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Are finite elements appropriate for use in molecular dynamic simulations?

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Abstract

The applicability of finite elements for molecular dynamic simulations depends on both the structure’s dimensions and the underlying force field type. Shell and continuum elements describe molecular structures only in an average sense, which is why they are not subject of this paper. In contrast, truss and beam elements are potentially attractive candidates when it comes to accurately reproducing the atomic interactions. However, special considerations are required for force fields that use not only two-body, but also multi-body potentials. For the example of bending and torsion energies it is shown how standard beam element models have to be extended to be equivalent to classical molecular dynamic simulations.

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... Subsequently, twisting problems for SWCNTs have been addressed by many researchers using both thin shell theory and MD, MM, and MSM methods (cf., [33,35,36]). Among these studies we mention [5][6][7][8]15,19,20,28,29,32], where the MM method was used, and [13], where the MD method was used. The cited authors used different force fields to describe covalent bonds of atoms: the Modified Morse force field [10] (cf., [5,15]), the Reactive Empirical Bond Order (REBO) force field, first generation [11], the second parameter set (cf., [6][7][8]19,20]), the REBO force field, second generation [12] (cf., [13]), the DREIDING force field [27] (cf., [28,29]), the MM3 force field [1] (cf., [32]). ...
... Among these studies we mention [5][6][7][8]15,19,20,28,29,32], where the MM method was used, and [13], where the MD method was used. The cited authors used different force fields to describe covalent bonds of atoms: the Modified Morse force field [10] (cf., [5,15]), the Reactive Empirical Bond Order (REBO) force field, first generation [11], the second parameter set (cf., [6][7][8]19,20]), the REBO force field, second generation [12] (cf., [13]), the DREIDING force field [27] (cf., [28,29]), the MM3 force field [1] (cf., [32]). We note that all these force fields describe, in varying degrees, the covalent bonds between the sp2 carbon nanoforms. ...
... This approach allows a reliable determination of the critical states of the deformation parameter (time), unlike studies in which criteria of this kind are not used. Note that among the above-cited studies of the buckling of nanostructures, a similar approach to determining the critical states and post-critical deformation modes in quasi-static problems is employed by Hollerer [19], Hollerer and Celigoj [20], Nasdala et al. [28,29], and Wackerfuß [34]. However, the dynamic stability criterion for discrete elastic systems based on determining quasi-bifurcation points (for details, see [25]) in nanostructure buckling problems appears to be first used in the present study. ...
... Experimentally obtained measurement data of the curing process is used to compare and validate the presented numerical crosslinking method with respect to the crosslinking procedure and the resulting network structure. The molecular modelling method is incorporated into the Molecular Dynamic Finite Element Method (MDFEM) [33][34][35] framework that can be used for deriving physical material properties [36,37]. ...
... This leads to a FE calculation, which is comparable to a MD simulation. For further general information on MD implementations into the FEM, the reader is referred to Nasdala et al. [33] and Wackerfuß [46]. All simulations were performed using the DREIDING force field, in its form published by Mayo et al. [47]. ...
Chapter
Reliable simulation of polymers on an atomistic length scale requires a realistic representation of the cured material. A molecular modeling method for the curing of epoxy systems is presented, which is developed with respect to efficiency while maintaining a well equilibrated system. The main criterion for bond formation is the distance between reactive groups and no specific reaction probability is prescribed. The molecular modeling is studied for three different mixing ratios with respect to the curing evolution of reactive groups and the final curing stage. For the first time, the evolution of reactive groups during the curing process predicted by the molecular modeling is validated with near-infrared spectroscopy data, showing a good agreement between simulation results and experimental measurements. With the proposed method, deeper insights into the curing mechanism of epoxy systems can be gained and it allows us to provide reliable input data for molecular dynamics simulations of material properties.
... Wernik and Meguid (2010) presented a nonlinear response of armchair and zigzag CNT under tensile and torsional loading conditions using an atomistic-based continuum modeling technique. Nasdala et al. (2012) introduced an extended beam macro element to analyze small and large deformations along with standard trusses, beams, and spring elements for MD simulations. ...
... There are three models for FE modeling of SWCNT using standard elements. A spring, which is the simplest FE, truss, and beam elements can be used for FE modeling (Nasdala et al., 2012). The nanotube is composed of a number of carbon atoms that have a hexagonal lattice form and are bonded together with covalent bonds, which are con- sidered in the simulation to be like a beam element. ...
Article
The residual stresses play a significant role in the mechanical properties and strengthening capability of nanocomposites. The present research aims to numerically investigate the residual stress relaxation in nanotube-reinforced polymers in response to mechanical tensile loading. The systems under study consist of the armchair and zigzag single-walled carbon nanotubes (SWCNT) embedded in a polymer matrix. The nanotubes and polymer matrix are assumed to be bonded by van der Waals interactions based on the Lennard-Jones (L-J) potential at the interface. The interactions between carbon atoms in the nanotube and nodes in the polymer matrix are modelled by equivalent springs. In order to evaluate the analysis of elastic-perfectly plastic using finite element (FE) modelling, first, relaxation of the plastic residual stresses on steel hemisphere in contact with a rigid flat surface was examined in a loading-unloading cycle and verified with available data. Afterwards, the residual stress relaxation in nanotubes with different space-frame structures was computed due to displacement-controlled loading. Finally, the stress state and the plastic residual stresses in the nanocomposite for different carbon nanotube content were analyzed and discussed during loading and unloading. Regarding the effect of tensile stress, it was revealed that nanotube structures have significant effects on the residual stresses created in the nanocomposite.
... Experimentally obtained measurement data of the curing process is used to compare and validate the presented numerical crosslinking method with respect to the crosslinking procedure and the resulting network structure. The molecular modelling method is incorporated into the Molecular Dynamic Finite Element Method (MDFEM) [33][34][35] framework that can be used for deriving physical material properties [36,37]. ...
... This leads to a FE calculation, which is comparable to a MD simulation. For further general information on MD implementations into the FEM, the reader is referred to Nasdala et al. [33] and Wackerfuß [46]. All simulations were performed using the DREIDING force field, in its form published by Mayo et al. [47]. ...
Article
Reliable simulation of polymers on an atomistic length scale requires a realistic representation of the cured material. A molecular modelling method for the curing of epoxy systems is presented, which is developed with respect to efficiency while maintaining a well equilibrated system. The main criterion for bond formation is the distance between reactive groups and no specific reaction probability is prescribed. The molecular modelling is studied for three different mixing ratios with respect to the curing evolution of reactive groups and the final curing stage. For the first time, the evolution of reactive groups during the curing process predicted by the molecular modelling is validated with near-infrared spectroscopy data, showing a good agreement between simulation results and experimental measurements. With the proposed method, deeper insights into the curing mechanism of epoxy systems can be gained and it allows us to provide reliable input data for molecular dynamics simulations of material properties.
... The MM simulator used in the present study is the PIONER homemade finite element code [72]. 5 Furthermore, to take into account the elementary internal forces resulting from the simulation of each type of elementary energy for the bond interactions (bond-stretching, bond-angle bending, bond torsion, and bond inversion) constituting the DREIDING force field, we introduce fictitious "finite" elements, for which the internal force vectors and the tangential stiffness matrices are determined (see also [63,64,[107][108][109]125]). ...
... In subsequent simulation studies [2] of the deformation, vibration, and buckling of SLGSs, we used the DREIDING force field (cf., [100], see also [107][108][109]125]) with the standard parameter set of the DREIDING force field (cf., [100]), but this parameter set does not give a sufficiently accurate reproduction of the mechanical moduli of graphene (cf., [78]). Previously (see [78]), we have presented a modified parameter set of the DREIDING force field, which improves the modeling of the mechanical moduli of graphene (the 2D Young's modulus Y , Poisson's ratio ν, and the bending rigidity modulus D) sufficient for simulating the deformation of SLGSs at small strains, but possibly large displacements and rotations of the graphene atomic lattice. ...
Article
Full-text available
Molecular mechanics/molecular dynamics (MM/MD) methods are widely used in computer simulations of deformation (including buckling, vibration, and fracture) of low-dimensional carbon nanostructures (single-layer graphene sheets (SLGSs), single-walled nanotubes, fullerenes, etc). In MM/MD simulations, the interactions between carbon atoms in these nanostructures are modeled using force fields (e.g., AIREBO, DREIDING, MM3/MM4). The objective of the present study is to fit the DREIDING force field parameters (see Mayo et al. J Phys Chem 94:8897–8909, 1990) to most closely reproduce the mechanical parameters of graphene (Young’s modulus, Poisson’s ratio, bending rigidity modulus, and intrinsic strength) known from experimental studies and quantum mechanics simulations since the standard set of the DREIDING force field parameters (see Mayo et al. 1990) leads to unsatisfactory values of the mechanical parameters of graphene. The values of these parameters are fitted using primitive unit cells of graphene acted upon by forces that reproduce the homogeneous deformation of this material in tension/compression, bending, and fracture. (Different sets of primitive unit cells are used for different types of deformation, taking into account the anisotropic properties of graphene in states close to failure.) The MM method is used to determine the dependence of the mechanical moduli of graphene (Young’s modulus, Poisson’s ratio, and bending rigidity modulus) on the scale factor. Computer simulation has shown that for large linear dimensions of SLGSs, the mechanical parameters of these sheets are close to those of graphene. In addition, computer simulation has shown that accounting for in-layer van der Waals forces has a small effect on the value of the mechanical moduli of graphene.
... In this chapter, a brief overview of the Molecular Dynamic Finite Element Method (MDFEM) is given. For a more detailed presentation, the reader is referred to earlier work of the senior authors of the present paper [9,10]. ...
... Even though it is possible to obtain the correct behavior with a certain superposition of rigidly and flexibly connected beam elements, as shown in [10], the MDFEM provides a more elegant method, using special 2-, 3-and 4-node elements for bond stretch and physical interactions, bending and torsion. By superposing those elements, various atomistic problems can be modeled. ...
... [9,51,89]), and a new force field LCBOPII which is an improved version of the REBO-2 force field has been developed [90]. In the present simulation study of the nonlinear deformation and buckling of SLGSs, we use the DREIDING force field [57] (as the authors of [70][71][72][73]). In this study, an attempt is made to modify the parameters of the DREIDING force field to bring the values of Young's modulus, Poisson's ratio, and the bending modulus of graphene closer to the values obtained in [3]. ...
