Graded Carbon Nanotube Research Articles

Vibration characteristics of zigzag FGM single-walled carbon nanotubes based on Ritz method with ring-stiffeners

In this study, Rayleigh–Ritz method is used for free vibrations of functionally graded carbon nanotubes based on Sander’s shell theory. The effects of various parameters for length-to-diameter ratio and height-to-diameter ratio for zigzag single-walled carbon nanotubes have been fully inspected. The frequency pattern with mass density per unit lateral area and in-plane rigidity has been calculated and found reverse order. The tube material is taken as functionally graded material and their material quantity is located by the polynomial volume fraction law. It is examined that the frequencies increase on increasing the ratio of height-to-diameter. Moreover, the impact of ring supports around the tube circumferential is studied for their various positions along tube axial length. These rings are restrained the radial displacement in the transverse direction.

Dynamic instability analysis of FG-CNTRC laminated conical shells surrounded by elastic foundations within FSDT

A comprehensive understanding of the dynamic instability of shell structure is critical to avoid resonance damage. On the basis of that, an accurate and analytical method for investigation the dynamic instability of laminated functionally graded carbon nanotube reinforced composite (FG-CNTRC) conical shell surrounded by the elastic foundations is presented in this work based on the first-order shear deformation theory. In the analysis, uniform or functionally graded distributions of reinforcements across the shell thickness are considered and the extended Voigt model is employed to estimate the CNTRC material properties. The governing equations of conical shell subjected to parametric excitation are established by the Hamilton's principle considering first order shear deformation shell theory. Then the Mathieu-Hill equations describing the parametric stability of conical shell are obtained by generalized differential quadrature (GDQ) method, and the Bolotin's method is utilized to obtain the first-order approximations of principal instability regions of shell structure. By comparing the numerical results with the existing solutions in open literature, the validity of the proposed theoretical model is verified. Finally, the influences of volume fractions and types of CNTs, lamination angle, elastic foundations stiffness and lamination angle on the dynamic stability of laminated FG-CNTRC conical shell have been investigated.

Stress distribution of a sector of cylindrical sandwich shell with FG-CNT core and piezoelectric face sheets subjected to blast pressure

ABSTRACT A novel layerwise theory along with higher-order shear deformation theory is studied to determine the stress distribution in a three-layer simply supported sector of cylindrical sandwich shell with piezoelectric face sheets and functionally graded carbon nanotube (FG-CNT) core. The out-of-shell displacement of the sandwich shell at each layer is assumed to be a quadratic polynomial function of the radial component in addition to a function of the coordinate components within the shell. The sandwich shell is subjected to the internal blast pressure so that the positive and negative phases of the pressure are considered. The 19 equations of motion have been derived by Hamilton’s principle and Maxwell’s static equations. The results indicate that the thickness, length and opening angle of the cylindrical shell have more effect on the stress distribution. In addition, the type of FG-CNT core has an effect on stress components in terms of time.

Time-depended stress analysis of a sector of the spherical sandwich shell with piezoelectric face sheets and FG-CNT core subjected to blast pressure

A novel Layerwise theory (LT) along with the higher-order shear deformation theory (HSDT) is investigated to determine the stress distribution in a three-layer sector of the spherical sandwich shell with piezoelectric face sheets and functionally graded carbon nanotube (FG-CNT) core. The simply-supported sandwich shell is subjected to the internal blast pressure so that the positive and the negative phases of the pressure are considered. The blast pressure is generated by a 1 Kg TNT hemispherical surface burst detonated. The out-of-shell displacement of the sandwich spherical shell at each layer is assumed to be a quadratic polynomial function of the radial component in addition to a function of the coordinate components within the shell. The nineteen equations of motion are derived by Hamilton's principle and Maxwell's static equations. By solving these nineteen equations, the stress distribution is calculated in the spherical sandwich shell in terms of time.

Static and free vibration analysis of functionally graded CNT reinforced sandwich plates using inverse hyperbolic shear deformation theory

In the present study, the static and free vibration analysis of functionally graded carbon nano-tubes reinforced (FG-CNTR) sandwich plates are studied in the framework of inverse hyperbolic shear deformation theory. The governing differential equations are derived using Hamilton’s principle and solved with the Navier’s solution technique. The analytical approach is used to obtain the deflections, stresses, natural frequencies, and corresponding mode shapes of FG-CNTR sandwich plates with different material properties, stacking sequences, span thickness ratios, core to face sheet thickness ratios, and loading conditions. Different types of reinforcement distribution such as uniformly distribution (UD) and functionally graded (FG) distribution of FG-O, FG-X, FG-/\, and FG-V are considered for the analysis. Also, the efforts are made to achieve the best possible arrangement for the stacking sequences and the appropriate reinforcement distribution that will produce improved static and free vibration responses for the FG-CNTR sandwich plates.

