Buckling Of Multi-walled Carbon Nanotubes Research Articles

Relationship between local buckling and atomic elastic stiffness in multi-walled carbon nanotubes under compression and bending deformations

This paper discusses unified criteria for the local buckling of multi-walled carbon nanotubes in both compression and bending deformations from a standpoint of atomic simulations. Defective and non-defective five-walled carbon nanotubes are subjected to compression and bending deformation by the molecular dynamics method using the adaptive intermolecular reactive empirical bond order potential. The atomic elastic stiffness of each atom, Bijα, is then evaluated during the deformations to discuss the onset of local buckling. The Bijα corresponds to the second-order derivatives of the atomic energy, i.e., the gradient of the local stress–strain surfaces in six-dimensional strain space. If Bijα has a negative eigenvalue, a local unstable path will exist in the direction of the strain. Under compression, the smallest eigenvalues, λ(1)α, of the Bijα for all atoms changes to a negative value long before buckling occurs, while the second-smallest eigenvalues, λ(2)α, of the Bijα of some atoms change to a negative value just prior to buckling. Under bending deformation, the change in the positivity of λ(2)α corresponds to a rippling deformation on the side of compressive bending stresses. A variation in the location of defects in the carbon nanotubes affects the peak stresses under compression and the peak moments under a bending deformation. However, for all models, local buckling occurs from the dense region of atoms whose λ(2)α<0 in some outer layers. This suggests that the atomic elastic stiffness is capable of acting as an evaluation criterion for local buckling in multi-walled carbon nanotubes.

Asymptotic estimates of buckling radial pressure for multi‐walled carbon nanotubes at different variants of boundary conditions

Buckling of short multi‐walled carbon nanotubes (MWCNTs) under external radial pressure is studied on the base of a multiple‐shell model. The modified Mushtari‐Donell‐Vlasov type equations taking into account the van der Waals (vdW) interaction forces between adjacent tubes are used as the governing ones. In contrast to a majority of available studies on buckling of MWCNTs, which consider only the simply supported boundary conditions, this paper based on the asymptotic approach allows for the study of the buckling behavior of MWCNTs with different variants of the boundary conditions at the tube edges. At first, the pre‐buckling membrane hoop stress‐resultants induced by radial pressure are determined for each wall. Then, introducing a small parameter defined as a thickness‐to‐radius ratio, the asymptotic solutions of the boundary value problem are constructed for different cases which depend on the outermost radius of a MWCNT. The relevance of the present approach is confirmed by good agreement between asymptotic estimates and exact values of the buckling radial pressure for simply supported double‐ and triple‐walled nanotubes determined on the base of the accepted shell model. In addition, the validity of the asymptotic estimates is justified by comparing theirs with existing data obtained on the base of the available multiple‐shell model taking into account the pressure dependence of the interlayer vdW forces. The influence of the outermost radius, aspect ratio and boundary conditions as well on the buckling radial pressure is analyzed in this study.

1113 多層CNTの圧縮と曲げ変形における局部座屈の原子弾性剛性係数による検討(OS8-3 材料力学と計算力学の新展開)

1113 多層CNTの圧縮と曲げ変形における局部座屈の原子弾性剛性係数による検討(OS8-3 材料力学と計算力学の新展開)

Positional Controlled Manipulation of the Carbon Nanotube Surface by Selective Screening

A scalable process has been demonstrated for the controlled functionalization and positional controlled buckling of multiwalled carbon nanotube (MWCNT) by selectively masking the nanotube surface with chemically inert and thermally stable ceramic material. A uniform thin layer of soluble ceramic precursor, polycarbosilane (PCS), was first coated on MWCNT by the modified solution-based coating technique. The discrete silicon carbide (SiC) islands were then produced on nanotube surface during annealing at high temperature. The size and distribution of SiC islands on the surface of nanotube could be topologically regulated by controlling the thickness of PCS layer on MWCNT. The modulated surface of MWCNTs was then exploited for the site specific functionalization by guiding the incoming functional groups to the exposed surface. By optimizing the openings of the exposed surface of MWCNTs, the degree of chemical modifications could be tailored make. The controlled screening of nanotube surface has also been utilized for the physical modifications like selective buckling of tube walls by impinging high-energy ions on MWCNTs where the position of bending under irradiation fluence could be customized by the degree of passivation of the nanotube surface.

