Symmetries of 2-lattices and second order accuracy of the Cauchy--Born Model

TL;DR

This paper demonstrates that the Cauchy--Born model for 2-lattices can achieve second order accuracy through a novel symmetry-based approach, improving continuum modeling of complex materials.

Contribution

It introduces a new symmetry-based method to attain second order accuracy in the Cauchy--Born model for 2-lattices and constructs a compatible energy density for multi-species systems.

Findings

Cauchy--Born model is second order accurate with a novel kinematic connection.

Identifies symmetries in multi-species models to develop a new energy density.

Applicable to materials like graphene, hcp metals, and shape memory alloys.

Abstract

We show that the Cauchy--Born model of a single-species 2-lattice is second order if the atomistic and continuum kinematics are connected in a novel way. Our proof uses a generalization to 2-lattices of the point symmetry of Bravais lattices. Moreover, by identifying similar symmetries in multi-species pair interaction models, we construct a new stored energy density, using shift-gradients but not strain gradients, that is also second order accurate. These results can be used to develop highly accurate continuum models and atomistic/continuum coupling methods for materials such as graphene, hcp metals, and shape memory alloys.