# A Coupled Refined Model of Atomistic and Continuum Parameters of Diatomic Covalent Bonds

## TL;DR

This paper improves the modeling of carbon nanostructures by linking atomistic properties to continuum mechanics in a thermodynamically stable way.

## Contribution

A new model using Timoshenko beam theory to accurately capture shear and bending in C–C bonds, avoiding nonphysical Poisson’s ratios.

## Key findings

- Classical Euler–Bernoulli beam models produce nonphysical Poisson’s ratios above 0.5 due to shear deformation neglect.
- A sixth-order equation links bond diameter, Poisson’s ratio, and force constants for thermodynamic stability.
- Physically consistent macroscopic parameters can be derived from atomistic force constants within a narrow stable range.

## Abstract

This study addresses the challenge of consistently transferring atomistic parameters of the C–C bond into phenomenological material characteristics within the framework of continuum mechanics. Particular attention is given to determining the effective transverse diameter of the covalent C–C bond in carbon nanostructures. The dependence of this diameter on Poisson’s ratio ν is examined, and the influence of the interatomic stiffness constants kr,kθandkτ
is systematically analyzed. Classical representative-volume models of the C–C bond based on the Euler–Bernoulli beam hypothesis violate thermodynamic stability conditions and lead to nonphysical Poisson’s ratio values exceeding 0.5, due to the neglect of shear deformation effects. To overcome this limitation, an approach based on Timoshenko beam theory is proposed, accounting for both bending and shear deformations. This approach enables estimation of energetically equivalent states between the phenomenological representative volume and the corresponding atomistic C–C bond model. As a result, a sixth-order algebraic equation is derived linking the effective bond diameter, the Poisson’s ratio, and the molecular mechanics force constants. Analysis of this equation reveals a narrow range of effective bond diameters and Poisson’s ratios for which thermodynamic stability conditions are satisfied. Within this range, physically consistent macroscopic material parameters can be directly expressed in terms of atomistic force constants.