Invariant-theoretic approach to nonlinear hyperelastic constitutive modeling of graphene

TL;DR

This paper introduces a novel invariant-theoretic hyperelastic model for graphene that inherently incorporates symmetry, simplifies elastic constant determination, and aligns well with ab initio and phonon-based stability predictions.

Contribution

It develops a symmetry-invariant hyperelastic model for graphene that reduces complexity and improves accuracy over previous models by embedding symmetry directly into the constitutive functions.

Findings

Model accurately predicts stress under various deformations.

Elastic stability limits match phonon calculation results.

Model conforms to ab initio energy and stress data.

Abstract

We develop a hyperelastic constitutive model for graphene --- describing in-plane deformations involving both large isotropic and deviatoric strains --- based on the invariant-theoretic approach to representation of anisotropic functions. The strain energy density function $\psi$ is expressed in terms of special scalar-valued functions of the 2D logarithmic strain tensor $ \mathbf E^{(0)}$ --- called the symmetry-invariants --- that remain invariant w.r.t. the material symmetry group of graphene, $\mathcal G = \mathcal C_{6v}$. Our constitutive model conforms to a larger set of \textit{ab initio} energies/stresses while introducing fewer elastic constants than previously-proposed models. In particular, when the strain energy is expressed in terms of symmetry invariants, the material symmetry group is intrinsically incorporated within the constitutive response functions; consequently, the elastic constants (of all orders) in the formulation are \textit{a priori} independent: i.e., no two or more elastic constants are related by symmetry. This offers substantial simplification in terms of formulation as it eliminates the need for identifying the independent elastic constants, a task which can become particularly cumbersome as higher-order terms in the strain energy density function are incorporated. We validate our constitutive model by computing (1) the stress and (2) the elastic stability limit for a set of homogeneous finite deformations --- comprising uniaxial stretch/stress along the armchair, and the zigzag directions; and equi-biaxial tension. The stress values predicted by the model are in good agreement with the directly-calculated \textit{ab initio} values. The elastic stability limits predicted by acoustic tensor analysis compare well with the predictions from phonon calculations carried out independently using linear response density functional perturbation theory.