Tensor equation of state for copper and aluminum

Abstract

Increasing the strain rate up to about 1/ns in the conditions of ultra-short-pulse powerful laser irradiation of thin metal foils of submicron thickness reveals very strong elastic precursors, for which decoupling of pressure and stress deviators is unreasonable. A nonlinear stress–strain relationship (tensor equation of state–TEOS) is required for establishing theoretical models of such dynamic processes and unsteady shock waves. We proposed an ANN-TEOS model, which adopts an artificial neural network (ANN) trained on the data of density functional theory (DFT) calculations for the cold curve and analytical form for thermal contributions fitted to molecular dynamics (MD) data. We showed the efficiency of feed-forward ANN in approximation of the cold curve. Besides, the range of applicability of Hooke’s law for approximation of the cold curve was examined. The DFT data were used to train a cold-curve ANN and to fit elastic constants of the Hooke’s law with nonlinear corrections for copper and aluminum. Whereas the ANN is applicable for a complex approximation within a wide range of deformed states, the Hooke’s law applicability is restricted to the strain level of about 0.1. MD simulations were used to construct and fit the thermal contributions. The developed ANN-TEOS model was applied to calculate the shock adiabats for both plastic shock wave (nearly hydrostatic omnidirectional loading) and elastic shock wave (uniaxial loading). Elastic shock Hugoniots lie above the plastic ones for both metals providing an opportunity for a shock wave to split into elastic precursor and plastic wave even for conditions, in which the plastic shock wave velocity exceeds the longitudinal sound speed. A simultaneous consideration of TEOS and plasticity model is required for the prediction of splitting and two-wave structure. Our comparison of several interatomic potentials with the stress sates calculated by means of DFT showed much higher precision of the classical force field in comparison with the examined machine-learning potentials for the considered problem of severe deformation. This result elucidates one more time the known problem of the restricted range of applicability of the machine-learning potentials and the need to include the required range of states in the training dataset for these potentials.