Use of two-point correlation functions in bounding the elastic properties of polycrystals

Much work has been done in this century to obtain predictions of a materials elastic properties from limited information regarding its microstructure. Attempts at approximating the elastic stiffness of polycrystals and composites has usually involved averaging or weighting the properties of the constituent phases or elements using the fraction of their occurrence in the material. More recent advances in statistical continuum theory have formulated bounds on the elastic stiffness through the application of perturbation expansions in the principles of minimum potential and complimentary energy. These formulations have required the knowledge of n-point correlation functions of lattice orientation which until recently were not available.Recent development of Orientation Imaging Microscopy (OIM) has allowed large sets of orientation measurements to be obtained with information regarding their spatial relationships. While this technology allows high-order correlation functions of lattice orientation to be constructed from the orientation data one and two-point statistics are most readily obtainable.This dissertation investigates the benefits derived from applying experimentally obtained one and two-point correlation functions to several bounding theories proposed in the mechanics literature. These calculated bounds will be compared to the macroscopic elastic response of the study material. In addition, the orientation data is examined to determine microstructural detail which can be correlated to improving the bounds on macroscopic material properties.After the statistical information was extracted and applied to the several theories, it was discovered that the contribution of the full two-point correlation functions do not significantly alter the bounds. The bounds can be significantly narrowed by considering the contribution of the 'principal value' of an integral involving a discontinuity. The resulting term uses only one-point statistics linked by a Dirac delta function. This correction narrows the bounds by approximately 75%. The measured response of the test material fell within these improved bounds. The bounds for different crystallographic textures were also calculated. The choice of processing methods and hence the resulting texture can greatly alter where the bounds lie.