Calculation of Effective Elastic Constants for Polycrystalline Materials

摘要

The average elastic properties of a polycrystalline material do not only depend on the anisotropic elastic properties of the individual grains, but also on the distribution of crystal orientations (represented by an orientation distribution function (ODF)), on the mutual misorientation of neighbouring grains, on the size of the grains and on their shape. We use a previously developed cluster model to calculate the influence of texture, an-isotropy, orientation correlation and grain shape, with the latter being confined to a few idealized cases. The model consists of several hundred uniformly shaped and equally sized grains. The individual orientations of the grains are picked at random by using a random generator or an appropriate modification when simulating a given ODF. To incorporate the correlation of the grain orientation we use a method already reported elsewhere. The displacement that occurs within the grains as one subjects the cluster to external forces is expanded in terms of grain referenced basis functions. The expansion coefficients are determined by requiring these expansions and the associated stresses to match continuously at the grain boundaries. Once the displacement is known, one can calculate the stresses and strains everywhere and form their spatial averages. The effective elastic constants are defined as interconnecting these averages.rnIt turns out that for a given ODF (e.g. for a random ODF) the effective elastic constants may vary within the entire range defined by the familiar Voigt and Reuss value if one assumes sizably different misorientation distributions functions (MODF). Our model is also suited for studying the influence of the grain shape on the elastic constants.