Molecular Structural Mechanics Method Research Articles

Mechanical Properties of Two-Dimensional Metal Nitrides: Numerical Simulation Study.

It is expected that two-dimensional (2D) metal nitrides (MNs) consisting of the 13th group elements of the periodic table and nitrogen, namely aluminium nitride (AlN), gallium nitride (GaN), indium nitride (InN) and thallium nitride (TlN), have enhanced physical and mechanical properties due to the honeycomb, graphene-like atomic arrangement characteristic of these compounds. The basis for the correct design and improved performance of nanodevices and complex structures based on 2D MNs from the 13th group is an understanding of the mechanical response of their components. In this context, a comparative study to determine the elastic properties of metal nitride nanosheets was carried out making use of the nanoscale continuum modelling (or molecular structural mechanics) method. The differences in the elastic properties (surface shear and Young's moduli and Poisson's ratio) found for the 2D 13th group MNs are attributed to the bond length of the respective hexagonal lattice of their diatomic nanostructure. The outcomes obtained contribute to a benchmark in the evaluation of the mechanical properties of AlN, GaN, InN and TlN monolayers using analytical and numerical approaches.

Advanced nonlinear buckling analysis of a compressed single layer graphene sheet using the molecular mechanics method

The standard molecular mechanics (MM) method with the DREIDING force field (see Mayo et al. The Journal of Physical Chemistry, 1990, 94: 8897–8909) and the molecular structural mechanics (MSM) method with Bernoulli–Euler beam elements are used to study the quasi-static nonlinear buckling and post-buckling behavior of a compressed nearly square single layer graphene sheet (SLGS) for different types of boundary conditions. The novelty of this study is the finding that well-calibrated parameter sets provide similar values of buckling forces/modes and similar post-buckling deformations for stable equilibrium configurations of SLGSs in simulations using the standard MM and MSM methods. In addition, the effect of accounting for non-bonded van der Waals (vdW) forces on the advanced post-buckling deformation modes of compressed SLGSs was studied for the first time. It is shown that the column-like post-buckling deformation modes of compressed SLGSs obtained without the inclusion of vdW interatomic forces are similar to the well-known ones for the planar Euler elastica, and the advanced post-buckling modes obtained with the inclusion of interatomic vdW forces are qualitatively different from the corresponding modes for the planar Euler elastica. In addition, the study shows the theoretical possibility of the existence of post-buckling out-of-plane equilibrium configurations whose stability is provided by attractive vdW forces.

Finite element buckling analysis of double-layered graphene nanoribbons

In this paper, the molecular structural mechanics method is applied to assess the critical buckling stress of double-layered graphene nanoribbons (DLGNRs) for different types of chirality, stacking modes and aspect ratios. In the developed atomistic model, the DLGNRs are considered as space-frame structures, in which the interlayer van der Waals (vdW) interactions and the in-plane covalent bonds are modeled as truss and beam elements, respectively. The beam sectional properties are obtained by establishing a conjunction between the structural and molecular mechanics. The Lennard-Jones potential is used to derive the elastic properties of the truss rod elements. The finite element method is employed for an eigenvalue-based buckling analysis. The results show that the critical buckling stress of the DLGNRs increases with increasing the mode shape number and decreases with increasing the aspect ratio for both the inphase and the antiphase modes for different types of chirality and stacking modes. Also, it is found that for the inphase modes, the critical buckling stress reaches its maximum value for the configuration with AA stacking mode and armchair chirality for all the aspect ratios. In the antiphase modes, the configuration with the highest stability is dependent on the aspect ratio. For the aspect ratios below 10, the configuration with AB stacking mode and armchair chirality has the highest stability; however, on the other hand, by increasing the aspect ratio above 10, the configuration with AB stacking mode and zigzag chirality represents the highest stability.

