A molecular mechanics study on size-dependent elastic properties of single-walled boron nitride nanotubes

We present an analytical study for the elastic properties of single-walled boron nitride nanotubes via a molecular mechanics model. Closed-form expressions for Young's modulus, Poisson's ratio and surface shear modulus are derived as functions of the nanotube diameter. The results are helix angle sensitive and comparable to those from ab initio calculations. This work is a first effort to establish analytical model of molecular mechanics for composite nanotubes and reveals the dissimilarities between size-dependent elastic properties of carbon and boron nitride nanotubes.     

Size-dependent elastic properties of a single-walled carbon nanotube via a molecular mechanics model

An analytical model based on a molecular mechanics approach is presented to relate the elastic properties of a single-walled carbon nanotube to its atomic structure. We derive closed-form expressions for elastic modulus and Poisson's ratio as a function of the nanotube diameter. Properties at different length scales are directly connected via these expressions. The analytically calculated elastic properties for achiral nanotubes using force constants obtained from experimental data of graphite are compared to those based on tight binding numerical calculations. This study represents a preliminary effort to develop analytical methods of molecular mechanics for applications in nanostructure modeling.     

Measuring elastic property of single-walled carbon nanotubes by nanoindentation: A theoretical framework

Nanoindentation is an effective technique for deducing the elastic property of single-walled carbon nanotubes (SWCNTs). Following an atomistic study of the nanoindentation mechanism, reverse analysis algorithms are proposed by utilizing the indentation force-depth data measured from the initial uniaxial compression and post-buckling regimes, respectively, which lead to stretching stiffness of 382 Pa m and 429 Pa m, that are very close to those in the literature. Parallel finite element simulations incorporating atomic interactions are also carried out, which closely duplicates the indentation response of SWCNTs in atomistic simulations. The numerical studies carried out in this paper may be used to guide the nanoindentation experiments, explain and extract useful data from the test, as well as stimulate new experiments.     

Homogeneous monotropic elastic rods: Normal uniform configurations and universal solutions

A number of simple solutions are obtained which are universal for an homogeneous monotropic elastic rod whose theory is based on a Cosserat-type model.

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Sommario Si ottengono alcune soluzioni semplici che sono universali per una trave omogenea monotropica la cui teoria è basata su un modello alla Cosserat.

     

A molecular based anisotropic shell model for single-walled carbon nanotubes

Single-walled carbon nanotubes (SWCNTs) are frequently modeled as isotropic elastic shells. However, there are obvious evidences showing that SWCNTs exhibit remarkable chirality induced anisotropy that should not be neglected in some cases. In this paper, we derive the closed-form expressions for the anisotropic elastic properties of SWCNTs using a molecular mechanics model. Based on these anisotropic elastic properties, we develop a molecular based anisotropic shell model (MBASM) for predicting the mechanical behavior of SWCNTs. The explicit expressions for the coupling of axial, circumferential, and torsional strains, the radial breathing mode frequency, and the longitudinal and torsional wave speeds are obtained. We show that the MBASM is capable of predicting the effects of size and chirality on these quantities. The efficiency and accuracy of the MBASM are validated by comparisons of the present results with the existing results.     

Wave propagation in carbon nanotubes embedded in an elastic matrix

This paper reports the results of an investigation into the characteristics of wave propagation in carbon nanotubes embedded in an elastic matrix, based on an exact shell model. Each of the concentric tubes of multi-walled carbon nanotubes is considered as an individual elastic shell and coupled together through the van der Waals forces between two adjacent tubes. The matrix surrounding carbon nanotubes is described as a spring element defined by the Winkler model. The effects of rotatory inertia and elastic matrix on the wave velocity, the critical frequency, and the amplitude ratio between two adjacent tubes are described and discussed through numerical examples. The results obtained show that wave propagation in carbon nanotubes may appear in a critical frequency at which the wave velocity changes suddenly; the elastic matrix surrounding carbon nanotubes debases the critical frequency and the wave velocity, and changes the vibration modes between two adjacent tubes; the rotatory inertia based on an exact shell model obviously influences the wave velocity at some wave modes. Finally, a comparison of dispersion solutions from different shell models is given. The present work may serve as a useful reference for the application and the design of nano-electronic and nano-drive devices, nano-oscillators, and nano-sensors, in which carbon nanotubes act as basic elements.     

