Stochastic FEM on nonlinear vibration of fluid-loaded double-walled carbon nanotubes subjected to a moving load based on nonlocal elasticity theory

This paper adopts stochastic FEM to study the statistical dynamic behaviors of nonlinear vibration of the fluid-conveying double-walled carbon nanotubes (DWCNTs) under a moving load by considering the effects of the geometric nonlinearity and the nonlinearity of van der Waals (vdW) force. The Young’s modulus of elasticity of the DWCNTs is considered as stochastic with respect to the position to actually characterize the random material properties of the DWCNTs. Besides, the small scale effects of the nonlinear vibration of the DWCNTs are studied by using the theory of nonlocal elasticity. Based on the Hamilton’s principle, the nonlinear governing equations of the fluid-conveying double-walled carbon nanotubes under a moving load are formulated. The stochastic finite element method along with the perturbation technique is adopted to study the statistical dynamic response of the DWCNTs. Some statistical dynamic response of the DWCNTs such as the mean values and standard deviations of the non-dimensional dynamic deflections are computed and checked by the Monte Carlo Simulation, meanwhile the effects of the nonlocal parameter, aspect ratio and the flow velocity on the statistical dynamic response of the DWCNTs are investigated. It can be concluded that the nonlocal solutions of the dynamic deflections get larger with the increase of the nonlocal parameters due to the small scale effect, and as the flow velocity increases, the maxima non-dimensional dynamic deflections of the DWCNTs get larger.     

Nonlinear free vibration of embedded double-walled carbon nanotubes with layerwise boundary conditions

This paper investigates the large-amplitude free vibration of a double-walled carbon nanotube (DWCNT) surrounded by an elastic medium in the presence of temperature change. Based on continuum mechanics, a nonlocal elastic beam model is employed in which nanotubes are coupled together via the van der Waals (vdW) interlayer interactions. The Pasternak foundation model and a nonlinear vdW model are utilized to describe the surrounding elastic medium effect and the vdW interlayer interactions, respectively. DWCNTs with different boundary conditions are analyzed utilizing the Timoshenko beam theory that considers the shear deformation and rotary inertia effects. The governing equations are derived from Hamilton’s principle; the Galerkin method is utilized to discretize the governing equations. The influences of the nonlocal parameter, spring constant, carbon nanotube aspect ratio, and temperature change on the nonlinear free vibration characteristics of a double-walled carbon nanotube with different boundary conditions are thoroughly investigated. It is deduced that the nonlocal parameter, spring constant, and the aspect ratio play significant roles for the value of the nonlinear frequency. Also, the temperature change and the type of boundary conditions have an effect on the nonlinear frequency.     

Electro-thermo-torsional buckling of an embedded armchair DWBNNT using nonlocal shear deformable shell model

Electro-thermo-torsional buckling response of a double-walled boron nitride nanotube (DWBNNT) has been investigated based on nonlocal elasticity and piezoelasticity theories. The effects of surrounding elastic medium such as the spring constant of the Winkler-type and the shear constant of the Pasternak-type are taken into account. The van der Waals (vdW) forces are considered between inner and outer layers of nanotube. According to the relationship between the piezoelectric coefficient of armchair boron nitride nanotubes (BNNTs) and stresses, the first order shear deformation theory (FSDT) is used. Energy method and Hamilton’s principle are employed to obtain coupled differential equations containing displacements, rotations and electric potential terms. The detailed parameter study is conducted to investigate the effects of nonlocal parameter, elastic foundation modulus, temperature change, piezoelectric and dielectric constants on the critical torsional buckling load. Results indicate that the critical buckling load decreases when piezoelectric effect is considered.     

