Fifth-degree elastic potential for predictive stress-strain relations and elastic instabilities under large strain and complex loading in Si

Materials under complex loading develop large strains and often transition via an elastic instability, as observed in both simple and complex systems. Here, we present Si I under large strain in terms of Lagrangian strain by an $5^{th}$-order elastic potential found by minimizing error relative to density functional theory (DFT) results. The Cauchy stress-Lagrangian strain curves for arbitrary complex loadings are in excellent correspondence with DFT results, including elastic instability driving Si I$\rightarrow$II phase transformation (PT) and the shear instabilities. PT conditions for Si I$\rightarrow$II under action of cubic axial stresses are linear in Cauchy stresses in agreement with DFT predictions. Such elastic potential permits study of elastic instabilities and orientational dependence leading to different PTs, slip, twinning, or fracture, providing a fundamental basis for continuum simulations of crystal behavior under extreme loading.