A differential approach to microstructure-dependent energy bounds for multiphase heterogeneous media
Liping Liu (Rutgers)
Prager Medal Symposium in honor of George Weng: Micromechanics, Composites and Multifunctional Materials
Wed 9:00 - 10:30
MacMillan 117
In this paper we present a new method of deriving microstructure-dependent bounds on the effective properties of general heterogeneous media. We first define and calculate the higher-order polarization tensors for multiphase heterogeneous media, and next derive a differential inequality on the energy with the initial condition given by the polarization tensors. Using the comparison theorem we obtain bounds on the energy induced by the inhomogeneities. These new bounds, taking into account of the average Eshelby tensors for homogeneous problems, are much tighter than the microstructure-independent bounds such as the classic Hashin-Shtrikman bounds on one hand, and on the other hand, recover the classic bounds by minimizing or maximizing the bounds over all possible average Eshelby tensors. Also, these bounds are applicable to non-well-ordered composites and multifunctional composites. It is anticipated that this new approach will be useful for the modeling and optimal design of a variety of heterogeneous media.