Kroner

where:
[k] = k 2 − k 1 (105)
[µ] = µ 2 − µ 1 (106)
We now proceed to derive the tensor A in Eq. 70. Putting Eq. 104 into Eq. 70
and using Eq. 11 we have:
I + P : [C] ≡ B = b ′ I ′ + b ′′ I ′′ (107)
b ′ = 1 + 3[k]p 1 (108)
b ′′ = 1 + 2[µ]p 2 (109)
Note, that because of the properties of fourth-order tensors described in Section
3:
which can be written as:
A = B −1 = 1 b ′ I′ + 1
b ′′ I′′ (110)
A = a ′ I ′ + a ′′ I ′′ (111)
a ′ = 1 b ′ = k 1
k 1 + 3[k]k 1 p 1
(112)
a ′′ = 1
b ′′ = µ 1
µ 1 + 2[µ]µ 1 p 2
(113)
Comparing Eqs. 112 and 113 with Markov’s [6] Eq. (4.58), we find that:
3k 1 p 1 = α 1 (114)
2µ 1 p 2 = β 1 (115)
Note that Markov has an error in his Eq. (4.58) for the basis tensor I ′′
ijkl which
he writes as:
which should be corrected to read:
1
2 (δ ijδ kl + δ il δ jk − 2 3 δ ijδ kl ) (116)
I ′′
ijkl = 1 2 (δ ikδ jl + δ il δ jk − 2 3 δ ijδ kl ) (117)
This concludes the derivation of tensor A for isotropic matrix and isotropic
inhomogeneity.
9 Self consistent scheme for cubic polycrystals
The goal is to derive an equation (Kroner’s cubic) that relates the shear modulus
of the aggregate of cubic crystals to the elastic coefficients of a cubic crystal.