On the molecular mechanics of single layer graphene sheets

The molecular structural mechanics (MSM) method is developed by applying beam elements to model bonded interactions between carbon atoms in the atomic lattices of single-layer graphene sheets (SLGSs). The novelty of the approach developed in this paper lies in the accurate adjustment of the geometric and material parameters of Bernoulli–Euler beam elements to simulate reference mechanical moduli (2D Young’s modulus, Poisson’s ratio, and bending rigidity modulus) of graphene. The MSM method with an advanced geometric and material parameter set of Bernoulli–Euler beam elements is implemented by means of the commercial MSC.Marc finite element (FE) code. We also employ the standard molecular mechanics (MM) method using the DREIDING force field (see Mayo et al. The Journal of Physical Chemistry, 1990, 94: 8897–8909). We implemented this force field in the homemade PIONER FE code using a modified parameter set which reproduces the mechanical moduli of graphene reasonably well. Computer simulations show that the free vibration frequencies and modes of SLGSs obtained using the standard MM and MSM methods converge. However, the buckling forces of compressed SLGSs obtained by the two methods provide acceptable convergence only for the lowest values of the critical forces.

Prediction of Mechanical Properties of Coiled Carbon Nanotubes by Molecular Structural Mechanics Based Finite Element Modelling

Carbon nanotubes (NTC) have very spectacular mechanical properties related to their nanometric structure, their perfect arrangement and their one-dimensional geometry. As with all materials, structural defects are inevitable and affects NTC properties. Among these defects, we distinguish the topological defects, the dislocations and the penta-hepta defect. But the presence of these defects is not totally harmful, because the existence of some structure like the coiled nanotube is the result of these defects. For this, in the first part of this work, the coiled carbon nanotube structure is studied, a method for the designing of this structure is proposed, the geometric parameters are detailed and the structural coefficients are determined. Therefore, a procedure for moving from a graphene sheet to a coiled nanotube is developed. Then, the second part of this study represents an attempt to calculate the spring constants of the spiral carbon nanotube. Mechanical properties of this material are investigated by means of molecular structural mechanics (MSM) method in ANSYS finite element code. The model serves as a link between the computational chemistry and the solid mechanics by substituting discrete molecular structures, with an equivalent-structural model. A coiled carbon nanotube has been modeled on the nanoscale by one-dimensional elements (3D beam). The results show a considerable influence of structural parameters (diameter, chirality, pitch and defect position) on the coiled nanotube mechanical properties.

Free vibration of super-graphene carbon nanotube networks via a beam element based coarse-grained method

A new beam element based coarse-grained model is developed to investigate efficiently the mechanical behavior of a large system of super-graphene carbon nanotube (SGCNT) networks with all boundaries clamped supported. The natural frequencies and mode shapes of the SGCNT networks made of single-walled carbon nanotubes (SWCNTs) with different diameters and lengths are obtained via the proposed coarse-grained model. The applicability of the coarse-grained model for the SGCNT networks is verified by comparison with the molecular structural mechanics model. The natural frequencies and associated mode shapes obtained via the coarse-grained model agree well with the results obtained from the molecular structural mechanics method, indicating that the coarse-grained model developed in this study can be applied for the dynamic prediction of the SGCNT networks.

Linear free vibration of graphene sheets with nanopore via Aifantis theory and Ritz method

This article aims to study the natural frequency of defective graphene sheets since the existence of cut-outs in plates may be essential on the basis of their desired functionality. A combination of the Aifantis theory and Kirchhoff thin plate hypothesis is used to derive governing equations of motion. The Ritz method is employed to derive discrete equations of motion. The molecular structural mechanics method is also employed to specify the effective length scale parameter. In the ‘numerical results’ Section, the effects of different parameters such as boundary conditions and diameter of the hole-to-side length ratio on the fundamental frequency of graphene sheets are studied.

