The second-order procedure: exact vs approximate results for isotropic, two-phase composites

摘要

Estimates for the effective behavior of statistically isotropic, two-phase, nonlinear elastic (or viscous) composites were given by Ponte Castafieda (l996; J. Mech. Phys. Solids 44, 827), making use of the second-order procedure', also introduced in that reference. The actual computation of these estimates took advantage of a simplifying hypothesis consisting of taking the average strain deviators in the phases to be proportional to the macroscopic strain deviator. In this paper, a more general analysis is presented that shows that while the 'proportional strain' hypothesis does hold for some special cases, it does not in general. In fact, the average strain deviators in the phases are found to be only co-axial with the macroscopic strain deviator. On the other hand, the proportional strain hypothesis, together with an approximate form of the relevant P tensor, is found to lead to fairly small errors, relative to the exact solution of the problem. The largest errors, which occur for large phase contrast and nonlinearity, are less than about two percent. Because of this. and given the significant simplifications achieved, the use of the proportionaI strain hypothesis and of the approximate form of the P tensor are recommended. Certainly, these approximations are much more accurate than the isotropic-type approximations that have been used in the past in the context of other homogenization procedures.