Differential effective medium theory and effective particle mapping

D-EMT [28,27] was chosen as the best candidate for the composite inclusion problem for the following reason. The accuracy of an EMT is often linked to how well its percolation properties match those of the experimental system being considered [19,30,29]. In D-EMT, the inclusions are always discontinuous, and the matrix is always continuous. Concrete has the same properties since the granular aggregate is fully surrounded by the cementing matrix material. Thus, the microstructures of the theory and the problem of interest are well matched. Furthermore, we want to be able to incorporate a range of sizes of particles into the theory in a controlled way. It is not clear how to do this in a self-consistent three-component model [21], but we will show that this is not difficult to achieve with the present approach.

The standard D-EMT is a two-phase theory, or rather two topological phases, since each inclusion can be a different phase through having different elastic properties. In the present case, the thin shell of disturbed material around each granular inclusion causes conceptual problems for D-EMT, since it introduces at least one more topological phase. To make use of D-EMT in this setting, the following question arises: Should this shell should be treated as part of the inclusion, or as part of the matrix?

Since the shell regions, disregarding possible overlaps between shells, will necessarily assume the same shape as the spherical inclusion particles, the option of making the shell regions part of the inclusions seems the most appropriate one. This is accomplished by mapping each spherical inclusion particle, together with its accompanying shell, into a single effective particle, with size sufficient to incorporate both and with uniform elastic properties. This idea is illustrated in Fig. 1, and will be developed more fully below. Thus, the effective medium theory that we develop will be for a material having a matrix that contains spherical composite inclusions.

Figure 1: The mapping of a composite inclusion particle into an effective particle whose diameter is that of the outer shell. The figure shows a cross-section of a sphere (or a circle) taken through the center.

In what follows in Section 2, first the standard two-phase D-EMT is described. Then the effective particle mapping is discussed, showing how the three phase model of concrete can be reduced to a two topological phase model, which is more readily handled with D-EMT. Then we show how the standard D-EMT can be modified so as to use the effective particles, producing a new D-EMT result.