Radial buckling of multi-walled carbon nanotubes under hydrostatic pressure

Radial buckling stresses of carbon nanotubes (CNTs) need to be studied in high-pressure resonance Raman scattering spectrum. In this work, the closed-form expression of the critical buckling stress of multi-walled carbon nanotubes (MWCNTs) under hydrostatic pressure is derived that can be conveniently employed. Using the derived formulae, the critical buckling stresses of single-walled carbon nanotubes and double-walled carbon nanotubes with different diameters are calculated. The results are in good agreement with other reported literatures. In addition, the critical buckling stresses of each layer of a quintuple-walled CNT in different buckling modes are predicted, showing that the buckling instability can occur not only in the outermost rolled layer, but also in other rolled layer of MWCNTs by considering different diameters and buckling modes.

Hard-to-Soft Transition in Radial Buckling of Multi-Concentric Nanocylinders

We investigate the cross-sectional buckling of multi-concentric tubular nanomaterials, which are called multiwalled carbon nanotubes (MWNTs), using an analysis based on thin-shell theory. MWNTs under hydrostatic pressure experience radial buckling. As a result of this, different buckling modes are obtained depending on the inter-tube separation d as well as the number of constituent tubes N and the innermost tube diameter. All of the buckling modes are classified into two deformation phases. In the first phase, which corresponds to an elliptic deformation, the radial stiffness increases rapidly with increasing N. In contrast, the second phase yields wavy, corrugated structures along the circumference for which the radial stiffness declines with increasing N. The hard-to-soft phase transition in radial buckling is a direct consequence of the core-shell structure of MWNTs. Special attention is devoted to how the variation in d affects the critical tube number Nc, which separates the two deformation phases observed in N -walled nanotubes, i.e., the elliptic phase for N Nc. We demonstrate that a larger d tends to result in a smaller Nc, which is attributed to the primary role of the interatomic forces between concentric tubes in the hard-to-soft transition during the radial buckling of MWNTs.

Axisymmetric compressive buckling of multi-walled carbon nanotubes under different boundary conditions

The paper studies the axisymmetric compressive buckling behavior of multi-walled carbon nanotubes (MWNTs) under different boundary conditions based on continuum mechanics model. A buckling condition is derived for determining the critical buckling load and associated buckling mode of MWNTs, and numerical results are worked out for MWNTs with different aspect ratios under fixed and simply supported boundary conditions. It is shown that the critical buckling load of MWNTs is insensitive to boundary conditions, except for nanotubes with smaller radii and very small aspect ratio. The associated buckling modes for different layers ofMWNTs are in-phase, and the buckling displacement ratios for different layers are independent of the boundary conditions and the length of MWNTs. Moreover, for simply supported boundary conditions, the critical buckling load is compared with the corresponding one for axial compressive buckling, which indicates that the critical buckling load for axial compressive buckling can be well approximated by the corresponding one for axisymmetric compressive buckling. In particular, for axial compressive buckling of double-walled carbon nanotubes, an analytical expression is given for approximating the critical buckling load. The present investigation may be of some help in further understanding the mechanical properties of MWNTs.

Thermal Buckling of Multiwalled Carbon Nanotubes Using a Semi-Analytical Finite Element Approach

In this article, a semi-analytical finite element approach based on the nonlocal elastic shell model is proposed to study the thermal buckling of multiwalled carbon nanotubes (MWCNTs). By taking nonlocality into consideration, the small scale effect is incorporated into the model presented herein. Based upon the Eringen nonlocal elasticity, the displacement field equations coupled by the van der Waals interaction force are derived. Comprehensive results for the thermal buckling of multiwalled carbon nanotubes are given. The influences of the small scale parameter and boundary conditions on the thermal instability of MWCNTs are examined. It is indicated that the possibility of the radial buckling mode of deformation due to radially elevated temperature is significantly higher than that of axial buckling mode of deformation due to uniformly reduced temperature when carbon nanotubes experience thermal loads.