Predicting bucking and vibration behaviors of functionally graded carbon nanotube reinforced composite cylindrical panels with three-dimensional flexibilities

A full understanding of the mechanical behaviors of a three-dimensional (3D) functionally graded carbon nanotube reinforced composite (FG-CNTRC) cylindrical panel is important for structural design of engineering composite components. In this paper, the buckling and free vibration studies of FG-CNTRC cylindrical panel with different patterns of CNT distribution are investigated using the three-dimensional theory of elasticity. The cylindrical panels are subjected to axial and circumferential initial stresses, which frequently occur in real engineer structures. The state space technique along the radial direction and the Fourier series expansion along the in-plane coordinate (are) employed to formulate the problem and solve it analytically. A parametric study is carried out to examine the effects of the CNT distribution pattern, CNT volume fraction, length to mid radius ratio, and mid radius to thickness ratio on the buckling and vibration behaviors of FG-CNTRC cylindrical panels. Some interesting findings are observed, which may help design the CNTRC cylindrical panel structures. Besides, the presented studies may serve as benchmarks for researchers to check the validity of their future research works.

Vibration characteristics and damping properties of functionally graded carbon nanotubes reinforced hybrid composite skewed shell structures under hygrothermal conditions

The present article deals with the vibration and damping characteristics of functionally graded carbon nanotubes reinforced hybrid composite skewed shell structure in different hygrothermal conditions. Carbon nanotube reinforced polymer as a matrix phase and carbon fibre as a reinforcing phase are used, and carbon fibre is graded with uniform distribution along the thickness direction for the shell panel according to the power law distribution. The Mori–Tanaka scheme and strength of materials are used to determine the mechanical properties of such functionally graded carbon nanotubes reinforced hybrid composite materials. Finite element modelling has been done by considering an eight-noded shell element with the transverse shear effect according to Mindlin’s hypothesis, and an oblique coordinate system is used for the functionally graded carbon nanotubes reinforced hybrid composite skewed shell structures. Damping is incorporated into such carbon nanotube–based hybrid skewed shell structure based on the Rayleigh damping model. A MATLAB-based in-house computer code has been developed for the proposed formulation and verified with published research work before using for the present dynamic analysis of functionally graded carbon nanotubes reinforced hybrid composite skewed shell structure under hygrothermal conditions. The effect of the carbon nanotube, carbon fibre, material distribution as per power law index and hygrothermal conditions on the damping behaviour of such functionally graded carbon nanotubes reinforced hybrid composite skewed shell structures have been studied. Furthermore, parametric studies are carried out for the first resonant frequency, absolute amplitude, settling time and carbon nanotube impact on the vibrational behaviour of different functionally graded carbon nanotubes reinforced hybrid composite skewed shell structures under different hygrothermal conditions.

Geometrically nonlinear analysis of doubly curved imperfect shallow shells made of functionally graded carbon nanotube reinforced composite (FG_CNTRC)

This article presents an analytical approach to investigate the geometrically nonlinear response of doubly curved imperfect shallow shells made of functionally graded carbon nanotube reinforced composite (FG_CNTRC) under in-plane and lateral loads. For all of the case studies, the edges of shells are assumed to be diaphragm supports, which are subjected to in-plane edge-loads. The governing equations are developed for thin shells based on the classical theory considering the geometric nonlinear terms using the Von Karman–Donnell sense. The Galerkin method is used to solve the extended nonlinear system of differential equilibrium and compatibility equations. The solution results in a closed-form formula representing the equilibrium path (load-deflection relationship) of the shell which is validated by numerical methods, such as finite element and mesh-free methods. The parametric studies are performed for doubly curved and cylindrical shells subjected to uniform external pressure and in-plane edge loads, respectively. The effects of various parameters such as volume fraction of FG_CNTRCs and geometrical properties on the equilibrium paths are examined in detail.

