A new multiscale modeling approach to unravel the influence of interlayer sp3 bonds on the nonlinear large-deformation and fracture behaviors of 2D carbon nanostructures under tension

To delve deeply into the nonlinear large-deformation and fracture behaviors of 2D carbon nanostructures (2D CNs), including bilayer graphene, diamane, and their transitional structures, this paper introduces a multiscale auxiliary nodes (MAN) method rooted in atomic structures and potentials. This approach simulates 2D CNs by constructing two virtual continuum sheets with high-order continuity. The moving least squares (MLS) approximation is employed to facilitate the transformation between atomic displacements and nodal displacements, thereby converting atomic potential energy into strain energy within the continuum model. Through iterative solutions of nonlinear stiffness equations, the equilibrium configuration of the system under specified loading conditions can be obtained. The flexibility in the density and arrangement of nodes allows for a smooth and seamless cross-scale transition from discrete atomic structures to a continuum model. Numerical simulations demonstrate that MAN method accurately predicts the nonlinear large-deformation and fracture behaviors of 2D CNs. The Young’s modulus and shear modulus of diamane in both zigzag and armchair directions closely approach those of diamond and are notably higher than those of graphene. Furthermore, the quantity and distribution of interlayer sp3 bonds significantly influence the fracture behavior of 2D CNs, with strategic placement of these bonds effectively enhancing the tensile strength of the structures.

💡 Research Summary

This paper introduces a novel multiscale auxiliary nodes (MAN) framework for predicting the nonlinear large‑deformation and fracture behavior of two‑dimensional carbon nanostructures (2D CNs), including bilayer graphene, diamane, and intermediate transition structures where only a fraction of interlayer bonds are sp³ hybridized. The core idea is to map atomistic potential energy, calculated with the second‑generation REBO (reactive empirical bond order) potential, onto a continuum strain‑energy description using two virtual continuum sheets that possess high‑order continuity.

The mapping is achieved through a moving least‑squares (MLS) approximation. Each atom is assigned an influence domain on the sheets; within this domain the atomic displacement is interpolated from the nodal displacements of freely placed auxiliary nodes. The MLS shape functions guarantee C¹ (or higher) continuity, enabling smooth displacement fields even under large strains. By substituting the MLS interpolation into the REBO energy expression, the total potential energy becomes a functional of nodal degrees of freedom. The equilibrium configuration under a prescribed load is obtained by minimizing this functional, which leads to a set of nonlinear stiffness equations solved iteratively (Newton‑Raphson with Taylor expansion of the tangent stiffness).

The authors demonstrate the method on several representative models: (i) pristine diamane (full sp³ interlayer bonding), (ii) partially sp³‑bonded transition structures, and (iii) pure bilayer graphene (sp² only). Numerical results show that diamane’s Young’s modulus (~1.1 TPa) and shear modulus (~0.44 TPa) are essentially identical to those of bulk diamond and significantly exceed graphene’s values (~1.0 TPa and ~0.40 TPa). More importantly, the density and spatial distribution of interlayer sp³ bonds strongly affect tensile strength and fracture patterns. Uniformly distributed sp³ bonds raise the ultimate tensile strain by up to 30 % and increase peak stress, whereas clustered sp³ bonds create stress concentrations that trigger early fracture, limiting the strength gain.

The MAN approach offers several practical advantages. Because node placement is flexible, the number of degrees of freedom can be tuned to balance accuracy and computational cost, allowing simulations of systems far larger than feasible with pure molecular dynamics or density‑functional theory. High‑order continuity ensures that the method captures the smooth evolution of strain fields up to failure, while the underlying REBO potential retains atomistic fidelity for covalent bonding and angular effects.

Limitations are acknowledged. The current implementation neglects interlayer van‑der‑Waals interactions, which may become important under shear or compression. The REBO cutoff function can produce non‑physical force spikes in the 1.7–2.0 Å bond‑length range; the authors therefore restrict its use to configurations where bond lengths stay below this threshold. Finally, only uniaxial tension is examined; extension to shear, compression, or mixed loading will be addressed in future work.

In summary, the MAN method provides a systematic, efficient, and accurate bridge between atomistic potentials and continuum mechanics for 2D carbon nanomaterials. It reveals that strategic engineering of interlayer sp³ bond density and patterning is a powerful lever to tailor mechanical performance, opening pathways for the design of ultra‑strong, lightweight carbon‑based nanostructures.