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The molecular dynamic finite element method (MDFEM)
Computers, Materials & Continua
Abstract
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.
... Following [4], the finite element approaches, which consider the atomic interaction in a discrete sense and do not use any kind of homogenization techniques (e.g. Cauchy-Born rule), can be classified into two groups: Firstly, the bond-related approaches [5][6][7][8][9][10][11][12], where each individual term of the interatomic potential energy is considered separately as a single finite element. And secondly, the atom-related approaches [3,4], where all terms of the internal energy that belong to a certain atom are considered as a finite element. ...
... In order to illustrate the procedure and to make clear the difference between type A and B, the atom-related lists E for the bond length terms, valence angle terms and dihedral angle terms are stated in the following. Assuming that the coordination number C = 3, one obtains the following lists (1,2,6), (1,3,7), (1,3,8), (1,4,9), (1,2,5,12), (1,2,6,13), (1,2,6,14), (1,3,7,15), (1,3,7,16), (1,3,8,17), (1,3,8,18), (1,4,9,19), (1,4,9,20), (1,4,10,21), (2,1,4,9), (2, 1, 4, 10), (3, 1, 2, 5), (3, 1, 2, 6), (3,1,4,9), (3, 1, 4, 10), (4, 1, 2, 5), (4, 1, 2, 6), ...
... In order to illustrate the procedure and to make clear the difference between type A and B, the atom-related lists E for the bond length terms, valence angle terms and dihedral angle terms are stated in the following. Assuming that the coordination number C = 3, one obtains the following lists (1,2,6), (1,3,7), (1,3,8), (1,4,9), (1,2,5,12), (1,2,6,13), (1,2,6,14), (1,3,7,15), (1,3,7,16), (1,3,8,17), (1,3,8,18), (1,4,9,19), (1,4,9,20), (1,4,10,21), (2,1,4,9), (2, 1, 4, 10), (3, 1, 2, 5), (3, 1, 2, 6), (3,1,4,9), (3, 1, 4, 10), (4, 1, 2, 5), (4, 1, 2, 6), ...
A mathematically rigorous methodology for embedding the governing equations of molecular dynamics in the formalism of the finite element method is presented. Only one generalized finite element type is needed to cover all different types of existing interatomic potentials. The finite element type is simply specified by two parameters characterizing the type of the interatomic potential to be considered. Built on this formulation a partitioned Runge–Kutta method—summarizing a wide range of explicit and implicit, single- and multi-stage, lower and higher order time integration schemes—is embedded in a unified manner. The required finite element residual vector and the related Jacobian matrix are stated explicitly. The related FE-mesh coincides with the neighborhood lists used in standard molecular dynamics enabling the use of common tools. The range, versatility and performance of the proposed finite element formulation have been demonstrated by means of several numerical examples.
... We first attempt to determine the parametersᾱ andD following [100]:ᾱ = 22.36 nm −1 andD = n · 0.48664 aJ. For graphene, we use n = 4/3 [108]; then,D = 0.648853 aJ. Using these values of the parametersᾱ andD and the value of the parameter k ba from [100], we obtain the following values of the parameters k r and k θ : k r = 648.853 ...
... where φ is the current value of the dihedral angle andk da is a specified parameter. It is assumed that for graphene small strains the contribution of both elementary potential energies, V da and V ia , can be represented as one elementary potential energy of (28) (cf., [108]). The harmonic approximation of the energyṼ da can be represented as ...
... wherek τ is the torsional stiffness of each of the four "levers" whose torsion leads to torsion of the entire bond considered [96]. 19 Nasdala et al. (cf., [108]) obtained the value of the coefficient k c = 0.6857 in solving the problem of the displacement of one atom of graphene out of the plane. In solving the problem of initial post-buckling deformation of a compressed SLGS (see [78]), we estimated the value of this coefficient as k c = 0.33. ...
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.
... [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. ...
... We determine the parameters a and D following [57]: a = 22.36 nm 21 and D = n Á 0:48664 aJ. For graphene, we use n = 4/3 [71]; then D = 0:648853 aJ. Using these values of the parameters a and D and the value of the parameter k ba from [57], we obtain the following values of the parameters k r and k u : ...
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.
... Alternatively, Nasdala et al. [35] proposed a 4-node element for the accurate treatment of mechanical torsional potentials, as well as a 2-node and a 3-node element for bond stretching and bond-angle bending potentials [36]. The latter work emphasises the advantages of element topologies, which are not explicitly dependent on rotational DoF, but rather use translational DoF only. ...
... The latter work emphasises the advantages of element topologies, which are not explicitly dependent on rotational DoF, but rather use translational DoF only. Additionally, Nasdala et al. [35,36] demonstrate the ease with which MDFEM elements perform conformational analyses as compared to structural elements and discuss time integrator schemes for dynamic simulations [36]. While obtaining the Jacobian (i.e. ...
... The latter work emphasises the advantages of element topologies, which are not explicitly dependent on rotational DoF, but rather use translational DoF only. Additionally, Nasdala et al. [35,36] demonstrate the ease with which MDFEM elements perform conformational analyses as compared to structural elements and discuss time integrator schemes for dynamic simulations [36]. While obtaining the Jacobian (i.e. ...
