Constitutive relation of an orthorhombic polycrystal with the shape coefficients

摘要

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 sm0l and the texture coefficients cmnl respectively. Assuming that Ceff in the constitutive relation depends on the shape coefficients sm0l and the texture coefficients cmnl, by the principle of material frame-indifference we derive an analytical expression for Ceff up to terms linear in sm0l and cmnl , and the expression would be applicable to the polycrystal whose texture is weak and whose crystalline mean shape has weak anisotropy. Ceff contains six unspecified material constants (λ, μ, c, s1, s2, s3), five shape coefficients (s020, s220, s400, s420, s440), and three texture coefficients (c400, c420, c440). The results based on the perturbation approach are used to determine the five material constants approximately. We also find that the shape coefficients s2m0 and s4m0 are all zero if the crystalline mean shape is a cuboid. Some examples are given to compare our computational results.