... 1. To derive expressions for the internal force vectors and tangential stiffness matrices for the Nbody potentials (2 N 4) of the DREIDING force field, treated as potential energies of fictitious 'finite' elements, in a form suitable for implementing in finite element codes. Note that similar approaches (i.e. using N-body potentials as potential energies of fictitious 'finite' elements) for the DREIDING force field were employed in [70][71][72][73], but the expressions for internal force vectors and tangential stiffness matrices of elements presented in these papers cannot be used directly for implementation in finite element codes. 2. To modify the parameter set of the DREIDING force field to improve the agreement of the simulated mechanical moduli (2D Young's modulus Y, Poisson's ratio n, and bending stiffness modulus D) of graphene with the well-known reference values of this material; to determine the size-dependence of the mechanical properties of SLGSs. ...
Article
Full-text available
This paper presents a quasi-static nonlinear buckling analysis of compressed single-layer graphene sheets (SLGSs) using the molecular mechanics method. Bonded interactions between carbon atoms are simulated using a modified parameter set of the DREIDING force field that leads to better agreement between simulated mechanical properties of graphene and reference literature data than the standard parameter set of this force field (see Mayo et al., J Phys Chem 1990; 94: 8897-8909). Identification of constraints of atoms of the SLGS edges with the boundary conditions of clamped and simply supported thin plates is made. The buckling loads and modes obtained by linear and nonlinear buckling analysis of a compressed quadratic SLGS with a side length of 6€‰nm are shown to be close to each other. In addition, it has been found by nonlinear buckling analysis that only equilibrium configurations with modes of initial post-buckling deformed configurations correlated with the one-half-wave column-like buckling mode have stable equilibrium configurations for clamped and simply supported SLGSs. As the edges of a simply supported SLGS approach each other, the geometry of this mode of post-buckling deformation with inclusion of the non-bonded van der Waals (vdW) interactions between carbon atoms becomes closer to the geometry of a single-walled carbon nanotube, and without inclusion of the vdW interactions, this mode has the geometry of a cylinder with a drop-shaped cross-section.
... The aforementioned limitations associated with using each of the structural FEM element types (e.g. springs, trusses and beams) are reviewed in-depth by Nasdala et al. [43]. Spring-based, trussbased and beam-based methods [27][28][29][30][31][32][33][34] may still result in good numerical results for mechanical A. A. R. WILMES AND S. T. PINHO properties of conventional geometries such as CNT or graphene under standard load conditions (e.g. ...
... residual force vector) and Hessian (i.e. tangent stiffness), Nasdala et al. [35,36,43] separate the components in the derivatives stemming from the force field potential from those of the geometric element topology; it appears this separation has been carried out for mathematical convenience rather than as an intended core component of the developed model. ...
Article
A new Molecular Dynamics Finite Element Method (MDFEM) with a coupled mechanical‐charge/dipole formulation is proposed. The equilibrium equations of Molecular Dynamics (MD) are embedded exactly within the computationally more favourable Finite Element Method (FEM). This MDFEM can readily implement any force field because the constitutive relations are explicitly uncoupled from the corresponding geometric element topologies. This formal uncoupling allows to differentiate between chemical‐constitutive, geometric and mixed‐mode instabilities. Different force fields, including bond‐order reactive and polarisable fluctuating charge–dipole potentials, are implemented exactly in both explicit and implicit dynamic commercial finite element code. The implicit formulation allows for larger length and time scales and more varied eigenvalue‐based solution strategies. The proposed multi‐physics and multi‐scale compatible MDFEM is shown to be equivalent to MD, as demonstrated by examples of fracture in carbon nanotubes (CNT), and electric charge distribution in graphene, but at a considerably reduced computational cost. The proposed MDFEM is shown to scale linearly, with concurrent continuum FEM multi‐scale couplings allowing for further computational savings. Moreover, novel conformational analyses of pillared graphene structures (PGS) are produced. The proposed model finds potential applications in the parametric topology and numerical design studies of nano‐structures for desired electro‐mechanical properties (e.g. stiffness, toughness and electric field induced vibrational/electron‐emission properties). Copyright © 2014 John Wiley & Sons, Ltd.
... However, in a straightforward application of truss and beam elements, some problems are inevitable. The material parameters for bond bending and torsion cannot be identified uniquely for each interaction, and there is no distinction between natural and equilibrium bond angles [16]. Furthermore, standard elements with constant parameters cannot be used to model the nonlinear force field potentials. ...
... Furthermore, the absence of rotational degrees of freedom (DOF) at the element nodes leads to a more computationally effective approach compared with the use of beam elements. The high accuracy and effectiveness of the MDFEM, as well as numerous successful applications of this method in a wide range of studies [16][17][18][19][20], inspired us to adopt the MDFEM for the analysis of CCNTs. In the following section, the force field potentials described by DREIDING [21] are investigated, and the basics of the MDFEM are briefly reviewed. ...
Article
The mechanical response of single-walled helically coiled carbon nanotubes (CCNTs) under large axial deformations is examined using a molecular dynamics finite element method. The 3D reference configuration of CCNTs is determined based on a 2D graphene layer using conformal mapping. Three sets of analyses are performed to fully describe the mechanical response of (n, n) CCNTs under elongation up to the bond breaking point and compression down to the solid length or the onset of buckling instability. First, the strain dependency of the mechanical properties of individual CCNTs during deformation is investigated by calculating the stress-strain curve and the spring constant of the CCNTs for the entire load range. Significant responses including brittle fracture under tension and buckling instability under compression are observed. Second, to examine the size dependence of the mechanical properties, several CCNTs with different geometric parameters are constructed, and their spring constant, fracture strain, fracture load, and energy storage density are determined. All CCNTs exhibit a superelasticity of 50-66%. A comparison between the mechanical properties of CCNTs and those of carbon nanotubes (CNTs) reveals that the fracture load and energy storage per atom of CCNTs is lower than that of the corresponding armchair CNTs.
... Special-purpose elements are introduced to meet specific requirements - [156,157] carbon nanostructures in GANS, which are applicable for different problems [128]. ...
Article
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Concrete is the most popular construction material in infrastructure projects due to its numerous natural advantages. Nevertheless, concrete constructions frequently suffer from low tensile strength and poor durability performance which are always urgent tasks to be solved. The concrete reinforced by various nanomaterials, especially graphene and its associated nanostructures (GANS), shows excellent chemical and physical properties for engineering applications. The influence of GANS on cement composites is a multiscale behavior from the nanoscale to the macroscale, which requires a number of efforts to reveal via numerical and experimental approaches. To meet this need, this study provides a comprehensive overview of the numerical modeling for GANS reinforced concrete in various scales. The background and importance of the topic are addressed in this study, along with the review of its methodologies, findings, and applications. Moreover, the study critically summarizes the performance of GANS reinforced concrete, including its mechanical behavior, transport phenomena, and failure mechanism. Additionally, the primary challenges and future prospects in the research field are also discussed. By presenting an extensive overview, this review offers valuable insights for researchers and practitioners interested in numerical simulation to advance concrete science and engineering.
... With regard to the MD simulation as part of the Capriccio method, it may be beneficial to investigate further approaches to model particle-based systems. Recently, a finite element based treatment of molecular dynamics has been published by Nasdala and co-workers [127,128]. The basic idea of this method, which is called "MDFEM", is to use a finite element framework to capture the interactions between the particles as they are introduced in Section 3.3. ...
Thesis
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In modern engineering applications, plastics play an important role for instance in the field of lightweight constructions or as substitutes for classical materials like wood, metal, or glass. Typically, they consist of organic polymers, which are long-chained molecules comprising numerous monomers as repeat units. In addition, polymers frequently contain fillers, plasticisers, or colourants to achieve and to adjust specific properties. In recent years, new techniques have been established to produce and to disperse filler particles in the range of nanometres, which corresponds to the typical dimensions of the monomers. Experiments reveal that these so-called “nanofillers” may significantly toughen polymers, improve their fatigue lifetime, and enhance control of their thermodynamical properties, even for low filler contents in terms of mass or volume. This cannot be explained by a simple rule of mixture, but is traced back to the very large ratio of surface to volume in case of nanofillers and to the associated processes at the molecular level. The effective design of such “nanocomposites” is demanding and often requires timeconsuming mechanical testing. For a better understanding of the relevant parameters and in order to improve the process of material development, it is beneficial to substitute “real” experiments by numerical simulations. To this end, sophisticated computation techniques are required that account for the specific processes taking place at the level of atoms and molecules. Particle-based strategies, as for instance employed in physical chemistry, are able to consider the atomistic structure in detail and thus permit to simulate material behaviour at atomistic length scales. However, it is still not possible to apply these techniques to large-scale systems relevant in engineering. There, the material behaviour of structures is typically described by continuum approaches, which, on the other hand, cannot account explicitly for the processes at the atomistic level. To overcome this, the present thesis proposes a novel coupling scheme to incorporate particle-based simulations into continuum-based methods. In particular, it links molecular dynamics as a standard tool in physical chemistry with the finite element method, which is nowadays widely used in engineering applications. This multiscale simulation approach has been developed jointly by the Theoretical Physical Chemistry Group at the Technische Universität Darmstadt and the Chair of Applied Mechanics at the Friedrich-Alexander-Universität Erlangen-Nürnberg. Thus, it bases upon expertise in atomistic simulation as well as continuum mechanics, whereby crucial modifications of established techniques in both fields had to be developed. Two sample systems, modelling pure polystyrene and a polystyrene-silica nanocomposite, are studied numerically and prove the suitability of the new approach. In this context, various parameters of the proposed method and its implementation are investigated. Based on this, a number of options to improve this multiscale technique are discussed and relevant issues for future research are summarised.
... The C-C bond length here is l 0 = 0.142 nm . Zigzag and armchair nano- Nasdala et al. [34] illustrated that the standard truss and beam elements can be represented atomic interactions accurately. Energy-equivalent model, resulting from the foundation of molecular and continuum mechanics, considers the mechanical properties of CNTs (i.e.; Young's modulus, shear (1.a) �� ⃗ C h = n� ⃗ a 1 + m� ⃗ a 2 , ...
Article
Full-text available
This paper aims to investigate the size scale effect on the buckling and post-buckling of single-walled carbon nanotube (SWCNT) rested on nonlinear elastic foundations using energy-equivalent model (EEM). CNTs are modelled as a beam with higher order shear deformation to consider a shear effect and eliminate the shear correction factor, which appeared in Timoshenko and missed in Euler–Bernoulli beam theories. Energy-equivalent model is proposed to bridge the chemical energy between atoms with mechanical strain energy of beam structure. Therefore, Young’s and shear moduli and Poisson’s ratio for zigzag (n, 0), and armchair (n, n) carbon nanotubes (CNTs) are presented as functions of orientation and force constants. Conservation energy principle is exploited to derive governing equations of motion in terms of primary displacement variable. The differential–integral quadrature method (DIQM) is exploited to discretize the problem in spatial domain and transformed the integro-differential equilibrium equations to algebraic equations. The static problem is solved for critical buckling loads and the post-buckling deformation as a function of applied axial load, CNT length, orientations and elastic foundation parameters. Numerical results show that effects of chirality angle, boundary conditions, tube length and elastic foundation constants on buckling and post-buckling behaviors of armchair and zigzag CNTs are significant. This model is helpful especially in mechanical design of NEMS manufactured from CNTs.
