Estimating Elastic Constants without invoking Cauchy-Born rule: A Displacement Field-Based Approach

The Cauchy-Born rule, which effectively imposes an affine deformation within the domain of a deformed crystalline structure when subjected to affine boundary displacements, is commonly invoked to estimate the structure's elastic constants under relevant symmetry considerations. An intriguing scenario arises, among others, in hexagonal crystals where boundary displacement along the crystallographic axis a (parallel to the x direction) results in non-zero shear stresses in the xy-plane, suggesting non-affine internal deformation. To resolve this, we present a novel methodology for determining the elastic constants of crystalline materials within the first-principles based density functional theory framework. For a range of boundary displacements chosen according to the crystal's symmetry, we first estimate the displacement field measured at discrete atomic positions, which is then used to derive the strain fields within the crystal. Elastic constants are subsequently obtained by minimizing the difference between the measured stresses and the stress response corresponding to the average strains over the range of deformed configurations. The methodology is applied to various structures, including dielectrics and insulators, and the results agree with available experimental data.