On the statistical distribution of elastic moduli polycrystals

A method for calculating the distributions of the components of stiffness and compliance tensors of cubic polycrystalite material from parameters of single crystals is developed. Their mathematical form is derived under the assumption of normal distributions for the members of the rotation group SO(3). Finally the impact of texture on the distributions of Young’s modulus E, shear modulus G and Poisson’s ratio ν analyzed. 1. The problem The elastic properties of polycrystalite material are described by the compliance tensor Sijkl or the stiffness tensor Cijkl (i, j, k, l = 1, 2, 3) [1, 2, 3]. Because of the symmetry of strain and stress tensor, and for energetical considerations both tensors can be expressed by symmetric matrices Sij and Cij (i, j = 1, 2, ..., 6), respectively, each containing 21 independent components [1]. In the sequel we focus on the compliance tensor, the components of which are termed elastic moduli. For anisotropic crystals the symmetry properties of the various crystal systems reduce the number of independent matrix elements. As an example, there are only three independent elements for a cubic crystal system [2, 3]. ∗Paper prepared in the framework of the Sonderforschungsbereich 747 “Mikrokaltumformen Prozesse, Charakterisierung, Optimierung”, project B2 “Verteilungsbasierte Simulation”, University of Bremen, supported by Deutsche Forschungsgemeinschaft