... Harik [6] proved that the structural characteristics of CNTs can be modelled as a beam for small radii and a cylindrical shell for large radii. Nasdala et al. [7] illustrated that the standard truss and beam elements can be represented atomic interactions accurately. Energy equivalent model, resulting from the foundation of molec-ular and continuum mechanics, considers the mechanical properties of CNTs (i.e; Young's modulus, shear modu-lus, and Poisson's ratio) as a material sizedependent by many researchers. ...
Article
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This paper presents a novel numerical procedure to predict nonlinear buckling and postbuckling stability of imperfect clamped–clamped single walled carbon nanotube (SWCNT) surrounded by nonlinear elastic foundation. Nanoscale effect of CNTs is included by using energy-equivalent model (EEM) which transferring the chemical energy between carbon atoms to mechanical strain energy. Young’s modulus and Poisson’s ratio for zigzag ( n, 0 ), and armchair ( n, n ) carbon nanotubes (CNTs) are presented as functions of orientation and force constants by using energy-equivalent model (EEM). Nonlinear Euler-Bernoulli assumptions are proposed considering mid-plane stretching to exhibit a large deformation and a small strain. To simulate the interaction of CNTs with the surrounding elastic medium, nonlinear elastic foundation with cubic nonlinearity and shearing layer are employed. The governing nonlinear integro-partial-differential equations are derived in terms of only the lateral displacement. The modified differential quadrature method (DQM) is exploited to obtain numerical results of the nonlinear governing equations. The static problem is solved for critical buckling loads and the postbuckling deformation as a function of applied axial load, curved amplitude, CNT length, and orientations. Numerical results show that the effects of chirality angle and curved amplitude on static response of armchair and zigzag CNTs are significant. This model is helpful especially in mechanical design of NEMS manufactured from CNTs.
... These degrees of freedom are not present in molecular mechanics methods and the stiffness matrix coefficients for these terms rely on various assumptions which are questionable in a molecular context. This leads to situations where pure bond torsions that occur in molecular mechanics are modelled can bemodelled as beam bending in AFEM, as shown by Nasdala et al. [41]. Complete neglect of the non-bonded interaction terms will always produce significant differences when comparing molecular mechanics minimisations, using potentials with bonded and non-bonded terms, with AFEM regardless of how well the bonded terms are reproduced. ...
Article
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We calculate the tensile and shear moduli of a series of boron nitride nanotubes and their piezoelectric response to applied loads. We compare in detail results form a simple molecular mechanics potential, the Universal Force Field, with those from the atomistic finite element method using both Euler-Bernoulli and Timoshenko beam formulations. The molecular mechanics energy minimisations are much more successful than those using the atomistic finite element method, and we analyse the failure of the latter approach both qualitatively and quantitatively.
... Continuum models and those based on finite element (FE) calculations, on the other extreme, can handle very large systems and, with adequate homogenization processes, can be linked to macroscopic models [24,25]. The major drawback recognized for beam-type FE models is probably that they rely on equivalent (homogenized) continuous representations, which sacrifice detailed atomic interactions; also, the parameters of standard beam models are frequently obtained from harmonic potentials and there is a lack of accurate dedicated bond force constants for each specific molecule; in addition, most of them use as initial coordinates what has been named "natural" coordinates, instead of equilibrium ones dictated from fundamental quantum mechanics [26][27][28]. Even with these limitations, continuous beam-type models have proven to be accurate enough and a very valuable tool for predicting effective elastic properties of nanostructures [19,[29][30][31], and especially useful when large dimensional scales are of interest. ...
Article
A hierarchical approach bridging the atomistic and nanometric scales is used to compute the elastic properties of boron nitride nanosheets and nanoribbons, examining the effect of sheet size, aspect ratio and anisotropy. The approach consists in obtaining the bond force (force field) constants by dedicated computations based on density functional theory (DFT) and using such constants as input for larger scale structural models solved by finite element analysis (FEA). The bond force constants calculated by DFT are 616.9 N/m for stretching, 6.27×10⁻¹⁹ Nm/rad² for in-plane rotation and 1.32×10⁻¹⁹ Nm/rad² for dihedral rotation. Using these constants, the elastic properties of boron nitride nanosheets and nanoribbons predicted by FEA are almost independent of the sheet size, but strongly dependent on their aspect ratio. The sheet anisotropy increases with increased aspect ratio, with nanoribbons of aspect ratios of 10 exhibiting a ratio of elastic moduli along both in-plane directions of 1.7.
... All simulations are carried out using the Molecular Dynamic Finite Element Method (MDFEM, for a detailed presentation see [28,29]). The MDFEM provides a framework for calculating classical molecular dynamics (MD) problems within the wellestablished Finite Element Method (FEM) (see, e.g., Wackerfuss [29] for an overview of molecular dynamics in the context of the FEM). ...
Article
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Boehmite nanoparticles show great potential in improving mechanical properties of fiber reinforced polymers. In order to predict the properties of nanocomposites, knowledge about the material parameters of the constituent phases, including the boehmite particles, is crucial. In this study, the mechanical behavior of boehmite is investigated using Atomic Force Microscopy (AFM) experiments and Molecular Dynamic Finite Element Method (MDFEM) simulations. Young’s modulus of the perfect crystalline boehmite nanoparticles is derived from numerical AFM simulations. Results of AFM experiments on boehmite nanoparticles deviate significantly. Possible causes are identified by experiments on complementary types of boehmite, that is, geological and hydrothermally synthesized samples, and further simulations of imperfect crystals and combined boehmite/epoxy models. Under certain circumstances, the mechanical behavior of boehmite was found to be dominated by inelastic effects that are discussed in detail in the present work. The studies are substantiated with accompanying X-ray diffraction and Raman experiments.
... [128,132,133] See Figure 2.4 for an illustration of how the MSM approach treats the C−C bond as an effective structural element as presented by Eberhardt and Wallmersperger. [124] Through recent extensions and improvements of the original MSM approaches, [124,[134][135][136][137] such as improving the underlying chemical force field description to ensure consistency in terms of energy, [124] the NCM approach was shown to be an effective compromise between atomistic and continuum techniques. This analysis was successfully utilized to model CNTs containing a few simple kinds of wall defects. ...
Thesis
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The intrinsic and scale-dependent properties of nanofibers (NFs), nanowires, and nanotubes have made them the focus of many application-specific nanostructured materials studies. However, various NF morphology and proximity effects can lead to > 1000x reductions in the performance of NF-based material architectures, such as bulk materials and structures comprised of scalable aligned NF arrays. The physical and chemical origins of these effects, along with the concomitant structure property mechanisms of materials comprised of aligned NFs, are not currently known and cannot be properly integrated into existing theories. This originates, in part, from an incomplete understanding of the morphology of real NF systems, particularly in three-dimensions. Through experiments, theory, and multi-scale simulation, this dissertation presents a framework capable of modeling the stochastic 3D morphology of a relevant NF system, carbon nanotubes (CNTs), assembled into aligned CNT (A-CNT) arrays. New descriptions of the multi-wall A-CNT morphology demonstrate that the CNT tortuosity, quantified via sinusoidal amplitude-wavelength waviness ratio (w), decreases significantly from w ~/~ 0:2 to 0:1 as the CNT volume fraction (Vf) is increased from Vf ~ 1 to 20%. Using these new relations, a 3D stochastic morphology description is presented, and used to quantify the mechanical behavior of A-CNT arrays, A-CNT polymer matrix nanocomposites (A-PNCs), and A-CNT carbon matrix nanocomposites (A-CMNCs) via a mechanics analysis that was previously applied to carbon nanocoils. Focusing on deformations in the A-CNT axial reinforcement direction, torsion and shear deformation mechanisms, which are governed by the low (< 1 GPa) intrinsic shear modulus of the CNTs, are shown to have an effective compliance contribution of > 90% in the experimental A-CNT w regime, and are inferred to be the physical mechanisms responsible for the previously observed ~ 100x increase in the ACNT effective indentation modulus as Vf is increased from ~ 1 to 20%. In the case of A-PNCs, the polymer matrix effectively eliminates the torsion compliance contribution, so that the observed ~ 2x enhancement in the effective axial elastic modulus of A-PNCs as Vf is increased from ~ 1 to 20% is explained. The geometry of the graphitic crystallites that comprise the pyrolytic carbon (PyC) matrix of A-CMNCs is found to not evolve significantly at pyrolysis temperatures of 1000 to 1400°C, and crystallite size estimates from Raman spectroscopy reveal that the Tuinstra-Koenig correlation disagrees with the sizes measured by x-ray diffraction, suggesting a new amorphization transition crystallite size of 6 nm instead of 2-3 nm. In the case of A-CMNCs, CNT reinforcement is shown to lower the energy barrier (inferred through the pyrolysis temperature) for meso-scale self-organization of the graphitic crystallites of the PyC matrix, while having no effect on the PyC matrix on the atomic scale. Mechanical property analysis and modeling indicates that the aerospace materials selection criterion of the A-CMNCs can be enhanced to >8 GPa x (g/cm3)-2 at Vf >20% (experimentally we observe a value of ~ 5 GPa x (g/cm3)-2 at Vf ~ 10%). A-CMNCs introduced in this work have the potential to outperform state-of-the-art superhard materials, such as diamond (~/~ 7:8 GPa x (g/cm3)-2) and cubic boron nitride (~/~ 5:2 GPa x (g/cm3)-2). Using the structure-property prediction tools developed in this thesis, precise tailoring and prediction of application-specific performance of aligned NF based architectures is enabled, and specific new understanding of A-CNT systems is established. Future paths of study that enable the design and manufacture of several classes of next-generation materials are recommended.
... The structural characteristics of nanotubes (NTs) make them similar to a beam for small radii and cylindrical shells for large radii [21]. Nasdala et al. [39] proved that the standard truss and beam elements are potentially attractive candidates to accurately reproducing the atomic interactions. The validity of the continuumbeam models for the constitutive behavior of CNTs is presented by Harik [22]. ...
Article
This paper investigates the effects of both size-dependency and material-dependency on the nonlinear static behavior of carbon nanotubes (CNTs). The energy-equivalent model (EEM) derived on the basis of molecular mechanics is exploited to describe the size-dependence of mechanical properties of CNTs, such as, Young's modulus, shear modulus and Poisson's ratio. Carbon nanotube is modeled as modified nonlocal Euler-Bernoulli and Timoshenko nanobeams with mid-plane stretching. To include the size-dependency and length scale effect of nanostructure, a nonlocal differential form of Eringen's model is proposed. The governing equilibrium equations for proposed beam theories are derived using the principle of virtual displacements, wherein the modified nonlinear von Karman strains are considered. A finite element model is developed to solve the nonlinear equilibrium equations. Numerical results are presented to show the effects of chirality angle, nonlocal parameter, moderate rotation, and boundary conditions of CNTs. These findings are helpful in mechanical design of high-precision devices and structures manufactured from CNTs.
