Use of classical variational principles to determine bounds for the effective bulk modulus in heterogeneous media

Abstract

Abstract.Bounds are here derived for the effective bulk modulus in heterogeneous media, denoted by k*, using the two standard variational principles of elasticity. As trial functions for the stress and strain fields we use perturbation expansions that have been modified by the inclusion of a set of multiplicative constants. The first order perturbation effect is explicitly calculated and bounds for k* are derived in terms of the correlation functions (/(0)k'(i)lc'(s)) and ([fc'(r)A/(s)/|u(0)]) where n' and k' are the fluctuating parts of the shear modulus n and the bulk modulus, fc, respectively. Explicit calculations are given for two phase media when fi'(x) = 0 and when the media are symmetric in the two phases. Results are also included for the dielectric problem when the media are composed of two symmetric phases.

Third-order thermo-mechanical properties for packs of Platonic solids using statistical micromechanics

“…Note that for an isotropic material, L q = 3K q Λ h + 2G q Λ s , where K q and G q are the bulk and shear moduli of material phase q, and the hydrostatic and shear projection tensors are defined as Λ h = Using variational principles, third-order bounds of the effective bulk (K L ≤ K e ≤ K U ) and shear moduli (G L ≤ G e ≤ G U ) have been derived by Beran & Molyneux [26], McCoy [27], and were later simplified and improved upon by Milton [25], Milton & Phan-Thien [28] and Gibiansky & Torquato [51]. The bounds of the effective moduli considered in this work are given in Milton & Phan-Thien [28] as…”

“…However, the sole microstructural information included in these bounds is the volume fractions (one-point probability functions) of the respective constituents. Three-point bounds for the effective permittivity were introduced by Beran [23], and these bounds were later simplified independently by Torquato [24] and Milton [25] as a function of the volume fraction c q and a microstructural parameter ζ q of each material phase q. Beran & Molneux [26], McCoy [27] and Milton & Phan-Thien [28] derived three-point bounds for the effective shear and bulk moduli incorporating c q , ζ q and an additional microstructural parameter η q . These microstructural parameters ζ q and η q depend on three-dimensional integrals involving one-, two-and three-point probability functions.…”

On the computation of the Hashin–Shtrikman bounds for transversely isotropic two-phase linear elastic fibre-reinforced composites

“…It is worth noting at this point that improved bounds, using more general polarizations leading to higher-order statistics, can also be derived [22][23][24]. However, it is often the case that such higher-order statistical correlation information for a given material is not known or difficult to determine accurately.…”

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

“…Their proposed bounds were later generalized by Walpole [65], Milton and Kohn [66] for anisotropic media, and by Zimmerman [67] to obtain bounds on the Poisson's ratio of the composites. Further improvements were achieved by using three point bounds in the works of Beran and Molyneux [68], Milton and Phan-Thien [69], and Torquato [70]. Employing the same approach, Rosen and Hashin [71]; Gibiansky and Torquato [72] derived bounds for the thermal expansion coefficient in thermoelastic problems and Bisegna and Luciano [73,74]; Hori and Nemat-Nasser [75] obtained bounds for the effective piezoelectric moduli in piezoelectric problems.…”