A quantum informed continuum model for ferroelectric materials

Abstract

Quantum mechanical stress relations are studied to guide the development of a continuum scale electromechanical constitutive model for ferroelectric materials. Stresses at the quantum scale are determined through the use of the Hellmann–Feynman theorem to obtain an electrostatic stress that depends on the electric quadrupole density as opposed to polarization dependent electrostriction. The result is integrated into a continuum model using a generalized set of electronic coordinate vector order parameters contained within a Lagrangian density formulation. The new constitutive model is shown to be consistent with both quantum based stress and classical phenomenological electrostriction. This conclusion is verified through a numerical study of lead titanate where calculations of energy, stress and polarization from density functional theory (DFT) are fit to continuum stored energy and electrostatic stresses. The numerical analysis includes uncertainty quantification using Bayesian statistics to gain further insight into material parameter uncertainty when approximating DFT calculations as a reduced-order continuum model.