... Several comprehensive presentations and reviews of AFEM/MDFEM and its implementations are available [24,25,29,30]. Nonetheless, MDFEM remains a non-consolidated method because formal derivations are scarce, with significant differences arising on the topologies of the required MDEFM-specific elements. ...
Conference Paper
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The recent rise of 2D materials, such as graphene, has expanded the interest in nano-electromechanical systems (NEMS). The increasing ability of synthesizing more exotic NEMS architectures, creates a growing need for a cost-effective, yet accurate nano-scale simulation method. Established methodologies like Molecular Dynamics (MD) trail behind synthesis capabilities because the computational effort scales quadratically. The equilibrium equations of MD are equivalent with those of the computationally more favourable Finite Element Method (FEM). However, current implementations exploiting this equivalence re-main limited due to the FEM iterative solvers requiring a large number of lengthy force field derivatives and specifically tailored element topologies. This paper proposes a formal deriva-tion of the merged Molecular Dynamic Finite Element Method (MDFEM) which establishes an uncoupling of the force field potentials from the element topologies. An implementation approach, which does not require manual derivations, is presented. Different non-linear MD force field potentials are implemented exactly within the FEM, at reduced computational costs. The proposed multi-scale and multi-physics compatible MDFEM is equivalent to to the MD as demonstrated by an example of brittle fracture in Carbon Nanotubes (CNT).
... The resulting Atomistic Finite Element Method (AFEM) [8], also named Molecular Dynamic Finite Element Method (MDFEM) [12], is both computationally more favourable than MD [8], and offers a significant increase in compatibility and integrability with larger scale continuum FEM simulations. Several comprehensive presentations and reviews of AFEM/MDFEM and its implementations are available [8,[12][13][14]. ...
Conference Paper
Full-text available
The recent rise of 2D materials, such as graphene, has expanded the interest in nanoelectromechanical systems (NEMS). The increasing ability of synthesizing more exotic NEMS architectures, creates a growing need for a cost-effective, yet accurate nano-scale simulation method. Established methodologies like Molecular Dynamics (MD) trail behind synthesis capabilities because the computational effort scales quadratically. The equilibrium equations of MD are equivalent with those of the computationally more favourable Finite Element Method (FEM). However, current implementations exploiting this equivalence remain limited due to the FEM iterative solvers requiring a large number of lengthy force field derivatives and specifically tailored element topologies. This paper proposes a merged Molecular Dynamic Finite Element Method (MDFEM) which does not require the manual derivation of these derivatives. Hence, implementing MDFEM-specific element topologies is straightforward and thus, different non-linear MD force field potentials can be solved exactly within the FEM, at reduced computational costs. The proposed multi-scale and multi-physics compatible MDFEM is equivalent to the MD, as demonstrated firstly by an example of brittle fracture in Carbon Nanotubes (CNT), and secondly by conformational analyses on Non-Equilibrium initial meshes of Pillared Graphene Structures (PGS).
Article
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.
Chapter
The complex effect of nanoparticles on an epoxy-based and anhydride cured DGEBA/Boehmite nanocomposite with different particle concentrations is considered in this chapter. A combination of X-ray scattering, calorimetry (fast scanning and temperature modulated calorimetry) and dielectric spectroscopy was employed to characterize the structure, vitrification kinetics and the molecular dynamics of the nanocomposites. Firstly, the unfilled polymer was found to be intrinsically heterogeneous, showing regions with different crosslinking density, indicated by two separate dynamic glass transitions. Moreover, the glass transition temperature decreases with increasing nanoparticle concentration, as a result of changes in the crosslinking density. In addition, it was shown that the incorporation of nanoparticles can result in simultaneous increase in the number of mobile segments for low nanoparticle concentrations and on the other hand, for higher loading degrees the number of mobile segments decreases, due to the formation of an immobilized interphase.
Chapter
Boehmite nanoparticles show great potential in improving mechanical properties of fiber reinforced polymers. In order to predict the properties of nanocomposites, knowledge about the material parameters of the constituent phases, including the boehmite particles, is crucial. In this study, theAtomic force microscopy mechanical behavior of boehmite is investigated using Atomic Force Microscopy (AFM) experiments and Molecular Dynamic Finite Element Method Molecular dynamic finite element method (MDFEM) simulations. The Young’s modulus of the perfect crystalline boehmite nanoparticlesBoehmite nanoparticle is derived from numerical AFM simulations. Results of AFM experiments on boehmite nanoparticles deviate significantly. Possible causes are identified by experiments on complementary types of boehmite, i.e. geological and hydrothermally synthesized samples, and further simulations of imperfect crystals and combined boehmite/epoxy models. Under certain circumstances, the mechanical behavior of boehmite was found to be dominated by inelastic effects that are discussed in detail in the present work. The studies are substantiated with accompanying X-ray DiffractionX-ray diffraction and RamanRaman spectroscopy experiments.
Chapter
The development of a physically based constitutive model for glass fiber reinforced boehmite nanoparticle-filled epoxy nanocomposites undergoing finite strain is investigated. The constitutive model allows capturing the main features of the stress-strain relationship of the nanocomposites, including the nonlinear hyperelastic, time-dependent and softening behavior. A methodological framework based on molecular dynamics simulations and experimental tests is proposed to identify the material parameters required for the model. The fiber-matrix interaction is characterized by a composite model, which multiplicatively decomposes the deformation gradient into a uniaxial deformation along the fiber direction and a subsequent shear deformation. The effect of the nanoparticles on the stress–strain response is taken into account through the adoption of a modulus enhancement model. The Eyring model parametrized using molecular simulations is used to describe the rate-dependent viscoelastic deformation under loading. The stress softening behavior is captured by a monotonically increasing function of deformation, so-called softening variable. The results show that the model predictions of stress-strain relationships are in good agreement with experimental data at different fiber and nanoparticle weight fractions. Finally, the constitutive model is implemented in the finite element analysis and examined by means of a benchmark example. Experimental–numerical validation confirms the predictive capability of the present modeling framework, which provides a suitable tool for analyzing fiber reinforced nanoparticle/epoxy nanocomposites.
Article
To analyze the experimentally observed failure process in nanoparticle/polymer nanocomposites, a variety of factors, including nonlocal characteristics of damage mechanism and nonlinear viscoelasticity, are required to be investigated. This work presents the development and numerical implementation of a finite deformation gradient-enhanced damage model for boehmite nanoparticle (BNP)/epoxy nanocomposites. The parameters identification of the nonlocal constitutive description is realized using a framework based on molecular simulations and experimental tests. In this context, molecular simulations are performed to parameterize the Argon model of viscoelasticity, while damage and nonlocal parameters are determined using experimental data obtained from compact-tension tests. The nonlocal constitutive model integrated into a nonlinear FE analysis is validated by comparing the numerical results of compact-tension tests of BNP/epoxy samples with experimental data. The experimental–numerical validation confirms the predictive capability of the modeling framework. The proposed procedure can be extended to other types of nanoparticle reinforced thermosetting polymers.
Article
Graphene and its associated nanostructures (GANS) have been widely investigated by means of experimental and numerical approaches over the last decade. GANS and GANS reinforced composite materials show exceptional promise towards superior mechanical and thermal properties along with limitless opportunity to tailor, control, design, modify and manipulate such properties. These attributes make graphene and its associated nanostructures as one of the most important future material technologies in aerospace, automotive, medical, civil and military sectors of the 21st century. Among the various numerical methods used to analyse GANS and GANS reinforced composite materials, the finite element method (FEM) plays a prominent role. The FEM has been the standard analysis and simulation method for conventional structural and mechanical problems over the past half a century. However, its growing role and impact in atomistic-scale numerical simulation in general, and GANS, in particular, is not well known within the wider scientific and engineering modelling and simulation research community. There is a compelling need to document the expansive use of the finite element method, its advantages, shortcomings, relevance and purpose in a way which is pertinent to both material science and numerical simulation researchers. This paper serves this need by discussing the current state of the art of finite element methodologies available to study GANS and GANS reinforced composites in the most comprehensive manner. A detailed description of the popular space frame based numerical simulation strategy widely used to represent GANS is given. An extensive survey is conducted on more than 600 research papers in order to examine the finite element predictions of the mechanical and thermal properties of graphene and its associated composite materials. These properties are selected in view of their direct relevance to crucial future technologies, such as high-performance automotive components, aerospace and bioengineering systems, energy technologies, and advanced therapeutic and surgical devices. Omissions of some fundamental mechanical and thermal modelling issues for GANS have been identified and insightful guidance towards future research directions to comprehensively address them is given. By reviewing a significant breadth of publications across several academic disciples, a large scatter in the numerical predictions of essential material constants arising from the differences in fundamental assumptions and approximations has been reported. The origin of such discrepancies has been identified, analysed and established. The paper further focuses on the idealization of nanostructures and nanocomposites by means of representative volume elements (RVEs). The need for this multiscale modelling strategy to mature in order to include the simultaneous description of different material length scales within multiphysics simulation problems has been discussed. This paper will serve as standalone reference material for future research works and will pave the way for novel investigations in the context of atomistic simulations and their potential applications to the development of next-generation engineering devices and cutting-edge technological applications.
Article
The accurate prediction of the complex material response of nanoparticle/epoxy nanocomposites for thermomechanical load cases is of great interest for engineering applications. In the present work, three main contributions with respect to multi-scale modelling of the viscoelastic damage behaviour of nanocomposites are presented. Firstly, a constitutive model for the viscoelastic damage behaviour at finite temperatures below the glass-transition temperature is proposed. The constitutive model captures the main characteristics of the material response including the non-linear hyperelasticity, softening behaviour and the effect of temperature. Secondly, the material model is calibrated using purely experimental results to evaluate the best capability of the model in reproducing the stress–strain response at different strain rates and temperatures. The calibrated model predicts the material behaviour across a range of nanoparticle weight fractions with good agreement with experimental results. Finally, a combined approach of experimental testing and molecular simulations is proposed to identify the parameters of the constitutive model. This study shows that the proposed simulation-based framework can be used to significantly reduce the number of experimental tests required for identification of material parameters without a significant loss of accuracy in the material response prediction. The predictive capability of the atomistically calibrated constitutive model is validated, with additional experimental results not used within the parameter identification, in terms of an accurate representation of the viscoelastic damage behaviour of nanoparticle/epoxy nanocomposites at finite temperatures. The present study underlines the capabilities of numerical molecular simulations intended for the characterisation of material properties with respect to physically based constitutive modelling and multi-scale approaches.
