非均匀材料细观结构的定向分布函数(Ⅰ)——定向分布函数和不可约张量

Orientation Distribution Functions for Microstructures of Heterogeneous Materials(Ⅰ)- Directional Distribution Functions and Irreducible Tensors

在最近研究非均匀材料的物理和力学性质的各种基于细观力学的方法中 ,定向分布函数(ODF)和晶体定向分布函数 (CODF)的概念起着重要的作用 ,它们分别定义在单位球面和旋转群上· 本文通过两部分的内容 ,用具有不可约张量系数的傅立叶展开对它们分别作了深入的研究· 群表示理论指出平方可积的定向分布函数可以展开为球谐函数的绝对收敛的傅立叶级数 ,而其中的球谐函数又能进一步用不可约张量表示· 这样一些不可约张量系数的基本重要性在于它们刻划了材料组元和缺陷的体积、形状、相、位置的宏观或全局影响· 第 (Ⅰ )部分对定义在 N维单位球上的定向分布函数的不可约张量Fourier展开的一般性质进行了研究 ,其中重点是构造二维和三维不可约张量的简单表示 ,以便于得到它们在各种点群 (完全正交群的子群 )对称性的约束形式 ;第 (Ⅱ )部分给出了晶体定向分布函数的不可约张量展开的显式表示 ,并且给出了不可约张量以及定向分布函数和晶体定向分布函数不可约张量展开在各种点群下的约束形式·

In this two_part paper, a thorough investigation is made on Fourier expansions with irreducible tensorial coefficents for orientation distribution functions (ODFs) and crystal orientation distribution functions (CODFs), which are scalar functions defined on the unit sphere and the rotation group, respectively. Recently it has been becoming clearer and clearer that concepts of ODF and CODF play a dominant role in various micromechanically_based approaches to mechanical and physical properties of heterogeneous materials. The theory of group representations shows that a square integrable ODF can be expanded as an absolutely convergent Fourier series of spherical harmonics and these spherical harmonics can further be expressed in terms of irreducible tensors. The fundamental importance of such irreducible tensorial coefficients is that they characterize the macroscopic or overall effect of the orientation distribution of the size, shape, phase, position of the material constitutions and defects. In Part (Ⅰ), the investigation about the irreducible tensorial Fourier expansions of ODFs defined on the N_dimensional (N_D) unit sphere is carried out. Attention is particularly paid to constructing simple expressions for 2_ and 3_D irreducible tensors of any orders in accordance with the convenience of arriving at their restricted forms imposed by various point_group (the synonym of subgroup of the full orthogonal group) symmetries. In the continued work (Part Ⅱ), the explicit expression for the irreducible tensorial expansions of CODFs is established. The restricted forms of irreducible tensors and irreducible tensorial Fourier expansions of ODFs and CODFs imposed by various point_group symmetries are derived.