Carbon Nanotube-reinforced Composite Beams Research Articles

Nonlinear free vibration analysis of elastically supported carbon nanotube-reinforced composite beam with the thermal environment in non-deterministic framework

AbstractThis paper deals with the investigation of nonlinear free vibration behavior of elastically supported carbon nanotube reinforced composite (CNTRC) beam subjected to thermal loading with random system properties. Material properties of each constituent’s material, volume fraction exponent and foundation parameters are considered as uncorrelated Gaussian random input variables. The beam is supported by a Pasternak foundation with Winkler cubic nonlinearity. The higher order shear deformation theory (HSDT) with von-Karman non-linearity is used to formulate the governing equation using Hamilton principle. Convergence and validation study is carried out through the comparison with the available results in the literature for authenticity and accuracy of the present approach used in the analysis. First order perturbation technique (FOPT),Second order perturbation technique (SOPT) and Monte Carlo simulation (MCS) methods are employed to investigate the effect of geometric configuration, volume fraction exponent, foundation parameters, distribution of reinforcement and thermal loading on nonlinear vibration characteristics CNTRC beam.The present work signifies the accurate analysis of vibrational behaviour influences by different random variables. Results are presented in terms of mean, variance (COV) and probability density function (PDF) for various aforementioned parameters.

Adaptive fuzzy sliding mode control for vibration suppression of a rotating carbon nanotube-reinforced composite beam

In this paper, the robust vibration control of a rotating carbon nanotube reinforced composite beam subjected to a temperature rise is studied. The governing mathematical partial differential equations are derived by Hamilton’s principle based on Euler–Bernoulli beam theory. The Galerkin method is then used to obtain the temporal ordinary differential equations and therefore to perform vibration analysis. For the purpose of vibration control, piezoelectric patches are used as sensors to measure the displacement of the beam and as actuators to implement control forces. A model-free adaptive fuzzy sliding mode controller is utilized to suppress the vibration of the rotating carbon nanotube reinforced composite beam. Since the plant’s state vector cannot be measured for control purposes and only the piezoelectric sensor’s output is available, a model-free adaptive fuzzy sliding mode observer is proposed here. Simulation studies demonstrate the effectiveness of the proposed method.

A simple quasi-3D sinusoidal shear deformation theory with stretching effect for carbon nanotube-reinforced composite beams resting on elastic foundation

A simple quasi-3D sinusoidal shear deformation theory with stretching effect for carbon nanotube-reinforced composite beams resting on elastic foundation

Thermo-mechanical vibration analysis of sandwich beams with functionally graded carbon nanotube-reinforced composite face sheets based on a higher-order shear deformation beam theory

ABSTRACTThis article proposes a higher-order shear deformation beam theory for free vibration analysis of functionally graded carbon nanotube-reinforced composite sandwich beams in a thermal environment. The temperature-dependent material properties of functionally graded carbon nanotube-reinforced composite beams are supposed to vary continuously in the thickness direction and are estimated through the rule of mixture. The governing equations and boundary conditions are derived by using Hamilton's principle, and the Navier solution procedure is used to achieve the natural frequencies of the sandwich beam in a thermal environment. A parametric study is led to carry out the effects of carbon nanotube volume fractions, slenderness ratio, and core-to-face sheet thickness ratio on free vibration behavior of sandwich beams with functionally graded carbon nanotube-reinforced composite face sheets. Numerical results are also presented in order to compare the behavior of sandwich beams including uniformly distributed carbon nanotube-reinforced composite face sheets to those including functionally graded carbon nanotube-reinforced composite face sheets.