Article
The precise knowledge of the temperature-dependent non-linear viscoelastic material behaviour of polymers is of great importance for engineering applications. The present work is a contribution to meet the challenge of bridging the inherently different time scales of molecular dynamics (MD) and experiments by providing a consistent comparison and assessment of viscoelastic theories. For this reason, the physically motivated theories for viscoelasticity of Eyring and Argon as well as the Cooperative model are evaluated with regard to their predictive capability for the characterisation of the viscous behaviour over a broad range of temperatures and strain rates. MD simulations of tensile tests are performed and the effect of strain rate and temperature on the yield stress is examined. The distinctive feature of this study is to demonstrate that viscoelastic theories can be successfully calibrated using only MD results. For a comparison to experimental data, we conduct tensile tests at three different strain rates and at three temperatures in the glassy regime. Experimental validation confirms the predictive capability of the Argon model, which can provide an accurate formulation of epoxy viscoelasticity for physically motivated constitutive models. The present study not only underlines the ability of MD simulations for identifying and characterising physical phenomena on the molecular level, but also shows that molecular simulations can substitute experimental tests for the characterisation of the viscoelastic material behaviour of polymers.
Article
The development of a physically based constitutive model for glass fiber reinforced boehmite nanoparticle-filled epoxy nanocomposites undergoing finite strain is investigated. The constitutive model allows capturing the main features of the stress-strain relationship of the nanocomposites, including the nonlinear hyperelastic, time-dependent and softening behavior. A methodological framework based on molecular dynamics simulations and experimental tests is proposed to identify the material parameters required for the model. The fiber-matrix interaction is characterized by a composite model, which multiplicatively decomposes the deformation gradient into a uniaxial deformation along the fiber direction and a subsequent shear deformation. The effect of the nanoparticles on the stress–strain response is taken into account through the adoption of a modulus enhancement model. The Eyring model parametrized using molecular simulations is used to describe the rate-dependent viscoelastic deformation under loading. The stress softening behavior is captured by a monotonically increasing function of deformation, so-called softening variable. The results show that the model predictions of stress-strain relationships are in good agreement with experimental data at different fiber and nanoparticle weight fractions. Finally, the constitutive model is implemented in the finite element analysis and examined by means of a benchmark example. Experimental–numerical validation confirms the predictive capability of the present modeling framework, which provides a suitable tool for analyzing fiber reinforced nanoparticle/epoxy nanocomposites.
Article
Experimental tests show that nonlinear viscoelasticity characterizes the mechanical behavior of boehmite nanoparticle (BNP)/epoxy nanocomposites. This paper presents the development and numerical implementation of a physically based constitutive model for BNP/epoxy nanocomposites undergoing finite strain. The proposed constitutive model allows capturing the main features of the stress-strain relationship of BNP/epoxy nanocomposites, including the nonlinear hyperelastic, time-dependent and softening behavior. The characterizing feature of this study is to propose a methodological framework based on molecular dynamics simulations and experimental tests to identify the material parameters for the model. Molecular simulations in conjunction with the Eyring viscosity theory are used to characterize the viscoelastic deformation of epoxy resins under loading. The concept of strain amplification is also adopted to account for the effect of nanoparticles on the stress–strain response of the nanocomposites. The stress softening behavior is captured by a monotonically increasing function of deformation, so-called damage variable. The results show that the model predictions of stress-strain relationships are in good agreement with experimental data at different BNP weight fractions. Finally, the constitutive model is implemented in the finite element analysis and examined by means of a benchmark example. Experimental–numerical validation confirms the predictive capability of the present modeling framework, which provides a suitable tool for analyzing BNP/epoxy nanocomposites.
Article
A new, modified molecular structural mechanics model for the determination of the elastic properties of carbon nanotubes is presented. It is designed specifically to overcome drawbacks in existing molecular structural mechanics models, which are not consistent with their underlying chemical force fields in terms of energy. As a result, modifications are motivated, developed and implemented in order to create a new, energy consistent molecular structural mechanics model. Hence, the new model leads to a better prediction of the material parameters for single wall carbon nanotubes, while the simple applicability of the approach is maintained. The results calculated for the elastic constants (Young's modulus, Poisson ratio) of armchair and zig-zag CNTs are given and discussed. Both elastic constants were found to be dependent on the chirality as well as on the carbon nanotube diameter. An asymptotic value of approximately 800 GPa was obtained for the Young's modulus and a value of approximately 0.28 for the Poisson ratio.
Article
Stretching (kr) and bending (k ) bond force constants appropriate to describe the bond stiffness of graphene and benzene are calculated using density functional theory. The effect of employing different exchange-correlation functionals for the calculation of k r and k  is discussed using the generalised gradient approximation (GGA) and the local density approximation (LDA). For benzene, kr = 7.93 mdyn Å-1 and k  = 0.859 mdyn Å rad-2 using LDA, while kr = 7.67 mdyn Å-1 and k  = 0.875 mdyn Å rad-2 using GGA. For graphene, kr = 7.40 mdyn Å-1 and k  = 0.769 mdyn Å rad-2 using LDA, while kr = 6.88 mdyn Å-1 and k  = 0.776 mdyn Å rad-2 using GGA. This means the difference between the bond force constants for benzene and graphene can be as large as ∼12%. The comparison between these two systems allows for elucidation of the effect of periodicity and substitution of carbon atoms by hydrogen in the stiffness of C–C bonds. This effect can be explained by a different redistribution of the charge density when the systems are subjected to strain. The parameters kr and k  computed here can serve as an input to molecular mechanics or finite element codes of larger carbon molecules, which in the past had frequently assumed the same bond force constants for graphene, benzene or carbon nanotubes.
Article
Molecular mechanics implemented in finite element codes is becoming a popular method to predict elastic properties of carbon nanotubes. However, the limits of application of this approach have not yet been systematically assessed. Therefore, this work investigates the accuracy and range of application of finite element analysis (FEA) to predict the elastic modulus and Poisson’s ratio of singlewall carbon nanotubes (SWCNTs), by comparing predictions of FEA to ab initio computations based on density functional theory (DFT). FEA predicts an elastic modulus which agrees well with DFT for SWCNTs with diameters larger than 0.8 nm. FEA underpredicts the Poisson’s ratio with respect to DFT, unless the FEA bond force constants previously computed for benzene are adjusted. The use of adequate values of bond force constants and the equilibrium configuration of atomic coordinates predicted by DFT as the input geometry for FEA improves its predictions. The significance of the numerical values chosen for the bond force constants in FEA is also discussed.
Article
The present work investigates Young’s modulus of hexagonal nanosheets and nanotubes based on dimensional analysis and molecular mechanics. Using second derivatives of the strain energy density revealed from molecular dynamics simulations at 0 K (i.e., molecular mechanics) with harmonic potentials for various combinations of force constants, Young’s modulus have been computed for single-walled armchair and zigzag nanotubes of different radii. This parametric study with the aid of dimensional analysis allows explicitly establishing Young’s modulus of (n, n) armchair and (n, 0) zigzag nanotubes as functions of the force constants, bond length and chiral index n. Proposed formulae are applied to estimate Young’s modulus of graphene, boron nitride, silicon carbide sheets and their nanotubes. The accuracy of the proposed formulae are verified and discussed with available data in the literature for these three sheets and their nanotubes.
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The molecular mechanics (MM) method is used to determine the frequencies and natural vibration shapes and to determine the buckling critical parameters and the postcritical deformation shapes of single-walled carbon nanotubes with twisted ends. The following two variants of the MM method are used: the standard MM method and the mixed method of molecular mechanics/molecular structure mechanics method (MM/MSM). Computer simulation shows that the MM/MSM method allows one to obtain acceptable values of frequencies and natural vibration shapes as well as of critical angles of twist, appropriate buckling modes, and postcritical deformation configurations of nanotubes compared with the same characteristics of nanotube free vibrations and buckling obtained by the standard MM method.
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A finite element model is developed to study the mechanical behavior of nano-structured materials. The model serves as a link between computational chemistry and solid mechanics by substituting discrete molecular structures with an equivalent-continuum model. The model reported here is a continuation of a previous model, which was developed by the authors and was applied to determine the effective stiffness of a graphene sheet. The present model is developed by bending the previously developed graphene sheet model around its vertical edge to form a single wall carbon nanotube, which, then, has been characterized to find its mechanical properties.
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In the present paper we intend to explain the discrepancies between theoretical and experimental values of mechanical properties for carbon nanotubes. In the numerical analysis pristine and defective carbon nanotubes are taken into account in order to find how the presence of defects affects on the mechanical properties of the analyzed structures. The imperfect carbon nanotube is modeled as the structure with one missing carbon atom that introduces five- and nine-membered rings referring to the perfect configuration of hexagons arrays.
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In this paper, the macroscopic elastic properties of carbon nanotube reinforced composites are evaluated through analysing the elastic deformation of a representative volume element (RVE) under various loading conditions. This RVE contains three components, i.e. a carbon nanotube, a transition layer between the nanotube and polymer matrix and an outer polymer matrix body. First, based on the force field theory of molecular mechanics and computational structural mechanics, an equivalent beam model is constructed to model the carbon nanotube effectively. The explicit relationships between the material properties of the equivalent beam element and the force constants have been set-up. Second, to describe the interaction between the nanotube and the outer polymer matrix at the level of atoms, the molecular mechanics and molecular dynamics computations have been performed to obtain the thickness and material properties of the transition layer. Moreover, an efficient three-dimensional eight-noded brick finite element is employed to model the transition layer and the outer polymer matrix. The macroscopic behaviours of the RVE can then be evaluated through the traditional finite element method. In the numerical simulations, the influences of various important factors, such as the stiffness of transition layer and geometry of RVE, on the final macroscopic material properties of composites have been investigated in detail.
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In molecular mechanics, the formalism of the finite element method can be exploited in order to analyze the behavior of atomic structures in a computationally efficient way. Based on the atom-related consideration of the atomic interactions, a direct correlation between the type of the underlying interatomic potential and the design of the related finite element is established. Each type of potential is represented by a specific finite element. A general formulation that unifies the various finite elements is proposed. Arbitrary diagonal- and cross-terms dependent on bond length, valence angle, dihedral angle, improper dihedral angle and inversion angle can also be considered. The finite elements are formulated in a geometrically exact setting; the related formulas are stated in detail. The mesh generation can be performed using well-known procedures typically used in molecular dynamics. Although adjacent elements overlap, a double counting of the element contributions (as a result of the assembly process) cannot occur a priori. As a consequence, the assembly process can be performed efficiently line by line. The presented formulation can easily be implemented in standard finite element codes; thus, already existing features (e.g. equation solver, visualization of the numerical results) can be employed. The formulation is applied to various interatomic potentials that are frequently used to describe the mechanical behavior of carbon nanotubes. The effectiveness and robustness of this method are demonstrated by means of several numerical examples. Copyright © 2008 John Wiley & Sons, Ltd.
