Particular expressions of upper and lower estimates for the macroscopic elastic bulk modulus of random cell tetragonal polycrystalline materials are derived and computed for a number of practical crystals. The cell-shape-unspecified bounds, based on minimum energy principles and generalized polarization trial fields, appear close to the simple bounds for specific spherical cell polycrystals. | Vietnam Journal of Mechanics, VAST, Vol. 38, No. 3 (2016), pp. 181 – 192 DOI: IMPROVED ESTIMATES FOR THE EFFECTIVE ELASTIC BULK MODULUS OF RANDOM TETRAGONAL CRYSTAL AGGREGATES Vuong Thi My Hanh, Pham Duc Chinh, Vu Lam Dong∗ Institute of Mechanics, Vietnam Academy of Science and Technology, Hanoi, Vietnam ∗ E-mail: vldong@ Received April 10, 2015 Abstract. Particular expressions of upper and lower estimates for the macroscopic elastic bulk modulus of random cell tetragonal polycrystalline materials are derived and computed for a number of practical crystals. The cell-shape-unspecified bounds, based on minimum energy principles and generalized polarization trial fields, appear close to the simple bounds for specific spherical cell polycrystals. Keywords: Variational bounds, effective elastic bulk modulus, random cell polycrystal, tetragonal crystal. 1. INTRODUCTION Macroscopic (effective) elastic moduli of polycrystalline materials depend on the elastic constants of the base crystal and aggregates’ microstructure, which is often of random and irregular nature. The first simple estimations for the effective moduli are Voigt arithmetic average and Reuss harmonic average. Using minimum energy and complimentary energy principles and constant strain and stress trial fields, respectively, Hill [1] established that Voigt and Reuss averages are upper and lower bounds on the possible values of the effective elastic moduli of orientation-unpreferable polycrystalline aggregates. Assuming that the shape and crystalline orientations of the grains within a random polycrystalline aggregates are uncorrelated and using their own variational principles, Hashin and Shtrikman [2] derived the respective second order bounds for the moduli that are significantly tighter than the first order Voigt-Reuss-Hill bounds. Using HashinShtrikman-type polarization trial fields, but comming directly from classical minimum energy principles, Pham [3–6] .