Imperfection sensitivity of thermal post-buckling behaviour of functionally graded carbon nanotube-reinforced composite beams

This paper investigates the imperfection sensitivity of thermal post-buckling behaviour of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beams subjected to in-plane temperature variation. The material properties of FG-CNTRCs are assumed to be graded in the thickness direction and temperature-dependent. A generic imperfection function is used to model various possible imperfections, including sine type, global and localized imperfections. The governing equations are derived based on the first-order shear deformation beam theory and von-Kármán geometric nonlinearity. The differential quadrature method in conjunction with modified Newton–Raphson technique is employed to determine the thermal post-buckling equilibrium path of imperfect FG-CNTRC beams. Thermal buckling is treated as a subset problem. A parametric study is conducted to examine the effects of imperfection mode, half-wave number, location and amplitude on their thermal post-buckling performance. The influences of distribution pattern and volume fraction of carbon nanotubes, boundary conditions and slenderness ratio are discussed as well. The results indicate that the thermal post-buckling is highly sensitive to the imperfection mode, half-wave number, location as well as its amplitude. It is also shown that the clamped-clamped FG-CNTRC beam is more sensitive to imperfections than those with other boundary conditions whereas other parameters do not substantially affect the imperfection sensitivity of thermal post-buckling behaviour.

Stability and vibration analyses of carbon nanotube-reinforced composite beams with elastic boundary conditions: Chebyshev collocation method

ABSTRACTThis article aims to investigate stability and vibration behavior of carbon nanotube-reinforced composite beams supported by classical and nonclassical boundary conditions. To include significant effects of shear deformation and rotary inertia, Timoshenko beam theory is used to formulate the coupled equations of motion governing buckling and vibration analyses of the beams. An effective mathematical technique, namely Chebyshev collocation method, is employed to solve the coupled equations of motion for determining critical buckling loads and natural frequencies of the beams with different boundary conditions. The accuracy and reliability of the proposed mathematical models are verified numerically by comparing with the existing results in the literature for the cases of classical boundary conditions. New results of critical buckling loads and natural frequencies of the beams with nonclassical boundary conditions including translational and rotational springs are presented and discussed in detail associated with many important parametric studies.

Imperfection sensitivity of postbuckling behaviour of functionally graded carbon nanotube-reinforced composite beams

The imperfection sensitivity of the postbuckling behaviour of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beams subjected to axial compression is investigated based on the first-order shear deformation beam theory with a von Kármán geometric nonlinearity. The material properties of FG-CNTRC are assumed to vary in the beam thickness direction and are estimated according to the extended rule of mixture. The differential quadrature method is employed to discretize the governing differential equations and the modified Newton-Raphson iterative technique is used to obtain the postbuckling equilibrium paths of FG-CNTRC beams with various imperfections. Parametric studies are carried out to examine the effects of imperfection modes, half-wave number, location, and amplitude on the postbuckling response of beams. The influences of CNT distribution pattern and volume fraction, boundary conditions, and slenderness ratio are also discussed. Numerical results in graphical form show that the postbuckling behaviour is highly sensitive to the imperfection amplitude. The imperfection mode and its half-wave number also moderately affect the imperfection sensitivity of the postbuckling response, whereas the effects of other parameters are much less pronounced.

Thermo-electro-mechanical postbuckling of piezoelectric FG-CNTRC beams with geometric imperfections

This paper presents thermo-electro-mechanical postbuckling analysis of geometrically imperfect functionally graded carbon nanotube-reinforced composite (FG-CNTRC) hybrid beams that are integrated with surface-bonded piezoelectric actuators. The material properties of FG-CNTRCs are assumed to be temperature-dependent and graded in the thickness direction. By using a generic imperfection function, various possible imperfections with different shapes and locations in the beam are considered. The theoretical formulations are based on the first-order shear deformation beam theory with von-Kármán nonlinearity. A differential quadrature approximation based iteration process is employed to obtain the postbuckling equilibrium path of piezoelectric FG-CNTRC hybrid beams under thermo-electro-mechanical loading. Parametric studies are conducted to examine the effect of geometric imperfection, distribution pattern and volume fraction of carbon nanotubes, temperature rise, actuator voltage, beam geometry and boundary conditions on the thermo-electro-mechanical postbuckling behaviour. The results show that the thermo-electro-mechanical postbuckling is considerably affected by the imperfection mode, half-wave number, location and amplitude, as well as the temperature rise and boundary conditions. The effect of applied actuator voltage is much less pronounced but tends to be relatively more noticeable as the slenderness ratio increases.

