Asymptotic estimates on scatter ranges for elastic properties of completely random polycrystals

Abstract

One observes that the shape and crystalline orientations of the anisotropic grains in a completely random polycrystal are uncorrelated, hence the polycrystal appears (we say, nearly) macroscopically homogeneous and isotropic, and has relatively (almost) definite macroscopic elastic properties. However, because of the polycrystalline irregular microgeometry, the macroscopic properties of the aggregate may be not unique (even in principle and for the large representative element limit), and the macroscopic homogeneity and isotropy hypotheses for it may be not exact (but approximate with some accuracy). With these statements we abandon the conventional strict uniqueness and exactness viewpoint and, in fact, adopt weaker but perhaps more realistic hypotheses that allow for small uncertainties. Our upper and lower bounds on elastic moduli of random polycrystals, though based on (approximate) statistical isotropy and symmetry hypotheses, can still provide asymptotic estimates on possible ranges for the properties of the aggregates with certain accuracy, provided the intervals between bounds are sufficiently small. The formal bounds are used to derive explicit estimates for the aggregates of tetragonal crystals (classes 4, 4 ̄ ,4/m ). The numerical results appear reliable, as the stated asymptotic condition is met.