Boundary condition-selective length dependence of the flexural rigidity of microtubules

Length-dependent flexural rigidity (FR) is observed experimentally for microtubules (MTs) subjected to certain boundary conditions. To shed some light on this unique feature, we have studied the FR of MTs with different boundary conditions. A molecular structural mechanics method is employed to accurately describe the real boundary conditions imposed on MTs in experiments. Some of component protofilaments of MTs are blocked at the ends while others are free. In addition, linked kinesin is treated as an elastic body rather than a rigid body. Our simulations show that for relatively long MTs having a length comparable to those measured in experiments the length-dependent rigidity is detected only for those with fixed–free and fixed–fixed ends, which is consistent with the experimental observation. To capture the physics leading to the above phenomenon, Timoshenko beam model is adopted accounting for both transverse shear effect (TSE) and imperfect boundary effect (IBE). Comparison between TSE and IBE indicates that the boundary condition-selective length-dependence achieved for the FR of relatively long MTs is primarily a result of the influence of IBE rather than TSE.

Calculating Mechanics Characteristics of Single-Walled Carbon Nanotube Materials by Finite Element Method

In this paper, we establish the mechanical property analysis of Single-walled Carbon Nanotubes (SWCNTs) modified beam element model based on the molecular structural mechanics method. Then we study the mechanical properties of their radial direction characteristics using the finite element software Abaqus. The model simulated the different bending stiffness with rectangular section beam elements C-C chemical force field. When the graphene curled into arbitrary chirality of SWCNTs spatial structure, the adjacent beam position will change the moment of inertia of the section of the beam. Compared with the original beam element model and the calculation results, we found that the established model largely reduced the overestimate of the original model of mechanical properties on the radial direction of the SWCNTs. At the same time, compared with other methods available in the literature results and the experimental data, the results can be in good agreement.

An improved molecular structure mechanics method and its application for graphene wrinkling

This paper proposes an improved molecular structure mechanics (iMSM) method to study the wrinkling characteristics of annular graphene sheet (AGS). An idea of variable cross-section beam, combined with 3-node Timoshenko beam, is introduced in the iMSM model for considering the flexibility of C-C bond and C-C joint simultaneously, and the stiffness matrix of beam element is deduced for later calculation. The parameters of equivalent beam are obtained by using an interior point optimization approach. Based on the proposed iMSM model, the wrinkling characteristics of AGS are simulated, which are verified by molecular dynamic (MD) simulation. A predicted model is used to evaluate the wrinkling level of AGS, which agrees well with the iMSM simulation, and then the effects of beam parameters on graphene wrinkling are also analyzed. The results indicate that our proposed model is reliable to the wrinkling analysis of AGS and may be further used for other researches on graphene.

Vibration analysis of defective graphene based on the molecular structural mechanics method

A molecular structural mechanics method has been implemented to investigate the vibrational characteristics of single-layer graphene (SLG) with defects. By adopting the lumped mass unit to replace carbon atoms, and the beam element with circular cross-section to mimic C–C covalent bonds, SLG is modeled as a space framework. The simulation results show that the chirality almost has no effect on the natural frequency and the vibration mode of SLG, while boundary conditions have great influences. The influences of defects with different number and location on the natural frequencies are also studied. It is concluded that vibration mode is insensitive to the vacancy defect, small hole and short flaw, but large holes and long flaws can affect the vibration characteristics. So the graphene sheet even with small defect effects might be selected as the nanosensor material as well as pristine graphene. The conclusions in this paper may provide some references for the design of nanosensor.

A modified molecular structure mechanics method for analysis of graphene

Based on molecular mechanics and the deformation characteristics of the atomic lattice structure of graphene, a modified molecular structure mechanics method was developed to improve the original one, that is, the semi-rigid connections were used to model the bond angle variations between the C-C bonds in graphene. The simulated results show that the equivalent space frame model with semi-rigid connections for graphene proposed in this article is a simple, efficient, and accurate model to evaluate the equivalent elastic properties of graphene. Though the present computational model of the semi-rigid connected space frame is only applied to characterize the mechanical behaviors of the space lattices of graphene, it has more potential applications in the static and dynamic analyses of graphene and other nanomaterials.

Free vibration analysis of microtubules based on the molecular mechanics and continuum beam theory.

