Prediction of nonlocal scaling parameter for armchair and zigzag single-walled carbon nanotubes based on molecular structural mechanics, nonlocal elasticity and wave propagation

Atomic vibration in the Carbon Nanotubes (CNTs) gives rise to non-local interactions. In this paper, an expression for the non-local scaling parameter is derived as a function of the geometric and electronic properties of the rolled graphene sheet in single-walled CNTs. A self-consistent method is developed for the linearization of the problem of ultrasonic wave propagation in CNTs. We show that (i) the general three-dimensional elastic problem leads to a single non-local scaling parameter (e(0)), (ii) e(0) is almost constant irrespective of chirality of CNT in the case of longitudinal wave propagation, (iii) e(0) is a linear function of diameter of CNT for the case of torsional mode of wave propagation, (iv) e(0) in the case of coupled longitudinal-torsional modes of wave propagation, is a function which exponentially converges to that of axial mode at large diameters and to torsional mode at smaller diameters. These results are valid in the long-wavelength limit. (C) 2011 Elsevier Ltd. All rights reserved.

Axial wave propagation in coupled nanorod system with nonlocal small scale effects

This article deals with the axial wave propagation properties of a coupled nanorod system with consideration of small scale effects. The nonlocal elasticity theory has been incorporated into classical rod/bar model to capture unique features of the coupled nanorods under the umbrella of continuum mechanics theory. Nonlocal rod model is developed for coupled nanorods. The strong effect of the nonlocal scale has been obtained which leads to substantially different wave behavior of nanorods from those of macroscopic rods. Explicit expressions are derived for wavenumber, cut-off frequency and escape frequency of nanorods. The analysis shows that the wave characteristics of nanorods are highly over estimated by the classical rod model, which ignores the effect of small-length scale. The studies also shows that the nonlocal scale parameter introduces certain band gap region in axial or longitudinal wave mode, where no wave propagation occurs. This is manifested in the spectrum cures as the region, where the wavenumber tends to infinite or wave speed tends to zero. The effect of the coupled spring stiffness is also capture in the present analysis. It has been also shown that the cut-off frequency increases as the stiffness of the coupled spring increases and also the coupled spring stiffness has no effect on escape frequency of the axial wave mode in the nanorod. This cut-off frequency is also independent of the nonlocal small scale parameter. The present study may bring in helpful insights while investigating multiple-nanorod-system-models for future nano-optomechanical systems applications. The results can also provide useful guidance for the study and design of the next generation of nanodevices that make use of the wave propagation properties of coupled single-walled carbon nanotubes or coupled nanorods. (C) 2011 Elsevier Ltd. All rights reserved.

Uncertainty Quantification in Ultrasonic Wave Based Detection of Delamination in Composite Structures

In this paper we consider the problem of guided wave scattering from delamination in laminated composite and further the problem of estimating delamination size and layer-wise location from the guided wave measurement. Damage location and region/size can be estimated from time of flight and wave packet spread, whereas depth information can be obtained from wavenumber modulation in the carrier packet. The key challenge is that these information are highly sensitive to various uncertainties. Variation in reflected and transmitted wave amplitude in a bar due to boundary/interface uncertainty is studied to illustrate such effect. Effect of uncertainty in material parameters on the time of flight are estimated for longitudinal wave propagation. To evaluate the effect of uncertainty in delamination detection, we employ a time domain spectral finite element (tSFEM) scheme where wave propagation is modeled using higher-order interpolation with shape function have spectral convergence properties. A laminated composite beam with layer-wise placement of delamination is considered in the simulation. Scattering due to the presence of delamination is analyzed. For a single delamination, two identical waveforms are created at the two fronts of the delamination, whereas waves in the two sub-laminates create two independent waveforms with different wavelengths. Scattering due to multiple delaminations in composite beam is studied.