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.
... 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. ...
... 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]. ...
... Disadvantageous is the comparatively high effort to generate simulation models. Sample applications of the MDFEM include carbon nanotubes [9] as well as elastomers [17]. ...
... Advanced computational techniques such as DFT calculations and MD simulations are timeconsuming. Molecular dynamic finite element methods, sometime known as atomic-scale finite element methods or atomistic finite element methods, have been developed to analyze nanostructured materials in a computationally efficient way [46][47][48][49][50]. In MDFEM, atoms and atomic displacements are considered as nodes and translational degrees of freedom (nodal displacements), respectively. ...
... Using the well-known Newton-Raphson method (see e.g. Press et al. [51]), and following the procedure in the previous work [46][47][48][49][50] ...
The uniaxial tensile mechanical properties of pristine and defective hexagonal boron nitride (BN) and silicon carbide (SiC) sheets are investigated through a molecular dynamics finite element method with Tersoff and Tersoff-like potentials. 2-Atom vacancy and 2 types of Stone-Wales defects are considered. It is found that uniaxial tensile stress-strain curves of defective and pristine sheets are almost identical up to fracture points. A centered single defect reduces significantly fracture stress and fracture strain from those of the corresponding pristine sheet. In contrast, Young׳s modulus is nearly unchanged by a single defect. One 2-atom vacancy in the sheet׳s center reduces 15-18% and 16-25% in fracture stress, and 32-34% and 32-48% in fracture strain of BN and SiC sheets, respectively. Reduction in fracture properties depends on the tensile direction as well as the orientation of Stone-Wales defects.
... The MD method has increasingly been incorporated in the Finite Element Method (FEM) framework [12][13][14][15][16][17][18][19][20][21][22][23] as the equilibrium equations of MD and FEM may be expressed in equivalent forms. The resulting Atomistic Finite Element Method (AFEM) [13], also referred to as Molecular Dynamic Finite Element Method (MDFEM) [24], is both computationally more favourable than MD [25], and offers a significant increase in compatibility and integrability with larger scale continuum FEM simulations. Structural mechanics elements (e.g. ...
... 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. ...
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 MD method has increasingly been incorporated in FEM [20][21][22][23][24][25][26][27][28][29][30][31] for structural simulations, as the equilibrium equations of MD and FEM can be expressed in equivalent forms. The resulting MDFEM, also referred to as Atomic-Scale Finite Element Method (AFEM) [21], is computationally more favourable than MD [21,24]. ...
... Several MDFEM, which use specifically derived element formulations capable of non-linearities, have been reported [20][21][22][23]32]. However, these element designs vary considerably in their topological complexity and their flexibility in accommodating different force fields, while their implementation does not readily extent to multi-physics. ...
Emerging Graphene-based structures hold the highest transformative potential for the composite landscape, with worldwide research developing a myriad of novel electronic, structural and coating applications. Progress in synthesis has led to Graphene, Carbon Nanotubes (CNT) and complex 3D Pillared Graphene (PGS) being precisely manufactured with dimensions up to, and well beyond the millimetre scale. The increased ability to synthesize nano-electro-mechanical-systems of varying architectures and dimensions requires a physically accurate and numerically efficient modelling method. Modelling large giant covalent network structures is computationally unaffordable with quantum mechanics or even molecular dynamics methods.
We derive a new method, embedding Molecular Dynamics (MD) exactly in the computationally more favourable Finite Element Method (FEM). This work presents the first MDFEM which is exactly equivalent to MD, but within FEM, for any force field including bond-order reactive or polarisable fields. By modelling buckling, failure, vibrational and piezoelectric behaviours, among others, this method can contribute to the design of Graphene composites, sensors and other multi-functional nano-materials.
... The MD method has increasingly been incorporated in the Finite Element Method (FEM) framework67891011 as the equilibrium equations of MD and FEM may be expressed in equivalent forms. 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 [8,121314. ...
... 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 [8,121314. 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. The available MDFEM element designs vary considerably in complexity and their implementation is often not straight-forward8910. ...
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).
... 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. ...
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.
... Boehmite exhibits an orthorhombic unit cell (see Fig. 3a), as reported by Bokhimi et al. [29] and experimentally confirmed through X-ray diffraction [30]. More information on Boehmite and its mechanical behavior can be found in [31], where the material was comprehensively investigated experimentally and numerically using the Molecular Dynamic Finite Element Method [32]. ...
The mechanical properties of nanocomposites are significantly influenced by interfacial interactions between nanoparticles and matrix. In this work, the elastic interphase properties of boehmite nanoparticle/epoxy composites are investigated using molecular dynamics simulations. The distinctive feature of this study is the characterization of the interphase properties thanks to the concept of atomic strain, which allows to capture the stiffness gradient at the interphase. The simulation results suggest that the size of the interphase region may not only be determined by the variation of the mass density, but also by an alteration of the polymer network structure close to the particle. A significant increase of the interphase stiffness is observed for a strong chemical bonding between boehmite and epoxy, while purely physical interactions lead to a slight reduction of the interphase stiffness compared to the bulk epoxy stiffness. Finite element analyses of representative volume elements of the nanocomposite show that the homogenized elastic properties are considerably influenced by the elastic interphase properties. The proposed simulation framework not only estimates elastic interphase properties of layered structures, but can be extended for studying the elastic properties of arbitrarily shaped contiguous subsections of molecular models.