Investigation of nanotube length effect on the reinforcement efficiency in carbon nanotube based composites

The main goal of this research is to study the tensile behavior of embedded short carbon nanotubes (CNTs) in a polymer matrix in presence of van der Waals (vdW) interaction as inter-phase region. A 3D finite element model of a unit cell consisting of capped carbon nanotubes, inter-phase and surrounding polymer is built. The unit cell is subjected to tensile load case to obtain longitudinal Young’s modulus of the investigated cell. A parametric study is carried out to investigate the effect of CNT’s length on reinforcement. It is observed that improvement in the Young’s modulus of CNT-composite is negligible for lengths smaller than 100 nm and saturation takes place in larger lengths on the order of 10 μm. Furthermore, a comparison between results obtained for short carbon nanotubes and long carbon nanotube is presented. The efficient length of CNT in form of (10, 10) is obtained at the order of 10 μm. Finally, it was shown that direct use of micromechanics equations for short fibers will overestimate the stiffness. However, employing effective stiffness of equivalent fiber comprising of CNT and its inter-phase instead of high modulus of CNT will lead us to more appropriate results, which are in an acceptable agreement with conventional semi-empirical micromechanics equations.     

Electro-thermo-mechanical buckling of DWBNNTs embedded in bundle of CNTs using nonlocal piezoelasticity cylindrical shell theory

Axial buckling analysis of double-walled Boron Nitride nanotubes (DWBNNTs) embedded in an elastic medium under combined electro-thermo-mechanical loadings is presented in this article. Virtual displacement method based on nonlocal cylindrical piezoelasticity continuum shell theory is employed to derive the equilibrium equations. Boron Nitride nanotube (BNNT) is assumed to be surrounded by a bundle of carbon nanotubes (CNTs) as elastic medium for reinforcement. The elastic medium is simulated as Winkler–Pasternak foundation, and adjacent layers interactions are assumed to have been coupled by van der Walls (vdW) force evaluated based on the Lennard–Jones model. The effects of parameters such as electric and thermal loads, elastic medium and small scale are investigated on the buckling behavior of the DWBNNTs. The electric field and its direction are found to have affected the magnitude of the critical buckling load. Moreover, an analysis is carried out to estimate the nonlocal critical electro-thermo-mechanical load for the axial buckling of embedded DWBNNTs.     

Free vibration analysis of carbon nanotubes by using three-dimensional theory of elasticity

This paper studies vibration behavior of single-walled carbon nanotubes based on three-dimensional theory of elasticity. To accounting for the size effect of carbon nanotubes, nonlocal theory is adopted to the shell model. The nonlocal parameter is incorporated into all constitutive equations in three dimensions. Governing differential equations of motion are reduced to the ordinary differential equations in thickness direction by using Fourier series expansion in axial and circumferential direction. The state equations obtained from constitutive relations and governing equations are solved analytically by making use of the state space method. A detailed parametric study is carried out to show the influences of the nonlocal parameter, thickness-to-radius ratio and length-to-radius ratio. Results reveal that excluding small-scale effects caused decreasing accuracy of natural frequencies. Furthermore, the obtained closed form solution can be used to assess the accuracy of conventional two-dimensional theories.     

Deriving Analytical Expressions for the Ideal Curves and Using the Curves to Obtain the Temperature Dependence of Equation-of-State Parameters

Different equations of state (EOSs) have been used to obtain analytical expressions for the ideal curves, namely, the Joule–Thomson inversion curve (JTIC), Boyle curve (BC), and Joule inversion curve (JIC). The selected EOSs are the Redlich–Kwong (RK), Soave–Redlich–Kwong (SRK), Deiters, linear isotherm regularity (LIR), modified LIR (MLIR), dense system equation of state (DSEOS), and van der Waals (vdW). Analytical expressions have been obtained for the JTIC and BC only by using the LIR, MLIR, and vdW equations of state. The expression obtained using the LIR is the simplest. The experimental data for the JTIC and the calculated points from the empirical EOSs for the BC are well fitted into the derived expression from the LIR, in such a way that the fitting on this expression is better than those on the empirical expressions given by Gunn et al. and Miller. No experimental data have been reported for the BC and JIC; therefore, the calculated curves from different EOSs have been compared with those calculated from the empirical equations. On the basis of the JTIC, an approach is given for obtaining the temperature dependence of an EOS parameter(s). Such an approach has been used to determine the temperature dependences of A ₂ of the LIR, a and b parameters of the vdW, and the cohesion function of the RK. Such temperature dependences, obtained on the basis of the JTIC, have been found to be appropriate for other ideal curves as well.     

