More information about the implemented homogenization method may be found below or in our papers
**(1) Titrian et al., MRS proceedings (2013)**,or open-access paper
**(2) Friák et al., Materials 5 (2012) 1853.**

The polycrystalline elastic response of multi-phase aggregates
can be determined within a self-consistent solution for the effective medium [1,
2] from (i) the elastic single-crystalline constants and (ii) the volumetric fractions of individual components. The concept was
generalized by Middya and Basu [1] to single-phase aggregates with non-cubic symmetries and further extended
by Middya *et al.* [2] to multi-phase composites. The basic steps of the effective medium
approach may be found below. A macroscopic effective medium that is elastically
homogeneous and contains microscopic fluctuations may be characterized by an
effective elastic constant *C*_{ijkl}* defined by

〈σ_{ij}(r)〉 = C*_{ijkl} 〈ε_{kl}(r)〉

(1)

where σ_{ij}(**r**) and ε_{kl}(**r**) are the local stress and strain fields at point
**r**, respectively, and the angular brackets denote ensemble averages.
Assuming the aggregate is in equilibrium, the local field of elastic stiffnesses C(**r**)
can be decomposed into an arbitrary constant part C_{0} and a fluctuating part δC(r).
The resulting local strain field, ε, can then be written (in a short-hand notation) as

ε = ε^{0} + GTε^{0},

(2)

where ε_{0} and G are the strain and modified Green′s function of the medium defined by C_{0}.
The T-matrix is given by

T = δC(I - G δC)^{-1}

(3)

where I is the unit tensor. Employing the local stress-strain relation and equations (1) and (2) we obtain

C* = C^{0} + 〈T〉(I + 〈GT〉)^{-1}

(4)

The exact evaluation of 〈T〉 and 〈GT〉 is unfortunately impossible for any realistic case.
However, by neglecting inter-granular correlations,
the T-matrix can be rearranged in terms of single-grain *t* matrices (*t _{α}*) for each grain

*α*

T =* Σ _{α} t_{α}* = τ

(5)

Inserting equation (5) into equation (4) gives

C* = C^{0} + 〈τ〉 (I + 〈Gτ〉)^{-1}

(6)

For a single phase polycrystal, the self-consistent solution of eq. 6 can be obtained
by choosing a C_{0} that satisfies the condition 〈τ〉 = 0.
For a multi-phase polycrystal, a solution to equation (6) can be found by
accounting for the volume fraction v^{i} and τ^{i} of each phase *i* [2] via

〈*Σi v ^{i} τ^{i}*〉 = 0

(7)

The actual equations for specific crystallographic classes may be found in Refs. [1,2].

References:

[1] T. R. Middya and A. N. Basu, *J. Appl. Phys.* 59, 2368 (1986).

[2] T. R. Middya, M. Paul, and A. N. Basu,* J. Appl. Phys.*59, 2376 (1986).

**Further reading:**

Information about Voigt method can be found in:

• W. Voigt, Lehrbuch der Kristallphysik, Teubner, Stuttgart, 1928.

A description of the Reuss method can be found in:

• A. Reuss and Z. Angew, Math. Mech., 9:49, 1929

More information about Voigt and Reuss Method can be found in:

• P.Louis and M. Robert, Variational Method of Determining Effective Moduli of Polycrystals:
(A) Hexagonal Symmetry, (B) Trigonal Symmetry, *J. Appl, Phys. *, 36 , 2879, 1965

• W. J. Peter, Hashin Shtrikman Bounds on the Effective Elastic Moduli of Polycrystals with Orthorombic Symmetry
*J. Appl.Phys. *, 50, 6290, 1979

Information about mechanical stability of input elastic constants can be found in:

• Ting, T.C.T., Anisotropic Elasticity, New York, Oxford, Oxford University Press, 1996