... Their assumption of charges of +3e and −3e, for the boron and nitrogen atoms respectively, is intuitive but it is not consistent with results from population analysis of ab initio quantum mechanical calculations [18,19]. Nasdala et al. developed a multi element approach (MDFEM) that uses different element types to represent the force terms in the underlying molecular mechanics model [20]. To account for different types of forces, the elements consist of two node spring elements to carry bond elongation, three node coupled bond elements to carry angular deformation and four node elements to carry torsion. ...
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.
... The MD enriched continuum method by Belytschko and Xiao (2003) and Xiao and Belytschko (2003) was also another pioneering approach to couple a potential energy Hamiltonian calculation conducted on a fine scale MD domain with a Lagrangian calculation on a coarse scale continuum domain with an overlapped bridging domain among the two representations. Recently, an implementation of interatomic potential laws within a displacement-based finite element (FE) formulation has also been proposed in Nasdala et al. (2010), with a rigorous implicit solution scheme, aiming at generating models where non-linear discrete and continuous systems can be suitably combined. ...
... So we were able to update the all-atom rubber model [9] to an explicit analysis which neither makes use of the Newton-Raphson procedure nor is limited by the bandwidth of any stiffness matrix since it requires only a mass matrix. As shown in [19], a 10,000 atom model leads to quite smooth load-deflection curves. The goal of this paper is to give an answer to some of the remaining questions: How many atoms are needed to get a representative rubber model, does the result depend on the loading velocity, the initial configuration or the strain amplitude, and what is the influence of the cross-linking density? ...
... Molecular dynamic finite element methods, sometime known as atomic-scale finite element methods or atomistic finite element methods, have been developed to analyze nanostructured materials in a computationally efficient way [43][44][45][46][47]. In MDFEM, atoms and atomic displacements are considered as nodes and translational degrees of freedom (nodal displacements), respectively. ...
Effects of vacancies and Stone-Wales defects on the mechanical properties of silicene are investigated through molecular dynamic finite element method with Tersoff potential. Young's modulus, Poisson's ratio and uniaxial tensile stress-strain curves are considered in the armchair and zigzag directions. It is found that pristine and lowly defective silicene sheets exhibit almost the same elastic nature up to fracture points. However, a single defect weakens significantly the silicene sheet, resulting in a considerable reduction in the fracture strength. One 2-atom vacancy in the sheet's center reduces 18-20 % in fracture stress and 33-35 % in fracture strain. The weakening effects of Stone-Wales defects vary with the tensile direction and the orientation of these defects.
... With the progress in computer technology, wide tri-dimensional discrete systems can nowadays be modeled by molecular dynamics (MD), accounting for nonlinear interatomic potential laws and nonlocal interactions among the discrete molecules. Attempts to couple MD with FEM have also been explored [7]. In this study, the focus regards the analysis of the ability of nonlocal molecular discrete systems to tolerate flaws. ...
Discrete systems are modeled as a network of nodes (particles, molecules, or atoms) linked by
nonlinear springs to simulate the action of van der Waals forces. Such systems are nonlocal if links connecting
non-adjacent nodes are introduced. For their topological characterization, a nonlocality index (NLI) inspired by
network theory is proposed. The mechanical response of 1D and 2D nonlocal discrete systems is predicted
according to finite element (FE) simulations based on a nonlinear spring element for large displacements
implemented in the FE programme FEAP. Uniaxial force-displacement responses of intact and defective
systems (with links or nodes removed) are numerically simulated. Strain localization phenomena, size-scale
effects and the ability to tolerate defects are investigated by varying the degree of nonlocality.
... Moreover, the model gives an effective interatomic potential between SxG atoms, it can be used for estimating third-order elastic constants using a phononbased technique ()() (Pham and Tahir Cagin 2010). Because of the asymptotic correspondence between the lattice and the continuum Green's functions, our model should be applicable to multiscale modeling of Si x Ge 1−x alloys using standard techniques (Tewary 2004;Tewary and Read 2004;Read and Tewary 2007;Nasdala, Kempe et al. 2010;Ojeda and Cagin 2010). For an excellent review of multiscale modeling in nanomechanics, see (Shen and Atluri 2004). ...
A semicontiuum Green's-function-based model is proposed for analysis of averaged mechanical characteristics of SixGe1-x. The atomistic forces in the model are distributed at discrete lattice sites, but the Green's function is approximated by the continuum GF in the far field and by the averaged lattice GF in the near field. Averaging is achieved by replacing Si and Ge atoms by identical hypothetical atoms that are x fraction Si and (1-x) fraction Ge. The parameters of the model are derived using the atomistic model from the interatomic potential between the hypothetical atoms. The interatomic potential is obtained from the radial embedded atom model proposed in an earlier paper. The parameters of the model potential are estimated partly by interpolation and partly by fitting the calculated and measured values of the cohesive energy and the lattice constant of SixGe1-x, as functions of x. The model is applied to calculate the elastic constants of SixGe1-x and the displacement and the strain field at the free surface of a semi-infinite alloy for different values of x due to a buried point defect. The elastic constants predicted by the model are used to calculate the curvature of a single crystal of Si with a 49 nm epitaxial film of Si(0.846)Ge(0.154). The calculated value (312.8 m) of the radius of curvature is in excellent agreement with the recently measured value (314.5 m) at our laboratory.
