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Elastic Constants

Computes the full second-order elastic constant tensor of a structure from stress-strain data, along with the isotropic mechanical properties derived from it.

Overview

ElasticConstantsAnalyzer applies a series of small Cartesian and shear deformations to the (optionally pre-relaxed) input cell, evaluates the stress response of each with the calculator, and fits the elastic tensor \(C_{ij}\) by least squares. Unlike CubicElasticConstantsAnalyzer, it is not restricted to cubic cells: the crystal system is detected automatically from the cell shape (triclinic through cubic), and only the independent \(C_{ij}\) components required for that symmetry are fit, e.g. 3 for cubic (\(C_{11}, C_{12}, C_{44}\)), 5 for hexagonal, 6 for tetragonal, up to 21 for triclinic.

From the fitted tensor, the analyzer also reports the standard polycrystalline-average mechanical properties: bulk and shear modulus (Voigt, Reuss, and Voigt-Reuss-Hill), Young's modulus, Poisson's ratio, Pugh's ratio, and Chen-Vickers hardness.

Theory

Elastic tensor fit

Generalized Hooke's law relates stress and strain (Voigt notation, indices 1-6) linearly:

\[ \sigma_i = \sum_{j=1}^{6} C_{ij}\,\varepsilon_j \]

For each of num_deform deformation magnitudes (up to max_deform, in percent for axial strain and degrees for shear) along every Cartesian/shear direction allowed by the cell's symmetry, the analyzer records the resulting strain \(\varepsilon\) (relative to the reference cell) and stress \(\sigma\) (with the ambient pressure of the reference cell subtracted out). Stacking these into one linear system per symmetry class and solving by least squares yields the independent \(C_{ij}\) for that Bravais lattice. Constants are reported in GPa.

Voigt, Reuss, and Hill averages

A real polycrystal is an aggregate of randomly oriented single-crystal grains, so it has no single well-defined bulk/shear modulus, only bounds. The Voigt average assumes uniform strain across grains (an upper bound):

\[ K_V = \tfrac{1}{9}\big[(C_{11}+C_{22}+C_{33}) + 2(C_{12}+C_{13}+C_{23})\big], \qquad G_V = \tfrac{1}{15}\big[(C_{11}+C_{22}+C_{33}) - (C_{12}+C_{13}+C_{23}) + 3(C_{44}+C_{55}+C_{66})\big] \]

The Reuss average assumes uniform stress instead (a lower bound), using the compliance tensor \(S = C^{-1}\):

\[ K_R = \big[(S_{11}+S_{22}+S_{33}) + 2(S_{12}+S_{13}+S_{23})\big]^{-1}, \qquad \frac{1}{G_R} = \tfrac{1}{15}\big[4(S_{11}+S_{22}+S_{33}) - 4(S_{12}+S_{13}+S_{23}) + 3(S_{44}+S_{55}+S_{66})\big] \]

The Voigt-Reuss-Hill (VRH) average, used as the "best estimate" polycrystalline modulus, is simply their arithmetic mean: \(K_{VRH} = (K_V+K_R)/2\), \(G_{VRH} = (G_V+G_R)/2\).

Derived mechanical properties

From \(K_{VRH}\) and \(G_{VRH}\):

\[ E = \frac{9 K_{VRH} G_{VRH}}{3K_{VRH}+G_{VRH}}, \qquad \nu = \frac{3K_{VRH}-2G_{VRH}}{2(3K_{VRH}+G_{VRH})}, \qquad k = \frac{G_{VRH}}{K_{VRH}} \]

\(E\) is Young's modulus (GPa) and \(\nu\) is Poisson's ratio. \(k\) is Pugh's ratio. Empirically, metals with \(k \lesssim 0.57\) tend to be ductile, and above that, brittle. Finally, the Chen-Vickers hardness estimate is

\[ H_v = 2\,(k^2 G_{VRH})^{0.585} - 3 \]

in GPa.

References