Numerical and exact models for free vibration analysis of cylindrical and spherical shell panels

The present paper shows a comparison between classical two-dimensional (2D) and three-dimensional (3D) finite elements (FEs), classical and refined 2D generalized differential quadrature (GDQ) methods and an exact three-dimensional solution. A free vibration analysis of one-layered and multilayered isotropic, composite and sandwich cylindrical and spherical shell panels is made. Low and high order frequencies are analyzed for thick and thin simply supported structures. Vibration modes are investigated to make a comparison between results obtained via the FE and GDQ methods (numerical solutions) and those obtained by means of the exact three-dimensional solution. The 3D exact solution is based on the differential equations of equilibrium written in general orthogonal curvilinear coordinates. This exact method is based on a layer-wise approach, the continuity of displacements and transverse shear/normal stresses is imposed at the interfaces between the layers of the structure. The geometry for shells is considered without any simplifications. The 3D and 2D finite element results are obtained by means of a well-known commercial FE code. Classical and refined 2D GDQ models are based on a generalized unified approach which considers both equivalent single layer and layer-wise theories. The differences between 2D and 3D FE solutions, classical and refined 2D GDQ models and 3D exact solutions depend on several parameters. These include the considered mode, the order of frequency, the thickness ratio of the structure, the geometry, the embedded material and the lamination sequence.     

A refined dynamic stiffness element for free vibration analysis of cross-ply laminated composite cylindrical and spherical shallow shells

An exact free vibration analysis of doubly-curved laminated composite shallow shells has been carried out by combining the dynamic stiffness method (DSM) and a higher order shear deformation theory (HSDT). In essence, the HSDT has been exploited to develop first the dynamic stiffness (DS) element matrix and then the global DS matrix of composite cylindrical and spherical shallow shell structures by assembling the individual DS elements. As an essential prerequisite, Hamilton’s principle is used to derive the governing differential equations and the related natural boundary conditions. The equations are solved symbolically in an exact sense and the DS matrix is formulated by imposing the natural boundary conditions in algebraic form. The Wittrick–Williams algorithm is used as a solution technique to compute the eigenvalues of the overall DS matrix. The effect of several parameters such as boundary conditions, orthotropic ratio, length-to-thickness ratio, radius-to-length ratio and stacking sequence on the natural frequencies and mode shapes is investigated in details. Results are compared with those available in the literature. Finally some concluding remarks are drawn.     

Hygrothermal deformation of orthotropic nanoplates based on the state-space concept

This paper deals with the investigation of the effect of hygrothermal conditions on the bending of nanoplates using Levy type solution model employing the state-space concept. The nanoplates are assumed to be subjected to a hygrothermal environment. The two-unknown function plate theory is used to derive the governing differential equations on the basis of Eringen's nonlocal elasticity theory. The governing equations contain the small scale effect as well as hygrothermal and mechanical effects. These equations are converted into a set of first-order linear ordinary differential equations with constant coefficients. Analytical solution of bending response for nanoplates under combinations of simply supported, clamped and free boundary conditions is obtained. Comparison of the results with those being in the open literature is made. The influences played by small scale parameter, temperature rise, the degree of moisture concentration, boundary conditions, plate aspect ratio and side-to-thickness ratio are studied.     

Green's functions for the stress intensity factor evolution in finite cracks in orthotropic materials

The elastodynamic response of an infinite orthotropic material with finite crack under concentrated loads is examined. Solution for the stress intensity factor history around the crack tips is found. Laplace and Fourier transforms are employed to solve the equations of motion leading to a Fredholm integral equation on the Laplace transform domain. The dynamic stress intensity factor history can be computed by numerical Laplace transform inversion of the solution of the Fredholm equation. Numerical values of the dynamic stress intensity factor history for some example materials are obtained. This solution can be used as a Green's function to solve dynamic problems involving fini te cracks.     