Mode-I stress intensity factors are estimated for 28 buckled two-dimensional hexagonal materials, including 6 mono-elemental (silicene, indiene, blue phosphorene, arsenene, antimonene and bismuthene) and 22 binary (CS, CSe, CTe, SiO, SiS, SiSe, SiTe, SiGe, GeO, GeS, GeSe, GeTe, SnO, SnS, SnSe, SnTe, SnGe, SnSi, InAs, InSb, GaAs and AlSb) two-dimensional materials. The crack-tip displacement field revealed from linear elastic fracture mechanics is adopted to find the stress intensity factor. Atomic-scale finite element method with Stillinger-Weber potentials is used to simulate the tensile tests. Mode-I stress intensity factors of these 28 two-dimensional materials appear ≤ 0.8 MPam, which are very small in comparison with boronitrene and graphene. Most of them exhibit their fracture toughness below 0.5 MPam. Our findings are helpful in the design of nano-devices with these two-dimensional materials.
The formation of interphases between inorganic nanofillers and polymer matrices is usually known to have a dominant impact on the overall properties of the nanocomposite. Sometimes, multiple chemical and mechanical interphases with different thicknesses coexist in one nanocomposite system. Particularly in thermosetting matrices such as epoxies, the effect of nanofillers on the properties of the matrix is not only limited to the immediate distances, but a long-range chemical alteration in polymer network occurs. In this chapter, first, we investigate the interphase properties of epoxy/boehmite model systems via a combination of atomic force microscopy approaches. Up to the resolution limitations of experimental approaches, long-range mechanical and chemical interphases are visualized via various physical and chemical property mapping of the surface. For investigating short-range interphases with sub-nanometer resolution, we propose a methodological framework to characterize the interphase properties based on molecular dynamic simulations.
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.
The performance of fiber reinforced plastics (FRP), in particular their outstanding mechanical properties, motivate research into possibilities of further enhancement due to their high lightweight potential. Nanotechnology opens a new horizon for further improvement. The research presented in this book shows a significant augmentation of the matrix dominated properties by means of nanoparticle reinforced matrices. The results clearly demonstrate the capabilities of these continuous fiber reinforced nanocomposites as well as the need to gain a comprehensive understanding of the acting principles and mechanisms depending on the material, the surface functionalisation of the particles and the dispersive properties of the matrix. The chapter gives an overview on the state of research as the starting point of the research work and the theses obtained, which contribute to a comprehensive understanding of the acting mechanisms of nano-scaled additives to polymer matrices of continuous fiber reinforced polymer composites with respect to improved matrix dominated properties, damage tolerance combined with unchanged processability.
The crack-tip displacement field and molecular dynamics finite element method with Tersoff potentials were used to find the mode-I stress intensity factors (SIF) of silicene, aluminum nitride (AlN), and silicon carbide (SiC) hexagonal sheets. Fracture properties of graphene and boronitrene are also included for comparison. It is found that K Ic t (K Ic is mode-I critical SIF and t is the sheet's thickness) of silicene, AlN, and SiC sheets are approximately 80, 66, and 47%; and 73, 64, and 45% smaller values of those of graphene for crack along the armchair and zigzag directions, respectively. The estimated fracture toughness of silicene is close to the experimental data of single-crystal silicon.
All the previous discrete systems and probably many others analyzed by the scientific Community share the common features of complex systems where the overall properties emerge from the non-local non-linear interactions between their basic network components and can only be predicted by numerically simulating the system response and the dynamic evolution of defects (cracks, node removal, link removal). Failure of the system is generally the result of a percolation of defects at different scales, which leads to complex redistribution of internal forces/flows. Understanding how the system is able to withstand perturbations, i.e., its resilience or flaw-tolerance, is a problem common to all of the disciplines mentioned in the previous section.
Phương phap phần tử hữu hạn nguyen tử (AFEM) được phat triển để xac định đặc trưng đan hồi của ống cac bon na no đơn lớp, ống boron nitride (BN), tấm BN va graphen. Sử dụng cung cac hằng số lực, kết quả mo đun đan hồi khi tinh bằng AFEM sai khac so với kết quả tinh bằng động lực học phân tử khoảng 5%. Ben cạnh đo, kết quả tinh bằng AFEM cũng rất phu hợp khi được so sanh với cac phương phap khac. Điều đo cho thấy AFEM la một phương phap đơn giản, nhanh chong xac định đặc trưng đan hồi cho cac vật liệu cấu truc na no
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.