Nonlinear Vibration Analysis of Functionally Graded Carbon Nanotubes Sandwich Cylindrical Panels

In this research, we investigate the nonlinear vibration of functionally graded carbon nanotubes (FG-CNTs) for simply supported sandwich cylindrical panels. The sandwich consisting of three layers formed of (FG-CNTs) and isotropic material as (CNT, ALMINUME, CNT). Mechanical properties of the sandwich media are acquired according to a refined rule of blend approach. The governing equations were derived using a first-order deformation theory (FOSDT). Four kinds of carbon nanotubes of sandwich cylindrical panels were analyzed. The volume fraction of CNTs is varied. The properties of nonlinear responses and free vibration are studied. The numerical approach employs the fourth-order Runge-Kutta and Galerkine procedure. Which conducted for the dynamic analysis of the panels to present the natural frequencies and non-linear dynamic response expression. The results show that; the natural frequencies and the nonlinear vibration amplitude decrease with the volume fraction and thickness ratio increase. The nonlinear vibration amplitude response increases when increasing the excitation force. The initial imperfection and the elastic foundation have a minor impact on the nonlinear vibration response of the panel. The Pasternak Foundation has a larger impact than the Winkler foundation. The structure formed of FG-CNT present an excellent choice for high-performance of engineering applications.

Free vibration analysis of FG-CNTRC conical shell panels using the kernel particle Ritz element-free method

The free vibration analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) conical shell is carried out by using element-free kp-Ritz method. Based on the first-order shear deformation shell theory, the approximate displacement field is expressed by the shape function of the nuclear particle, and the governing equation is established. The material properties of conical shell plates are determined by rule of mixtures and change with the thickness. In this work, the convergence is studied in terms of the number of nodes, and the accuracy of the proposed method is verified by comparing the current solution with the solution in the literature from different aspects. The effects of volume fraction, boundary conditions, distribution patterns of CNTs, half vertex angle and radius thickness ratio on their free vibration frequency characteristics are discussed in detail.

Static and free vibration analysis of functionally graded CNT reinforced composite plates using trigonometric shear deformation theory

In the present study, static and free vibration analysis of single walled functionally graded carbon nano-tube (CNTs) reinforced composite plate is carried out in the framework of trigonometric shear deformation theory. Trigonometric shear deformation theory full fills the traction free boundary condition at the top and bottom of the plate due to which it does not require shear correction factor. The Hamilton’s principle is used to derive governing differential equations and the Navier’s solution technique is used to obtain closed-form solution. The analytical approach is used to analyse the effect of the different span thickness ratios, volume fraction of CNTs, and distribution of CNTs on the deflection, stresses, natural frequencies, and corresponding mode shapes of CNTs reinforced composite plate. Four different types of CNTs reinforcement distribution such as, uniformly distribution (UD) and three types of functionally graded (FG) i.e. FG-O, FG-X and FG-V are selected here for the analysis. Here, the efforts are made to achieve the best possible arrangement for the reinforcement distribution that will produce the improved static and free vibration responses for the functionally graded CNTs reinforced composite plate.

Thermal vibration and buckling analysis of functionally graded carbon nanotube reinforced composite quadrilateral plate

This paper investigates the thermal vibration and buckling of the functionally graded carbon nanotube reinforced composite (FG-CNTRC) quadrilateral plate using the meshless method. Four distributions of the reinforcements along the thickness direction are considered, which are the uniform distributions (UD), FG-V, FG-O and FG-X. The corresponding effective material properties including Young's modulus, mass density, Poisson's ratio and thermal expansion coefficients are estimated by the rule of mixture with the CNT efficiency parameters accounting for the size-dependence. The first-order shear deformation theory (FSDT) with the consideration of thermal effects is employed. Based on Hamilton's principle and the moving least square (MLS) approximation, the discrete governing equations for the vibration of the FG-CNTRC plate are derived. The free vibration of the regular and irregular plates with and without the temperature effect are considered respectively, and the corresponding natural frequencies and mode shapes are obtained by the eigenvalue problem. The stability analysis of the uniaxial and biaxial mechanical and thermal buckling are also conducted subsequently. The effects of the volume fraction, distribution pattern, geometrical characteristics and temperature on the buckling behaviors of the CNTRC quadrilateral plate are discussed in detail.

