A first-principles method to calculate fourth-order elastic constants of solid materials

Abstract

A first-principles method is presented to calculate elastic constants up to the fourth order of crystals with the cubic and hexagonal symmetries. The method relies on the numerical differentiation of the second Piola-Kirchhoff stress tensor and a density functional theory approach to calculate the Cauchy stress tensor for a list of deformed configurations of a reference state. The number of strained configurations required to calculate the independent elastic constants of the second, third, and fourth order is 24 and 37 for crystals with the cubic and hexagonal symmetries, respectively. Here, we present conceptual aspects of our method, we provide technical details of its implementation and use, we assess its accuracy, and we discuss several applications. In particular, this method is applied to five crystalline materials with the cubic symmetry (diamond, silicon, aluminum, silver, and gold) and two metals with the hexagonal close packing structure (beryllium and magnesium). Our results are compared to available experimental data and previous computational studies. Calculated linear and nonlinear elastic constants are also used in the context of nonlinear elasticity theory to predict values of volume and bulk modulus over an interval of pressures. These predictions are compared to results obtained from density functional theory calculations to assess the reliability of our method.