This study performs a series of Molecular Dynamics (MD) and Molec-ular Statics (MS) simulations to investigate the mechanical properties of single-walled carbon nanotubes (SWCNTs) under a uniaxial tensile strain. The simula-tions focus specifically on the effects of the nanotube helicity, the nanotube diame-ter and the percentage of vacancy defects on the bond length, bond angle and tensile strength of zigzag and armchair SWCNTs. In this study, a good agreement is ob-served between the MD and MS simulation results for the stress-strain response of the SWCNTs in both the elastic and the plastic deformation regimes. The MS simulations reveal that in the plastic deformation regime, the tensile strength of the armchair and zigzag SWCNTs increases with an increasing wrapping angle. In addition, it is shown that the tensile strength reduces significantly at larger val-ues of the nanotube diameter. Moreover, it is observed that the tensile strength of both SWCNTs reduces as the percentage of defects within the nanotube structure increases. Finally, it is found that the results obtained from the molecular statics method are relatively insensitive to instabilities in the atomic structure, particularly in the absence of thermal fluctuations, and are in good agreement with the predic-tions obtained from the molecular dynamics method.
We have calculated the effects of structural distortions of armchair carbon nanotubes on their electronic and electrical properties. We found that the bending of the nanotubes decreases their transmission function in certain energy ranges and leads to an increased electrical resistance. Electronic structure calculations show that these energy ranges contain localized states with significant σ-π hybridization resulting from the increased curvature produced by bending. Twisting strongly affects the electronic structure of nanotubes (NTs). Normally metallic armchair (n,n) NT’s develop a band gap which initially scales linearly with twisting angle and then reaches a constant value. This saturation is associated with a structural transition to a flattened helical structure. The computed values of the twisting energy and of the band gap are strongly affected by allowing structural relaxation in the twisted structures. Finally, our calculations show that the large contact resistances observed for single-wall NT’s are likely due to the weak coupling of the NT to the metal in side bonded NT-metal configurations.
The concept of collocation, originally used by Wilson in the development of dissipative algorithms for structural dynamics, is systematically generalized and analysed. Optimal schemes within this class are developed and compared with a recently proposed family of dissipative algorithms, called a methods. The α methods are found to be superior on the basis of standard measures of dissipation and dispersion.
It is pointed out that the tendency to overshoot is an important and independent factor which should be considered in an evaluation of an implicit scheme. The basis for studying overshoot is discussed and the optimal collocation and α methods are compared. It is found that pathological overshooting is an inherent property of collocation schemes, whereas the overshooting characteristics of the α methods are good.
Due to the limitation of fabrication technologies nowadays, structural or atomistic defects are often perceived in carbon nanotubes (CNTs) during the manufacturing process. The main goal of the study aims at providing a systematic investigation of the effects of atomistic defects on the nanomechanical properties and fracture behaviors of single-walled CNTs (SWCNTs) using molecular dynamics (MD) simulation. Furthermore, the correlation between local stress distribution and fracture evolution is studied. Key parameters and factors under investigation include the number, type (namely the vacancy and Stone-Wales defects), location and distribution of defects. Results show that the nanomechanical properties of the CNTs, such as the elastic modulus, ultimate strength and ultimate strain, are greatly affected by the defects and also their percentage and type. It is also found that the CNTs present a brittle fracture as the strain attains a critical value, and in addition, the fracture crack tends to propagate along the high tensile stress concentration area. Moreover, the distribution pattern of defects is another driving factor affecting the nanomechanical properties of the CNTs and the associated fracture evolutions.
This article provides a review of the computational nanomechanics, from the ab initio methods to classical molecular dynamics simulations, and multi-temporal and spatial scale simulations. The recent improvements and developments are briefly discussed. Their applications in nanomechanics and nanotubes are also summarized.
A multi-walled carbon nanotube is modeled as a multiple-elastic cylindrical structure. The numerical-analytical method is adopted to analyze the characteristics of harmonic waves propagating along an anisotropic carbon nanotube. Each wall of the carbon nanotube is divided into three-nodal-line layer elements. The deflections of two adjacent tubes are coupled through the van der Waals. The governing equation of element is obtained from Hamilton's principle. A set of system equation of dynamics equilibrium for the entire structure is obtained by the assembling of all the elements. From solution of the eigenvalue equations, the dispersive characteristics, group velocities of multi-walled carbon nanotubes are achieved, and these properties of the six characteristic wave surfaces are also obtained.
The concept of the micropolar theory is employed to investigate vibration behaviors of carbon nanotubes. The constitutive relation has been deduced from the two-dimensional analysis of the microstructure of the carbon nanotube. Van der Waals interactions are simulated by a weak spring model. Hamilton’s principle is employed to obtain dynamics equations of the multi-walled carbon nanotube. Numerical examples for both single-walled and double-walled carbon nanotubes are presented and the significant difference in vibration behaviors between them has been distinguished. Numerical results show that fundamental frequencies for the cantilever single-walled carbon nanotube decreases with increase of the aspect ratio of them, and the fundamental frequencies of the double-walled carbon nanotube are lower than those of the single-walled carbon nanotube with the same inner diameter and length. The first four natural frequencies for the double-walled carbon are coaxial.
