Fifth-degree elastic energy for predictive continuum stress-strain relations and elastic instabilities under large strain and complex loading in silicon

Materials under complex loading develop large strains and often phase transformation via an elastic instability, as observed in both simple and complex systems. Here, we represent a material (exemplified for Si I) under large Lagrangian strains within a continuum description by a 5(th)-order elastic energy 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 the elastic instability driving the Si I -> II phase transformation (PT) and the shear instabilities. PT conditions for Si I -> II under action of cubic axial stresses are linear in Cauchy stresses in agreement with DFT predictions. Such continuum elastic energy permits study of elastic instabilities and orientational dependence leading to different PTs, slip, twinning, or fracture, providing a fundamental basis for continuum physics simulations of crystal behavior under extreme loading.