Thermal Buckling and Postbuckling Analysis of Functionally Graded Carbon Nanotube-Reinforced Composite Beams

Thermal buckling and postbuckling of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beams are investigated in this paper based on Timoshenko beam theory within the framework of von Kármán geometric nonlinearity. The material properties of FG-CNTRCs are assumed to be temperature-dependent and vary in the beam thickness direction. The governing equations are derived by employing Hamilton’s principle then discretized by using differential quadrature (DQ) method. An iterative scheme is used to obtain the critical buckling temperature and nonlinear thermal postbuckling equilibrium path of the FG-CNTRC beam. Numerical results are presented for FG-CNTRC beams hinged or clamped at both ends, with particular focuses on the effects of the volume fraction of carbon nanotubes (CNTs), slenderness ratio, and end supports on the thermal buckling and postbuckling characteristics.

Nonlinear vibration of functionally graded carbon nanotube-reinforced composite beams with geometric imperfections

The nonlinear vibration of imperfect shear deformable functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beams is studied in this paper based on the first-order shear deformation beam theory and von Kármán geometric nonlinearity. A one-dimensional imperfection model in the form of the product of trigonometric and hyperbolic functions are used to describe the various possible geometric imperfections such as sine type, global, and localized imperfections. The governing equations are derived by employing the Ritz method and then solved by an iteration procedure. Special attention is given to the influences of imperfection mode, location, and amplitude on the nonlinear behaviour. The linear vibration is also discussed as a subset problem. Numerical results in tabular and graphical forms show that the nonlinear vibration behaviour of imperfect FG-CNTRC beams is considerably sensitive to sine type and global imperfections (except for G2-mode), whereas the effect of localized imperfection is much less pronounced. It is also observed that whether the FG-CNTRC beam exhibits the “hard-spring” or “soft-spring” vibration behaviour is largely dependent on the initial imperfection mode, its amplitude as well as the vibration amplitude.

On bending, buckling and vibration responses of functionally graded carbon nanotube-reinforced composite beams

In this work, a trigonometric refined beam theory for the bending, buckling and free vibration analysis of carbon nanotube-reinforced composite (CNTRC) beams resting on elastic foundation is developed. The significant feature of this model is that, in addition to including the shear deformation effect, it deals with only 3 unknowns as the Timoshenko beam (TBM) without including a shear correction factor. The single-walled carbon nanotubes (SWCNTs) are aligned and distributed in polymeric matrix with different patterns of reinforcement. The material properties of the CNTRC beams are assessed by employing the rule of mixture. To examine accuracy of the present theory, several comparison studies are investigated. Furthermore, the effects of different parameters of the beam on the bending, buckling and free vibration responses of CNTRC beam are discussed.

Elasticity Solution of Free Vibration and Bending Behavior of Functionally Graded Carbon Nanotube-Reinforced Composite Beam with Thin Piezoelectric Layers Using Differential Quadrature Method

Based on the theory of elasticity, bending and free vibrational analyses of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beam embedded in piezoelectric layers are carried out, using the state-space differential quadrature method (DQM). Applying the DQM to the governing differential equations and boundary conditions along the longitudinal directions, new state equations about state variables at discrete points are derived. By using the state-space technique across the thickness direction, semi- analytical closed form solutions are derived. The method is validated by comparing numerical results for beams without piezoelectric layers. Both the direct and the inverse piezoelectric effects are investigated and the influence of piezoelectric layers on the mechanical behavior of beam is studied. Furthermore, the effects of CNT volume fraction, kind of CNT distribution, length to thickness ratio and edge boundary conditions on the mechanical behavior of the beams are examined.

