Buckling analysis of embedded nanotubes using gradient continuum theory

The buckling of nanotubes embedded in an elastic matrix is modeled within the framework of Timoshenko beams. Both a stress gradient and a strain gradient approach are considered. The energy variational approach is adopted to obtain the critical buckling loads. The dependences of the buckling load on the nonlocal parameter, the stiffness of the surrounding elastic matrix, and the transverse shear stiffness of the nanotubes are obtained. The results show a significant dependence of critical buckling load on the nonlocal parameter and the stiffness of the surround matrix. The Euler beam model, which neglects the shear stiffness of the nanotubes, over-predicts the critical buckling load. It is also found that the strain gradient model provides the lower bound and the stress gradient model provides the upper bound for the critical buckling load of nanotubes. In addition to mechanical buckling, thermally induced buckling of the nanotubes embedded in an elastic matrix is also studied. All results are expressed in closed-form and therefore are easy to use by materials scientists and engineers for the design of nanotubes and their composites.     

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.     

Nonlinear dynamical response of embedded fluid-conveyed micro-tube reinforced by BNNTs

In the present study, nonlinear dynamical behavior and stability of an embedded fluid conveying smart composite micro-tube under imposed electric potential and thermal loadings have been investigated. The composite matrix is the poly-vinylidene fluoride (PVDF) reinforced by double-walled boron nitride nanotubes (DWBNNTs). Composite structure is modeled based on piezoelectric fiber reinforced composite (PFRC) theory and a representative volume element has been considered for predicting the elastic, piezoelectric, dielectric and thermal properties of the smart composite tube. The fluid flow is assumed to be inviscid, irrotational and incompressible. Formulation presented here is based on Euler–Bernoulli beam model with von-Kármán geometric nonlinearity and nonlocal elasticity theory. The interactions between smart composite micro-tube and surrounding elastic media are simulated by Pasternak foundation model. The discretized governing equations of motion are directly obtained by minimizing the energy of the system. As a result, the eigen-values and eigenvectors (mode shapes) are to be obtained by the state-space matrix which is then solved by an iterative method to obtain nonlinear frequencies of smart composite tube. The results significantly show that imposing positive electric potential increases nonlinear stability of the system. In addition, it is concluded that applying electric and thermal loadings can be used as well as controlling parameters to improving stability of the smart composite micro-tube.     

Surface stress and agglomeration effects on nonlocal biaxial buckling polymeric nanocomposite plate reinforced by CNT using various approaches

In this article, the surface stress effect on the biaxial critical buckling load of nonlocal polymeric nanocomposite rectangular plate reinforced by carbon nanotubes (CNTs) is presented. Various approaches such as Eshelby–Mori–Tanaka, the extended mixture rule, Halpin–Tsai, and micromechanical are used to determine the effective material properties of polymeric nanocomposite plate. The governing equations of equilibrium are obtained by using Hamilton’s principle. The Navier’s method is considered to obtain the biaxial critical buckling load of polymeric nanocomposite rectangular plate for simply supported boundary conditions. A detailed parametric study is conducted to explain the effects of aspect ratio, elastic foundation, surface stress and agglomeration on the biaxial buckling of nanocomposite plate. The results show that surface stress effect plays an important role at nanoscale. Also, the biaxial critical buckling load decreases with increasing the CNTs volume fraction in the inclusion (agglomeration effect). The results of this research can be used for micro-electro-mechanical and nano-electro-mechanical devices.     

Vibration and buckling analysis of nanotubes (nanofibers) embedded in an elastic medium using Doublet Mechanics

In the present study, vibration and buckling of nanotubes (nanofibers) embedded in an elastic medium are studied. A length scale-dependent theory called Doublet Mechanics (DM) is used in the formulation. In this theory, discrete microstructure of solids is considered in the formulation and using a bottom-up approach macro level strains and stresses are obtained from microlevel strains and stresses. Taylor series expansion of the microlevel displacement is used in the definition of the micro strains. The number of terms in the Taylor series describes the microstructure of the considered solids. In this study, nanotube fibers are assumed as an Euler–Bernoulli beam embedded in an elastic medium. Simply supported and clamped boundary conditions are considered at the edges of the beams. Free vibration frequencies and critical buckling loads are obtained and compared with the classical elasticity results. It is shown that scale-dependent DM can be used at the nanolength scale.     

