Elastic behavior of particulate composites

Moving to a larger scale, let us now look at the influence of microstructure on elastic behavior. As indicated in the previous section, the elastic constants are a fundamental property of single crystals through the geometry and stiffness of the atomic bonds. Thus, one may expect elastic behavior to be controlled simply by the choice of material. By using composite materials, however, one can control the final set of elastic properties with some precision, i.e., by mixing phases with different elastic constants. Clearly, it is useful to be able to predict the elastic constants of a composite from those of its constituents. This has been accomplished for many types of composite microstructures. For this section, however, the emphasis will be on (elastically) isotropic composites, i.e., composites contain- 

Hashin and Shtrikman have determined bounds using basic elasticity energy theorems rather than stress and strain averaging and their predictions can be written in the form 

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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 ... 

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