Standard D-EMT

In the usual D-EMT [28], when a particle with elastic moduli K_p and G_p is embedded in a matrix with elastic moduli K_m and G_m, the dilute limit is used to generate an approximate equation that can be solved for the effective elastic moduli [28]. In the dilute limit, the value of c, the volume fraction of particles, is small enough so that the particles do not influence each other. The effective elastic moduli, K and G, are then given by [31]:

$$K = K_{m} + K_{m} k(K_p,K_m,G_m)\: c + O(c^2),$$ (1)

$$G = G_{m} + G_{m} g(G_p,K_m,G_m)\: c + O(c^2),$$ (2)

where k and g are dimensionless coefficients. These coefficients are often called the dilute limit slopes or the intrinsic moduli [32], and are functions of both the shape of the particle and the moduli shown. The higher order terms in the c expansion come from interactions between particles, and so are negligible in the dilute limit.

For circular particles in a 2-D matrix and spherical particles in a 3-D matrix, these dilute limits are known. For circular particles in 2-D, the values of k and g are:

$$k = \frac{(K_{m} + G_{m}) (K_p - K_{m})}{K_{m} (K_p + G_{m})},$$ (3)

$$g = \frac{2 (K_{m} + G_{m}) (G_p - G_{m})}{G_{m} (K_{m}+G_p) + G_p (K_{m}+G_{m})}.$$ (4)

For spherical particles in 3-D, the values of k and g are:

$$k = \frac{(K_{m} + \frac{4}{3} G_{m}) (K_p - K_{m})}{K_{m} (K_p + \frac{4}{3} G_{m})},$$ (5)

$$g = \frac{5 (K_{m} + \frac{4}{3} G_{m}) (G_p - G_{m})}{3 G_{m} (K_{m}+\frac{8}{9} G_{m}) + 2 G_p (K_{m}+2 G_{m})}.$$ (6)

The dilute limits are now used to generate approximate differential equations suitable to estimate the elastic moduli when arbitrary amounts of the included phase are introduced into the matrix. Suppose that a non-dilute volume fraction c of particles have been placed in the matrix. The effective elastic moduli of the entire composite system are now K = K() and G = G($G=G(\phi)$), where $\phi = 1 - c$ = 1 - c is the matrix volume fraction. The resulting system of matrix plus particles is treated as a homogeneous material. Suppose then that more particles are added by removing a differential volume element, dV, from the homogeneous material, and replacing it by an equivalent volume of the inclusion phase. The new elastic moduli, K + dK and G + dG, are given in the dilute limit by

$$K + d K = K + K\: k(K_p,K,G) {dV \over V},$$ (7)

$$G + d G = G + G\: g(G_p,K,G) {dV \over V},$$ (8)

where V is the total volume and k(K_p,K,G) and g(G_p,K,G) are the same quantities as those in eqs. (1) and (2), but with the replacement K_m K and G_m G . This is the key approximation that is made in order to generate the standard two-component D-EMT. The variable dV/V is now playing the role of the dilute volume fraction c in eqs. (1) and (2). When the volume element dV is removed, only a fraction $\phi$ is actually matrix material, so that the actual change in the matrix volume fraction, d$d\phi$, is given by

$$d\phi = - \phi {dV \over V}.$$ (9)

Making this substitution, eqs. (7) and (8) reduce to the coupled set of equations

$$\frac{dK}{d\phi} = - kK / \phi,$$

$$\frac{dG}{d\phi} = - gG / \phi.$$ (10)

These equations are coupled via the k and g terms, which depend on the values of K and G for the matrix at the given value of $\phi$.

The above has been written for a single size inclusion. For a size distribution of inclusion particle diameters $\{b_j\}$, the theory is only slightly more complicated. Some composites might also have different elastic moduli for different sizes of particles. A general way of characterizing the inclusion size distribution is by specifying the diameter of each type, d_j, j = 1,M, where M is the number of different kinds of particles, and f_j is the fraction of the total inclusion volume that is taken up by the j-th kind of particle, with $\sum_j f_j = 1$. The elastic moduli of the j-th kind of particle is given by K_j and G_j.

It makes some difference, mathematically, to the final results just how (i.e., in what order) the various inclusion types are mixed into the composite [15]. One might put all the smallest particles in first, then all of the next size particle, and so on, with the largest particles the last to be put in, or vice versa. We have chosen to assume the inclusion distribution is maintained as a fixed quantity throughout the mixing process, and have ignored other procedures. This means that the complete size distribution of inclusions is used each time particles are mathematically added. In the case of concrete, one can picture putting all the sand and rocks into a large container, throughly mixing the inclusions, then scooping out the mixture into the cement paste matrix. The way this affects the D-EMT is seen in the dilute limit, or the values of k and g, which become $\langle k \rangle = \sum_j f_j k_j$ and $\langle g \rangle = \sum_j f_j g_j$. These slopes are first averaged over the inclusion particle size distribution before being used in the formula for the dilute limit. The D-EMT is then built up the same way as for a single kind of particle, but using the average slopes.