On the Principle of a Geometric Mean of Even-Rank Symmetric Tensors for Textured Polycrystals

A practicable and simple averaging procedure of even-rank tensors is described, realizing an idea of Aleksandrov & Aizenberg [Dokl. Akad. Nauk SSSR, (1967), 167, 1028–1031]. It possesses the properties of a geometric mean, identically obeying the physical condition { E} = ( {E⁻¹})⁻¹. The orientation distribution function f(g) enters the calculations in the form of the well known arithmetic mean. The general case is completed by the consideration of specific (twice-symmetric) fourth-rank elastic tensors. Calculations with real and modelled orientation distributions lead to results close to those of much more complicated self-consistent schemes.