Automatic energy-momentum conserving time integrators for hyperelastic waves

An energy-momentum conserving time integrator coupled with an automatic finite element algorithm is developed to study longitudinal wave propagation in hyperelastic layers. The Murnaghan strain energy function is used to model material nonlinearity and full geometric nonlinearity is considered. An automatic assembly algorithm using algorithmic differentiation is developed within a discrete Hamiltonian framework to directly formulate the finite element matrices without recourse to an explicit derivation of their algebraic form or the governing equations. The algorithm is illustrated with applications to longitudinal wave propagation in a thin hyperelastic layer modeled with a two-mode kinematic model. Solution obtained using a standard nonlinear finite element model with Newmark time stepping is provided for comparison. (C) 2012 Elsevier B.V. All rights reserved.

Instability of Coupled Longitudinal and Transverse Electromagnetic Modes of Wave‐Propagation in Bounded Streaming Plasmas

The instability of coupled longitudinal and transverse electromagnetic modes associated with long wavelengths is studied in bounded streaming plasmas. The main conclusions are as follows: (i) For long waves for which O (k 2)=0, in the absence of relative streaming motion of electrons and ions and aωp/c<0.66, the whole spectrum of harmonic waves is excited due to finite temperature and boundary effects consisting of two subseries. One of these subseries can be identified with Tonks-Dattner resonance oscillations for the electrons, and arises primarily due to the electrons with frequencies greater than the electrostatic plasma frequency corresponding to the electron density in the midplane in the undisturbed state. The other series arises primarily due to ion motion. When aωp/c>0.66, in addition to the above spectrum of harmonic waves, the system admits an infinite number of growing and decaying waves. The instability associated with these modes is found to arise due to the interaction of the waves inside the plasma with the external electromagnetic field. (ii) For modes with comparatively shorter wavelengths for which O (k3)=0, the coupling due to finite temperature sets in, and it is found that the two series of harmonic waves obtained in (i) deriving energy from the transverse modes also become unstable. Thus, for these wavelengths the system admits three sets of growing and decaying modes, first two for all values of aωp/c and the third for (aωp/c) > 0.66. (iii) The presence of streaming velocities introduces various other coupling mechanisms, and we find that even for the wavelengths for which O (k2)=0, we get three sets of growing and decaying waves. The numerical values for the growth rates show that the streaming velocities enhance the growth rates of instability significantly.

Longitudinal Magnetic Field Effect on Nonlocal Ultrasonic Vibration Analysis of Single-Walled Carbon Nanotubes Based on Wave Propagation Approach

This paper studies the effect of longitudinal magnetic field on ultrasonic vibration in single walled carbon nanotubes (CNTs) based on nonlocal continuum medium theory. Governing partial differential equations of CNTs are derived by considering the Lorentz magnetic forces applied on CNTs induced by a longitudinal magnetic field through Maxwell equations. The vibration characteristics of CNTs under a longitudinal magnetic field are obtained by solving the governing equations via wave propagation approach. The effects of longitudinal magnetic field on vibration of CNTs are discussed through numerical experiments. The present analysis show that vibration frequencies of CNTs drops dramatically in the presence of the magnetic field for various circumferential wavenumbers. Such effect is also observed for various boundary conditions of the CNT. New features for the effect of longitudinal magnetic field on ultrasonic vibration of CNTs, presented in this paper are useful in the design of nano-drive device, nano-oscillator and actuators and nano-electron technology, where carbon nanotubes act as basic elements.

Wave propagation in single-walled carbon nanotube under longitudinal magnetic field using nonlocal Euler-Bernoulli beam theory

In the present work, the effect of longitudinal magnetic field on wave dispersion characteristics of equivalent continuum structure (ECS) of single-walled carbon nanotubes (SWCNT) embedded in elastic medium is studied. The ECS is modelled as an Euler-Bernoulli beam. The chemical bonds between a SWCNT and the elastic medium are assumed to be formed. The elastic matrix is described by Pasternak foundation model, which accounts for both normal pressure and the transverse shear deformation. The governing equations of motion for the ECS of SWCNT under a longitudinal magnetic field are derived by considering the Lorentz magnetic force obtained from Maxwell's relations within the frame work of nonlocal elasticity theory. The wave propagation analysis is performed using spectral analysis. The results obtained show that the velocity of flexural waves in SWCNTs increases with the increase of longitudinal magnetic field exerted on it in the frequency range: 0-20 THz. The present analysis also shows that the flexural wave dispersion in the ECS of SWCNT obtained by local and nonlocal elasticity theories differ. It is found that the nonlocality reduces the wave velocity irrespective of the presence of the magnetic field and does not influences it in the higher frequency region. Further it is found that the presence of elastic matrix introduces the frequency band gap in flexural wave mode. The band gap in the flexural wave is found to independent of strength of the longitudinal magnetic field. (C) 2011 Elsevier Inc. All rights reserved.

