Perturbation approach to elastic constitutive relations of polycrystals
This paper focuses on a statistical continuum theory of polycrystals and on the determination of their effective stiffness tensors. It is known that the orientation distribution function (ODF) is used to describe the probability density of a crystallite with some orientation in SO(3). Since single crystals are anisotropic and the elastictity stiffness tensor of any crystallite in a polycrystal is orientation-dependent, the effective elastic stiffness tensor of the polycrystal depends on the orientational distribution of crystallites in the polycrystal. In this paper, the ergodic hypothesis is employed (i. e. the volume averages of stress, strain, elastic stiffness tensor of a polycrystal are equivalent to their corresponding ensemble). The author derives simple expression for effective stiffness tensor with quadratic texture dependence by the perturbation method. Decomposing the stress, strain and displacement fields, and also stiffness tensor into their ensemble average parts and perturbation parts, the author gives a formula for the perturbation displacement field, in which Green's function of an orthorhombic aggregate of cubic crystallites is included, and then derives a formula for effective stiffness tensors with quadratic texture dependence. For Voigt and Reuss' models, and Man theory, the information on the relevant microstructure is restricted to the ODF of a polycrystal. The author finds that this restriction is valid if the orientations of all crystallites in a polycrystal are uncorrelated and any crystallite orientation is unrelated to its shape and size, otherwise the effect of the \(n\)-point orientation correlation function to effective stiffness tensor is included. Also, the author considers the effect of grain shapes on effective elastic stiffness tensors of polycrystals. Numerical examples relate to copper, steel and aluminum crystals.
scientific article; zbMATH DE number 2136851
scientific article