Nonlinear forced vibration analysis of functionally graded carbon nanotube-reinforced composite Timoshenko beams

This research deals with the forced vibration behavior of nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs) based on the Timoshenko beam theory along with von Kármán geometric nonlinearity. For the carbon-nanotube reinforced composite (CNTRC) beams, uniform distribution (UD) and three types of functionally graded (FG) distribution patterns of SWCNT reinforcements are considered. It is assumed that the material properties of FG-CNTRC beams are graded in the thickness direction and estimated through the rule of mixture. The nonlinear governing equations and corresponding boundary conditions are derived based on the Hamilton principle and discretized by means of the generalized differential quadrature (GDQ) method. After that, a Galerkin-based numerical technique is employed to reduce the set of nonlinear governing equations into a time-varying set of ordinary differential equations of Duffing type. Since the nanobeam responds periodically to harmonic excitations, a set of periodic differential matrix operators is introduced to discretize the Duffing equations on the time domain using the derivatives of a periodic base function. The vectorized form of final nonlinear parameterized equations is then solved through the use of pseudo-arc length continuum technique. Numerical results are presented to examine the effects of different parameters such as nanotube volume fraction, slenderness ratio, dimensionless damping parameter, dimensionless transverse force, CNT distributions and boundary conditions on the natural frequencies and frequency responses of FG-CNTRC beams.

Thermal bifurcation buckling of piezoelectric carbon nanotube reinforced composite beams

The nonlinear thermal bifurcation buckling behavior of carbon nanotube reinforced composite (CNTRC) beams with surface-bonded piezoelectric layers is studied in this paper. The governing equations of piezoelectric CNTRC beam are obtained based on the Euler–Bernoulli beam theory and von Kármán geometric nonlinearity. Two kinds of carbon nanotube-reinforced composite (CNTRC) beams, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The material properties of FG-CNTRC beam are assumed to be graded in the thickness direction. The SWCNTs are assumed aligned, straight and with a uniform layout. Exact solutions are presented to study the thermal buckling behavior of beams made of a symmetric single-walled carbon nanotube reinforced composite with surface-bonded piezoelectric layers. The critical temperature load is obtained for the nonlinear problem. The effects of the applied actuator voltage, temperature, beam geometry, boundary conditions, and volume fractions of carbon nanotubes on the buckling of piezoelectric CNTRC beams are investigated.

Nonlinear analysis of nanotube-reinforced composite beams resting on elastic foundations in thermal environments

This paper studies the behaviors of large amplitude vibration, nonlinear bending and thermal postbuckling of nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs) resting on an elastic foundation in thermal environments. Two types of carbon nanotube-reinforced composite (CNTRC) beams, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The material properties of FG-CNTRCs are assumed to be graded in the beam thickness direction, and are estimated through a micromechanical model. The motion equations of a CNTRC beam on an elastic foundation are derived based on a higher order shear deformation beam theory. The thermal effects are also included in the motion equations and the material properties of CNTRCs are assumed to be temperature-dependent. Numerical studies are carried out for the nonlinear vibration, nonlinear bending and thermal postbuckling of CNTRC beams resting on Pasternak elastic foundations under different thermal environmental conditions. It is found that a CNTRC beam with intermediate CNT volume fraction does not necessarily have intermediate nonlinear frequencies, buckling temperatures and thermal postbuckling strengths. Thermal postbuckling path of unsymmetric FG-CNTRC beams is no longer the bifurcation type.

Analytical solutions for bending, buckling and vibration responses of carbon nanotube-reinforced composite beams resting on elastic foundation

The objective of the present paper is to investigate the bending, buckling and vibration behaviors of carbon nanotube-reinforced composite (CNTRC) beams. The beams resting on the Pasternak elastic foundation, including a shear layer and Winkler spring, are considered. The single-walled carbon nanotubes (SWCNTs) are aligned and distributed in polymeric matrix with different patterns of reinforcement. The material properties of the CNTRC beams are estimated by using the rule of mixture. Various shear deformation theories are employed to deal with the problems. The mathematical models provided in this paper are numerically validated by comparison with some available results. New results of bending, buckling and vibration analyses of CNTRC beams based on several higher-order shear deformation theories are presented and discussed in details. Several aspects of beam types, spring constant factors, carbon nanotube volume fraction, etc., are taken into investigation.