Like any other geometric structure or building, carbon nanotubes may break down due to either material failure or structural failure. In this paper, it is shown that the failure mechanism-of carbon nanotubes not only depends on the type and direction of loading but also on the location and number of defects. For the finite element simulations we use a new 4-node finite element without rotational degrees of freedom based on the force field method. For the examples shown here, mainly a single-walled (10,10) armchair nanotube with different Stone-Wales defects, the material parameters are directly taken from the DREIDING force field. For carbon nanotubes subject to tension a kind of material failure, i. e. a breaking of bonds, can be observed. For carbon nanotubes subject to bending, an interesting question is whether they fail due to a breaking of bonds in the tension zone, which would be similar to the tension experiment, or due to a snap-through of bonds in the compression zone. From our FE simulations, it can be concluded that neither of these two failure mechanisms, but local buckling in the compression zone can be observed. From a mechanical point of view, however, it is not a pure bifurcation problem because the buckles are formed relatively slowly which corresponds more to a snap-through problem. For carbon nanotubes subject to torsion, we have to distinguish between bifurcation problems which are the case for defect-free nanotubes and snap-through problems which can be observed for those with defects. In all cases the Stone-Wales defects are responsible for a reduction of the maximum load, about 10 % for tension and bending, and up to 30 % for torsion.
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.
This paper deals with the ground state of an interacting electron gas in an external potential v(r). It is proved that there exists a universal functional of the density, Fn(r), independent of v(r), such that the expression Ev(r)n(r)dr+Fn(r) has as its minimum value the correct ground-state energy associated with v(r). The functional Fn(r) is then discussed for two situations: (1) n(r)=n0+n(r), n/n01, and (2) n(r)= (r/r0) with arbitrary and r0. In both cases F can be expressed entirely in terms of the correlation energy and linear and higher order electronic polarizabilities of a uniform electron gas. This approach also sheds some light on generalized Thomas-Fermi methods and their limitations. Some new extensions of these methods are presented.
Self-consistent calculations of the Cu+ ion have been carried out using five different methods of approximating the Hartree-Fock exchange. These calculations have been compared with Hartree's Cu+ calculation to test the accuracy of the various approximations and to interpret their interrelations. The best results were obtained from two quite different methods. The first, suggested by Liberman with modifications which we have introduced, uses a different local exchange potential for each orbital and gives a very good approximation to the Hartree-Fock method, but with considerable computational difficulty. The second amounts to multiplying the local potential proportional to the ⅓ power of the electronic charge density, suggested by the senior author in 1951, by a constant factor α chosen to minimize the total energy. This second method is much simpler to apply than the first and gives very nearly as good orbitals, as well as a very good total energy, but gives poor one-electron energies for the x-ray levels. The reasons for the different results are discussed. The latter method, which has been empirically arrived at by a number of the workers in the energy-band field, is probably the most useful one for practical calculation.
An exact solution is obtained for the Schroedinger equation representing the motions of the nuclei in a diatomic molecule, when the potential energy function is assumed to be of a form similar to those required by Heitler and London and others. The allowed vibrational energy levels are found to be given by the formula E(n)=Ee+hω0(n+12)−hω0x(n+12)2, which is known to express the experimental values quite accurately. The empirical law relating the normal molecular separation r0 and the classical vibration frequency ω0 is shown to be r03ω0=K to within a probable error of 4 percent, where K is the same constant for all diatomic molecules and for all electronic levels. By means of this law, and by means of the solution above, the experimental data for many of the electronic levels of various molecules are analyzed and a table of constants is obtained from which the potential energy curves can be plotted. The changes in the above mentioned vibrational levels due to molecular rotation are found to agree with the Kratzer formula to the first approximation.
An empirical interatomic potential is introduced, which gives a convenient and relatively accurate description of the structural properties and energetics of carbon, including elastic properties, phonons, polytypes, and defects and migration barriers in diamond and graphite. The potential is applied to study amorphous carbon formed in three different ways. Two resulting structures are similar to experimental a-C, but another more diamondlike form has essentially identical energy. The liquid is also found to have unexpected properties.
We describe a computer program we have been developing to build models of molecules and calculate their interactions using empirical energy approaches. The program is sufficiently flexible and general to allow modeling of small molecules, as well as polymers. As an illustration, we present applications of the program to study the conformation of actinomycin D. In particular, we study the rotational isomerism about the D-Val-, L-Pro, and L-Pro-Sar amide bonds as well as comparing the energy and structure of the Sobell model and the x-ray structure of actinomycin D.
Das Variationsprinzip ∫δ¯Ψ(L − E) Ψdτ=0 (L=Energieoperator) liefert, bekanntlich die Wellengleichung im Konfigurationsraum. Es wird gezeigt, daß der Ansatz Ψ=ψ1 (χ1) ψ2 (χ1)... ψN(χN) (N=Anzahl der Elektronen) zu den Gleichungen der Hartreeschen Theorie des „selfconsistent field“ führt. Dieser Ansatz hat aber nicht die richtige Symmetrie. In dem wichtigen Spezialfall der „völligen Entartung des Termsystems“ kann aber Ψ durch ėin Produkt zweier Determinanten [Formel (50) des Textes] approximiert werden. Die entsprechende Rechnung wird durchgeführt. Die Gleichungen, die sich für ψi(χ) ergeben, enthalten „Austauschglieder“ und können als Eulersche Gleichungen eines dreidimensionalen Variationsproblems mit der Energie als Wirkungsintegral [Formel (93)] aufgefaßt werden. Die Gleichungen sind nicht wesentlich komplizierter als die von Hartree, dürften aber viel genauere Resultate ergeben. Zum Schluß wird eine Formel für die Intensitäten angegeben, die- Glieder enthält, welche einer „Umgruppierung“ der inneren Elektronen bei einem Quantensprung entsprechen.