Axial buckling of multi-walled carbon nanotubes and nanopeapods

In this paper, we investigate both pre- and post-buckling behaviors of multi-walled carbon nanotubes and multi-walled carbon nanopeapods by incorporating into the applied forces of a prescribed beam equation both van der Waals interactions between the adjacent walls of the nanotubes and the interactions between the fullerenes and the inner wall of the nanotube. Two beam theories are employed. First, we utilize Donnell’s equilibrium equation to derive an axial stability condition for the multi-walled carbon nanotubes and multi-walled carbon nanopeapods. We then determine analytically the critical forces for single-walled and double-walled nanotubes and nanopeapods. Given the outer nanotube of a fixed radius, we observe that the critical force and strain derived from the axial buckling stability criterion decrease as a result of the molecular interactions between the adjacent layers of the nanotubes and the molecular interactions between the embedded fullerenes and the inner carbon nanotube, which is in agreement with existing literature. Next, we utilize an Euler–Bernoulli beam equation incorporating the curvature effect to obtain the post-buckled axial bending displacement for the multi-walled nanotubes and nanopeapods. We find that the interactions between molecules generate an inward force, which tends to resist any applied forces. While the inward force induced by the fullerenes to the inner wall of the nanotube vanishes as we increase the applied force, the inward force induced by the layers increases as the applied force increases. The main contribution of this paper is the incorporation of both van der Waals interactions and the curvature effect into prescribed beam theories to accurately measure the critical forces and the buckled displacements of multi-walled nanotubes and nanopeapods subject to a small external force. Our analysis is relevant to future nano devices, such as biological sensors and measuring devices for small forces arising from electrical charges or Casimir forces.

Torsional buckling analysis of multiwalled carbon nanotubes subjected to thermoelectromechanical loadings

The theoretical torsional buckling analysis of multiwalled carbon nanotubes (MWCNTs) under thermoelectromechanical loadings has been studied, which is of great significance to the application of carbon nanotubes in the electrical and thermal condition. Based on the Donnell shell theory, the theoretical governing equations of torsional buckling for MWCNTs subjected to thermoelectromechanical loadings have been established. The effects of torsional load, applied electrical and temperature loads, surrounding elastic medium and van der Waals forces between two adjacent nanotubes are taken into account at the same time. The critical buckling condition has been derived in terms of the buckling modes and the parameters that indicate the effects of thermoelectrical change, surrounding elastic medium and the van der Waals forces. Numerical results in general case are obtained for the thermal and electrical effects on torsional buckling of MWCNTs. It is shown that the critical buckling load under certain electrical and temperature field only depends on one factor, that is, the wavenumber of torsional buckling modes. Again, the critical buckling torques of MWCNTs change with the electrical and thermal field. Furthermore, the electrical field has much more powerful influence on the critical buckling torques than the thermal field.

Nonlocal elasticity effect on column buckling of multiwalled carbon nanotubes under temperature field

Based on nonlocal theory of thermal elasticity mechanics, an elastic multiple column model is developed for column buckling of MWNTs with large aspect ratios under axial compression coupling with temperature change. The present model treats each of the nested tubes as an individual column interacting with adjacent nanotubes through the intertube van der Waals forces. The thermal effect is incorporated in the formulation. In particular, an explicit expression is derived for the critical axial strain of a double-walled carbon nanotube which clearly demonstrates that small scale effects contribute significantly to the thermo-mechanical behavior of multiwalled carbon nanotubes and cannot be ignored.