Microscopic expression for frequency and wave vector dependent dielectric constant of a dipolar liquid

A microscopic expression for the frequency and wave vector dependent dielectric constant of a dense dipolar liquid is derived starting from the linear response theory. The new expression properly takes into account the effects of the translational modes in the polarization relaxation. The longitudinal and the transverse components of the dielectric constant show vastly different behavior at the intermediate values of the wave vector k. We find that the microscopic structure of the dense liquid plays an important role at intermediate wave vectors. The continuum model description of the dielectric constant, although appropriate at very small values of wave vector, breaks down completely at the intermediate values of k. Numerical results for the longitudinal and the transverse dielectric constants are obtained by using the direct correlation function from the mean‐spherical approximation for dipolar hard spheres. We show that our results are consistent with all the limiting expressions known for the dielectric function of matter.

Ultrasonic wave characteristics of a monolayer graphene on silicon substrate

The present work deals with an ultrasonic type of wave propagation characteristics of monolayer graphene on silicon (Si) substrate. An atomistic model of a hybrid lattice involving a hexagonal lattice of graphene and surface atoms of diamond lattice of Si is developed to identify the carbon-silicon bond stiffness. Properties of this hybrid lattice model is then mapped into a nonlocal continuum framework. Equivalent force constant due to Si substrate is obtained by minimizing the total potential energy of the system. For this equilibrium configuration, the nonlocal governing equations are derived to analyze the ultrasonic wave dispersion based on spectral analysis. From the present analysis we show that the silicon substrate affects only the flexural wave mode. The frequency band gap of flexural mode is also significantly affected by this substrate. The results also show that, the silicon substrate adds cushioning effect to the graphene and it makes the graphene more stable. The analysis also show that the frequency bang gap relations of in-plane (longitudinal and lateral) and out-of-plane (flexural) wave modes depends not only on the y-direction wavenumber but also on nonlocal scaling parameter. In the nonlocal analysis, at higher values of the y-directional wavenumber, a decrease in the frequency band gap is observed for all the three fundamental wave modes in the graphene-silicon system. The atoms movement in the graphene due to the wave propagation are also captured for all the tree fundamental wave modes. The results presented in this work are qualitatively different from those obtained based on the local analysis and thus, are important for the development of graphene based nanodevices such as strain sensor, mass and pressure sensors, atomic dust detectors and enhancer of surface image resolution that make use of the ultrasonic wave dispersion properties of graphene. (C) 2011 Elsevier Ltd. All rights reserved.

Wave dispersion in a cellular composite with modulated microstructure

In this paper we report a modeling technique and analysis of wave dispersion in a cellular composite laminate with spatially modulated microstructure, which can be modeled by parameterization and homogenization in an appropriate length scale. Higher order beam theory is applied and the system of wave equations are derived. Homogenization of these equations are carried out in the scale of wavelength and frequency of the individual wave modes. Smaller scale scattering below the order of cell size are filtered out in the present approach. The longitudinal dispersion relations for different values of a modulation parameter are analyzed which indicates the existence of stop and pass band patterns. Dispersion relations for flexural-shear case are also analyzed which indicates a tendency toward forming the stop and pass bands for increasing values of a shear stiffness modulation parameter. The effect the phase angle (θ) of the incident wave indicates the existence more number of alternative stop bands and pass bands for θ = 45°.