Static, dynamic and natural frequency analyses of functionally graded carbon nanotube annular sector plates resting on viscoelastic foundation

In this paper, static, dynamic and natural frequency responses of composite annular sector plate with carbon nanotubes (CNTs) reinforcements resting on viscoelastic foundation are investigated. The carbon nanotubes are considered to have uniform or functionally graded pattern in the plate thickness. Kelvin-voight model is used to model the viscoelastic foundation. To model the problem, Hamilton principle based on first shear deformation plate theory and finite element method are applied. The mechanical properties of annular sector plate composed of CNTs and polymer matrix are evaluated by using the rule of mixtures. The numerical results are obtained for investigating the effect of various factors such as distribution and volume fraction of CNTs, boundary conditions, stiffness and damping coefficients of viscoelastic foundation and sector angles on natural frequency, static and dynamic transient responses of the plate.

Analysis of forced and free vibrations of composite porous core sandwich cylindrical shells and FG-CNTs reinforced face sheets resting on visco-Pasternak foundation under uniform thermal field

The radial basis function and Newmark-beta methods were used to investigate the vibrational behavior of composite sandwich cylindrical shells with porous core and functionally graded carbon nanotube. The porous core material is a kind of open-cell metal foam, whose, depending on the porosity coefficient and the core thickness, mechanical properties vary. The Hamilton principle was used to extract its motion equations. In this cylindrical sandwich shell, temperature variations are uniformly distributed and resting on the visco-Pasternak foundation with different boundary conditions. Parametric studies were presented to investigate the influence of FG porosity, temperature change, visco-Pasternak foundation, volume fraction, geometrical properties, and distribution of carbon nanotubes on natural frequencies and response dynamics of the sandwich composite cylindrical shell. The numerical outcomes express the distribution of CNTs and volume fraction play a pivotal role in the natural rates and dynamic response of the cylindrical sandwich shell. It is also perceived that there is an increase in the cylindrical shell deformation amplitude with an increase in temperature, damping, and core porosity.

Nonlinear deflection analysis of CNT/magneto-electro-elastic smart shells under multi-physics loading

In this article, a first attempt has been made to address the nonlinear deflection problem of magneto-electro-elastic shells reinforced with carbon nanotubes subjected to multiphysics loads like mechanical, electric and magnetic loads. In this regard, a mathematical model based on higher-order shell theory, von-Karman’s nonlinearity is derived using finite element platform. The true flexural behavior of the functionally graded carbon nanotube reinforced magneto-electro-elastic shell under the action of individual/combined multiphysics loading is captured through higher-order nonlinear terms. Meanwhile, the results of this article emphasize on understanding the influence of coupling fields in conjunction with material and geometrical parameters on the deflection of functionally graded carbon nanotube reinforced magneto-electro-elastic shells. In addition, the study provides an intensive and meticulous insight into the deflection of curved shells with respect to different shell geometry, carbon nanotube distributions, carbon nanotube volume fractions, aspect ratio, thickness ratio, shallowness ratio, and electro-magnetic loads. The study is believed to pave way for extensive further studies on these classes of materials which can prove to be instrumental in various avenues of aerospace research and development.

Nonlinear Torsional Buckling of Functionally Graded Carbon Nanotube Orthogonally Reinforced Composite Cylindrical Shells in Thermal Environment

This paper deals with the nonlinear large deflection torsional buckling of functionally graded carbon nanotube (CNT) orthogonally reinforced composite cylindrical shells surrounded by Pasternak’s elastic foundations with the thermal effect. The shell is made by two layers where the polymeric matrix is reinforced by the CNTs in longitudinal and circumferential directions for outer and inner layers, respectively. The stability equation system is obtained by combining the Donnell’s shell theory, von Kármán nonlinearity terms, the circumferential condition in average sense and three-state solution form of deflection. The critical torsional buckling load, postbuckling load-deflection and the load-end shortening expressions are obtained by applying the Galerkin procedure. The effects of temperature change, foundation parameters, geometrical properties and CNT distribution law on the nonlinear behavior of cylindrical shell are numerically predicted. Especially, the effect of orthogonal reinforcement in comparison with longitudinal and circumferential reinforcement on the torsional buckling behavior of shells is observed.