Mechanical buckling of functionally graded material cylindrical shells surrounded by Pasternak elastic foundation

In this study, the mechanical buckling of functionally graded material cylindrical shell that is embedded in an outer elastic medium and subjected to combined axial and radial compressive loads is investigated. The material properties are assumed to vary smoothly through the shell thickness according to a power law distribution of the volume fraction of constituent materials. Theoretical formulations are presented based on a higher-order shear deformation shell theory (HSDT) considering the transverse shear strains. Using the nonlinear strain–displacement relations of FGMs cylindrical shells, the governing equations are derived. The elastic foundation is modelled by two parameters Pasternak model, which is obtained by adding a shear layer to the Winkler model. The boundary condition is considered to be simply-supported. The novelty of the present work is to achieve the closed-form solutions for the critical mechanical buckling loads of the FGM cylindrical shells surrounded by elastic medium. The effects of shell geometry, the volume fraction exponent, and the foundation parameters on the critical buckling load are investigated. The numerical results reveal that the elastic foundation has significant effect on the critical buckling load.     

Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory

A single-elastic beam model has been developed to analyze the thermal vibration of single-walled carbon nanotubes (SWCNT) based on thermal elasticity mechanics, and nonlocal elasticity theory. The nonlocal elasticity takes into account the effect of small size into the formulation. Further, the SWCNT is assumed to be embedded in an elastic medium. A Winkler-type elastic foundation is employed to model the interaction of the SWCNT and the surrounding elastic medium. Differential quadrature method is being utilized and numerical solutions for thermal-vibration response of SWCNT is obtained. Influence of nonlocal small scale effects, temperature change, Winkler constant and vibration modes of the CNT on the frequency are investigated. The present study shows that for low temperature changes, the difference between local frequency and nonlocal frequency is comparatively high. With embedded CNT, for soft elastic medium and larger scale coefficients (e₀a) the nonlocal frequencies are comparatively lower. The nonlocal model-frequencies are always found smaller than the local model-frequencies at all temperature changes considered.     

Dynamic stability analysis of multi-walled carbon nanotubes with arbitrary boundary conditions based on the nonlocal elasticity theory

Dynamic stability of embedded multi-walled carbon nanotubes (MWCNTs) in an elastic medium and thermal environment and subjected to an axial compressive force is studied based on the nonlocal elasticity and Timoshenko beam theory. The developed nonlocal beam model has the capability to consider the small scale effects. The generalized differential quadrature method is employed to discretize the dynamic governing differential equations of MWCNTs with various end supports. A parametric study is conducted to investigate the influences of static load factor, temperature change, nonlocal parameter, slenderness ratio, and spring constant of elastic medium on the dynamic stability characteristics of MWCNTs.     

Small scale effect on the thermal buckling of orthotropic arbitrary straight-sided quadrilateral nanoplates embedded in an elastic medium

As a first endeavor, the small scale effect on the thermal buckling characteristic of orthotropic arbitrary straight-sided quadrilateral nanoplates embedded in an elastic medium is investigated. The surrounding elastic medium is modeled as the two-parameter elastic foundation. The formulation is derived using the classical plate theory (CPT) in conjunction with the nonlocal elasticity theory. The solution procedure is based on the transformation of the governing equations from physical domain to computational domain and then discretization of the spatial derivatives by employing the differential quadrature method (DQM) as an efficient and accurate numerical tool. The fast rate of convergence of the method is shown and the results are compared against existing results in literature. Then, the influence of small scale parameter in combination with the elastic medium parameters, geometrical shape and the boundary conditions on the thermal buckling load of the nanoplates is investigated.     

Axial buckling analysis of single-walled carbon nanotubes in thermal environments via the Rayleigh–Ritz technique

Axial buckling characteristics of single-walled carbon nanotubes (SWCNTs) including thermal environment effect are studied in this paper. Eringen’s nonlocal elasticity equations are incorporated into the classical Donnell shell theory to establish a nonlocal elastic shell model which takes small-scale effects into account. The Rayleigh–Ritz technique is implemented in conjunction with the set of beam functions as modal displacement functions to consider the four commonly used boundary conditions namely as simply supported–simply supported, clamped–clamped, clamped–simply supported, and clamped-free in the buckling analysis. Selected numerical results are presented to demonstrate the influences of small scale effect, aspect ratio, thermal environment effects and boundary conditions in detail. It is found that the value of aspect ratio has different effects on the critical axial buckling loads of SWCNTs in low and high temperature environments. Also, it is observed that the difference between the thermal axial buckling responses of SWCNTs relevant to various boundary conditions is more prominent for higher values of nonlocal elasticity constant.     

