Elastic properties and constitutive behaviour of graphene at finite temperature and large deformation

Abstract

In the earlier reported theoretical studies, the bending modulus of the graphene sheet predicted with the inclusion of the dihedral energy term is observed to be almost double as compared to those reported without considering the dihedral energy term. The bending modulus reported considering dihedral energy term is observed to be more accurate when compared with the quantum mechanical calculations. The findings of these studies for bending modulus (with/without considering dihedral energy terms) are reported without considering the effect of finite temperature. In the present work, a multiscale computational framework incorporating the dihedral energy term and finite temperature in the constitutive model is presented for the first time. A temperature-dependent bond length is expressed in terms of strain and curvature tensor through the temperature-related quadratic-type Cauchy-Born rule. The total Helmholtz free energy of the unit cell is expressed as a sum of the potential energy of the unit cell and thermal energy due to the thermal vibrations of atoms arising from the finite temperature. The atomic interactions are modelled through the Tersoff-Brenner potential considering different sets of empirical parameters, including the authors’ recently proposed set of empirical parameters. The stress/moment tensors and the tangent constitutive matrix are obtained from the derivatives of the Helmholtz free energy density with strains and curvatures. The thermal properties, elastic properties, and tangent stiffness coefficients for armchair and zigzag graphene with/without dihedral energy term at finite strain, curvature, and temperature are investigated and discussed in detail. The proposed multiscale model facilitates the accurate prediction of the constitutive behaviour of the graphene in a thermal environment with remarkable improvements for the prediction of bending modulus due to the inclusion of the dihedral energy term at finite temperature.