An efficient size-dependent shear deformable shell model and molecular dynamics simulation for axial instability analysis of silicon nanoshells

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Highlights

  • Development an efficient size-dependent shear deformable shell model including surface effects.

  • Molecular dynamics simulation of the nonlinear instability of silicon cylindrical nanoshells.

  • Comparison of the results of the classical and non-classical shell models with those of MD simulation.

Abstract

Understanding the size-dependent behavior of structures at nanoscale is essential in order to have an effective design of nanosystems. In the current investigation, the surface elasticity theory is extended to study the nonlinear buckling and postbuckling response of axially loaded silicon cylindrical naoshells. Thereby, an efficient size-dependent shear deformable shell model is developed including the size effect of surface free energy. A boundary layer theory of shell buckling in conjunction with a perturbation-based solution methodology is employed to predict the size dependency in the buckling loads and postbuckling behavior of silicon nanoshells having various thicknesses. After that, on the basis of the Tersoff empirical potential, a molecular dynamics (MD) simulation is performed for a silicon cylindrical nanoshell with thickness of four times of silicon lattice constant, the critical buckling load and critical shortening of which are extracted and compared with those of the developed non-classical shell model. It is demonstrated that by taking the effects of surface free energy into account, a very good agreement is achieved between the results of the developed size-dependent continuum shell model and those of MD simulation.

Graphical abstract

Introduction

Due to continuous evolution of nano electro mechanical systems (NEMS), structures at the scale of microns and sub-microns are viewed as the chief components which cause to promote greatly the advancement of several nanosystems and nanodevices [1], [2], [3], [4], [5], [6]. According to experimental studies, it has demonstrably indicated that the mechanical response of nanostructures is affected by some size effects. Because the classical continuum theory is a scale independent theory, it has not the capability to consider this inherent size-dependent behavior of structures at nanoscale. In this respect, some unconventional continuum theories have been developed to accommodate the size-dependency observed in nanoscaled structures [7], [8], [9], [10], [11], [12], [13], [14].

Because of different local environmental positions, atoms of a free surface experience different surrounding conditions compared to those of bulk of a material which leads to special mechanical properties instinct from the bulk. Regarding nanostructures in which the ratio of surface area to the bulk volume is considerable, the surface free energy plays an important role in the mechanical characteristics. There are so many computational and experimental researches in which the influence of surface stress effects on behavior of the structures at small scale has been studied. Radi [15] described the material behavior with the indeterminate theory of couple surface elasticity to investigate the problem of a stationary semi-infinite crack in an elastic solid. Zhang et al. [16] investigated the effect of surface/interface stress on the macroscopic plastic behaviors of nanocomposites using a new second-order moment nonlinear micromechanics theory.

Gurtin et al. [17] presented a model for vibrational analysis of microbeams including surface elasticity influences represented by applying distributed traction and compressive axial force. Then for an isotropic surface, Gurtin and Murdoch [18], [19] proposed a continuum model of surface elasticity namely as Gurtin-Murdoch elasticity theory in which the Laplace-Young equation extended to solid materials. The model has the capability to incorporate the surface stress effects into the mechanical response of nanostrustures which has been applied in many studies conducted for various problems about mechanical behavior of the structures at nanoscale. For example, Kiani [20] studied the surface effects on the free transverse vibrations and dynamic instability of current-carrying nanowires subjected to longitudinal magnetic field. Sahmani et al. [21] investigated the free vibration response of third-order shear deformable nanobeams made of functionally graded materials around the postbuckling domain incorporating the effects of surface free energy. Also, Sahmani et al. [22] predicted the nonlinear postbuckling characteristics of circular nanoplates in the presence of surface energy effects. Kasirajan et al. [23] presented a nonlocal nonlinear finite element formulation for Timoshenko beam theory including surface stress effects. Kiani [24] investigated thermo-elasto-dynamic behavior of axially functionally graded non-uniform nanobeams in the presence of surface free energy effects. Wang et al. [25] proposed a novel two-dimensional theory of piezoelectric nanoplates and boundary conditions including surface and nonlocal effects. Miandoab et al. [26] performed a comprehensive analysis on surface free energy effect on the dynamic behavior of electrostatically actuated nano-resonator. Sahmani et al. [27], [28], [29] examined the surface free energy effects on the nonlinear instability of nanoshells under axial compressive load, or hydrostatic pressure, or combination of them.