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A new structural mechanics model is developed to closely duplicate the atomic configuration and behaviours of single-walled carbon nanotubes (SWCNTs). The SWCNTs are effectively represented by a space frame, where primary and secondary beams are used to bridge the nearest and next-nearest carbon atoms, to mimic energies associated with bond stretching and angle variation, respectively. The elastic properties of the frame components are generalized from molecular dynamics (MD) simulation based on an accurate ab initio force field, and numerical analyses of tension, bending, and torsion are carried out on nine different SWCNTs. The space-frame model also closely duplicates the buckling behaviours of SWCNTs in torsion and bending. In addition, by repeating the same process with continuum shell and beam models, new elastic and section parameters are fitted from the MD benchmark experiments. As an application, all three models are employed to study the thermal vibration behaviours of SWCNTs, and excellent agreements with MD analyses are found. The present analysis is a systematic structural mechanics attempt to fit SWCNT properties for several basic deformation modes and applicable to a variety of SWCNTs. The continuum models and fitted parameters may be used to effectively simulate the overall deformation behaviours of SWCNTs at much larger length- and timescales than pure MD analysis.
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In order to understand the underlying mechanisms of inelastic material behavior and nonlinear surface interactions, which can be observed on macro-scale as damping, softening, fracture, delamination, frictional contact etc., it is necessary to examine the molecular scale. Force fields can be applied to simulate the rearrangement of chemical and physical bonds. However, a simulation of the atomic interactions is very costly so that classical molecular dynamics (MD) is restricted to structures containing a low number of atoms such as carbon nanotubes. The objective of this paper is to show how MD simulations can be integrated into the finite element method (FEM) which is used to simulate engineering structures such as an aircraft panel or a vehicle chassis. A new type of finite element is required for force fields that include multi-body potentials. These elements take into account not only bond stretch but also bending, torsion and inversion without using rotational degrees of freedom. Since natural lengths and angles are implemented as intrinsic material parameters, the developed molecular dynamic finite element method (MDFEM) starts with a conformational analysis. By means of carbon nan-otubes and elastomeric material it is demonstrated that this pre-step is needed to find an equilibrium configuration before the structure can be deformed in a succeeding loading step.
Conference Paper
In this paper a new method is introduced for carbon nanotubes modeling. It combines features of Molecular Mechanics and Finite Element Analysis. This method is based on the development of a new finite element, whose internal energy is determined by the semi-empirical Brenner molecular potential model; all quantities are calculated analytically in order to gain more accuracy. The method is validated through comparisons to results provided by other researchers and are obtained either by experimental procedures or theoretical predictions. The bending and shearing of CNTs is also simulated.
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The authors report the parameters for a new generic force field, DREIDING, that they find useful for predicting structures and dynamics of organic, biological, and main-group inorganic molecules. The philosophy in DREIDING is to use general force constants and geometry parameters based on simple hybridization considerations rather than individual force constants and geometric parameters that depend on the particular combination of atoms involved in the bond, angle, or torsion terms. Thus all bond distances are derived from atomic radii, and there is only one force constant each for bonds, angles, and inversions and only six different values for torsional barriers. Parameters are defined for all possible combinations of atoms and new atoms can be added to the force field rather simply. This paper reports the parameters for the nonmetallic main-group elements (B, C, N, O, F columns for the C, Si, Ge, and Sn rows) plus H and a few metals (Na, Ca, Zn, Fe). The accuracy of the DREIDING force field is tested by comparing with (i) 76 accurately determined crystal structures of organic compounds involving H, C, N, O, F, P, S, Cl, and Br, (ii) rotational barriers of a number of molecules, and (iii) relative conformational energies and barriers of a number of molecules. The authors find excellent results for these systems.
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A finite element technique is used to mimic radial deformation of single‐walled carbon nanotubes under hydrostatic pressure. The elastic deformation of nanotubes is modeled via elastic beams. Properties of the beam element are evaluated by considering characteristics of the covalent bonds between the carbon atoms in a hexagonal lattice. Applying the beam model in a three dimensional space, the elastic properties of the nanotube in the transverse direction are evaluated. The effects of diameter and wall thickness on the radial and circumferential elastic moduli of zigzag and armchair nanotubes are considered. Results are in good agreement with molecular structural mechanics data in the literature.
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An idea of ``spatial periodic strain'' is proposed and an equilibrium relationship is established for the mechanics of zigzag single-walled carbon nanotubes (SWCNTs). An efficient approach is presented to investigate mechanical properties of zigzag SWCNT, and its validity is demonstrated by comparing its calculation results with existing results.
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This paper studies effects of rotary inertia and shear deformation on transverse wave propagation in individual carbon nanotubes (CNTs) within terahertz range. Detailed results are demonstrated for transverse wave speeds of doublewall CNTs, based on Timoshenko-beam model and Euler-beam model, respectively. The present models predict some terahertz critical frequencies at which the number of wave speeds changes. The effects of rotary inertia and shear deformation are negligible and transverse wave propagation can be described satisfactorily by the existing single-Euler-beam model only when the frequency is far below the lowest critical frequency. When the frequency is below but close to the lowest critical frequency, rotary inertia and shear deformation come to significantly affect the wave speed. Furthermore, when the frequency is higher than the lowest critical frequency, more than one wave speed exists and transverse waves of given frequency could propagate at various speeds that are considerably different than the speed predicted by the single-Euler-beam model. In particular, rotary inertia and shear deformation have a significant effect on both the wave speeds and the critical frequencies especially for CNTs of larger radii. Hence, terahertz transverse wave propagation in CNTs should be better modeled by Timoshenko-beam model, instead of Euler-beam model.
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This paper reports a study of the elastic behavior of multi-walled carbon nanotubes (MWCNTs). The nested individual layers of an MWCNT are treated as single-walled frame-like structures and simulated by the molecular structural mechanics method. The interlayer van der Waals forces are represented by Lennard–Jones potential and simulated by a nonlinear truss rod model. The computational results show that the Young's moduli and shear moduli of MWCNTs are in the ranges of 1.05±0.05 and 0.40±0.05 TPa, respectively. Results indicate that the tube diameter, tube chirality and number of tube layers have some noticeable effects on the elastic properties of MWCNTs. Furthermore, it has been demonstrated that the inner layers of an MWCNT can be effectively deformed only through the direct application of tensile or shear forces, not through van der Waals interactions.
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In this paper, an atomic-scale finite-element (AFE) model is proposed for single-walled carbon nanotubes (SWCNTs), which are considered to behave like space-frame structures when subjected to loadings. To create the AFE models, three-dimensional beam elements are used to model the bonds between carbon atoms as loading-carrying elements, while the nodes are placed at the locations of carbon atoms to connect the loading-carrying elements. The material properties of beam elements can be determined by using a linkage between molecular and continuum mechanics. In order to evaluate the AFE model and its performance, the influence of tube wall thickness on Young's modulus of SWCNTs is investigated. It is found that the selection of wall thickness significantly affects the magnitude of the Young's modulus. For the values of wall thickness used in this study, the obtained values of Young's modulus agree well with the corresponding theoretical results. Furthermore, the results also illustrate that Young's modulus is inversely proportional to the wall thickness. The presented results demonstrate that the proposed AFE model can be used as a valuable tool for studying the mechanical behaviour of carbon nanotubes.
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In this paper, the influence of various vacancy defects on the critical buckling loads and strains in carbon nanotubes under axial compression is investigated via a new structural model in ABAQUS software. The necessity of desirable conditions and expensive tests for experimental methods, in addition to the time expenditure required for atomic simulations, are the motivation for this work, which, in addition to yielding accurate results, avoids the obstacles of the previous methods. In fact, this model is a combination of other structural models designed to eliminate the deficiencies inherent in individual approaches. Because the present model is constructed in the CAE space of ABAQUS, there is no need to program for different loading and boundary conditions. A nonlinear connector is considered for modeling of stretching and torsional interactions, and a nonlinear spring is used for modeling of the angle variation interactions. A Morse potential is employed for stretching and bending potentials, and a periodic type of bond torsion is used for torsion interactions. The effect of different types of vacancy defects at various locations on the critical buckling loads and strains is studied for zigzag and armchair nanotubes with various aspect ratios (Length/Diameter). Comparison of our results with those of buckling of shells with cutouts indicates that vacancy defects in the carbon nanotubes can most likely be modeled as cutouts of the shells. Finally, results of the present structural model are compared with those from molecular dynamics (MD) simulation and show good agreement between our model and the MD model.
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The objective of the study is to develop a totally new theory and structural mechanics model for phonon dispersion analysis of carbon nanotubes. The fundamental theory and computational algorithm for phonon dispersion analysis of carbon nanotubes are developed based on the symplectic theory and algorithm established in applied mechanics in recent years. Carbon nanotubes are simulated by two kinds of structural mechanics models, i.e. the conventional sub-structure model and the inter-belt model. The variational principle for wave dissipation analysis of periodic structure is given on the basis of the symplectic-mathematical theory. Numerical examples are carried out to demonstrate the validity of the theory and algorithm developed. By the comparison of the results obtained by the two kinds of structural mechanics models, it can be found that the inter-belt model has more advantages than the conventional sub-structure model in the calculation of phonon spectra of nanotubes. As a basic research work, the present study illustrates well the potential of the symplectic-mathematical theory as well as the inter-belt model and is valuable for the further research in computational nanomechanics.
Article
In this paper, two different approaches for modeling the behaviour of carbon nanotubes are presented. The first method models carbon nanotubes as an inhomogeneous cylindrical network shell using the asymptotic homogenization method. Explicit formulae are derived representing Young’s and shear moduli of single-walled nanotubes in terms of pertinent material and geometric parameters. As an example, assuming certain values for these parameters, the Young’s modulus was found to be 1.71TPa, while the shear modulus was 0.32TPa. The second method is based on finite element models. The inter-atomic interactions due to covalent and non-covalent bonds are replaced by beam and spring elements, respectively, in the structural model. Correlations between classical molecular mechanics and structural mechanics are used to effectively model the physics governing the nanotubes. Finite element models are developed for single-, double- and multi-walled carbon nanotubes. The deformations from the finite element simulations are subsequently used to predict the elastic and shear moduli of the nanotubes. The variation of mechanical properties with tube diameter is investigated for both zig-zag and armchair configurations. Furthermore, the dependence of mechanical properties on the number of nanotubules in multi-walled structures is also examined. Based on the finite element model, the value for the elastic modulus varied from 0.9 to 1.05TPa for single and 1.32 to 1.58TPa for double/multi-walled nanotubes. The shear modulus was found to vary from 0.14 to 0.47TPa for single-walled nanotubes and 0.37 to 0.62 for double/multi-walled nanotubes.