Prediction of Young’s modulus of graphene sheets and carbon nanotubes using nanoscale continuum mechanics approach

Analytical formulations are presented to predict the elastic moduli of graphene sheets and carbon nanotubes using a linkage between lattice molecular structure and equivalent discrete frame structure. The obtained results for a graphene sheet show an isotropic behavior, in contrast to limited molecular dynamic simulations. Young’s modulus of CNT represents a high dependency of stiffness on tube thickness, while dependency on tube diameter is more tangible for smaller tube diameters. The presented closed-form solution provides an insight to evaluate finite element models constructed by beam elements. The results are in a good agreement with published data and experimental results.     

Thermoelastic dynamic solution of a multilayered spherically isotropic hollow sphere for spherically symmetric problems

Summary. The thermoelastic dynamic solution of a multilayered spherically isotropic hollow sphere in the state of spherical symmetry is obtained. By the method of superposition, the displacement is divided into two parts: one is quasi-static and the other is dynamic. The quasi-static solution is first derived in an explicit form by using the transfer matrix method. Then by introducing a new dependent variable, the governing equations, boundary conditions as well as the initial conditions for the dynamic solution are rewritten, and the dynamic solution is obtained by the separation of variables method coupled with the initial parameter method as well as the orthogonal expansion technique. The present method is suitable for a multilayered spherically isotropic hollow sphere consisting of arbitrary layers and subjected to arbitrary spherically symmetric thermal loads. Numerical results are finally presented and discussed.     

Curvature effects on pressure-induced buckling of empty or filled double-walled carbon nanotubes

Summary The curvature effects of interlayer van der Waals (vdW) forces on pressure-induced buckling of empty or filled double-walled carbon nanotubes (DWNTs) are studied for various radii, length-to-radius ratios, end conditions and internal-to-external pressure ratios. The analysis is based on a double-elastic shell model and assumes that the interlayer vdW pressure at a point between the inner and outer tubes depends not only on the change of the interlayer spacing, but also on the change of the curvatures of the inner and outer tubes at that point. Here the role of filling substances inside DWNTs is modeled by a uniformly distributed internal pressure. The present work aims to study the curvature effects on critical radial pressure. An explicit formula is obtained for the external buckling pressure of empty or filled DWNTs. The critical value of external pressure is estimated with various internal-to-external pressure ratios. It is shown that the curvature effects play a more significant role in buckling problems under radial pressure for small radii DWNTs than under pure axial stress. Our results show that loading transfer through vdW forces prior to buckling is important for the pressure-induced buckling of DWNTs rather than axially compressed buckling.     

Dynamic SIF of interface crack between two dissimilar viscoelastic bodies under impact loading*

In this paper, the transient dynamic stress intensity factor (SIF) is determined for an interface crack between two dissimilar half-infinite isotropic viscoelastic bodies under impact loading. An anti-plane step loading is assumed to act suddenly on the surface of interface crack of finite length. The stress field incurred near the crack tip is analyzed. The integral transformation method and singular integral equation approach are used to get the solution. By virtue of the integral transformation method, the viscoelastic mixed boundary problem is reduced to a set of dual integral equations of crack open displacement function in the transformation domain. The dual integral equations can be further transformed into the first kind of Cauchy-type singular integral equation (SIE) by introduction of crack dislocation density function. A piecewise continuous function approach is adopted to get the numerical solution of SIE. Finally, numerical inverse integral transformation is performed and the dynamic SIF in transformation domain is recovered to that in time domain. The dynamic SIF during a small time-interval is evaluated, and the effects of the viscoelastic material parameters on dynamic SIF are analyzed.     

Nonlinear thermo-elastic bending of functionally graded carbon nanotube-reinforced composite plates resting on elastic foundations by dynamic relaxation method

In this study, the nonlinear thermo-elastic bending analysis of a functionally graded carbon nanotube-reinforced composite plate resting on two parameter elastic foundations is investigated. The material properties of the carbon nanotube-reinforced composite plates are assumed to be temperature dependent and graded in the thickness direction. The nonlinear formulations are based on a first-order shear deformation plate theory and large deflection von Karman equations. A dynamic relaxation method is employed to solve the plate nonlinear partial differential equations. The effects of volume fraction of carbon nanotubes, thermal gradient, temperature dependency, elastic foundation, boundary conditions, plate width-to-thickness ratio, aspect ratio, and carbon nanotubes distribution are studied in detail.     

