## - Elastic Properties of Lattice with Particle Overlay

The simplest approach for calculating the Young’s modulus of a composite is to revert to analytical models. Quite basic are the parallel and series models. In the parallel model two layers of material are loaded in the direction of the layers. Assuming that the two layers have different Young’s moduli (i.e., Ea for the aggregate layer and Em for the matrix layer), the Young’s modulus of the composite can be computed with

= +E E V E Vm m a a (5.1)

where Vm and Va refer to the matrix and aggregate volume fractions, respectively. In the series model the two layers are loaded perpendicular to the main layer direction, and we obtain

= +E V E

V E

a (5.2)

In Figure 5.1 the parallel model is a straight line connecting Ey for 0% and 100% aggregate fraction; the series model comes near the lowest curvilinear line, which is the lower Hashin bound. The upper curved line is the upper Hashin bound that is explained below. Experimental results, for example, those obtained by Wittmann, Sadouki, and Steiger (1993) on mortar, are well in between these extremes, which are considered as the absolute upper and lower bounds for the Young’s modulus of a 2-phase composite. Deviations appear when the aggregate volume fraction exceeds 50%. We return to these deviations in Section 5.2.