Asymptotic estimates on uncertainty of the elastic moduli of completely random trigonal polycrystals

Because of the uncertain irregular microgeometry, a random polycrystalline aggregate (the shape and crystalline orientations of the constituent anisotropic grains of which are uncorrelated) may have elastic moduli scattered over some, though small, intervals, while the conventional macroscopic homogeneity and isotropy hypotheses for it may be considered only as approximate with corresponding uncertainty. Our formal bounds are proposed to provide the estimates on those uncertainties in asymptotic sense. Explicit expressions for the aggregates of trigonal crystals (classes 3m, 32, ) are derived.