First-Principles Calculation of Third-Order Elastic Constants via Numerical Differentiation of the Second Piola-Kirchhoff Stress Tensor

Abstract

A general method is presented to calculate from first principles the full set of third-order elastic constants of a material of arbitrary symmetry. The method here illustrated relies on a plane-wave density functional theory scheme to calculate the Cauchy stress, and numerical differentiation of the second Piola-Kirchhoff stress tensor to evaluate the elastic constants. It is shown that finite difference formulas lead to a cancellation of the finite basis set errors, whereas simple solutions are proposed to eliminate numerical errors arising from the use of Fourier interpolation techniques. Applications to diamond, silicon, aluminum, magnesium, graphene, and a graphane conformer give results in excellent agreement with both experiments and previous calculations based on fitting energy density curves, demonstrating both accuracy and generality of our new methodology to investigate nonlinear elastic behaviors of materials.

The generalized quasiharmonic approximation via space group irreducible derivatives

“…The strain measure parameterizes the non-rotational component of the deformation of the lattice vectors, and there are an infinite number of strain measures [50,51]. The Lagrangian strain measure is commonly used in the context of nonlinear elastic constants [20][21][22][23][24], and it is appealing given that the conjugate stress (i.e. the second Piola-Kirchoff stress) is inherently symmetric and it is straightforward to change reference frames [52].…”

“…The advantage of the Taylor series approach is that the elastic energy is numerically exact up to some order in strain, so long as the derivatives are faithfully computed. While nonlinear elastic constants have been computed from first-principles [20][21][22][23][24] and have been invoked in the early QHA literature [3,25], we are not aware of their use in modern QHA calculations.…”

Nonlinear elasticity of prestressed single crystals at high pressure and various elastic moduli

“…Examples of elastic moduli are given up to third-order in [53][54][55] and fourth-order in [56,57] at small strains (e.g., 0.02-0.03) and up to fifth-order for finite strains (0.37 for Si I) in [58]. Knowledge of the above elastic constants allows one to determine elastic moduli C 0 (E 0 ) according to Eq.…”

Ab initio calculations of third-order elastic coefficients

“…This study demonstrates how to evaluate strain effects on SOEC based on TOE theory based on ab initio calculations. The development of ab initio-based methods to compute TOEC is an active research area [e.g., [15][16][17][18][19]. We adopt the popular approach which expands the strain energy vs. the Lagrangian strain [e.g., 15,[17][18][19].…”