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Monte Carlo-Based Finite Element Method for the Study of Randomly Distributed Vacancy Defects in Grap 时间:2018-10-10 22:49   来源:未知   作者:admin   点击: 次   Abstract:This paper proposed an effective stochastic finite element method for the study of randomly distributed vacancy defects in graphene sheets. The honeycomb lattice of graphene is represented by beam finite elements. The simulation results of the pristine graphene are in accordance with literatures. The randomly dispersed vacancies are propagated and performed in graphene by integrating Monte Carlo simulation (MCS) with the beam finite element model (FEM). The results present that the natural frequencies of different vibration modes decrease with the augment of the vacancy defect amount. When the vacancy defect reaches 5%, the regularity and geometrical symmetry of displacement and rotation in vibration behavior are obviously damaged. In addition, with the raise of vacancy defects, the random dispersion position of vacancy defects increases the variance in natural frequencies. The probability density distributions of natural frequencies are close to the Gaussian and Weibull distributions.
1. Introduction
  Graphene, discovered in 2004 [1], is a two-dimensional (2D) carbon nanomaterial composed of a hexagonal honeycomb lattice. Based on its superior properties, such as good flexibility and high thermal and electrical conductivity [2, 3], graphene has a huge potential in the field of aerospace, microelectronics, microelectromechanical systems, and nanocomposites [4, 5]. For applications and development in different fields, the analysis of stress states, vibration behavior, and stability characteristics attract much attention of researchers and designers.
Graphene has an atomic-scale honeycomb lattice consisting of carbon atoms and sp2 hybrid orbitals. Owing to its small size in nanometer scale, it is difficult to have precise measurement of mechanical properties in physical experiments for graphene [6–12]. In the investigation of nanomaterials, theoretical and numerical methods are powerful substitutes and helpful supplements. Among the theoretical methods, tight-binding molecular dynamics [13], density function theory [14–16], and molecular dynamics (MD) simulation [17–19] are more frequently used atomistic-based approaches. Besides, size-dependent continuum theories are the other promising methods [20–24], which consist of the nonlocal elasticity theory [20], strain gradient theory [21–23], and modified couple stress theory [24]. However, atomistic-based methods are expensive for systems with large amounts of atoms, while classical size-dependent methods face difficulties on analyzing the vacancy defects with random distributions.
  Among the material property corresponding parameters, the intrinsic strength and Young’s modulus of graphene monolayers vary [2] in a large interval. It has been found that, owing to the appearance of defects in the microstructure of nanomaterials, the evident deviation and nonnegligible fluctuation occur in numerical simulations and physical experiments [25–27]. Therefore, attempts and struggles are required for studying and simulating the defects in graphene to explain the fluctuation in simulations or experiments reasonably.
  The Monte Carlo simulation (MCS) is a reliable sampling method with wide applications in engineering and research [28–31]. With sufficiently large sampling space, a satisfactory accuracy can be achieved in the numerical simulation. The results of MCS are usually considered as an exact solution or comparison standard [32, 33]. This study utilizes the Monte Carlo-based finite element model (MC-FEM) to effectively discuss the influences of vacancy defects in the vibration behavior of graphene. MC-FEM has the advantages of convenient programming, high computational accuracy, and good convergence. In addition, MC-FEM can be used to overcome the difficulty of propagation of the unpredictable and stochastic vacancy defects in graphene. This study provides a fresh idea and perspective, which are meaningful and important to inspire researchers in nanomaterials and related fields.