On the third-order bounds of the effective shear modulus of two-phase composites

Formulation of variational bounds for properties of inhomogeneous media constitutes one of the most fundamental parts of mechanics. The earliest work on multiphase media is the so-called Voigt’s upper bound and Reuss’ lower bound, corresponding to the simple rule of mixture or first-order bounds. The second-order bounds were formulated by Hashin and Shtrikman for macroscopically isotropic random composites. The third-order bounds of the bulk modulus were derived by Beran, which contain a pair of third-order bulk parameters. The third-order bounds of the shear modulus first derived by McCoy were improved by Milton and Phan-Thien, which further involve a pair of third-order shear parameters. In this study, by applying the stochastic variational principle of Xu (2009) the third-order bounds of the shear modulus are derived in an analytically most extensive trial function space. By further modifying Milton’s definition of shear parameters, the third-order bounds are finalized into a symmetric form, exactly like the Beran’s bounds of the bulk modulus. Since the bounds of the shear modulus play an essential role in plasticity theory of composites, the finalization of the third-order bounds also paves the way for further formulation of variational principles and bounds of nonlinear composites.