Dynamic Stability of Functionally Graded Carbon Nanotube-Reinforced Composite Beams

This article presents a dynamic stability analysis of functionally graded nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs) based on Timoshenko beam theory. The material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are assumed to vary in the thickness direction and are estimated through the rule of mixture. The differential quadrature method is employed to convert the governing differential equations into a linear system of Mathieu-Hill equations from which the boundary points on the unstable regions are determined by Bolotin's method. Free vibration and elastic buckling are also discussed as subset problems. A parametric study is conducted to investigate the influences of nanotube volume fraction, slenderness ratio, and end supports on the dynamic stability characteristics of FG-CNTRC beams. Numerical results for composite beams reinforced by uniformly distributed carbon nanotube are also provided for comparison.

Large amplitude vibration of carbon nanotube reinforced functionally graded composite beams with piezoelectric layers

Large amplitude free vibration of functionally graded carbon nanotube reinforced composite (CNTRC) beams with surface-bonded piezoelectric layers subjected to a temperature change and an applied voltage is studied in this paper. The governing equations of the piezoelectric CNTRC beam are derived based on Euler–Bernoulli beam theory, von Kármán geometric nonlinearity and the physical neutral surface concept. Both uniformly distribution (UD) and functionally graded (FG) distribution patterns of the single-walled carbon nanotube (SWCNT) reinforcements are considered. It is assumed that the material properties of the FG-CNTRC beam vary in the thickness direction, and that the SWCNTs are aligned and straight. Galerkin procedure is used to obtain the second order nonlinear ordinary equation in time with cubic nonlinear term. Multiple times scales method is then employed to determine the nonlinear free vibration characteristics of the beam clamped at both ends. The effects of the applied voltage, temperature change, beam geometry, the volume fraction and distribution pattern of the SWCNTs on the linear and nonlinear frequencies of the piezoelectric CNTRC beams are investigated through a comprehensive parametric study.

Free vibrations and buckling analysis of carbon nanotube-reinforced composite Timoshenko beams on elastic foundation

This study deals with free vibrations and buckling analysis of nanocomposite Timoshenko beams reinforced by single-walled carbon nanotubes (SWCNTs) resting on an elastic foundation. The SWCNTs are assumed to be aligned and straight with a uniform layout. Four different carbon nanotubes (CNTs) distributions including uniform and three types of functionally graded distributions of CNTs through the thickness are considered. The rule of mixture is used to describe the effective material properties of the nanocomposite beams. The governing equations are derived through using Hamilton's principle and then solved by using the generalized differential quadrature method (GDQM). Natural frequencies and critical buckling load are obtained for nanocomposite beams with different boundary conditions. Effects of several parameters, such as nanotube volume fraction, foundation stiffness parameters, slenderness ratios, CNTs distribution and boundary conditions on both natural frequency and critical buckling load are investigated. The results indicate that the above-mentioned parameters play a very important role on the free vibrations and buckling characteristics of the beam.

Nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams

This paper investigates the nonlinear free vibration of functionally graded nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs) based on Timoshenko beam theory and von Karman geometric nonlinearity. The material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are assumed to be graded in the thickness direction and estimated though the rule of mixture. The Ritz method is employed to derive the governing eigenvalue equation which is then solved by a direct iterative method to obtain the nonlinear vibration frequencies of FG-CNTRC beams with different end supports. A detailed parametric study is conducted to study the influences of nanotube volume fraction, vibration amplitude, slenderness ratio and end supports on the nonlinear free vibration characteristics of FG-CNTRC beams. The results for uniformly distributed carbon nanotube-reinforced composite (UD-CNTRC) beams are also provided for comparison. Numerical results are presented in both tabular and graphical forms to investigate the effects of nanotube volume fraction, vibration amplitude, slenderness ratio, end supports and CNT distribution on the nonlinear free vibration characteristics of FG-CNTRC beams.