Article
Based on a method of molecular structural mechanics (MSM), the effect of environmental temperature on elastic properties of armchair and zigzag single-walled carbon nanotubes is investigated. Single-walled carbon nanotubes with different chiral vectors are considered as a molecular structural mechanics model, which is composed of the discrete molecular structures through the carbon-to-carbon bonds. By considering the effect of environmental temperature on force constant values of the bonds stretching, bonds angle bending and torsional resistance, the corresponding basic parameters of a truss of the single-walled carbon nanotubes are obtained in different environmental temperatures, respectively. Nanoscale structural mechanics simulation for the elastic properties of single-walled carbon nanotubes in different environmental temperatures reveals that the elastic modulus of single-walled carbon nanotubes decreases significantly with the increase of environmental temperature. It is noted that the Young's modulus of single-walled carbon nanotubes is more sensitive to environmental temperature than the shear modulus.
Article
Boron-nitride nanotubes can be thought of as rolled sheets of plane hexagonal boron-nitride. In this paper a computationally efficient modeling approach is pursued. The honeycomb-like structure of the lattice is exploited and a special finite element is developed based on this hexagonal pattern. The internal energy is calculated using semi-empirical molecular mechanics functions and energy minimization algorithms are applied in order to obtain the equilibrium state under various loading conditions. Results are found to be in agreement with data found in the open literature. The introduced modeling approach provides a computationally efficient way to analyze nanotubes without the need of large-scale simulations, while it does not require lattice periodicity and structural perfection.
Article
A partitioned-domain multiscale method is a computational framework in which certain key regions are modeled atomistically while most of the domain is treated with an approximate continuum model (such as finite elements). The goal of such methods is to be able to reproduce the results of a fully atomistic simulation at a reduced computational cost. In recent years, a large number of partitioned-domain methods have been proposed. Theoretically, these methods appear very different to each other making comparison difficult. Surprisingly, it turns out that at the implementation level these methods are in fact very similar. In this paper, we present a unified framework in which fourteen leading multiscale methods can be represented as special cases. We use this common framework as a platform to test the accuracy and efficiency of the fourteen methods on a test problem; the structure and motion of a Lomer dislocation dipole in face-centered cubic aluminum. This problem was carefully selected to be sufficiently simple to be quick to simulate and straightforward to analyze, but not so simple to unwittingly hide differences between methods. The analysis enables us to identify generic features in multiscale methods that correlate with either high or low accuracy and either fast or slow performance. All tests were performed using a single unified computer code in which all fourteen methods are implemented. This code is being made available to the public along with this paper.
Article
In this paper, to investigate the buckling characteristics of carbon nanotubes, an equivalent beam model is first con-structed. The molecular mechanics potentials in a C–C covalent bond are transformed into the form of equivalent strain energy stored in a three dimensional (3D) virtual beam element connecting two carbon atoms. Then, the equivalent stiff-ness parameters of the beam element can be estimated from the force field constants of the molecular mechanics theory. To evaluate the buckling loads of multi-walled carbon nanotubes, the effects of van-der Waals forces are further modeled using a newly proposed rod element. Then, the buckling characteristics of nanotubes can be easily obtained using a 3D beam and rod model of the traditional finite element method (FEM). The results of this numerical model are in good agree-ment with some previous results, such as those obtained from molecular dynamics computations. This method, designated as molecular structural mechanics approach, is thus proved to be an efficient means to predict the buckling characteristics of carbon nanotubes. Moreover, in the case of nanotubes with large length/diameter, the validity of Euler's beam buckling theory and a shell model with the proper material properties defined from the results of present 3D FEM beam model is investigated to reduce the computational cost. The results of these simple theoretical models are found to agree well with the existing experimental results.
Article
This paper investigates the hyperelastic behaviour of single wall carbon nanotubes (SWCNTs) by means of a finite element-based lattice approach. A one-term incompressible Ogden-type hyperelastic model is chosen to describe the mechanical response of SWCNTs under tensile loading. In order to determine the material constants of the model, numerical tests are conducted on a representative arrangement of carbon atoms, establishing equality between the Ogden strain–energy and the variation of the Tersoff–Brenner interatomic potential. The material constants determined here are then used in numerical simulations carried out on SWCNTs models. A good predictive capability of the present model is found when the obtained results are compared to published data. A first conclusion obtained from the present work suggests that a value of 0.147 nm for the Csingle bondC bond equivalent diameter is suitable for the hyperelastic description of SWCNTs. A second conclusion reveals a prediction of 0.51 for the breaking strain of SWCNTs under tension, which is in excellent agreement with results obtained from molecular dynamics simulations and continuum theory.
Article
A formulation for the equivalent mechanical properties (Young's modulus, shear modulus and Poisson's ratio) of the C–C bond of CNTs under tensile and bending small deformations is derived. The C–C bonds behave like structural elements with negligible lateral deformation when stretched. The results from the model have been applied to structural mechanics truss assemblies representing single wall nanotube and nanoropes, providing good comparison with existing numerical and experimental results available.
Article
A finite element simulation technique for estimating the mechanical properties of multi-walled carbon nanotubes is developed. In the present modeling concept, individual carbon nanotube is simulated as a frame-like structure and the primary bonds between two nearest-neighboring atoms are treated as beam elements, the beam element properties are determined via the concept of energy equivalence between molecular dynamics and structural mechanics. As to the simulation of the interlayer van der Waals force which has intrinsic nonlinearity and complicated applying region, a simplifying method is proposed that the interlayer pressure caused by van der Waals force instead of the force itself is to be considered, and we make use of the linear part of the interlayer pressure near the equilibrium condition to avoid the nonlinearity in problem, then linear spring elements whose stiffness is determined by equivalent force concept can be utilized to simulate the interlayer van der Waals force such that significant modeling and computing effort is saved in performing the finite element analysis. Numerical examples for estimating the mechanical properties of nanotubes, such as axial and radial Young’s modulus, shear modulus, natural frequency, buckling load, etc., are presented to illustrate the accuracy of this simulation technique. By comparing to the results found in the literature and the possible analytical solutions, it shows that the obtained mechanical properties of nanotubes by the present method agree well with their comparable results. In addition, the relations between these mechanical properties and the nanotube size are also discussed.
Article
A critical review on the validity of different experimental and theoretical approaches to the mechanical properties of carbon nanotubes for advanced composite structures is presented. Most research has been recently conducted to study the properties of single-walled and multi-walled carbon nanotubes. Special attention has been paid to the measurement and modeling of tensile modulus, tensile strength, and torsional stiffness. Theoretical approaches such as molecular dynamic (MD) simulations, finite element analysis, and classical elastic shell theory were frequently used to analyze and interpret the mechanical features of carbon nanotubes. Due to the use of different fundamental assumptions and boundary conditions, inconsistent results were reported. MD simulation is a well-known technique that simulates accurately the chemical and physical properties of structures at atomic-scale level. However, it is limited by the time step, which is of the order of 10−15 s. The use of finite element modeling combined with MD simulation can further decrease the processing time for calculating the mechanical properties of nanotubes. Since the aspect ratio of nanotubes is very large, the elastic rod or beam models can be adequately used to simulate their overall mechanical deformation. Although many theoretical studies reported that the tensile modulus of multi-walled nanotubes may reach 1 TPa, this value, however, cannot be directly used to estimate the mechanical properties of multi-walled nanotube/polymer composites due to the discontinuous stress transfer inside the nanotubes.
Article
The research in the paper proposes the effective in-plane stiffness and bending rigidity of armchair and zigzag carbon nanotubes (CNTs) through the analysis of a representative volume element (RVE) of the graphene layer via continuous elastic models. The bonds in the RVE are modeled with corresponding stretching and rotating springs. The in-plane stiffness of CNT in the equivalent elastic plate model is first obtained by equating the energy stored in the RVE and the strain energy induced in the continuous plate model under a uniaxial stretching subjected to CNT. The corresponding bending rigidity of CNT is derived by considering the inversion contributing to the bending resistance of the graphene sheet from a large deflection analysis in the elastic plate model of CNT. The results show that the in-plane stiffness of zigzag nanotube is more sensitive to the size of the tube than that of armchair nanotube. Besides, the in-plane stiffness of armchair nanotube is almost twice as big as that of zigzag nanotube at small size. Furthermore, the results also show that the effect of axial deformation on the derivation of bending rigidity is more obvious in armchair nanotube than that in zigzag nanotube at bigger curvature of bending. In addition, the explanation on the result that the bending rigidity of zigzag nanotubes is bigger than that of armchair tubes is provided from a mechanics analysis. In the end, the effect of radius on the bending rigidity is discussed in the research. It is hoped that the research in this paper may provide a benchmark on the derivation of mechanical properties of CNT from continuum models.
Article
The aim of this paper is to propose a Single Walled Carbon Nanotube (SWCNT) finite element (FE) model, based on the use of non-linear and torsional spring elements, to evaluate its mechanical properties. The choice of the spring elements to build the FE model, was based on the observation that other elements as beam, truss or shell are not very applicable because of the complex interaction of many atoms and the absence of rotational degrees of freedom. Moreover, it was also possible to model the bond interaction without introducing any non-physical variable, such as area and inertia of atoms linkage when using beam elements.With the proposed model, the influence of tube diameter and chirality on the Young’s modulus of SWCNTs was investigated. In particular, armchair, zig-zag and chiral nanotubes, with different size, were tested under uniaxial load.The results show that good agreement was achieved with existing experimental results. The presented results demonstrate that the proposed FE model may also provide a valuable numerical tool for the prediction of the strength behaviour of single walled carbon nanotubes.
Article
A three-dimensional finite element (FE) model for armchair, zigzag and chiral single-walled carbon nanotubes (SWCNTs) is proposed. The model development is based on the assumption that carbon nanotubes, when subjected to loading, behave like space-frame structures. The bonds between carbon atoms are considered as connecting load-carrying members, while the carbon atoms as joints of the members. To create the FE models, nodes are placed at the locations of carbon atoms and the bonds between them are modeled using three-dimensional elastic beam elements. The elastic moduli of beam elements are determined by using a linkage between molecular and continuum mechanics.In order to evaluate the FE model and demonstrate its performance, the influence of tube wall thickness, diameter and chirality on the elastic moduli (Young's modulus and shear modulus) of SWCNTs is investigated. The investigation includes armchair, zigzag and chiral SWCNTs. It is found that the choice of wall thickness significantly affects the calculation of Young's modulus. For the values of wall thickness used in the literature, the obtained values of Young's modulus agree very well with the corresponding theoretical results and many experimental measurements. Dependence of elastic moduli to diameter and chirality of the nanotubes is also obtained. With increased tube diameter, the elastic moduli of the SWCNTs increase. The Young's modulus of chiral SWCNTs is found to be larger than that of armchair and zigzag SWCNTs. The presented results demonstrate that the proposed FE model may provide a valuable tool for studying the mechanical behavior of carbon nanotubes and their integration in nano-composites.