The LCAO, or Bloch, or tight binding, approximation for solids is discussed as an interpolation method, to be used in connection with more accurate calculations made by the cellular or orthogonalized plane-wave methods. It is proposed that the various integrals be obtained as disposable constants, so that the tight binding method will agree with accurate calculations at symmetry points in the Brillouin zone for which these calculations have been made, and that the LCAO method then be used for making calculations throughout the Brillouin zone. A general discussion of the method is given, including tables of matrix components of energy for simple cubic, face-centered and body-centered cubic, and diamond structures. Applications are given to the results of Fletcher and Wohlfarth on Ni, and Howarth on Cu, as illustrations of the fcc case. In discussing the bcc case, the splitting of the energy bands in chromium by an antiferromagnetic alternating potential is worked out, as well as a distribution of energy states for the case of no antiferromagnetism. For diamond, comparisons are made with the calculations of Herman, using the orthogonalized plane-wave method. The case of such crystals as InSb is discussed, and it is shown that their properties fit in with the energy band picture.
We present a new scheme to extract numerically "optimal" interatomic potentials from large amounts of data produced by first-principles calculations. The method is based on fitting the potential to ab initio atomic forces of many atomic configurations, including surfaces, clusters, liquids and crystals at finite temperature. The extensive data set overcomes the difficulties encountered by traditional fitting approaches when using rich and complex analytic forms, allowing to construct potentials with a degree of accuracy comparable to that obtained by ab initio methods. A glue potential for aluminium obtained with this method is presented and discussed.
A new molecular mechanics force field, the Universal force field (UFF), is described wherein the force field parameters are estimated using general rules based only on the element, its hybridization, and its connectivity. The force field functional forms, parameters, and generating formulas for the full periodic table are presented.
CHARMM (Chemistry at HARvard Macromolecular Mechanics) is a highly flexible computer program which uses empirical energy functions to model macromolecular systems. The program can read or model build structures, energy minimize them by first- or second-derivative techniques, perform a normal mode or molecular dynamics simulation, and analyze the structural, equilibrium, and dynamic properties determined in these calculations. The operations that CHARMM can perform are described, and some implementation details are given. A set of parameters for the empirical energy function and a sample run are included.
Aus der Schrdingerschen Gleichung lt sich durch eine kurze elementare Rechnung ohne Veruachlssigung die Beziehung
m\fracd2 dt2 \smallint \smallint \smallint dt.YY* .x = \smallint \smallint \smallint dt.YY* ( - \frac¶V¶x )m\frac{{d^2 }}{{dt^2 }}\smallint \smallint \smallint d\tau .\Psi \Psi * .x = \smallint \smallint \smallint d\tau .\Psi \Psi * \left( { - \frac{{\partial V}}{{\partial x}}} \right)
ableiten, die fr ein kleines und klein bleibendes Wellenpaket (m von der Ordnung 1 g) besagt, da die Beschleunigung seiner Lagekoordinaten im Sinne der Newtonschen Bewegungsgleichungen zur rtlichen Kraft -V/x pat.
Ranges of validity for the continuum-beam models, the length-scale effects and continuum assumptions are analyzed in the framework of scaling analysis of NT structure. Two coupled criteria for the applicability of the continuum models are presented. Scaling analysis of NT buckling and geometric parameters (e.g., diameter and length) is carried out to determine the key non-dimensional parameters that control the buckling strains and modes of NT buckling. A model applicability map, which represents two classes of NTs, is constructed in the space of non-dimensional parameters. In an analogy with continuum mechanics, a mechanical law of geometric similitude is presented for two classes of beam-like NTs having different geometries. Expressions for the critical buckling loads and strains are tailored for the distinct groups of NTs and compared with the data provided by the molecular dynamics simulations. Implications for molecular dynamics simulations and the NT-based scanning probes are discussed.
The molecular structure of a material determines its mechanical, thermal and chemical properties. Thus, to better understand characteristic mechanical properties like damping behavior or softening, in principle, one just has to model the interactions of a sufficiently large number of atoms. Various force field approaches have been proposed for that purpose, which are based on molecular-dynamic simulations, or rather quantum-mechanical ab initio calculations. They provide the potential energy of a structure in dependence of sort and number of chemical and physical bonds. In general, the different energy forms can be represented by nonlinear normal, bending and torsional springs which suggests the use of a finite element code. However, standard finite elements like truss, beam or shell elements are not very applicable because of the interaction of many atoms and, considered from a mechanical perspective, the absence of rotational degrees of freedom. For example, a bending of beam elements would lead to unrealistic constraints of neighboring molecular groups. In order to overcome this disadvantage, a new 4-node finite element is introduced, which uses only translational degrees of freedom and therefore is capable of representing the different energy forms exactly.