A molecular structural mechanics (MSM) method has been implemented to investigate the free vibration of microtubules (MTs). The emphasis is placed on the effects of the configuration and the imperfect boundaries of MTs. It is shown that the influence of protofilament number on the fundamental frequency is strong, while the effect of helix-start number is almost negligible. The fundamental frequency is also found to decrease as the number of the blocked filaments at boundaries decreases. Subsequently, the Euler-Bernoulli beam theory is employed to reveal the physics behind the simulation results. Fitting the Euler-Bernoulli beam into the MSM data leads to an explicit formula for the fundamental frequency of MTs with various configurations and identifies a possible correlation between the imperfect boundary conditions and the length-dependent bending stiffness of MTs reported in experiments.

Buckling of microtubules: An insight by molecular and continuum mechanics

The molecular structural mechanics method has been extended to investigate the buckling of microtubules (MTs) with various configurations. The results indicate that for relative short MTs the shear deformation effect, rather than the nonlocal effect, is mainly responsible for the limitation of their widely used Euler beam description and the observed length-dependence of their bending stiffness. In addition, the configuration effect of MTs is also studied and considered as an explanation for the large scattering of the critical buckling force and bending stiffness observed in existing experiments. This configuration effect is also found to mainly originate from the geometry of the MTs and is mainly determined by the protofilament number.

A NUMERICAL INVESTIGATION ON THE FREE VIBRATION OF CARBON NANOPEAPODS AS VARIABLE FREQUENCY BEAM RESONATORS

This study aims at investigation of the resonance frequencies of carbon nanopeapods constructed by a single wall carbon nanotube and encapsulated buckyball molecules ( C 60). A nanopeapod can be used as a nanoscale variable frequency beam resonator according to the number and positions of the encapsulated fullerenes. Using the molecular structural mechanics method the covalence bonds are simulated by equivalent beam elements and the van der Waal interactions between the buckyballs and nanotube are modeled as linear springs. Also, an equivalent beam model is proposed for the nanopeapod with sectional properties which are obtained by the molecular structural mechanics model. The beam-like modes of free vibrations are obtained using both models and the effect of position and number of buckyballs on the resonance frequencies are investigated.

Elastic Properties of Carbon Nanotubes

The geometric structure of carbon nanotubes (CNTs) can be considered to be the curling of graphene (graphite sheet) (Gao & Li, 2003; Shen, 2004). The physical parameters of carbon in graphite are widely adopted in the molecule dynamics (MD) simulations or other theoretical studies of CNTs, such as the bond energy and length of C-C, the bond angle of C-C-C, etc (Belytschko et al., 2002; Mayo et al., 1990; Xin et al., 2007, 2008). Previous researches have shown that the elastic modulus of small CNTs changes a lot when the radius varies. However, the elastic modulus of CNTs fairly approaches that of graphene, while the radius of CNTs is large enough. Therefore, it is necessary to have a clear understanding of the mechanical properties of graphene for the further realization of the properties of CNTs. Numerous researchers carried out experiments to measure the effective elastic modulus of CNTs (Krishnan et al.,1998; Poncharal et al., 1999). They reported the effective Young’s modulus of CNTs ranging from 0.1 to 1.7 TPa, decreasing as the diameter increased, and the average was about 1.0~1.2 TPa. MD simulations have provided abundant results for the understanding of the buckling behavior of CNTs. The Young’s modulus of the CNTs was predicted about 1.0~1.2 TPa through various MD methods (Jin & Yuan, 2003; Li & Chou, 2003; Lu, 1997). Hu et al. (Hu et al., 2007) proposed an improved molecular structural mechanics method for the buckling analysis of CNTs, based on Li and Chou’s model (Hwang et al., 2010) and Tersoff-Brenner potential (Brenner, 1990). Due to the different methods employed on various CNTs in these researches, the reported data scattered around an average of 1.0 TPa. The elastic properties were also discussed in the theoretical analysis by Govinjee and Sackman (Govinjee and Sackman, 1999) based on Euler beam theory, which showed the size dependency of the elastic properties at the nanoscale, which does not occur at continuum scale. Harik (Harik, 2001) further proposed three non-dimensional parameters to validate the beam assumption, and the results showed that the beam model is only proper for CNTs with small radius. Liu et al (Liu et al., 2001) reported the decrease of the elastic modulus of CNTs with increase in the tube diameter. Shell model was also used in some researches (Wang et al., 2003), to study axially compressed buckling of multi-walled CNTs. And studies by Sudak (Sudak, 2003) reported that the scale effect of CNTs should not be ignored. Wang et al (Wang et al., 2006) investigated the buckling of CNTs and the results showed that the critical buckling load drawn with the classical continuum theory is higher than that

