| The properties of a polycrystal depend on its microstructure. In the mesoscale, the polycrystal is taken as aggregates of tiny crystallites. The microstructure of the polycrystal includes grain orientations and grain boundary arrangement.; The micromechanics of polycrystals becomes interesting only because we can measure the orientations of crystallites by orientation imaging microscopy or by X-ray diffraction. The statistical information on the microstructure can be described mathematically by the n-point orientation correlation function (n-OCF). The main challenge at hand is to develop schemes by which we could predict or estimate the elastic properties of the polycrystal from the measured statistical data. The effective elastic stiffness tensor Ceff of the polycrystal gives the relationship between the volume average stress and the volume average strain of the polycrystal. In this dissertation, we develop a statistical continuum theory on the polycrystal for the determination of Ceff. We are also interested in finding upper bounds and lower bounds of C eff under a given 1-OCF (or ODF, orientation distribution function).; The formulae of Ceff under the Voigt model, the Reuss model, or Man's theory account for the effect of texture coefficients of 1-OCF only up to linear terms. Empirical experience has so far suggested that Man's formula would work well for materials such as aluminum, whose single crystal has weak anisotropy. On the other hand, it is doubtful whether the same formula, which allows the effective stiffness tensor Ceff to depend only linearly on the texture coefficients, would suffice for materials such as copper, whose single crystal is strongly anisotropic. With this problem in mind, we derive a formula for C eff with quadratic texture dependence.; For the Voigt model, the Reuss model, the self-consistent method, and Man's formula, the relevant microstructure information is restricted to the orientation distribution function of the polycrystal. If the orientations of crystallites in a polycrystal were uncorrelated and were unrelated to their shapes and sizes, the restriction would be valid; otherwise, we need to consider the influence of the n-OCF's (n = 1,2,…) to Ceff. Our work is to derive a formula for Ceff, which includes the effects of the 1-OCF and 2-OCF. Then, for a given ODF, we obtain an upper bound and a lower bound on C eff. |