Small-scale effect on torsional buckling of multi-walled carbon nanotubes

The small-scale effect on the torsional buckling of multi-walled carbon nanotubes coupled with temperature change is investigated in this paper. A nonlocal multiple-shell model for the multi-walled carbon nanotubes surrounded an elastic medium under torsional and thermal loads is established, and then general solutions are obtained from the governing equations. The influence of the nonlocal effect on critical shear force and change in temperature is investigated. It is demonstrated that the critical shear force could be overestimated by the classical continuum theory and the nonlocal effect on critical buckling force decreases as the change in temperature increases at room or low temperature but increases as the change in temperature increases at higher temperature. Meanwhile, the effect of small size-scale is dependent on the buckling mode under different thermal environments. It is also shown that the innermost radius and the number of layer can affect the small-scale effect on critical change in temperature and buckling shear force. When the ratio of tube length and outmost radius are given, the critical shear force in each layer decreases and the nonlocal effect on the critical shear force becomes weaker as the innermost radius and the layer number increase.

Axial buckling of multiwall carbon nanotubes with heterogeneous boundaries

The finite element method has been employed to study the effects of different boundary conditions on the axial buckling of multiwall carbon nanotubes (MWCNTs). Unlike previous works, both homogeneous and heterogeneous end constraints are considered for the constituent tubes of various MWCNTs comprising shell-type (i.e., the length-to-diameter ratio L/D&amp;lt;10), beam-type (i.e., L/D&amp;gt;10), and the two different types of constituent tubes. The results show that clamping the individual tubes of simply supported or free MWCNTs exerts a variety of influences on their buckling behaviors depending on the type of the MWCNTs, the position, and the number of the clamped tubes. Clamping the outermost tube can enhance the critical buckling strain up to four times of its original value and can shift the buckling modes of those MWCNTs consisting both shell- and beam-type tubes. In contrast, little difference can be observed when simply supported ends of MWCNTs are replaced by free ends or vice versa. Explicit buckling mode shapes obtained using the finite element method for various physically realistic cases have been shown in the paper.

Influence of Multilayer vdW Interaction on Torsional Buckling of Multi-walled Carbon Nanotube under Thermal Fields

Torsional buckling of multi-walled carbon nanotube embedded in an elastic matrix subjected to thermal fields is studied using a continuum-based multilayered shell model. The model takes into account the multilayer van der Waals (vdW) interaction between any two layers of the MWCNT as radius dependent function. The elastic matrix surrounding the outermost tube is modeled as Pasternak foundation to account for not only the normal stresses, but also the shear stress between the outermost tube and the surrounding medium. The effect of inner tube radius, number of tubes, stiffness of surrounding elastic medium and temperature change on the critical torsional load is examined. Numerical results show that the vdW interaction will lead to a higher critical torsional load and at room or lower temperature the critical torsional load for infinitesimal buckling of a multi-walled carbon nanotube increases as the value of temperature change increases, while at a temperature higher than room temperature the critical torsional load for infinitesimal buckling of a multi-walled carbon nanotube decreases as the value of temperature change increases.

Dynamic column buckling of multi-walled carbon nanotubes under axial impact load

This paper studies the dynamic column buckling of multi-walled carbon nanotubes (MWNTs) under axial impact load. The analysis is based on the continuum mechanics model and a simplified model for the van der Waals forces between adjacent layers. By introducing initial imperfections for MWNTs and applying the method of preferred mode, a buckling condition is derived for the buckling load and associated buckling mode. In particular, explicit expressions are obtained for double-walled carbon nanotubes (DWNTs). Finally, numerical calculations are worked out for a DWNT and a five-layer MWNT with different length-to-radius ratios.

Torsional buckling of multi-walled carbon nanotubes under combined axial and radial loadings

This paper investigates the torsional buckling of multi-walled carbon nanotubes under combined axial and radial loadings based on the continuum mechanics model. In particular, an explicit expression is obtained for the torsional buckling of double-walled carbon nanotubes (DWNTs) under combined loadings. Numerical results show that axial tensile stress or internal pressure will make DWNTs resist higher critical shear membrane force, while axial compressive stress or external pressure will lead to a lower critical shear membrane force. Further, for torsional buckling of DWNTs coupling with small axial stress and internal pressure or external pressure, the effect of the axial stress, internal pressure or external pressure on the critical shear membrane force is linear, and the associated buckling wave numbers are unique and the same as that under corresponding pure torque.

