Self-Consistent Calculations - Elasticity of Multi-phase Aggregates

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 Cijkl* defined by

〈σij(r)〉 = C*ijkl 〈εkl(r)〉


where σij(r) and εkl(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 C0 and a fluctuating part δC(r). The resulting local strain field, ε, can then be written (in a short-hand notation) as

ε = ε0 + GTε0,


where ε0 and G are the strain and modified Green′s function of the medium defined by C0. The T-matrix is given by

T = δC(I - G δC)-1


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

C* = C0 + 〈T〉(I + 〈GT〉)-1


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 (tα) for each grain α

T = Σα tα = τ


Inserting equation (5) into equation (4) gives

C* = C0 + 〈τ〉 (I + 〈Gτ〉)-1


For a single phase polycrystal, the self-consistent solution of eq. 6 can be obtained by choosing a C0 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 vi and τi of each phase i [2] via

Σi vi τi〉 = 0


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

[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