Thermal postbuckling analysis of FG-CNTRC plates with various shapes and temperature-dependent properties using the VDQ-FEM technique

Using the ideas of variational differential quadrature and finite element methods (VDQ and FEM), a novel numerical approach is developed in this paper to investigate the postbuckling behavior of plates with arbitrary shape under the action of thermal load within the framework of higher-order shear deformation theory (HSDT). It is considered that the plates are made of functionally graded carbon nanotube reinforced composite (FG-CNTRC). By the proposed approach, the thermal postbuckling behavior of plates with arbitrary-shaped cutout can be modeled. The rule of mixture is utilized to obtain the effective material properties of nanocomposite which are considered to be temperature-dependent. To implement the proposed method that can be named as VDQFEM, the variational temperature-dependent formulation of problem is first developed. This formulation is presented using novel vector-matrix relations with the aim of utilizing in programming in an efficient way. In the context of VDQFEM, the space domain of plate is first transformed into a number of finite elements. In the next step, the VDQ discretization technique is implemented within each element. Then, the assemblage procedure is performed to derive the set of matricized governing equations which is finally solved by means of the pseudo arc-length continuation algorithm. One of the main novelties of present approach is proposing an efficient way based on mixed-formulation to accommodate the continuity of first-order derivatives on the common boundaries of elements for the used higher-order shear deformable plate model. Quadrilateral plate, skew plate, annular plate, square plate with rectangular hole and rectangular plate with circular hole under different boundary conditions including CCCC, SSSS, CSCS and CCSS (S: simply-supported, C: clamped) are considered in the numerical example. First, validation studies are presented in terms of critical buckling temperature of CNTRC rectangular plates with various CNT distribution patterns. It is shown that the proposed method has the advantages of standard VDQ technique including fast convergence rate, being locking-free and simple implementation. Moreover, it is capable of considering polygon and concave domains. Also, the effects of geometrical properties, volume fraction and distribution pattern of CNTs and temperature-dependency of material properties on the thermal postbuckling of plates are studied.

Investigating nonlinear vibrations of multi-scale truncated conical shell segments with carbon nanotube/fiberglass reinforcement using a higher order conical shell theory

This research deals with the nonlinear vibration analysis of functionally graded carbon nanotubes and fiber-reinforced composite truncated conical shell segments based upon third-order shear deformation theory. A detailed procedure for obtaining material properties of the multi-scale carbon nanotube/fiber-reinforced composite based on the three-dimensional Mori–Tanaka scheme has been provided. The truncated conical shell segments have been reinforced by distributed carbon nanotubes in the thickness direction according to uniform, linear, and nonlinear functions. The nonlinear equations have been solved via both Galerkin’s technique and Jacobi elliptic function method. Based on the numerical results, the effects of diverse carbon nanotube distribution, fiber volume, fiber orientation, and semi-vertex and open angles of the segment on vibrational frequencies of the truncated conical shell have been studied.

Longitudinal modeling and properties tailoring of functionally graded carbon nanotube reinforced composite beams: A novel approach

Carbon Nanotubes (CNTs) are considered a promising reinforcement for engineering advanced structures, such as aerospace applications. It became inevitable to tailor structures not only by conventional geometric but also by mechanical properties tailoring, to acquire the maximum load-carrying capacity. Therefore, this research presented a novel Finite Element (FE) modeling of Axially Functionally Graded CNT Reinforced Composite (AFG-CNTRC) beams, based on Timoshenko beam theory. However, an obstacle is raised due to properties discontinuity produced between consecutive elements in axial gradation. A solution is proposed based on the ladder-pattern concept. It defined a locally specific value at each element, satisfying the overall required distribution. It followed by a convergence test, static, and dynamic analysis with comprehensive parametric studies. The obtained results showed that the model is convergent, accurate, free of shear-locking, and suitable for axial gradation. Moreover, this study introduced an advanced technique for axial properties tailoring, utilizing privileges of CNT and aggregation. The longitudinal tailoring depends on both the CNT orientation angle and gradation index, which played a crucial role in response control and adjusting beam stiffness together with strength. This approach could solve enormous engineering problems, especially at critical cross-sections design and constraint response, without affecting the total weight.

Nonlinear dynamics and chaos of a nanocomposite plate subjected to electro–thermo–mechanical loads using Flügge–Lur’e–Bryrne theory

The study investigates nonlinear vibration of functionally graded carbon nanotube–reinforced polymer plate subjected to electro–thermo–mechanical loads resting on elastic foundations by an analytical approach. Uniform and functionally graded carbon nanotubes are used to reinforce through the thickness of the plate. Furthermore, a piezoelectric layer is perfectly bonded to the top of the plate to act as an actuator. The equations of motion are derived by the Flügge–Lur’e–Bryrne theory and then applying the Galerkin method to solve differential equations. The dynamical responses such as time history, phase plane graphs, and Poincaré map are carried out in this study. In addition, the effects of environment, materials, and geometry are scrutinized. The obtained results are also compared and validated with those of other studies.