Man （Nondestr Test Eval 15：191-214, 1999） derived the constitutive relation of a weakly-textured orthorhombic aggregate of cubic crystallites with effects of microstructure and initial stress. In this paper, a computational expression on the integration ∫SO（3） Q^× D^1m0dg is given. Then, by means of the computational expression, the general constitutive relation of a weakly-textured anisotropic polycrystal with the consideration of microstructure and initial stress is derived. As special cases of our general constitutive relation, two constitutive relations are given for an isotropic polycrystal and a weakly-textured anisotropic aggregate of cubic crystallites. The acoustoelastic tensor of the reference cubic crystal is derived to determine the material constants of the polycrystal. Two examples are given for understanding the physical meaning of the texture coefficients and the constitutive relations.
By the nonlinear optimization theory, we predict the yield function of single BCC crystals in Hill＇s criterion form. Then we give a formula on the macroscopic yield function of a BCC polycrystal Ω under Sachs＇ model, where the volume average of the yield functions of all BCC crystallites in Ω is taken as the macroscopic yield function of the BCC polycrystal. In constructing the formula, we try to find the relationship among the macroscopic yield function, the orientation distribution function （ODF）, and the single BCC crystal＇s plasticity. An expression for the yield stress of a uniaxial tensile problem is derived under Taylor＇s model in order to compare the expression with that of the macroscopic yield function.
多晶体中的晶粒取向分布可通过取向分布函数(orientation distribution function,ODF)表示.取向分布函数(ODF)可在Wigner D-函数基下展开,其展开系数称为织构系数.利用Clebsch-Gordan表达式推导出立方晶粒各向异性集合多晶体的弹性张量显表达式,该弹性张量表达式包含3个材料常数和9个织构系数.为了织构系数的超声波测定,给出了这9个织构系数与超声波速之间的关系式,并通过一个算例来验证这个关系式.
Some physical properties of crystals differ in direction n because crystal lattices are often anisotropic. A polycrystal is an aggregate of numerous tiny crystallites. Unless the polycrystal is an isotropic aggregate of crystallites, the physical properties of the polycrystal vary with n. The direction-dependent functions （DDF） for crystals and polycrystals are introduced to describe the variations of the physical properties in direction n. Until now there are few papers dealing systematically with relations between the DDF and the crystalline orientation distribution. Herein we give general expressions of the DDF for crystals and polycrystals. We discuss the applications of the DDF in describing the physical properties of crystals and polycrystals.
An orthorhombic polycrystal is an orthorhombic aggregate of tiny crystallites. In this paper, we study the effect of the crystalline mean shape on the constitutive relation of the orthorhombic polycrystal. The crystalline mean shape and the crystalline orientation arrangement are described by the crystalline shape function （CSF） and the orientation distribution function （ODF）, respectively. The CSF and the ODF are expanded as an infinite series in terms of the Wigner D-functions. The expanded coefficients of the CSF and the ODF are called the shape coefficients s^lm0 and the texture coefficients c^lmn respectively. Assuming that Ceff in the constitutive relation depends on the shape coefficients s^lm0 and the texture coefficients c^lmn by the principle of material frame-indifference we derive an analytical expression for C^eff up to terms linear in s^lmo and c^lmn and the expression would be applicable to the polycrystal whose texture is weak and whose crystalline mean shape has weak anisotropy. C^cff contains six unspecified material constants （λ, μ, c, s1, s2, s3）, five shape coefficients （s^2 00, s^2 20, s^4 00, s^4 20, s^4 40）, and three texture coefficients （c^4 99,c^4 20, c^4 40）, The results based on the perturbation approach are used to determine the five material constants approximately. We also find that the shape coefficients 2 and a s^2mo and s^4m0 are all zero if the crystalline mean shape is a cuboid. Some examples are given to compare our computational results.