Dynamic torsional buckling of multi-walled carbon nanotubes embedded in an elastic medium

In this paper the dynamic torsional buckling of multi-walled carbon nanotubes (MWNTs) embedded in an elastic medium is studied by using a continuum mechanics model. By introducing initial imperfections for MWNTs and applying the preferred mode analytical method, a buckling condition is derived for the buckling load and associated buckling mode. In particular, explicit expressions are obtained for embedded double-walled carbon nanotubes (DWNTs). Numerical results show that, for both the DWNTs and embedded DWNTs, the buckling form shifts from the lower buckling mode to the higher buckling mode with increasing the buckling load, but the buckling mode is invariable for a certain domain of the buckling load. It is also indicated that, the surrounding elastic medium generally has effect on the lower buckling mode of DWNTs only when compared with the corresponding one for individual DWNTs.

Modeling of Nanoimprinting of Metals by Nanotube Arrays

AbstractMolecular dynamics (MD) simulations are reported for buckling and interfacial friction of single- and multi-wall carbon nanotubes (CNT) with interwall sp3 bonds imprinted on copper substrates. A small perturbation of mechanical vibration is applied to the systems in the nanoimprinting. The imprinting capabilities of multi-wall CNTs are much better than single-wall CNTs. While the single-wall CNT is insensitive to the vibration, the indentation force and buckling of multi-wall CNTs with interwall sp3 bonding is dependent on the amplitude of the perturbation, providing a way to controlling the interfacial friction. There is an optimal amplitude, at which the buckling and friction force of the CNTs are minimized in nanoimprinting.

Combined torsional buckling of multi-walled carbon nanotubes coupling with axial loading and radial pressures

This paper reports the results of an investigation on combined torsional buckling of multi-walled carbon nanotubes (MWNTs) under combined torque, axial loading and radial pressures based on the continuum mechanics model, which takes into account the effect of the van der Waals interaction between adjacent tubes. A buckling condition is derived for determining the critical buckling torque and associated buckling mode. In particular, for combined torsional buckling of double-walled carbon nanotubes, an explicit expression is obtained and some detailed results are demonstrated. According to the innermost radius-to-thickness ratio, MWNTs are classified into three types: thin, thick, and (almost) solid. Numerical results are worked out for the critical buckling torque and associated buckling mode for all the three types of MWNTs subjected to various axial stresses (axial tensile stresses or axial compressive stresses), internal pressures, and external pressures. It is shown that, the axial tensile stress or the internal pressure will make the MWNTs resist higher critical buckling torque, while the axial compressive stress or external pressure will lead to a lower critical buckling torque. The effect of axial stress (axial tensile stress or axial compressive stress) on the critical buckling torque of MWNTs is very small for all the three types of MWNTs, while the effect of the internal pressure or external pressure is related to the types of MWNTs, which is strong for the thin MWNTs, moderate for the thick MWNTs, and small for the solid MWNTs. Numerical results also indicate that, the associated buckling mode is unique and dependent on the structure of MWNTs. Especially, for combined torsional buckling of MWNTs with very small axial stress and radial pressures, the buckling mode is just the one for the corresponding pure torsional buckling.

Mechanical buckling of multi-walled carbon nanotubes: The effects of slenderness ratio

Buckling strengths, in terms of compressive strain, of single-, double- and triple-walled carbon nanotubes (CNTs) are investigated to study the effects of slenderness ratio ( S R ) via the molecular dynamics (MD) simulations with the Tersoff potential. Under constant ratio of slenderness, the CNTs with small S R behave like a continuum shell object. For large S R 's, multi-walled CNTs exhibit the characteristics of the Euler columns. In addition, smaller nanotubes possess higher buckling-resistance. The buckling strength of multi-walled nanotubes is controlled by the size of their outermost shell.