Modulus Of Single-walled Carbon Nanotubes Research Articles

Theoretical predictions of size-dependent Young's and shear moduli of single-walled carbon nanotubes

In this study, based on the Force-Heat Equivalence Energy Density Principle, the energy density stored in Single-walled carbon nanotubes (SWCNTs) with different diameters is assumed as the same when SWCNTs with different diameters undergo the same elastic strain under a uniaxial external tensile loading, which is composed of strain energy density and the equivalent curvature energy density. Then, theoretical models without the thickness of SWCNTs for characterizing size-dependent Young's and shear moduli of SWCNTs are established considering the effect of curvature energy. To verify the proposed models, the Young's and shear modulus of SWCNTs in different diameters are predicted, and good agreement is obtained with the available theoretical and numerical simulation results. Besides, the predictions by our models are normalized by graphene values and found the same trends as other scholars. This work contributes to providing a new method to estimate the size-dependent elastic moduli of SWCNTs.

Determination of fundamental coupled torsional—radial frequency of single-walled carbon nanotubes

Research indicates that single-walled carbon nanotubes have a unique coupled torsional–radial vibration as one of their fundamental modes. Determination of their vibration frequency is required for efficient use of single-walled carbon nanotube in nano-electromechanical systems. However, there is no mathematical expression for these frequencies and their dependence on single-walled carbon nanotube geometry is unknown. This article examines the effect of diameter, length, and chirality on the fundamental coupled torsional–radial vibration frequency of single-walled carbon nanotube using molecular–structural–mechanics–approach, finite element analysis, and regression analyses. Consequently, a first-ever mathematical form of this frequency is derived. The form quickly and accurately predicts these frequencies at 1.5% in-sample, and 7.2% out-sample mean absolute percentage error. single-walled carbon nanotubes’ fundamental coupled torsional–radial vibration frequency is found independent of diameter and inversely proportional to length where the proportionality constant depends on chirality. The coupling of modes and the similarity of the frequency form with cylindrical shell suggest that single-walled carbon nanotube behave like thin shells in these vibrations. A form for effective circumferential shear modulus of single-walled carbon nanotube is also derived. This modulus is found to depend only on the chirality where achiral single-walled carbon nanotubes have higher values than chiral single-walled carbon nanotubes. Proposed mathematical forms can be used for characterization of single-walled carbon nanotubes, determination of single-walled carbon nanotubes’ effective shear modulus, and tuning operational frequency of single-walled carbon nanotube-based nano-electromechanical systems.

Effect of interphase, curvature and agglomeration of SWCNTs on mechanical properties of polymer-based nanocomposites: Experimental and numerical investigations

In this study, the elastic modulus of single-walled carbon nanotubes (SWCNTs)/epoxy nanocomposite was studied using the 3D finite element method and compared with experimental results to investigate the effect of SWCNTs interphase, curvature, and agglomeration on the prediction of the elastic modulus. Nanocomposite specimens containing 0.1, 0.3, and 0.5 wt% SWCNTs were fabricated to obtain SWCNTs/epoxy elastic modulus. The elastic modulus increased until SWCNTs was incorporated up to 0.3 wt% and after that, the trend of increasing elastic modulus declined. TEM images showed that in higher contents of filler, there were some local SWCNTs agglomerations within the composites which caused a dropped in elastic modulus of specimens containing 0.5 wt% SWCNTs. Also, six different 3D representative volume element (RVE) of SWCNTs/epoxy including incorporated cylindrical, cylindrical with agglomeration, curved-cylindrical, cylindrical with interphase, cylindrical with interphase and agglomeration and curved-cylindrical with agglomeration SWCNTs in the epoxy matrix have been generated using Digimat-FE and their elastic modulus evaluated by Digimat-FE solver. The numerical results cleared that the simplest cylindrical RVE has the greatest discrepancy with experimental results which showed the necessity of consideration of three important parameters including SWCNTs interphase, curvature, and agglomerations. By considering SWCNTs interphase and agglomeration the difference of numerical and experimental results decreased so that in specimens containing 0.1 wt% SWCNTs the error was only 6.8%. Also, the best results obtained from RVE of curved-cylindrical with agglomeration in specimen containing 0.1 wt% SWCNTs with only 4.1% error which showed the importance of considering SWCNTs agglomeration and curvature for modeling of nanocomposites.

