime}$ and $c^{\prime}$ that involve only f_j, c, and $\alpha_j = a_j^3/b_j^3$:

$$c^{\prime} = c \sum_{j=1}^M f_j \alpha_j$$ (21)

$$f_j^{\prime} = \frac{f_j \alpha_j}{ \sum_{i=1}^M f_i \alpha_i}.$$ (22)

It should again be noted that while the value of c was for non-overlapping inclusion particles, the value of $c^{\prime}$ is for the volume occupied by each inclusion particle and its surrounding shell, where the shell layers are assumed to not overlap. In a real concrete, these shells do overlap, which can cause percolation phenomena [37]. This treatment of the shell volume fraction is another approximation of the D-EMT method. In the numerical results described in the next section chosen to check the D-EMT, the shell regions actually do not overlap, and, therefore, are consistent with the assumptions built into the theory. An overlapping shell example is also chosen to show the minor difference this factor makes.

In summary, a D-EMT calculation is performed as follows. First, all the different kinds of composite inclusions are mapped to effective particles, with new moduli and sizes. Next, the inclusion particle size distribution is used to compute $c^{\prime}$ and $f_j^{\prime}$. Finally, the differential equations in eqs. (10) are solved numerically using a 4th order Runge-Kutta method, where the slopes $\langle k \rangle$ or $\langle g \rangle$ are averaged over the effective particle size distribution $f_j^{\prime}$. Note that for spherical particles, the new diameters of the effective particles, a_j, do not come in explicitly into any of the equations for k and g, but only in the definitions of $f_j^{\prime}$ and $c^{\prime}$. For many materials, including concrete, the inclusion particle size distribution is given by a sieve analysis, where partial volume fractions f_j refer to the amount lying inside a certain diameter range. This case can be easily converted to the one considered here, by dividing each range into several points, and dividing up the volume in that range appropriately. The actual FORTRAN software used to calculate the D-EMT results in this paper is freely available [38].