This paper gives expressions for the overall average elastic constants and thermal
expansion coefficients of a polycrystal in terms of its single crystal components. The
polycrystal is assumed to be statistically homogeneous, isotropic, and perfectly
disordered. Upper and lower bounds for the averages are easily found by
assuming a uniform strain or stress. The upper bound follows from Voigt’s
assumption that the total strain is uniform within the polycrystal while the lower
bound follows from Reuss’ original assumption that the stress is uniform. A
self-consistent estimate for the averages can be found if it is assumed that the
overall response of the polycrystal is the same as the average response of each
crystallite. The derivation method is based on Eshelby’s theory of inclusions and
inhomogeneities. We define an equivalent inclusion, which gives an expression for the
strain disturbance of the inhomogeneity when external fields are applied.
The equivalent inclusion is then used to represent the crystallites. For the
self-consistent model the average response of the grains has to be the same as
the overall response of the material, or the average strain disturbance must
vanish. The result is an implicit equation for the average polycrystal elastic
constants and an explicit equation for the average thermal expansion coefficients.
For the particular case of cubic symmetry the results can be reduced to a
cubic equation for the self-consistent shear modulus. For lower symmetry
crystals it is best to calculate the self-consistent bulk and shear modulus
numerically.