Modified Nonlocal Strain Gradient Elasticity for Nano-Rods and Application to Carbon Nanotubes

Nowadays, the modified nonlocal strain gradient theory provides a mathematically well-posed and technically reliable methodology to assess scale effects in inflected nano-structures. Such an approach is extended in this paper to investigate the extensional behavior of nano-rods. The considered integral elasticity model, involving axial force and strain fields, is conveniently shown to be equivalent to a nonlocal differential problem equipped with constitutive boundary conditions. Unlike treatments in the literature, no higher-order boundary conditions are required to close the nonlocal problem. Closed-form solutions of elastic nano-rods under selected loadings and kinematic boundary conditions are provided. As an innovative implication, Young’s moduli of Single-Walled Carbon Nanotubes (SWCNT) weare assessed and compared with predictions of Molecular Dynamics (MD). New benchmarks for numerical analyses were also detected.

Hyperbolic heat conduction and thermoelastic solution of functionally graded CNT reinforced cylindrical panel subjected to heat pulse

As a first attempt, the combined application of the differential quadrature method (DQM) and the Newton Raphson method is used to solve the non-Fourier heat conduction equations to obtain temperature, displacements, and stresses in a cylindrical panel made of functionally graded, carbon nanotubes (CNTs) reinforced composite. The heat conduction of a domain made of nanocomposites subjected to heat generation is simulated with a finite heat wave speed. Furthermore, based on the three-dimensional elasticity theory, the thermoelastic analysis of the nanocomposite cylindrical panel subjected to the transient heat conduction is presented. The dynamic Young's modulus of Single-Walled Carbon Nanotubes (SWCNT) can be expressed as a function of loading rate and environmental temperature. All material properties such as heat capacity (Cp), thermal relaxation time (τ), density (ρ) and thermal conductivity (K) are considered as a function of both temperature and CNT volume fraction. The hyperbolic heat conduction is solved to obtain temperature in the spatial and temporal domains. Then by implementing the obtained temperature in thermoelastic equations of the cylindrical panel, the displacements and stresses will be obtained at each time step. The proposed method marches in the time direction block by block. In each block, there are several time levels, and the numerical results at these time levels are obtained simultaneously. Through this way, the numerical solution at the (n + 1)th time level depends on the solutions at its previous levels from the 1st to the nth levels. The final results in the temporal domain are obtained using the Newton–Raphson method. Accuracy of the present solution is confirmed by comparing with some available results in the literature. A detailed numerical study is conducted to examine the effects of heat wave speed and heat flux, CNT volume fraction and the geometrical parameters on the deflection of the cylindrical panel.

Closed form solution for a nonlocal strain gradient rod in tension

A size-dependent integral elasticity model is developed for a small-scaled rod in tension based on the nonlocal strain gradient theory. The integral rod model contains a nonlocal parameter and a material length scale parameter to incorporate the scaling effects of nonlocal stress and microstructure-dependent strain gradient. In comparison to size-dependent differential models, the developed integral rod model is both self-consistent and well-posed. The governing equations and boundary conditions for the nonlocal strain gradient rod in tension are derived by employing the principle of virtual work. In addition to the classical natural and essential boundary conditions, non-classical natural and essential boundary conditions are present for the integral rod model. The closed-form solutions for predicting the displacement and reduced Young’s modulus are derived for four types of boundary conditions. It is shown explicitly that the integral rod model can exert stiffness-softening and stiffness-hardening effects by considering various values of size-dependent parameters and boundary conditions. It is found that, the developed rods with four different boundary conditions can predict the scaling effects of the Young’s modulus of single-walled carbon nanotube, and the scaling effects are more sensitive to the size-dependent parameters (the material length scale parameter and the nonlocal parameter) in comparison with the non-classical boundary conditions.