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.
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].
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.
A general methodology to develop hyper-elastic membrane models applicable to crystalline films one-atom thick is presented. In this method, an extension of the Born rule based on the exponential map is proposed. The exponential map accounts for the fact that the lattice vectors of the crystal lie along the chords of the curved membrane, and consequently a tangent map like the standard Born rule is inadequate. In order to obtain practical methods, the exponential map is locally approximated. The effectiveness of our approach is demonstrated by numerical studies of carbon nanotubes. Deformed configurations as well as equilibrium energies of atomistic simulations are compared with those provided by the continuum membrane resulting from this method discretized by finite elements.
The hyperconjugative result of bond stretching in alkenes has been studied with MM4. A low-temperature crystallographic study of 1,2-diarylindane[a]indane has been carried out, together with ab initio (MP2/6-31G*) calculations on model systems. The results are well reproduced with a force field designed to explicitly include hyperconjugation (MM4), and they show beyond doubt that hyperconjugative bond elongations exist both in theory and by experiment. © 1996 by John Wiley & Sons, Inc.
The excellent set of properties of carbon nanotube and carbon nanotube-based nanostructures has been established by various studies. However the claimed property values and trends have not been unanimously agreed upon. Using state of the art molecular dynamics and ab initio methods, we have extensively studied the mechanical, thermal and structural properties of carbon nanotubes and carbon nanotube based nanostructures. Additionally this study aims to address the approaches used in various studies to assess the validity and influence of various definitions used for determining the physical properties as reported in earlier experiments and theoretical calculations. We have come up with equations, which quantitatively address the wide differences in trend and values of nanotube axial modulus available across the literature. Applying a novel bond rearrangement scheme, we have found similar values in twist modulus of zigzag and armchair nanotubes. This opposes the claim of difference that was shown to be valid only at finite limit in our study. We have shown that the contribution of van der Waals energy in a multi-wall nanotube is powerful enough to make it hexagonal in shape but negligible in affecting the axial modulus. These insights will also help in designing micromechanics model of materials made from carbon nanotube or nanotube like structures. In particular, we have calculated the mechanical properties (young modulus, bending modulus and twist modulus) of isolated and bundled nanotubes, single and multi-wall nanotubes and sin-le and multi-wall carbon nanotube based tort. We also report studies on thermal variation of moduli and thermal expansion of nanotubes. The result obtained by first principles calculation based interatomic potential agrees well with the experimental results.
From a theory of Hohenberg and Kohn, approximation methods for treating an inhomogeneous system of interacting electrons are developed. These methods are exact for systems of slowly varying or high density. For the ground state, they lead to self-consistent equations analogous to the Hartree and Hartree-Fock equations, respectively. In these equations the exchange and correlation portions of the chemical potential of a uniform electron gas appear as additional effective potentials. (The exchange portion of our effective potential differs from that due to Slater by a factor of 23.) Electronic systems at finite temperatures and in magnetic fields are also treated by similar methods. An appendix deals with a further correction for systems with short-wavelength density oscillations.
The equation of motion of a system of 864 particles interacting through
a Lennard-Jones potential has been integrated for various values
of the temperature and density, relative, generally, to a fluid state.
The equilibrium properties have been calculated and are shown to
agree very well with the corresponding properties of argon. It is
concluded that, to a good approximation, the equilibrium state of
argon can be described through a two-body potential.
A search for low-energy helical and near-helical conformations of the tandemly repeated peptide (Asn-Ala-Asn-Pro)9 was undertaken by minimization of the CHARMM potential energy function from eight starting conformations; the latter were obtained from the two low-energy conformations of this repeated peptide found by Gibson and Scheraga, Proc. Natl. Acad. Sci. USA 83, 5649-5653 (1986), and the single conformation found by Brooks et al., Proc. Natl. Acad. Sci. USA 84, 4470-4474 (1987), and from modifications of these three conformations. The same eight starting conformations, as determined by dihedral angles, were used for minimizations of the AMBER and ECEPP potentials. Comparison of the final conformations by least-squares superposition of their C(α) atoms, and by inspection of the parameters of the ideal helix or coiled coil that most closely matched the coordinates of their C(α) atoms in a least-squares sense, showed that: (1) energy minimization, starting from the same conformation but using any two different potentials, could lead to final conformations whose resemblance to each other varied from acceptable to highly unsatisfactory; (2) the ordering of the final energy-minimized conformations, and the energy differences between them, were quite different for all three potentials; (3) the extent of agreement or disagreement between pairs of conformations generated using CHARMM and AMBER, CHARMM and ECEPP, or AMBER and ECEPP, respectively, was not significantly different.
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.
We present a unified scheme that, by combining molecular dynamics and density-functional theory, profoundly extends the range of both concepts. Our approach extends molecular dynamics beyond the usual pair-potential approximation, thereby making possible the simulation of both covalently bonded and metallic systems. In addition it permits the application of density-functional theory to much larger systems than previously feasible. The new technique is demonstrated by the calculation of some static and dynamic properties of crystalline silicon within a self-consistent pseudopotential framework.