Exact closed-form solution for non-local vibration and biaxial buckling of bonded multi-nanoplate system

In recent years, nonlocal elasticity theory is widely used for the analytical and computational modeling of nanostructures. This theory, developed by Eringen, has shown to be practical for the vibration and buckling analysis of nanoscale structures and reliable for predesign procedures of nano-devices. This paper considers buckling and dynamic analysis of multi-nanoplate systems. This type of system can be relevant to composite structures embedded with graphene sheets. Exact solutions for the natural frequencies and buckling loads of multi-nanoplate systems have been proposed by considering that the multi-nanoplate system is embedded within an elastic medium. Nonlocal elasticity theory is utilized for the mathematical establishment of the system. The solutions of the homogenous system of differential equations are obtained using the Navier’s method and trigonometric method. An asymptotic analysis is proposed to show the influence of increasing number of nanoplates in the system. Analytical expressions are validated with existing results in the literature for some special cases. Numerical results based on the analytical expressions is shown to quantify the effects of the change in nonlocal parameter, stiffness coefficients of the elastic mediums and the number of layers on the natural frequencies and buckling load.     

Nonlocal effect on the free vibration of short nanotubes embedded in an elastic medium

In this paper, the small size effect on the free vibration behavior of finite length nanotubes embedded in an elastic medium is investigated. The problem is formulated based on the three-dimensional (3D) nonlocal elasticity theory. Since the 3D nonlocal constitutive relations in a cylindrical coordinate system are used, in addition to displacement components, the stress tensor components are chosen as degrees of freedom. The surrounding elastic medium is modeled as the Winkler’s elastic foundation. The differential quadrature method as an efficient and accurate numerical tool in conjunction with the series solution is used to discretize the governing equations. Very fast rate of convergence of the method is demonstrated. The effects of the nonlocal parameter together with the other geometrical parameters and also the stiffness parameter of the elastic medium on the natural frequencies are studied.     

Mechanical buckling of two-dimensional functionally graded cylindrical shells surrounded by Winkler–Pasternak elastic foundation

In this research, buckling analysis of a two-dimensional, functionally graded, cylindrical shell that has been embedded in an outer elastic medium in the presence of combined axial and transverse loading based on third-order shear deformation shell theory is numerically investigated. Variations of the shell properties are considered to be continuous through length and thickness. Winkler–Pasternak foundation and simply supported boundary conditions have been applied. The problem has been solved using the generalized differential quadrature method. Geometrical, load, and foundation parameters beside functionally graded power indexes effects on the critical buckling load have been studied.     

Nonlinear buckling response of embedded piezoelectric cylindrical shell reinforced with BNNT under electro–thermo-mechanical loadings using HDQM

Nonlinear buckling response of a composite cylindrical shell made of polyvinylidene fluoride (PVDF), is investigated. A two-dimensional smart model surrounded by an elastic foundation subjected to combined electro–thermo-mechanical loading is considered. The nonlinear strain terms based on Donnell’s theory are taken into account using the first shear deformation theory. The Hamilton’s principle is employed to obtain coupled differential equations, containing displacement and electric potential terms. Harmonic differential quadrature method (HDQM) is applied to obtain the critical buckling load for clamped supported mechanical and free electric potential boundary conditions at both ends of the smart cylinder. Results indicate that the critical buckling load increases when piezoelectric effect is considered.     

Nonlocal buckling of double-nanoplate-systems under biaxial compression

This paper reports an analytical study on the buckling of double-nanoplate-system (DNPS) subjected to biaxial compression using nonlocal elasticity theory. The two nanoplates of DNPS are bonded by an elastic medium. Nonlocal plate theory is utilized for deriving the governing equations. An analytical method is used for determining the buckling load of DNPS under biaxial compression. Difference between nonlocal uniaxial and biaxial buckling in DNPS is shown. Both synchronous and asynchronous buckling phenomenon of biaxially compressed DNPS is highlighted. Study shows that the small-scale effects in biaxially compressed DNPS increases with increasing values of nonlocal parameter for the case of synchronous modes of buckling than in the asynchronous modes of buckling. The buckling load decrease with increase of value of nonlocal parameter or scale coefficient. In biaxial compression higher buckling modes are subjected to higher nonlocal effects in DNPS. Further the study shows that the increase of stiffness parameter brings uniaxial and biaxial buckling phenomenon closer while increase of aspect ratio widen uniaxial and biaxial buckling phenomenon.     

