Particulate composites elastic behavior

Name the two bounding approaches used to describe the constitutive elastic behavior of particulate composites. What assumption needs to be made about the geometry of the second phase in these approaches ... 

As it is known [13, 14], the scale effects are often found at the study of different materials mechanical properties. The dependence of failure stress on grain size for metals (Holl-Petsch formula) [15] or of effective filling degree on filler particles size in case of polymer composites [16] are examples of such effect. The strong dependence of elasticity modulus on nanofiller particles diameter is observed for particulate-filled elastomeric nanocomposites [5], Therefore, it is necessary to elucidate the physical grounds of nano- and micromechanical behavior scale effect for polymer nanocomposites. 

M-J. Pan, Microcracking Behavior of Particulate Titanium Diboride-Silicon Carbide Composites and Its Influence on Elastic Properties, Ph.D. Thesis, The Pennsylvania State University, 1994. 

Particles can have quite a variety of geometries, but they should be of approximately the same dimension in all directions (equiaxed). For effective reinforcement, the particles should be small and evenly distributed throughout the matrix. Furthermore, the volume fraction of the two phases influences the behavior mechanical properties are enhanced with increasing particulate content. Two mathematical expressions have been formulated for the dependence of the elastic modulus on the volume fraction of the constituent phases for a two-phase composite. These rule-of-mixtures equations predict that the elastic modulus should fall between an upper bound represented by... 

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