Analytical solutions for non-Newtonian fluid flows in pipe-like domains

This paper deals with some unsteady unidirectional transient flows of an Oldroyd-B fluid in unbounded domains which geometrically are axisymmetric pipe-like. An expansion theorem of Steklov is used to obtain exact solutions for flows satisfying no-slip boundary conditions. The well known solutions for a Navier-Stokes fluid, as well as those corresponding to a Maxwell fluid and a second grade one, appear as limiting cases of our solutions. The steady state solutions are also obtained for t→∞.     

Analytical and numerical solutions of the Local Inertial Equations

Neglecting the convective terms in the Saint-Venant Equations (SVE) in flood hydrodynamic modelling can be done without a loss in accuracy of the simulation results. In this case the Local Inertial Equations (LInE) are obtained. Herein we present two analytical solutions for the Local Inertial Equations. The first is the classical instantaneous Dam-Break Problem and the second a steady state solution over a bump. These solutions are compared with two numerical schemes, namely the first order Roe scheme and the second order MacCormack scheme. Comparison between analytical and numerical results shows that the numerical schemes and the analytical solution converge to a unique solution. Furthermore, by neglecting the convective terms the original numerical schemes remain stable without the need for adding entropy correction, artificial viscosity or special initial conditions, as in the case of the full SVE.     

Analytical solutions for anisotropic bimaterials under several boundary conditions on the interface

Summary Analytical solutions are proposed for the stress and displacement fields induced by in-plane loading of a bimaterial under several boundary conditions. The two joined orthotropic layers forming the bimaterial are allowed to possess different types of anisotropy with their planes of elastic symmetry arbitrarily inclined with respect to the horizontal.     

Explicit solution of the radial breathing mode frequency of single-walled carbon nanotubes

On the basis of a molecular mechanics model, an analytical solution of the radial breathing mode (RBM) frequency of single-walled carbon nanotubes (SWCNTs) is obtained. The effects of tube chirality and tube diameter on the RBM frequency are investigated and good agreement between the present results and existing data is found. The present analytical formula indicates that the chirality and size dependent elastic properties are responsible for the effects of the chirality and small size on the RBM frequency of an SWCNT. The project supported by the National Natural Science Foundation of China (10402019), Shanghai Rising-Star Program (05QMX1421), Shanghai Leading Academic Discipline Project (Y0103), and Key Project of Shanghai Committee of Science and Technology (04JC14034).     

Deformation of metallic single-walled carbon nanotubes in electric field based on elastic theory

The electromechanical properties of metallic single-walled carbon nanotubes (SWCNTs) in the electric field are demonstrated with a column shell model in this paper.A hemisphere model is introduced to determine the charge distribution and the local electric field in SWCNTs.By treating the SWCNT as an elastic column shell,the analytical solutions of the charged SWCNT’s axial strain and the radial strain are obtained.SWCNTs with a larger aspect ratio show greater deformation.The greatest radial deformation appears at the end of the tube.The significant axial strain can be induced in CNTs with a large length (around 100 nm) even though the applied electric field is not strong enough.When SWCNTs are fixed at both ends,the radius of SWCNTs becomes small along the axial position.     

Nonlocal shell model for elastic wave propagation in single- and double-walled carbon nanotubes

This paper investigates the transverse and torsional wave in single- and double-walled carbon nanotubes (SWCNTs and DWCNTs), focusing on the effect of carbon nanotube microstructure on wave dispersion. The SWCNTs and DWCNTs are modeled as nonlocal single and double elastic cylindrical shells. Molecular dynamics (MD) simulations indicate that the wave dispersion predicted by the nonlocal elastic cylindrical shell theory shows good agreement with that of the MD simulations in a wide frequency range up to the terahertz region. The nonlocal elastic shell theory provides a better prediction of the dispersion relationships than the classical shell theory when the wavenumber is large enough for the carbon nanotube microstructure to have a significant influence on the wave dispersion. The nonlocal shell models are required when the wavelengths are approximately less than 2.36×10⁻⁹ and 0.95×10⁻⁹ m for transverse wave in armchair (15,15) SWCNT and torsional wave in armchair (10,10) SWCNT, respectively. Moreover, an MD-based estimation of the scale coefficient e₀ for the nonlocal elastic cylindrical shell model is suggested. Due to the small-scale effects of SWCNTs and the interlayer van der Waals interaction of DWCNTs, the phase difference of the transverse wave in the inner and outer tube can be observed in MD simulations in wave propagation at high frequency. However, the van der Waals interaction has little effect on the phase difference of transverse wave.     