On a thickness free expression for the shear modulus of carbon nanotubes

The thickness of carbon nanotubes is an important issue for the characterization and design of these structures. In this article, thickness free expressions for the shear modulus of single-walled carbon nanotubes have been developed by finite element simulations on the minimum potential energy circle. As a part of this work, some equations have been obtained to define the relation between the thickness and the shear modulus, which are in good agreement with previous studies. Moreover, these expressions are in good agreement with both continuum and quantum mechanics and capable to support “Yakobson's paradox,” that the scattering data for the elastic properties of carbon nanotubes are due to the not-well-defined thickness for these structures. Furthermore, these expressions can provide a tool for the prediction of the shear modulus of single-walled carbon nanotubes in regards to any thickness assumption when the experimental investigations are too difficult to realize.

Thermoelastic response of CNT reinforced cylindrical panel resting on elastic foundation using theory of elasticity

Abstract Based on the three-dimensional elasticity theory, the investigation of the free vibration response of a carbon nanotube-reinforced cylindrical panel resting in elastic foundation in thermal environments is presented. The response of the elastic medium is formulated by the Winkler/Pasternak model. The cylindrical panel has been reinforced by carbon nanotube in the radial direction and the material properties are temperature dependent and estimated by the extended rule of mixture. Dynamic Young’s modulus of single-walled carbon nanotubes can be expressed a function of loading rate and environmental temperature. Differential quadrature method is being utilized and natural frequencies of cylindrical panel are obtained. An accuracy of the present solution is confirmed by comparing with some available results in the article. A detailed numerical study is conducted to examine the effects of temperature rise, carbon nanotube volume fraction, elastic foundations and the geometrical parameters on the deflection of the cylindrical panels.

The mechanical property of single-walled carbon nanotube under combined electromechanical loading

ABSTRACTThe mechanical properties and failure process of single-walled carbon nanotube (SWCNT) under combined electric field and tensile loading are investigated using the semi-empirical quantum mechanical method. The local and global structural deformation and variation of mechanical properties of SWCNT under different directions and intensity of external electric field are discussed systematically. It is shown that the electric field induced deformation in the radial and axial directions of the SWCNT are strongly dependent on the direction of electric field. The analysis of mechanical properties shows that the structure stiffness, tensile strength and failure strain of the SWCNT all decrease with the increase of the field intensity, which is particularly evident under the longitudinal electric field. The Young's modulus of SWCNTs vary with the tube diameter and are affected by the electric field. The increase of the length of the tubes intensifies the charge concentration at the tube ends under the electric field and lead to the decrease of mechanical properties of SWCNTs. The failure process of SWCNTs under the coupling effect of electric field and tensile loading is found to be controlled by the field strength and also affected by the electric charge accumulation.

Numerical simulation study of the elastic properties of single-walled carbon nanotubes containing vacancy defects

The mechanical behaviour of non-chiral and chiral single-walled carbon nanotubes containing different percentage (up to 10%) and types of vacancy defects is studied under tensile, bending and torsional loading. A three-dimensional finite element model is used in order to evaluate the corresponding rigidities and, subsequently, Young's and shear moduli and Poisson's ratio. The three rigidities decrease with the increase of the percentage of vacancies. Also, the Young's and shear moduli and the Poisson's ratio of single-walled carbon nanotubes are sensitive to the presence of vacancy defects in nanotube: elastic moduli decrease and the Poisson's ratio increases with increasing of the percentage of vacancies. The moduli of single-walled carbon nanotubes with 10.0% of vacancy defects, when compared with the values obtained for perfect nanotubes, are of about 43% for the Young's modulus and of about 33% for the shear modulus. On the contrary, the Poisson's ratio increases of about 4 times, compared with that obtained for the perfect nanotube.

Effect of Chiral Angle on Tensile Behavior Modeling of Single-Walled Carbon Nanotubes

The effect of chiral angle independent from diameter on tensile behavior of single-walled carbon nanotubes (SWCNTs) is studied. Two methods in finite element continuum mechanics modeling of SWCNTs are used. The role of chiral angle and diameter are investigated independently on linear and nonlinear tensile behavior of SWCNTs by modeling several types of nanotubes. Nonlinear tensile behavior is more influenced by chiral angle than the linear Young's modulus of SWCNTs. On the other hand, the effect of diameter is found to be more obvious on Young's modulus than nonlinear tensile behavior of SWCNTs.

