First-principles calculation of higher-order elastic constants from divided differences

Abstract

A method is presented to calculate from first principles the higher-order elastic constants of a solid material. The method relies on finite strain deformations, a density functional theory approach to calculate the Cauchy stress tensor, and a recursive numerical differentiation technique homologous to the divided differences polynomial interpolation algorithm. The method is applicable as is to any material, regardless its symmetry, to calculate elastic constants of, in principle, any order. Here, we introduce conceptual framework and technical details of our method, we discuss sources of errors, we assess convergence trends, and we present selected applications. In particular, our method is used to calculate elastic constants up to the 6$^{th}$ order of two crystalline materials with the cubic symmetry, silicon and gold. To demonstrate general applicability, our method is also used to calculate the elastic constants up to the 5$^{th}$ order of $α$-quartz, a crystalline material belonging to the trigonal crystal system, and the second- and third-order elastic constants of kevlar, a material with an anisotropic bonding network. Higher order elastic constants computed with our method are validated against density functional theory calculations by comparing stress responses to large deformations derived within the continuum approximation.

Figures (11)

• Figure 1: Schematic diagram illustrating the numerical basis of our method to calculate HOECs, in the specific case of elastic constants with equal indices (equal to $1\rightarrow xx$) up to the 4$^{th}$. From top to bottom, the PK2 stress tensor is sampled at equispaced finite strains around the reference state (gray square). Then, the first-order finite difference operator in Eq. \ref{['cfd2a']} (inset, bottom right) is applied recursively (Eq. \ref{['opera']}), first to calculate elastic constants of 2$^{nd}$ order for reference and deformed states (light blue rectangles), and then to calculate derivatives of increasing order on a reducing number of deformed states, until the elastic constant of 4$^{th}$ order is calculated for only the reference state.
• Figure 2: Polynomial of degree 10 (solid black line) reproducing a stress-strain curve, $P(\mu)$, encompassing mechanical instabilities (black arrows). Colored symbols show the data points used to calculate derivatives of $P(\mu)$ at $\mu=0$ up to 7$^{th}$ order using Eqs. \ref{['opera']} and \ref{['cfd2a']}. Exact values of $P(\mu)$ at equispaced strains are shown using red circles, whereas non-exact values including a random error of up to $\pm 0.01$ (GPa) are shown using blue squares. Derivatives of $P(\mu)$ obtained by using the two sets of points are shown in the inset using the same symbols, together with exact results (black discs). $[C^{(n)}]_{log10}$ stands for $\mathop{\mathrm{sgn}}\nolimits{[C^{(n)}]} \times \log{|C^{(n)}|}$.
• Figure 3: Components of the PK2 stress tensor of the deformed configurations used to calculate the elastic constants up to the 6$^{th}$ of the (a) diamond phase of silicon and (b) fcc gold. Values obtained by using Eq. \ref{['pk2nu']} are plotted against DFT results. Normal and shear components are shown with blue and green circles, respectively. $\Delta_{ave}$ and $\Delta_{max}$ are the average and maximum absolute deviations between the two sets of values, respectively.
• Figure 4: Pressure and independent SOECs of silicon versus hydrostatic (Lagrangian) strain. DFT results (colored circles) are compared to values (black solid lines) obtained by using a nonlinear elastic constitutive framework (Eqs. \ref{['nle']} and \ref{['pk2nu']}) and higher-order elastic constants computed with the present method.
• Figure 5: Components of the Cauchy stress tensor of Si versus shear strain in the $yz$ plane. Colored circles show DFT results, whereas black lines show values derived from nonlinear elastic constitutive equations (Eqs. \ref{['nle']} and \ref{['pk2nu']}) employing elastic constants up to the 6$^{th}$ (solid) and 3$^{rd}$ order (dashed).