Analysis of nonlinear vibrations of double-walled carbon nanotubes conveying fluid

This paper investigates the effect of the geometric nonlinearity and the nonlinearity of van der Waals (vdW) force on the transverse vibration of the double-walled carbon nanotubes conveying fluid and the interaction between two types of nonlinearities. By using the Hamilton’s principle, the nonlinear governing equations of the double-walled carbon nanotubes conveying fluid are deduced. The effects of two types of nonlinearities on the coaxial and noncoaxial vibrations of the double-walled carbon nanotubes conveying fluid are discussed in numerical examples. The results show that the effect of geometric nonlinearity on the amplitude–frequency properties can be neglected if two types of nonlinearities are simultaneously considered. Compared with the uncoupling, the coupling between the longitudinal and transverse vibrations has little effect on the amplitude–frequency properties with considering two types of nonlinearities simultaneously. However, the coupling has significant effect on the amplitude–frequency properties with only considering the geometric nonlinearity.     

A coupled bending-torsion model for electrostatically actuated torsional nano/micro-actuators with considering influence of van der Waals force

In the current paper, a coupled two degree of freedom model which considers both bending and torsion of the supporting torsion beams is presented for electrostatically actuated torsional nano/micro-actuators under the effect of van der Waals (vdW) force. Newton’s second law is utilized for finding the normalized equations governing the static behavior of the actuator. The implict function theorem is then utilized for finding the equations governing the pull-in state of the actuator. The related results show that torsion model considerably overestimates the pull-in parameters of the nano/micro-actuator. The concept of the instability mode is introduced, and it is shown that when the ratio of the bending stiffness to the torsion stiffness of the supporting torsion beams is relatively low, the dominant instability mode of the actuator would be the bending mode and otherwise the dominant instability mode would be the torsion mode. It is also observed that the presence of the vdW force can significantly reduce the pull-in angle and pull-in deflection of the nano/micro-actuator. The presented results also show that the vdW force can lead to considerable reduction in the pull-in voltage of the actuator. The equilibrium behavior of the actuator is studied, and it is observed that the vdW force and also bending of the supporting torsion beams greatly reduce the maximum allowable voltage which can be applied to the actuator. Results of this paper can be used for successful design of electrostatically actuated torsional nano/micro-actuators where the size of the actuator is sufficiently small, and as a result, the vdW force plays a major role in the system.     

Ion-exchange iron sorption by carbon nanotubes and nanofibers

We have studied room-temperature equilibrium in systems containing an aqueous Fe(II) or Fe(III) salt solution and carbon nanofibers or carbon nanotubes with various contents of functional groups. The sorption capacity of the sorbents has been determined as a function of contact time, sorbent weight to solution volume ratio, salt concentrations in solution, solution pH, and sorbent “solubility” (degree of functionalization). Equilibrium data have been described by the Langmuir and Freundlich equations, and the sorption kinetics have been represented by a first-order or pseudo-second-order equation. We have demonstrated that the sorption process can be accelerated by physical activation of the system.     

Some anomalies of the numerical solutions of the Euler equations

Summary The numerical solution of the unsteady Euler equations for compressible flow over a circular cylinder is obtained using standard numerical techniques. The equations, written in cylindrical coordinates, are discretized on an orthogonal grid via central differences for spatial derivatives, using a simple second order artificial viscosity form and a special treatment of the boundary conditions. Backward differences in time are employed resulting in a large system of nonlinear difference equations at each step. A direct solver (LAPACK), based on an efficient Gaussian elimination procedure for banded matrices, is used to solve the linearized system of equations. The stability of the nonunique solutions of the steady Euler equations is investigated. It is demonstrated that the symmetric solutions, with zero circulation, are not stable. For a certain Mach number range, a periodic solution is obtained where the shock oscillation persists. If a periodic circulation (within a certain frequency range) is enforced in the far field, an irregular solution emerges with unpredictable shock motions. For such a solution, the Lyapunov exponent is shown to be greater than zero, indicating the appearance of chaos.