Article
This paper presents a structural mechanics approach to modeling the deformation of carbon nanotubes. Fundamental to the proposed concept is the notion that a carbon nanotube is a geometrical frame-like structure and the primary bonds between two nearest-neighboring atoms act like load-bearing beam members, whereas an individual atom acts as the joint of the related load-bearing beam members. By establishing a linkage between structural mechanics and molecular mechanics, the sectional property parameters of these beam members are obtained. The accuracy and stability of the present method is verified by its application to graphite. Computations of the elastic deformation of single-walled carbon nanotubes reveal that the Young’s moduli of carbon nanotubes vary with the tube diameter and are affected by their helicity. With increasing tube diameter, the Young’s moduli of both armchair and zigzag carbon nanotubes increase monotonically and approach the Young’s modulus of graphite. These findings are in good agreement with the existing theoretical and experimental results.
Article
The bending mechanical property of carbon nanotubes are numerically investigated in this paper. An advanced finite element analysis package, ABAQUS, is used to simulate the formation of rippling which is the appearance of wavelike distortion on the inner arc of the bent nanotubes, caused by the severe anisotropy of carbon nanotubes and a relatively large deformation. A non-linear bending moment–curvature relationship is obtained, which shows the tangential stiffness greatly decreases when rippling appears. This result can be used to explain the phenomenon and conclusion of the resonant experiment measuring the Young's modulus of carbon nanotubes, in which the Young's modulus calculated using linear theory is found to sharply decrease as the diameter increases [Science 283 (1999) 1513]. Here an analytical method is adopted to conduct a vibration analysis using a bi-linear bending constitution simplifying from the non-linear bending moment–curvature relationship, and the effective Young's modulus have been calculated for multi-walled carbon nanotubes of various sizes. The result carried out in the paper is similar to the measuring result is given by Poncharal et al. [Science 283 (1999) 1513].
Article
This paper reports the elastic buckling behavior of carbon nanotubes. Both axial compression and bending loading conditions are considered. The modeling work employs the molecular structural mechanics approach for individual nanotubes and considers van der Waals interaction in multi-walled nanotubes. The effects of nanotube diameter, aspect ratio, and tube chirality on the buckling force are investigated. Computational results indicate that the buckling force in axial compression is higher than that in bending, and the buckling forces for both compression and bending decrease with the increase in nanotube aspect ratio. The trends of variation of buckling forces with nanotube diameter are similar for single-walled and double-walled carbon nanotubes. Compared to a single-walled nanotube of the same inner diameter, the double-walled carbon nanotube shows a higher axial compressive buckling load, which mainly results from the increase of cross-sectional area, but no enhancement in bending load-bearing capacity. The buckling forces of nanotubes predicted by the continuum beam or column models are significantly different from those predicted by the atomistic model.
Article
The multiscale simulation is important to the development of nanotechnology and to the study of materials and systems across multiple length scales. In order to develop an efficient and accurate multiscale computation method within a unified theoretical framework, we propose an order-N atomic-scale finite element method (AFEM). It is as accurate as molecular mechanics simulations, but is much faster than the widely used order-N2 conjugate gradient method. The combination of AFEM and continuum finite element method provides a seamless multiscale computation method suitable for large scale static problems.
Article
A method has been proposed for developing structure-property relationships of nano-structured materials. This method serves as a link between computational chemistry and solid mechanics by substituting discrete molecular structures with equivalent-continuum models. It has been shown that this substitution may be accomplished by equating the molecular potential energy of a nano-structured material with the strain energy of representative truss and continuum models. As important examples with direct application to the development and characterization of single-walled carbon nanotubes and the design of nanotube-based structural devices, the modeling technique has been applied to two independent examples: the determination of the effective-continuum geometry and bending rigidity of a graphene sheet. A representative volume element of the chemical structure of graphene has been substituted with equivalent-truss and equivalent-continuum models. As a result, an effective thickness of the continuum model has been determined. The determined effective thickness is significantly larger than the inter-planar spacing of graphite. The effective bending rigidity of the equivalent-continuum model of a graphene sheet was determined by equating the molecular potential energy of the molecular model of a graphene sheet subjected to cylindrical bending (to form a nanotube) with the strain energy of an equivalent-continuum plate subjected to cylindrical bending.
Article
Two types of 2-D nano-scale finite elements, the chemical bond element and the Lennard–Jones element, are formulated based upon inter-atomic and inter-molecular force fields. A nano-scale finite element method is employed to simulate polymer field deformation. This numerical procedure includes three steps. First, a polymer field is created by an off-lattice random walk, followed by a force relaxation process. Then, a finite element mesh is generated for the polymer field. Chemical bonds are modeled by chemical bond elements. If the distance between two non-bonded atoms or monomers is shorter than the action range of the Lennard–Jones attraction (or repulsion), a Lennard–Jones element is inserted between them. Finally, external load and boundary conditions are applied and polymer chain deformation is simulated step by step. During polymer deformation, failed Lennard–Jones bond elements are removed and newly formed Lennard–Jones elements are inserted into the polymer field during each loading step. The process continues until failure occurs. Two examples are presented to demonstrate the process. Stress–strain curves of polymer fields under unidirectional tensile load are derived. Continuum mechanical properties, such as modulus and polymer strength, are determined based upon the stress strain curve. Further, throughout the deformation process one observes polymer chain migration, nano-scale void generation, void coalescence and crack development.
Article
Analytical formulations are presented to predict the elastic moduli of graphene sheets and carbon nanotubes using a linkage between lattice molecular structure and equivalent discrete frame structure. The obtained results for a graphene sheet show an isotropic behavior, in contrast to limited molecular dynamic simulations. Young’s modulus of CNT represents a high dependency of stiffness on tube thickness, while dependency on tube diameter is more tangible for smaller tube diameters. The presented closed-form solution provides an insight to evaluate finite element models constructed by beam elements. The results are in a good agreement with published data and experimental results.
Article
Carbon nanotubes (CNTs) have attracted considerable attention in scientific communities due to their remarkable mechanical, thermal and electrical properties (high stiffness, high strength, resilience, etc.). In particular, mechanical properties of single wall nanotubes (SWNTs) have a Young’s modulus of about 1 TPa if normalized to their diameter showing why they are widely considered as reinforcing elements in advanced low weight composite structures. The determinations of mechanical properties of SWNT are currently investigated both experimentally and theoretically. However, to determine CNTs mechanical properties in a direct experimental way is a challenging and not economical task because of the technical difficulties and the costs involved in the manipulation of nanoscale objects. Due to the handling difficulty, estimation of mechanical properties using computer simulations are being performed by several author with different approaches.In this work a Finite Element Model of SWNTs based on molecular mechanics theory is proposed to evaluate mechanical properties as Young’s modulus, ultimate strength and strain. The novelty of the model lies on the use of non-linear and torsional spring elements, to evaluate SWNTs mechanical properties and tensile failure. With this approach, it was possible to model the bond interaction without making any assumption on non-physical variable, i.e. area, inertia of atom interaction when using beam approach. With the proposed model it was able to understand the evolution of the tensile failure of nanotubes. Moreover, it is important to point out that while most of the fracture evolution studies use molecular dynamics theory and technique, the proposed approach leaded to a minor computational time with the possibility to simulate large atoms system. The calculated mechanical properties show good agreement with existing other work and experimental results.
Article
The preparation of a new type of finite carbon structure consisting of needlelike tubes is reported. Produced using an arc-discharge evaporation method similar to that used for fullerene sythesis, the needles grow at the negative end of the electrode used for the arc discharge. Electron microscopy reveals that each needle comprises coaxial tubes of graphitic sheets ranging in number from two up to about 50. On each tube the carbon-atom hexagons are arranged in a helical fashion about the needle axis. The helical pitch varies from needle to needle and from tube to tube within a single needle. It appears that this helical structure may aid the growth process. The formation of these needles, ranging from a few to a few tens of nanometers in diameter, suggests that engineering of carbon structures should be possible on scales considerably greater than those relevant to the fullerenes.
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An empirical many-body potential-energy expression is developed for hydrocarbons that can model intramolecular chemical bonding in a variety of small hydrocarbon molecules as well as graphite and diamond lattices. The potential function is based on Tersoff's covalent-bonding formalism with additional terms that correct for an inherent overbinding of radicals and that include nonlocal effects. Atomization energies for a wide range of hydrocarbon molecules predicted by the potential compare well to experimental values. The potential correctly predicts that the pi-bonded chain reconstruction is the most stable reconstruction on the diamond \{111\} surface, and that hydrogen adsorption on a bulk-terminated surface is more stable than the reconstruction. Predicted energetics for the dimer reconstructed diamond \{100\} surface as well as hydrogen abstraction and chemisorption of small molecules on the diamond \{111\} surface are also given. The potential function is short ranged and quickly evaluated so it should be very useful for large-scale molecular-dynamics simulations of reacting hydrocarbon molecules.
Article
The orphan glutamate receptor delta2 is selectively expressed in Purkinje cells and plays a crucial role in cerebellar functions. Recently, ataxia in the hotfoot mouse ho4J was demonstrated to be caused by a deletion in the delta2 receptor gene (Grid2) removing the N-terminal 170 amino acids of the delta2 receptor. To understand how delta2 receptors function, we characterized mutations in eight additional spontaneously occurring hotfoot alleles of Grid2. The mouse Grid2 gene consists of 16 exons, spanning approximately 1.4 Mb. Genomic DNA analysis showed that seven hotfoot mutants had a deletion of one or more exons encoding the N-terminal domain of delta2 receptors. The exception is ho5J, which has a point mutation in exon 12. Deletions in ho7J, ho9J, ho11J and ho12J mice result in the in-frame deletion of between 40 and 95 amino acids. Expression of constructs containing these deletions in HEK293 cells resulted in protein retention in the endoplasmic reticulum or cis-Golgi without transport to the cell surface. Coimmunoprecipitation assays indicated that these deletions also reduce the intermolecular interaction between individual delta2 receptors. These results indicate that the deleted N-terminal regions are crucial for oligomerization of delta2 receptors and their subsequent transport to the cell surface of Purkinje cells. The relatively large size of the Grid2 gene may be one of the reasons why many spontaneous mutations occur in this gene. In addition, the frequent occurrence of in-frame deletions within the N-terminal domain in hotfoot mutants suggests the importance of this domain in the function of delta2 receptors.