On the Orientation Average Based on Central Orientation Density Functions for Polycrystalline Materials

The present work continues the investigation first started by Lobos et al. (J. Elast. 128(1):17–60, 2017) concerning the orientation average of tensorial quantities connected to single-crystal physical quantities distributed in polycrystals. In Lobos et al. (J. Elast. 128(1):17–60, 2017), central orientation density functions were considered in the orientation average for fourth-order tensors with certain index symmetries belonging to single-crystal quantities. The present work generalizes the results of Lobos et al. (J. Elast. 128(1):17–60, 2017) for the orientation average of tensors of arbitrary order by presenting a clear connection to the Fourier expansion of central orientation density functions and of the general orientation density function in terms of tensorial texture coefficients. The closed form of the orientation average based on a central orientation density function is represented in terms of the Fourier coefficients (referred to as texture eigenvalues) and the central orientation of the central orientation density function. The given representation requires the computation of specific isotropic tensors. A pragmatic algorithm for the automated generation of a basis of isotropic tensors is given. Applications and examples are presented to show that the representation of the orientation average offers a low-dimensional parametrization with major benefits for optimization problems in materials science. A simple implementation in Python 3 for the reproduction of all examples is offered through the GitHub repository https://​github.​com/​mauricio-fernandez-l/​centralODF-average.