Computing elastic moduli on 3-D X-ray computed tomography image stacks

A numerical task of current interest is to compute the effective elastic properties of a random composite material by operating on a 3D digital image of its microstructure obtained via X-ray computed tomography (CT). The 3-D image is usually sub-sampled since an X-ray CT image is typically of order 1000³ voxels or larger, which is considered to be a very large finite element problem. Two main questions for the validity of any such study are then: can the sub-sample size be made sufficiently large to capture enough of the important details of the random microstructure so that the computed moduli can be thought of as accurate, and what boundary conditions should be chosen for these sub-samples? This paper contributes to the answer of both questions by studying a simulated X-ray CT cylindrical microstructure with three phases, cut from a random model system with known elastic properties. A new hybrid numerical method is introduced, which makes use of finite element solutions coupled with exact solutions for elastic moduli of square arrays of parallel cylindrical fibers. The new method allows, in principle, all of the microstructural data to be used when the X-ray CT image is in the form of a cylinder, which is often the case. Appendix A describes a similar algorithm for spherical sub-samples, which may be of use when examining the mechanical properties of particles. Cubic sub-samples are also taken from this simulated X-ray CT structure to investigate the effect of two different kinds of boundary conditions: forced periodic and fixed displacements. It is found that using forced periodic displacements on the non-geometrically periodic cubic sub-samples always gave more accurate results than using fixed displacements, although with about the same precision. The larger the cubic sub-sample, the more accurate and precise was the elastic computation, and using the complete cylindrical sample with the new method gave still more accurate and precise results. Fortran 90 programs for the analytical solutions are made available on-line, along with the parallel finite element codes used.     

Exact solutions for radial deformations of a functionally graded isotropic and incompressible second-order elastic cylinder

We find closed-form solutions for axisymmetric plane strain deformations of a functionally graded circular cylinder comprised of an isotropic and incompressible second-order elastic material with moduli varying only in the radial direction. Cylinder's inner and outer surfaces are loaded by hydrostatic pressures. These solutions are specialized to cases where only one of the two surfaces is loaded. It is found that for a linear through-the-thickness variation of the elastic moduli, the hoop stress for the first-order solution (or in a cylinder comprised of a linear elastic material) is a constant but that for the second-order solution varies through the thickness. The radial displacement, the radial stress and the hoop stress do not depend upon the second-order elastic constant but the hydrostatic pressure and hence the axial stress depends upon it. When the two elastic moduli vary as the radius raised to the power two or four, the radial and the hoop stresses in an infinite space with a pressurized cylindrical cavity equal the pressure in the cavity. For an affine variation of the elastic moduli, the hoop stress in an internally loaded cylinder made of a linear elastic isotropic and incompressible material at the point is the same as that in a homogeneous cylinder. Here R_in and R_ou equal, respectively, the inner and the outer radius of the undeformed cylinder and R the radial coordinate of a point in the unstressed reference configuration.     

Coarse-grained potentials of single-walled carbon nanotubes

We develop the coarse-grained (CG) potentials of single-walled carbon nanotubes (SWCNTs) in CNT bundles and buckypaper for the study of the static and dynamic behaviors. The explicit expressions of the CG stretching, bending and torsion potentials for the nanotubes are obtained by the stick-spiral and the beam models, respectively. The non-bonded CG potentials between two different CG beads are derived from analytical results based on the cohesive energy between two parallel and crossing SWCNTs from the van der Waals interactions. We show that the CG model is applicable to large deformations of complex CNT systems by combining the bonded potentials with non-bonded potentials. Checking against full atom molecular dynamics calculations and our analytical results shows that the present CG potentials have high accuracy. The established CG potentials are used to study the mechanical properties of the CNT bundles and buckypaper efficiently at minor computational cost, which shows great potential for the design of micro- and nanomechanical devices and systems.     