Energy loss due to adhesion in longitudinal impact of elastic cylinders

Adhesive interaction between impacting bodies can cause energy loss, even in an otherwise elastic impact. Adhesion force induces tensile stress in the bodies, which modifies the stress wave profile and influences the restitution behavior. We investigate this effect by developing a finite element framework, which incorporates a Lennard-Jones-type potential for modeling the adhesive interaction between volume elements. With this framework, the classical problems in contact mechanics can be revisited without the restrictive surface-force approximation. In this paper, we study the longitudinal impact of an elastic cylinder on a rigid half-space with adhesion. In the absence of adhesion, this problem reduces to the impact between two identical cylinders in which there is no energy loss. Adhesion causes a fraction of energy in the stress waves to remain in the cylinder as residual stress waves. This apparent loss in kinetic energy is shown to be a unique function of maximum tensile strain energy. We have developed a 1-D model in terms of interaction force parameters, velocity and material properties to estimate the tensile stain energy. We show that this model can be used to predict practically important phenomena like capture wherein the impacting bodies stick together. (C) 2013 Elsevier Masson SAS. All rights reserved.

Comparative study of different nonconserving time integrators for wave propagation in hyperelastic waveguides

This paper addresses the formulation and numerical efficiency of various numerical models of different nonconserving time integrators for studying wave propagation in nonlinear hyperelastic waveguides. The study includes different nonlinear finite element formulations based on standard Galerkin finite element model, time domain spectral finite element model, Taylor-Galerkin finite element model, generalized Galerkin finite element model and frequency domain spectral finite element model. A comparative study on the computational efficiency of these different models is made using a hyperelastic rod model, and the optimal computational scheme is identified. The identified scheme is then used to study the propagation of transverse and longitudinal waves in a Timoshenko beam with Murnaghan material nonlinearity.

Exact N-wave solutions of generalized Burgers equations

Exact N-wave solutions for the generalized Burgers equation u(t) + u(n)u(x) + (j/2t + alpha) u + (beta + gamma/x) u(n+1) = delta/2u(xx),where j, alpha, beta, and gamma are nonnegative constants and n is a positive integer, are obtained. These solutions are asymptotic to the (linear) old-age solution for large time and extend the validity of the latter so as to cover the entire time regime starting where the originally sharp shock has become sufficiently thick and the viscous effects are felt in the entire N wave.

Orientational relaxation in a random dipolar lattice: Wave-number and frequency dependence

In order to understand the role of translational modes in the orientational relaxation in dense dipolar liquids, we have carried out a computer ''experiment'' where a random dipolar lattice was generated by quenching only the translational motion of the molecules of an equilibrated dipolar liquid. The lattice so generated was orientationally disordered and positionally random. The detailed study of orientational relaxation in this random dipolar lattice revealed interesting differences from those of the corresponding dipolar liquid. In particular, we found that the relaxation of the collective orientational correlation functions at the intermediate wave numbers was markedly slower at the long times for the random lattice than that of the liquid. This verified the important role of the translational modes in this regime, as predicted recently by the molecular theories. The single-particle orientational correlation functions of the random lattice also decayed significantly slowly at long times, compared to those of the dipolar liquid.

Wave propagation in elastic solids undergoing damage and growth process

Modeling and analysis of wave propagation in elastic solids undergoing damage and growth process are reported in this paper. Two types of diagnostic problems, (1) the propagation of waves in the presence of a slow growth process and (2) the propagation of waves in the presence of a fast growth process, are considered. The proposed model employs a slow and a fast time scale and a homogenization technique in the wavelength scale. A detailed analysis of wave dispersion is carried out. A spectral analysis reveals certain low-frequency bands, where the interaction between the wave and the growth process produces acoustic metamaterial-like behavior. Various practical issues in designing an efficient method of acousto-ultrasonic wave based diagnostics of the growth process are discussed. Diagnostics of isotropic damage in a ductile or quasi-brittle solid by using a micro-second pulsating signal is considered for computer simulations, which is to illustrate the practical application of the proposed modeling and analysis. The simulated results explain how an estimate of signal spreading can be effectively employed to detect the presence of a steady-state damage or the saturation of a process.