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 .

中文翻译：

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基于中心取向密度函数的多晶材料取向平均

目前的工作继续了 Lobos 等人首先开始的调查。(J. Elast. 128(1):17–60, 2017 ) 关于与分布在多晶中的单晶物理量相关的张量的取向平均值。在 Lobos 等人中。(J. Elast. 128(1):17-60, 2017 )，中心取向密度函数被考虑在四阶张量的取向平均值中，该张量具有属于单晶量的某些指数对称性。目前的工作概括了 Lobos 等人的结果。(J. Elast. 128(1):17–60, 2017 ) 通过提出与中心取向密度函数的傅立叶展开和用张量表示的一般取向密度函数的清晰联系，获得任意阶张量的取向平均值纹理系数。基于中心取向密度函数的取向平均值的闭合形式用傅立叶系数（称为纹理特征值）和中心取向密度函数的中心取向表示。给定的表示需要计算特定的各向同性张量。给出了自动生成各向同性张量基的实用算法。提供了应用程序和示例，以表明取向平均值的表示提供了低维参数化，对材料科学中的优化问题具有主要优势。通过 GitHub 存储库 https://github.com/mauricio-fernandez-l/centralODF-average 提供了 Python 3 中用于复制所有示例的简单实现。