A new method for characterizing the interphase regions of carbon nanotube composites

The elastic properties of a carbon nanotube (CNT) reinforced composite are affected by many factors such as the CNT–matrix interphase. As such, mechanical analysis without sufficient consideration of these factors can give rise to incorrect predictions. Using single-walled carbon nanotube (SWCNT) reinforced Polyvinylchloride (PVC) as an example, this paper presents a new technique to characterize interphase regions. The representative volume element (RVE) of the SWCNT–PVC system is modeled as an assemblage of three phases, the equivalent solid fiber (ESF) mimicking the SWCNT under the van der Waals (vdW) forces, the dense interphase PVC of appropriate thickness and density, and the bulk PVC matrix. Two methods are proposed to extract the elastic properties of the ESF from the atomistic RVE and the CNT-cluster. Using atomistic simulations, the thickness and the average density of interphase matrix are determined and the elastic properties of amorphous interphase matrix are characterized as a function of density. The method is examined in a continuum-based three-phase model developed with the aid of molecular mechanics (MM) and the finite element (FE) method. The predictions of the continuum-based model show a good agreement with the atomistic results verifies that the interphase properties of amorphous matrix in CNT-composites could be approximated as a function of density. The results show that ignoring either the vdW interaction region or the interphase matrix layer can bring about misleading results, and that the effect of internal walls of multi-walled carbon nanotubes (MWCNTs) on the density and thickness of the dense interphase is negligible.     

Crack in a 2D beam lattice: Analytical solutions for two bending modes

We consider an infinite square-cell lattice of elastic beams with a semi-infinite crack. Symmetric and antisymmetric bending modes of fracture under remote loads are examined. The related long-wave asymptotes corresponding to a continuous anisotropic bending plate are also considered. In the latter model, the symmetric mode is characterized by the square-root type singularity, whereas the antisymmetric mode results in a hyper-singular field. A solution for the continuous plate with a finite crack is also presented. These closed-form continuous solutions describe the fields in the whole plane. The main goal is to establish analytical connections between the ‘macrolevel’ state, defined by the continuous asymptote of the lattice solution, and the maximal bending moment in the crack-front beam, that is, to determine the resistance of the lattice with an initial crack to the crack advance. The solutions are obtained in the same way as for mass-spring lattices. Considering the static problems we use the discrete Fourier transform and the Wiener-Hopf technique. Monotonically distributed bending moments ahead of the crack are determined for the symmetric mode, and a self-equilibrated transverse force distribution is found for the antisymmetric mode. It is shown that in the latter case only the crack-front beam resists to the fracture development, whereas the forces in the other beams facilitate the fracture. In this way, the macrolevel fracture energy is determined in terms of the material strength. The macrolevel energy release is found to be much greater than the critical strain energy of the beam, especially in the hyper-singular mode. In both problems, it is found that among the beams surrounding the crack the crack-front beam is maximally stressed, and hence its strength defines the strength of the structure.     

Vacancy effect on the elastic constants of layer-structured nanomaterials

Layer-structured nanomaterials where alternating layers of nanocrystallites meet along high angle grain boundaries constitute a special category of nanomaterials. In the present study we investigated the effect of the presence of a vacancy on the elastic constants of such materials by the use of atomistic simulation methods. The calculations were performed on a model system where atoms interact via a Lennard–Jones potential and the elastic constants were obtained in the frame of homogeneous deformations, for nanocrystallite layer widths ranging from 2.24 up to 37.12 nm. The results show that the favoured position of the vacancy is located in the GB core. The state of relaxation of the structure is an important factor that affects the obtained results. In both the unrelaxed and relaxed structures results converge to a given value after the 5th (3 1 0) layer. This value seems to depend on the size of the nanocrystallite L and approaches the bulk value above a given size L. It is also concluded that in the case of a relaxed system there is a smoother variation of the system energy and elastic constant as a function of the distance of the vacancy from the GB plane when the size L increases. The way that external stresses are applied on the system affects the values of the obtained elastic properties, with the elastic constants related to the characteristic directions of the grain boundary being the most affected ones. These findings are of particular interest for fabrication methods of nanostructured materials, experimental methods for the measurement of their elastic properties as well as multiscale modelling schemes.     

Analytical solutions for cohesive zone models

Solutions are given for a cantilever beam specimen using a beam on elastic foundation model to incorporate various cohesive zone traction laws. These included both positive and negative linear slopes and constant stresses. Negative slopes give rise to multiple solutions. However, all the solutions give very similar results for energy release rate and beam root rotation confirming insensitivity to the form of the traction law. The use of these solutions to analyse peeling is discussed.