Vibration characteristics of wrinkled single-layered graphene sheets

A pseudo beam model with 3-node beam element is proposed to simulate the modal behavior of the wrinkled single-layered graphene sheets (SLGS) based on molecular structural mechanics (MSM) method. The wrinkling characteristics are simulated using a direct perturbed-force technique originated from the continuum theory. The primary bifurcation and the secondary wrinkling behaviors are investigated so as to obtain the characteristics of the formation and the evolution of the wrinkles. The wrinkled shape, stress and effective mass are introduced into the modal model using the updated geometry technique. The effects of the wrinkles on the vibration characteristics of the SLGS are then interpreted by comparing with the unwrinkled case. The effects of the aspect ratio on the modal behavior of wrinkled SLGS are further studied. Based on the MSM simulation results and the continuum thin plate model, a prediction model is proposed to obtain the natural frequency of wrinkled SLGS. The predictions agree well with the MSM simulations. The results and observations are good references to design a wrinkled highly frequency nano-devices.

Mechanical properties of double-layered graphene sheets

In this paper, the molecular structural mechanics method is employed to calculate the mechanical properties of a double-layered carbon graphene sheet more accurately. For this purpose, covalent bonds are modeled using nonlinear beam elements and van der Waals interactions are replaced by nonlinear truss elements. Morse potential and Lennard–Jones potential equations are used to simulate the covalent bonds and van der Waals interactions, respectively. For each atom, van der Waals forces are considered with respect to all the other atoms located in its cut-off radius. In addition to in-plane mechanical properties of single and double-layered graphene sheets some out-of-plane properties like the thickness-wise stiffness and shear modulus are studied. The results indicate that Young’s modulus of a double-layered carbon graphene sheet decreases linearly with strain while Poisson’s ratio is independent from it. Also it is noted that the thickness-wise stiffness significantly increases while the distance between the two layers declines however the shear modulus is independent from shear strain.

The Prediction of Mechanical Properties of Graphene by Molecular Mechanics and Structural Mechanics Method

Based on a new molecular structural mechanics model, the effective in-plane mechanical properties of monolayer graphene sheet is analytically analyzed in this paper. The energy equivalence between the basic cell of the atomic structure of graphene and the corresponding basic cell, defined in the homogenization of periodic cellular media, of its equivalent periodic framed structure is used to determine the mechanical properties of the equivalent structural members representing the C-C bonds of graphene. The resulting relationship between the mechanical parameters of the equivalent structural members and the force constants defined in molecular mechanics are different from those used in other molecular structural mechanics models. And these mechanical parameters yield more accurate effective mechanical properties of graphene, especially the Poisson ratio, than the existing molecular structural mechanics models.

Size- and shape-dependent effective properties of single-walled super carbon nanotubes via a generalized molecular structure mechanics method

A super carbon nanotube (SCNT) is a cylinder rolled up from a super graphene (SG) layer which is built by connecting Y-type single walled carbon nanotube (SWCNT) junctions. The unique structure of the SCNT results in two determinants of radius of the SCNT, the number of junctions along the circumferential direction of the SCNT and length of the constituent SWCNTs. The mechanical behaviors of the SCNTs are investigated by using a generalized molecular structure mechanics (MSMs) method which is generated by combining the MSM method with the failure criterion of the carbon–carbon bonds. The results show that number of junctions along the circumferential direction of the SCNTs and length of the constituent SWCNTs both have obvious effect on deformation process of the SCNTs. Moreover, the size- and shape-dependence of the Young’s modulus and Poisson’s ratio of the SCNTs are calculated by the generalized MSM method. Exact relations between the mechanical properties of the SCNTs and their size which can reasonably fit the computed results are provided.