One of the prevalent ways to estimate the efficiency of non-classical continuum models is to compare their results with those reported by molecular dynamics (MD) simulations. In recent years, several studies have been carried out for this purpose. For instance, Wang et al. [30] performed MD simulations for buckling response of GaN nanowires with different orientations under uniaxial compressive load. Jing et al. [31] carried out MD simulations on the buckling behavior of silicon nanowires subjected to axial compression. Khademolhosseini et al. [32] predicted the torsional buckling behavior of single-walled carbon nanotubes via nonlocal shell model and MD simulations to extract the proper value of nonlocal parameter. Ansari et al. [33] analyzed the free vibration response of single-layered graphene sheets using MD simulations and non-classical size-dependent plate model. Shen et al. [34] employed a nonlocal higher-order shear deformable plate model to explore the nonlinear vibrations of bilayer graphene sheets, the result of which are compared with those of MD simulations. Sahmani and Fattahi [35], [36], [37] utilized MD simulations in conjunction with nonlocal plate model for nonlinear instability analysis of zirconia and 3D metallic carbon nanosheets. Mehralian et al. [38] calibrated a developed nonlocal strain gradient shell model via comparison of the obtained critical buckling loads with those of MD simulations for carbon nanotubes.

In the current investigation, an efficient size-dependent shear deformable shell model based on surface elasticity is compared with MD simulation associated with the buckling and postbuckling response of axially loaded silicon nanoshell. To this end, the surface elasticity theory of Gurtin-Murdoch is utilized within the framework of the first-order shear deformation shell theory. Subsequently, the boundary layer theory of shell buckling together with an improved perturbation solution methodology is put to use to solve the problem. At the end, MD simulation is performed to predict the critical buckling load and critical shortening of a silicon cylindrical nanoshell, the results of which are compared with those of the classical and non-classical shell models.

Section snippets

Development of an efficient size-dependent shell model

As depicted in Fig. 1, a cylindrical nanoshell with the length L, thickness h, and mid-surface radius R is considered. For the sake of surface free energy effects, the nanoshell is decomposed into three parts including a bulk part and two additional thin surface layers (inner and outer layers). Corresponding to the bulk part, the material properties are Young’s modulus E and Poisson’s ratio ν. The inner and outer surface layers neglecting their thickness are assumed to have surface elasticity

Boundary layer-type governing equations

In order to carry out the analysis more conveniently, the following dimensionless quantities are definedX=πxL,Y=yR,β=LπR,η=L2π2h2,ε=π2RhL2{a11*,a12*,a55*,d11*,d12*,d55*,e11*}={A11*A110,A12*A110,A55*A110,D11*A110h2,D12*A110h2,D55*A110h2,E11*A110h2}W=ϵhw,F=ϵ2fA110h2,{Ψx,Ψy}=ϵ2Lπh{ψx,ψy},τ¯=τsA110Px=σxxR2A110,δx=ΔxR2Lhin which A110=(λ+2μ)h. Moreover, the derivative operators are defined in Appendix A.

As a result, the dimensionless governing differential equations associated with the size-dependent

Molecular dynamics simulation

In the current study, one of the most powerful and efficient tools for analysis of systems consisting of large number of atoms is employed based on MD simulations incorporating empirical potential performed using LAMMPS (Large scale Atomic/Molecular Massively Parallel Simulator) [45]. The simulation are performed for a silicon cylindrical nanoshells with R/h = 12 and L/R = 2. In addition, due to the limitation in number of atoms for the simulated structure, the nanoshell thickness is equal to the

Numerical results and discussion

In this section, the nonlinear axial buckling and postbuckling behavior of silicon nanoshells predicted by the classical and non-classical shell models are presented. Afterwards, the critical buckling load and critical shortening obtained by MD simulation are compared with those of the classical and non-classical shell models. The material properties of a nanoshell made of silicon are given in Table 1. Also, in all of the preceding numerical results, the values of length and radius of

Conclusion

According to the authors' knowledge, a comparison between the results of a size-dependent shear deformable shell model based on the Gurtin-Murdoch elasticity theory and those of MD simulation relevant to the buckling and postbuckling response of axially loaded silicon nanoshell was conducted for the first time in the literature. An efficient non-classical shell model through implementation of the well-known Gurtin-Murdoch elasticity theory was developed having an excellent capability to take

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