Self-consistent scheme and graded disorder in polycrystal elasticity

TLDR: In this paper, a recursion formula is established which gives the relevant upper (+) and lower (-) bounds for the tensor Ceff of effective elastic moduli of the overall grade n material.

Abstract: Using elasticity as an illustration a recursion formula is established which gives the relevant upper (+) and lower (-) bounds C(+or-n), (n=2,3,4... infinity ) for the tensor Ceff of effective elastic moduli of the overall grade n material. It follows from this formula that for n to infinity both C(+n) and C(-n) converge towards the well known self-consistent tensor Csc which therefore represents a material which, in a statistical sense, is perfectly homogeneous, isotropic and disordered. All these results also have implications for other material properties and for disordered structures other than polycrystals. Since the self-consistent approach is isomorphic to the coherent potential approximation, some of the conclusions should also be relevant for other areas of solid state physics.