A new multiscale formulation for the electromechanical behavior of nanomaterials

We present a new multiscale, finite deformation, electromechanical formulation to capture the response of surface-dominated nanomaterials to externally applied electric fields. To do so, we develop and discretize a total energy that combines both mechanical and electrostatic terms, where the mechanical potential energy is derived from any standard interatomic atomistic potential, and where the electrostatic potential energy is derived using a Gaussian-dipole approach. By utilizing Cauchy–Born kinematics, we derive both the bulk and surface electrostatic Piola–Kirchhoff stresses that are required to evaluate the resulting electromechanical finite element equilibrium equations, where the surface Piola–Kirchhoff stress enables us to capture the non-bulk electric field-driven polarization of atoms near the surfaces of nanomaterials. Because we minimize a total energy, the present formulation has distinct advantages as compared to previous approaches, where in particular, only one governing equation is required to be solved. This is in contrast to previous approaches which require either the staggered or monolithic solution of both the mechanical and electrostatic equations, along with coupling terms that link the two domains. The present approach thus leads to a significant reduction in computational expense both in terms of fewer equations to solve and also in eliminating the need to remesh either the mechanical or electrostatic domains due to being based on a total Lagrangian formulation. Though the approach can apply to three-dimensional cases, we concentrate in this paper on the one-dimensional case. We first derive the necessary formulas, then give numerical examples to validate the proposed approach in comparison to fully atomistic electromechanical calculations.

► Analytic expression for 1D Piola-Kirchhoff electrostatic bulk and surface stress tensors.
► Single governing equation for mechanics and electrostatics.
► Finite element electromechanical governing equations.