Estimation of the Young’s modulus of single-walled carbon nanotubes under electric field using tight-binding method

In this paper, the Young’s modulus of single-walled carbon nanotubes (SWCNTs) under uniaxial strain is studied by using the tight-binding method in the nearest neighbor approximation while the curvature effect take into account. Also, the effect of an external electric field on the Young’s modulus of carbon nanotubes is investigated. The strain dependent tight-binding parameters are obtained by comparing the tight-binding band structure of graphene with DFT calculations. In the next step, total energy and then the Young’s modulus of single-walled carbon nanotubes are calculated. Our results indicate the dependence of Young’s modulus on the tube diameter and the external electric field. By applying the axial electric field, the dependency of Young’s modulus to the diameter becomes less important. Also, by increasing the axial electric field, the Young’s modulus decreases. The axial electric field becomes much more important in the armchair nanotubes.

Shell-like instability of large diameter single-walled carbon nanotubes conveying fluid

The instability of large diameter single-walled carbon nanotubes (SWCNTs) conveying fluid is investigated based on the molecular mechanics. Using the modal expansion for structural displacements, the governing equations of coupled fluid-structural dynamics of SWCNTs are derived. The natural frequencies and mode shape of the SWCNTs are obtained based on the molecular structural mechanics to account for the effect of chirality and discrete nature of SWCNTs. The results show that the vibrational behavior of large diameter SWCNTs conveying fluid is size dependent, but the effect of chirality is negligible. The obtained results are compared with the equivalent continuum-based model in the literature and in some cases remarkable deviations between the results are observed. Accordingly, proper values for the equivalent wall thickness and Young’s modulus of SWCNTs are proposed to adopt the results of continuum-based model with the molecular mechanics predictions.

Research on the Elastic Modulus of Single-Walled Carbon Nanotubes

The calculation of elastic modulus of carbon nanotubes is an essential prerequisite for future Micro-nano devices fabrication that based on carbon nanotubes. The interaction potential of C-C covalent bond in single-walled carbon nanotubes (SWCNTs) is described by Terrsoff-Brenner potential in this paper. The elastic modulus of SWCNTs is ranged from 1.1078TPa to 1.23789TPa with the diameter increases from 0.3176nm to 2.0626nm by molecular dynamics (MD) simulation. The critical diameters of armchair and zigzag nanotubes are 1nm and 0.6nm.The elastic modulus changes with the diameter of nanotubes increasing clearly when the diameter smaller than the critical diameter, but the value of elastic modules tends to convergence when the diameter of nanotubes larger than the critical diameter.

An accurate spring-mass model for predicting mechanical properties of single-walled carbon nanotubes

An accurate spring-mass model, in the context of a three-dimensional finite element formulation, is developed for estimating Young’s and shear modulus of single-walled carbon nanotubes (SWCNTs). Lumped mass elements are placed at the atom locations and appropriate spring-type elements are defined as interconnections between the atoms in order to simulate the inter-atomic interactions. Based on the variation of the Brenner potential function, a simple way of computing the force constants used in the model developed is proposed. The obtained results for Young’s and shear modulus of SWCNTs for various kinds are graphically illustrated. Further, the influences of changes in the nanotube radius on the mechanical behavior are examined. The numerical results show good agreement with other published results in the literature.

A molecular dynamics study of the variations in the elastic properties of single-walled carbon nanotubes with tube radius, length and chirality

Carbon nanotube (CNT) is promising to revolutionise several fields in material science and is a major component of nanotechnology. In this study, molecular dynamics simulation is used to investigate the elastic properties of single-walled carbon nanotubes. Comprehensive numerical calculations are carried out for different armchair and zigzag carbon nanotubes with various geometric dimensions to assess the effects of tube radius, tube length and tube chirality on the elastic properties of single-walled carbon nanotubes (SWCNTs). The results show that, the axial Young’s modulus of both armchair and zigzag SWCNTs decreases with increasing tube radius which is also in good agreement with experimental observation. The results also reveal that, the value of the axial Young’s modulus of the SWCNTs is independent of the tube length and tube chirality.

