Effective elastic properties of two dimensional multiplanar hexagonal nanostructures

Abstract

A generalized analytical approach is presented to derive closed-form formulae for the elastic moduli of hexagonal multiplanar nano-structures. Hexagonal nano-structural forms are common for various materials. Four dierent classes of materials (single layer) from a structural point of view are proposed to demonstrate the validity and prospective application of the developed formulae. For example, graphene, an allotrope of carbon, consists of only carbon atoms to form a honeycomb like hexagonal lattice in a single plane, while hexagonal boron nitride (hBN) consists of boron and nitrogen atoms to form the hexagonal lattice in a single plane. Unlike graphene and hBN, there are plenty of other materials with hexagonal nano-structures that have the atoms placed in multiple planes such as stanene (consists of only Sn atoms) and molybdenum disulde (consists of two dierent atoms: Mo and S). The physics based high-delity analytical model developed in this article are capable of obtaining the elastic properties in a computationally ecient manner for wide range of such materials with hexagonal nano-structures that are broadly classied in four classes from structural viewpoint. Results are provided for materials belonging to all the four classes, wherein a good agreement between the elastic moduli obtained using the proposed formulae and available scientic literature is observed.

Probing the shear modulus of two-dimensional multiplanar nanostructures and heterostructures

“…After a concise discussion of the structural classification of nanomaterials, the equivalent elastic properties of the atomic bonds are described; thereby the closed-form expressions of the shear modulus are derived. The approach for obtaining the equivalent elastic properties of atomic bonds is wellestablished in scientific literature [32,41,43,47,48]. Therefore, the main contributing of this work lies in development of the analytical formulae for shear modulus of monoplanar and multiplanar hexagonal nanostructures and nano-heterostructures.…”

“…On the basis of structural configuration, monolayer two-dimensional materials can be classified in four different classes as shown in figure 1(d-g) [32]. For example, graphene [42] consists of a single type of atom (carbon) to form a hexagonal honeycomb-like lattice structure in one single plane, while there is a different class of materials that possess hexagonal monoplanar nanostructure with different constituent atoms such as hBN [44], BCN [56] etc.…”

“…The strain energies due to pure bending moment M causing a slope of ∆φ at the end points of the beam [32] can be written as…”

“…For understanding the structural performance of the nanostructures and nano-heterostructures (intended to be utilized as nanoelectromechanical systems such as resonators or nanosensors) from a mechanical strength view-point, it is of utmost importance to evaluate their Young's moduli, Poisson's ratios and shear modulus. While closed-form analytical expressions are reported in literature for Young's moduli and Poisson's ratios of multiplanar structural forms and nanoheterostructures [31,32], there is no such efficient formulae available yet for the shear modulus of nanostructure and nano-heterostructures. Shear modulus assumes a vital role in evaluating the resonance frequency of the vibration modes involving torsion.…”

Effective mechanical properties of multilayer nano-heterostructures

“…Molecular mechanics based analytical closed form formulae are presented by many researchers for materials having hexagonal nano-structures in a single layer such as graphene, hBN, stanene, MoS 2 etc. 7 , 8 , 31 – 33 . This approach of mechanical property characterization for single-layer nanostructures is computationally very efficient, yet accurate and physically insightful.…”