6. Evaluations and Bounds on Elastic Moduli of Heterogeneous Materials

This chapter is concerned with composites and polycrystals, consisting of two or more distinct phases that have known stiffnesses L _r defined in the fixed overall coordinate system of a representative volume V. Phase volume fractions C _r \( \Sigma_{{r = 1}}^n\,{c_r} = 1, \) are no longer small, hence evaluation of both overall properties and local fields must reflect interactions between individual phase volumes. Spatial distribution of the phases in V is statistically homogeneous, as described in Sect. 3.​2.​2, and perfect bonding is assumed at all interfaces. Of interest are derivations of upper and lower bounds on the overall stiffness \( {L} = {{L}^{\text{T}}} \) and compliance \( {M} = {{L}^{{ - 1}}} \) of the aggregate, and of estimates of phase volume averages of strain and stress fields, caused in the heterogeneous system by application of uniform overall strain \( {{\varepsilon }^0} \) or stress \( {{\sigma }^0} \). Those are sought in terms of known volume fractions, elastic moduli, shape and alignment of the constituent phases, Sects. 6.1 and 6.2.