A theoretical investigation of thermal effects on vibrational behaviors of single-walled carbon nanotubes

Thermal effects on the vibrational behaviors and dynamic Young’s modulus of single-walled carbon nanotubes (SWCNTs) are investigated through both constant temperature molecular dynamics (MD) simulation and modified molecular structural mechanics (MMSM) modeling. The MD simulation incorporates a modified Nosé-Hoover thermostat model to control the system temperature. In the MMSM modeling, the covalent and nonbonded interactions between carbon atom pairs are modeled with the second generation force field and the Lennard–Jones potential, respectively, where the covalent bonds are treated as Euler–Bernoulli beam and the temperature-dependent bond length and angle are determined through the Badger’s rule and MD simulation. The results derived from these two approaches are compared with each other and the published theoretical and experimental data. Results show that the dynamic Young’s modulus of the SWCNTs tends to be smaller than the published static one obtained from uniaxial tensile tests, and their natural frequency and dynamic Young’s modulus would decrease with increasing temperature. Moreover, a comparable frequency ratio of the first two flexural modes is achieved by these two approaches. The frequency ratio is highly dependent on their aspect ratio but independent of temperature, and would converge to the literature experimental data (about 6.1–6.2) as the aspect ratio becomes very large.

Investigation of the thermo-mechanical properties of single-walled carbon nanotubes based on the temperature-related higher order Cauchy–Born rule

In the present study, a nanoscale quasi-continuum constitutive model for predicting the thermal-mechanical properties of single-walled carbon nanotubes (SWCNTs) and graphene sheet is established based on the interatomic potential and the temperature-related higher order Cauchy–Born rule. Helmholtz free energy is used as the corresponding thermodynamic potential. It is a function of some temperature-dependent lattice parameters that can be determined through an energy minimization process. As an application of the proposed quasi-continuum model, temperature dependency and curvature effect of the specific heat, the coefficient of thermal expansion (CTE) and the Young’s modulus of SWCNTs are investigated systematically. Numerical results obtained show the effectiveness of the proposed constitutive model.

Predicting the Young’s Modulus of Single-Walled Carbon Nanotubes using Finite Element Modeling

A finite element simulation technique for estimating the mechanical properties of Single-Walled Carbon Nanotube (SWCNT), polymer composites is developed. In the present modeling work, individual carbon nanotube is simulated as a frame-like structure and the primary bonds between two nearest-neighboring atoms are treated as 3D beam elements. The beam element nonlinear properties are determined via the concept of energy equivalence between molecular dynamics and structural mechanics using Modified Morse potential. Young’s modulus of SWCNTs is estimated to illustrate the accuracy of this simulation technique. Results show that the obtained mechanical properties of nanotubes by the present method are in good agreement with their comparable results.

A semi-continuum finite element approach to evaluate the Young’s modulus of single-walled carbon nanotube reinforced composites

This paper describes a micromechanical finite element approach for the estimation of the effective Young’s modulus of single-walled carbon nanotube reinforced composites. These composite materials consist of aligned carbon nanotubes that are uniformly distributed within the matrix. Based on micromechanical theory, the Young’s modulus of the nanocomposite is estimated by considering a representative cylindrical volume element. Within the representative volume element, the reinforcement is modeled according to its atomistic microstructure while the matrix is modeled as a continuum medium. Spring-based finite elements are employed to simulate the discrete geometric structure and behavior of each single-walled carbon nanotube. The load transfer conditions between the carbon nanotubes and the matrix are modeled using joint elements of changeable stiffness that connect the two materials, simulating the interfacial region. The proposed model has been tested numerically and yields reasonable results for variable stiffness values of the joint elements. The effect of the interface on the performance of the composite is investigated for various volume fractions. The numerical results are compared with experimental and analytical predictions.