First-principles calculation of higher-order elastic constants using exact deformation-gradient tensors

We propose a general and easy-to-use method of the ab initio calculation of the higher-order elastic constants, which is based on the analytical formulas for the deformation-gradient tensors as functions of the Lagrangian strain. The method allows for elimination of the truncation errors in the Taylor expansion series of the deformation gradients and is particularly useful to calculate the fourth-order elastic constants, where large strains have to be applied. It also facilitates the calculation of the Lagrangian stress, which is helpful in determination of the strain-stress relations. To demonstrate the application of our approach, we derive the analytic formulas for the deformation gradients as functions of the Lagrangian strain tensors, which are used in calculations of the third-order elastic constants in trigonal crystals and the fourth-order elastic constants in cubic crystals. Then, we perform the ab initio calculations of the fourth-order elastic constants in face-centered-cubic aluminum. We discuss the results obtained using the strain-energy and strain-stress relations and analyze the errors of the fourth-order elastic constants which would be incurred when approximating the deformation gradients by the Taylor polynomials. We show that the relatively small truncation errors in the Taylor expansion series of the deformation gradients can cause significant deviations of the fourth-order elastic constants. This effect is larger for the strain-energy method than for the strain-stress approach. We find that in both methods, the deviations are particularly significant for the $C_{1155}$, $C_{1266}$, $C_{4455}$, $C_{1255}$, and $C_{1456}$ elastic constants and are mainly caused by the truncation errors in the nondiagonal elements of the Taylor expansion series of the deformation gradients.

Figure 1

The free-energy density ${\rho }_{0}E$ as a function of the parameter $\eta$ for the types of deformations used in the strain-energy method (Table 3). Symbols correspond to the calculated ab initio data while the solid lines represent the fourth-order polynomial fits. For a clear illustration of the ab initio results, every fourth computed data point is shown.

Figure 2

The selected elements of the tensors $\mathit{t}$ as functions of the parameter $\eta$ for the types of deformations used in the strain-stress method (Table 3). Symbols correspond to the calculated ab initio data while the solid lines represent the third-order polynomial fits. For a clear illustration of the ab initio results, every fourth computed data point is shown.

Figure 3

The truncation errors, $\mathit{\delta }{\tilde{\mathit{J}}}_C-\mathit{\delta }{\tilde{\mathit{J}}}_J$, in the Taylor expansion series of the deformation gradients ${\tilde{\mathit{J}}}_C-{\tilde{\mathit{J}}}_J$, obtained for $\eta ={\eta }_{\max }$, as a function of $m$.

Figure 4

The FOECs obtained using the strain-energy method with the deformation gradients ${\tilde{\mathit{J}}}_A-{\tilde{\mathit{J}}}_K$ approximated by their $m\mathrm{th}$-degree Taylor polynomials (squares) as a function of $m$. The results obtained using the exact ${\tilde{\mathit{J}}}_A-{\tilde{\mathit{J}}}_K$ tensors are represented by the solid lines while the dashed lines illustrate a tolerance of 1% error.

Figure 5

The FOECs obtained using the strain-stress method with the deformation gradients ${\tilde{\mathit{J}}}_A-{\tilde{\mathit{J}}}_K$ approximated by their $m\mathrm{th}$-degree Taylor polynomials (squares) as a function of $m$. The results obtained using the exact ${\tilde{\mathit{J}}}_A-{\tilde{\mathit{J}}}_K$ tensors are represented by the solid lines while the dashed lines illustrate a tolerance of 1% error.