Vibration analysis of embedded nanotubes using nonlocal continuum theory

Vibration of nanotubes embedded in an elastic matrix is investigated by using the nonlocal Timoshenko beam model. Both a stress gradient and a strain gradient approach are considered. The Hamilton’s principle is adopted to obtain the frequencies of the nanotubes. The dependencies of frequency on the stiffness and mass density of the surrounding elastic matrix, the nonlocal parameter, the transverse shear stiffness and the rotary inertia of the nanotubes are obtained. The results show a significant dependence of frequencies on the surrounding medium and the nonlocal parameter. The frequencies are over-predicted by using the Euler beam model that neglects the shear stiffness and rotary inertia of the nanotubes. It is also found that the lower bound and the upper bound for the frequencies of nanotubes are, respectively, provided by the strain gradient model provides and the stress gradient theory. Explicit formulas for the frequency are obtained and therefore are easy to use by material scientists and engineers for the design of nanotubes and nanotubes based composites.     

Nonlinear torsional buckling and postbuckling of eccentrically stiffened FGM cylindrical shells in thermal environment

The main aim of this paper is to investigate the nonlinear buckling and post-buckling of functionally graded stiffened thin circular cylindrical shells surrounded by elastic foundations in thermal environments and under torsional load by analytical approach. Shells are reinforced by closely spaced rings and stringers in which material properties of shell and the stiffeners are assumed to be continuously graded in the thickness direction. The elastic medium is assumed as two-parameter elastic foundation model proposed by Pasternak. Based on the classical shell theory with von Karman geometrical nonlinearity and smeared stiffeners technique, the governing equations are derived. Using Galerkin method with three-term solution of deflection, the closed form to find critical torsional load and post-buckling load–deflection curves are obtained. The effects of temperature, stiffener, foundation, material and dimensional parameters are analyzed.     

Thermal buckling analysis of nanoplates based on nonlocal elasticity theory with four-unknown shear deformation theory resting on Winkler–Pasternak elastic foundation

The thermal buckling analysis of nanoplates is based on nonlocal elasticity theory with four-unknown shear deformation theory resting on Winkler–Pasternak elastic foundation. The nanoplate is assumed to be under three types of thermal loadings, namely uniform temperature rise, linear temperature rise, and nonlinear temperature rise through the thickness. The theory involves four unknown variables with small-scale effects, as against five in the case of other higher-order theories and first-order shear deformation theory. Closed-form solution for theory was also presented. Results are presented to discuss the influences of the nonlocal parameter, aspect ratio, side-to-thickness ratio, and elastic foundation parameters on the thermal buckling characteristics of analytical rectangular nanoplates.     

Eccentric compression stability of multi-walled carbon nanotubes embedded in an elastic matrix

This paper reports the results of an investigation on the eccentric compression stability of multi-walled carbon nanotubes embedded in an elastic matrix. Based on continuum modeling, a multilayer shell model is presented for the eccentric compression buckling of multi-walled carbon nanotubes embedded in an elastic matrix, in which the effect of van der Waals forces between two adjacent tubes is taken into account. The critical bending moment and the eccentric compression mode for three types of multi-walled carbon nanotubes with different layer numbers and ratios of radius to thickness are calculated. Results obtained show that the eccentric compression buckling mode corresponding the critical bending moment is unique, and is different from the purely axial compression buckling of an individual multi-walled carbon nanotube. For different types of multi-walled carbon nanotubes, the effect of matrix stiffness on the critical bending moment of multi-walled carbon nanotubes under eccentric compression loading is obviously different, and is dependent on the innermost radius and layer numbers of the multi-walled carbon nanotubes. The critical bending stress exerted on the center tubes of nearly solid multi-walled carbon nanotubes does not change as the ratio of the axial compression loading to the bending membrane force increases. The new features and meaningful numerical results in this paper are helpful for the application and the design of nanostructures in which multi-walled carbon nanotubes act as basic elements.     

Buckling analysis of single walled carbon nanotube on Winkler foundation using nonlocal elasticity theory and DTM

In the present work differential transformation method (DTM) is used to predict the buckling behaviour of single walled carbon nanotube (SWCNT) on Winkler foundation under various boundary conditions. Four different boundary conditions namely clamped–clamped, simply supported, clamped hinged and clamped free are used to study the critical buckling loads. Effects of (i) size of SWCNT (ii) nonlocal parameter and (iii) Winkler elastic modulus on nonlocal critical buckling loads are being investigated and discussed. The DTM is implemented for the nonlocal SWCNT analyses and this yields results with high degree of accuracy. Further, present method can be applied to linear and nonlinear problems.