Elastic Properties and Nonlinear Elasticity of the Noncarbon Hexagonal Lattice Nanomaterials Based on the Multiscale Modeling

Abstract

This study presents the elastic properties and nonlinear elasticity of the two-dimensional noncarbon nanomaterials of hexagonal lattice structures having molecular structure XY. Four nitride-based and two phosphide-based two-dimensional nanomaterials, having graphene-like hexagonal lattice structure, are considered in the present study. The four empirical parameters associated with the attractive and repulsive terms of the Tersoff-Brenner potential are calibrated for noncarbon nanomaterials and tested for elastic properties, nonlinear constitutive behaviour, bending modulus, bending and torsional energy. The mathematical identities for the tangent constitutive matrix in terms of interatomic potential function are derived through an atomistic-continuum coupled multiscale framework of the extended version of Cauchy-Born rule. The results obtained using newly calibrated empirical parameters for cohesive energy, bond length, elastic properties, and bending rigidity are compared with those reported in the literature through experimental investigations and quantum mechanical calculations. The continuum approximation is attained through the finite element method. Multiscale evaluations for elastic properties and nonlinear stretching of the nanosheets under in-plane loads are also compared with those obtained from atomistic simulations.

Numerical Evaluation of the Elastic Moduli of AlN and GaN Nanosheets

“…As can be seen in Figure 13a and Table 8, the E sx and E sy moduli obtained for AlNNSs in the MD simulation study with Tersoff potential of Le [8] were in very good agreement with those evaluated by Jafaria et al [61], who used ab initio calculations performed with the Quantum ESPRESSO (QE) package. The current E sx and E sy values were about 15% higher than those by Le [8] and Jafaria et al [61], which in turn were nearly 5% higher than the E sx and E sy values calculated by Singh et al [71], who used MS simulations employing TB potential function to describe the interactions between Al and N atoms. The lowest surface Young's moduli in the zigzag and armchair directions were reported by Luo et al [67], who used the Vienna ab initio simulation package (VASP) for ab initio DFT calculations.…”

“…GaNNSs, calculated by Equation ( 15), from the shear were nearly constant for all square AlN and GaN nano modulus values obtained for GaNNSs showed greater The average surface shear modulus of AlNNSs, G higher than that of GaNNSs, which was G sxy = 0.026 T As was established previously, among 2D materia pounds), BN nanosheets exhibited superior mechanica phene [73,81]. The surface Young's and shear moduli o E sy = 0.324 TPa•nm, and G sxy = 0.065 TPa•nm [71], w than the E sx,y and G sxy values calculated for AlNNS should be taken into account in the assembly of heter and in the design of nanodevices, where the mechani important.…”

“…Singh et al [71] and Luo et al [67] obtained similar surface Young's moduli values, contrary to what was observed for AlNNSs. E sx and E sy values by Luo et al [67] and Singh et al [71] were ≈ 20% higher than the respective moduli calculated by Giannopoulos et al [77], who used spring elements to model the Ga-N bond under the NCM/MSM approach, and Rouhi et al [70], who employed TB potential in their MD simulation study, with the exception of the E sy value found in this last work. The lowest E sx and E sy values were reported by Le [8], despite his method providing satisfactory results for AlNNSs.…”

“…To understand the effect of the aspect ratio on Al As was established previously, among 2D materials in the 13th group (nitride compounds), BN nanosheets exhibited superior mechanical properties, close to those of graphene [73,81]. The surface Young's and shear moduli of the BNNSs, E sx = 0.334 TPa•nm, E sy = 0.324 TPa•nm, and G sxy = 0.065 TPa•nm [71], were nearly 2 and 2.5 times greater than the E sx,y and G sxy values calculated for AlNNSs and GaNNSs, respectively. This should be taken into account in the assembly of heterostructures involving 2D nitrides and in the design of nanodevices, where the mechanical strength of the components is important.…”

“…Rouhi et al [70] performed MD simulations with the Tersoff-Brenner (TB) potential function to study the mechanical properties of GaNNSs. Singh et al [71] used the TB potential to model the interactions between Al and N atoms in their molecular statics (MS) simulation study on determining the elastic constants of AlNNSs. Le [8] investigated the tensile properties of AlN and GaN nanosheets employing MD simulation with Tersoff potentials.…”