Kroner

2 Introduction
This is a detailed review of the derivation of Kroner’s cubic [2], which is used
to calculate the macroscopic shear modulus of an aggregate of cubic crystals.
The macroscopic modulus is assumed to apply to an isotropic ensemble of an
aggregate of cubic crystals in perfectly randomized orientations.
The first major part of this derivation is based on the treatment of equilibrium
by Gubernatis [5] and the calculation of the Green tensor function as
derived by Weinberger et al [4], all leading to an integral equation for strains
by Gubernatis [5].
The second major part is based on Markov’s [6] treatment for deriving Kroner’s
cubic from this integral equation. The third major part compares measured
shear moduli with the values predicted by Kroner’s cubic with Hashin’s bounds.
But first, a few words about fourth-order tensors.
3 Properties of fourth-order tensors
Fourth-order tensors are designated with such script (C), while second-order
tensors are designated without such script (C). Markov [6] defines the composition
of two fourth-order tensors C : B as:
(C : B) ijkl = C ijrs B srkl (1)
For a fourth-order tensor B with only minor symmetry:
the identity tensor I
B ijkl = B jikl (2)
B ijkl = B jilk (3)
has the property:
Since:
I ijkl = 1 2 (δ ikδ jl + δ il δ jk ) (4)
B : I = I : B = B (5)
B ijrs I srkl = 1 2 B ijrs(δ sk δ rl + δ sl δ rk ) = 1 2 (B ijlk + B ijkl ) = B ijkl (6)
I ijrs B srkl = 1 2 (δ irδ js + δ is δ jr )B srkl = 1 2 (B jilk + B ijkl ) = B ijkl (7)
Consider isotropic fourth-order tensors B with both major and minor symmetries:
B ijkl = B jikl = B ijlk = B klij (8)