Homogenization and Materials Design of Anisotropic Multiphase Linear Elastic Materials Using Central Model Functions

It is shown in the present work that for linear elastic multiphase materials the orientation average of a stiffness tensor can be given in a closed form if central model functions are taken into consideration for the description of the material orientation distribution. This holds for arbitrary numbers of material constituents and for arbitrary degree of anisotropy of any of the constituents. The first-order bounds of Voigt and Reuss, the geometric average of Aleksandrov and Aisenberg and, specifically, the Hashin–Shtrikman bounds are given in closed forms depending on the single crystal stiffnesses of the material constituents, on their volume fractions and, for elasticity, on a central orientation and two scalar parameters per central model function. The central orientation and the two scalar parameters, referred to as texture eigenvalues in the present work, reflect the complete influence of the corresponding central model function on the elastic properties. The set of all admissible texture eigenvalues is derived. The closed form expressions can be used, e.g., for efficient and low dimensional homogenization procedures taking the crystallographic texture into account or, e.g., for the determination of estimates for unknown single crystal behavior. It is further shown, that central crystallite orientation distributions exist which conserve the anisotropy of the single crystal behavior but invert the direction of the anisotropy. The closed form expressions documented in the present work are used for an exemplary materials design problem in order to determine favorable volume fractions and texture eigenvalues for the orientation distributions of considered constituents for given prescribed anisotropic properties.