Functionally Graded Carbon Nanotube-reinforced Research Articles

A unified impact response analysis for FG-CNTRC and 3D-GrF sandwich composite sector- rectangular coupled plates

This paper proposes a general method for analyzing the dynamic response characteristics of sector- rectangular coupled plates subjected to external impacts. This method combines the method of reverberation-ray matrix (MRRM) with the strip transfer function method, referred to as the strip method of reverberation-ray matrix (S-MRRM). The traditional MRRM has some limitations for the dynamic modeling of the sector structure and also requires the Lévy-type exact solution, leading to simply supported boundary conditions in certain direction of the structure. The present method can not only address the aforementioned limitations but also maintain the efficiency advantage of the wave method for solution. The investigated structure consists of sandwich panels composed of functionally graded carbon nanotube reinforced (FG-CNTRC) panels and the three-dimensional graphene foam (3D-GrF) core. The structure is discretized into several strip elements by the strip transfer function method, and the whole dynamic model of the structure is obtained through matrix assembly. The motion differential equation in matrix form is derived by using Hamilton principle, followed by obtaining the matrix-form wave number equation. Upon deriving the wave solution of the structure based on this equation, the transient and steady-state responses of the structure are obtained by employing MRRM. Convergence analysis is conducted on these calculation results, followed by comparison with FEM results to validate its prediction accuracy. Finally, through a comprehensive parametric study, the impacts of various factors such as CNTs distribution form, CNTs volume fraction, 3D-GrF pore distribution form, 3D-GrF porosity, and sandwich panel thickness ratio on both transient and steady-state responses of these structures were investigated, thereby providing guidance for engineering applications.

Free vibration analysis of FG‐GPL and FG‐CNT hybrid laminated nano composite truncated conical shells using systematic differential quadrature method

AbstractIn the present study, the free vibration of functionally graded graphene platelet‐reinforced (FG‐GPLs) and functionally graded carbon nanotube‐reinforced (FG‐CNTs) hybrid laminated nanocomposite truncated conical shells and panels are analyzed. Multi‐layers truncated conical shell and panel of pure FG‐CNTs, pure FG‐GPLs and hybrid CNTs‐GPLs reinforcement were evaluated. In light of its high accuracy in the calculation of thin and thick shells, a third‐order shear deformation theory is adopted. The governing equations and boundary conditions is derived using Hamilton's principle and is solved numerically using the systematic differential quadrature method (DQM) which uses Kronecker delta function. The effective mechanical properties of the CNT‐reinforced nanocomposite layers are estimated using the rule of mixtures, whereas those of the GPL‐reinforced nanocomposite layers is calculated using the Halpin‐Tsai micromechanical model. Convergence and accuracy evaluation of the presented study are confirmed and a number of parameters, including the CNTS volume fraction, GPLs mass fraction, distribution patterns (i.e., Uniform distribution (UD), Functionally graded O‐distribution (FG‐O), Functionally graded X‐distribution (FG‐X), Functionally graded V‐distribution (FG‐V) and FG‐A), different boundary conditions and the vertex angle of the cone, are investigated. The results obtained in this article for the combination of two different materials, GPLs and CNTs, showed that in controlling the natural frequency of the system, without changing the percentage of fiber and only by changing the arrangement, very wonderful results can be achieved.

Far-field blast responses of sandwich arbitrary polygonal reinforced plate system

This paper presents a spectral element model for investigating the dynamic response characteristics of the arbitrary polygonal built-up reinforced plate system subjected to far-field blast load. In the researched system, the plate elements are made of sandwich materials with functionally graded carbon nanotube reinforced (FG-CNTRC) panels and three-dimensional graphene foam (3D-GrF) core. The stiffness matrix assembly method is incorporated into the spectral element method (SEM) for establishing the overall dynamic formulas of arbitrary assembled plates. Thereinto, the governing equations of motion of plate elements are derived by the Hamilton's principle based on the simple first-order shear deformation theory (S-FSDT) retaining only four displacement variables, which provides an advantage to maintain interelement continuity conditions completely. After that, the natural frequencies and transient responses of the systems with different built-up forms are solved by introducing a golden section search algorithm and Fourier series expansion of the blast load. The admirable predictive accuracy of the present model is verified by comparing the obtained results and the reference solution from finite element method (FEM). Moreover, the effects of the CNT distribution form, CNT volume fraction, 3D-GrF pore distribution form, 3D-GrF porosity, the thickness ratio of the sandwich panel, TNT mass and explosion radius on the transient vibration properties of the system are revealed through a detailed parametric study.

On the frequency characteristics of rotating combined conical-conical shells made of FG-CNTRC composite materials under thermal environments

The current paper is focused on the frequency features of spinning coupled functionally graded carbon nanotube reinforced (FG-CNTRC) cone-cone shells undergoing high temperatures. The physic of the structures is based on two conical shell segments with different semi-vertex angles that are merges via the ideal bonding at the junctures. Simultaneous consideration of environmental thermal effects and rotation about the structure’s axis in dealing with conditions near and also far from critical speeds and buckling temperatures could add the novelty of the available paper. The frequency inspections include the study of free vibration behavior, frequency bifurcation of traveling waves and also identifying instability regions with regard to critical thermal buckling and critical speed phenomena. A two-step procedure based on first order shear deformation theory (FSDT) and von Kármán assumptions is employed to access the system’s formulation. Firstly, the initial thermal stress resultants are captured upon static step and then utilized through the dynamic step. Afterward, generalized differential quadrature (GDQ) method is implemented on the set of governing equations. After verifying the precision of results, some parametric studies are provided to investigate the effects of temperature and rotary speed elevations, radius and position of conjunction, CNT volume fraction and distribution schemes on the dynamic attributes of the spinning structure.

Thermal Buckling and Postbuckling Behaviors of Couple Stress and Surface Energy-Enriched FG-CNTR Nanobeams

Small-sized structural elements such as beams, plates, and shells are usually used as nanomechanical resonators, nanoscale mass sensors, nanoelectromechanical actuators, and nanoenergy harvesters. At the nanoscale, the structures usually possess a high surface area-to-bulk volume ratio, leading to the free energy related to surface atoms becoming considerable compared to that of the bulk part. Earlier reports indicated several physical reasons for size-dependent phenomena, e.g., nonlocal stress, surface energy, and couple stress. To provide an in-depth insight into the mechanical behavior of small-scale structures, size-dependent continuum models including two or more physical factors have attracted the attention of the academic community. This research analyzes the thermal buckling and postbuckling characteristics of functionally graded carbon nanotube-reinforced (FG-CNTR) nanobeams with a tri-parameter, nonlinear elastic foundation and subjected to a uniform temperature rise. Chen-Yao’s surface energy theory and Yang’s symmetrical couple stress theory are combined to capture two types of size effects in nanobeams. The postbuckling model is formulated based on the Euler–Bernoulli deformation hypothesis and Euler–Lagrange equation. Using a two-step perturbation technique, the related postbuckling equilibrium path is determined. In numerical analysis, the impacts of surface energy, couple stress, elastic foundation, boundary conditions, geometric factor, layout type, and volume fraction of CNTs on the thermal buckling and postbuckling behaviors of nanobeams are revealed. It is indicated that considering couple stress or surface energy can lead to a significant increase in the postbuckling stability of nanobeams compared to the case in which it is not considered. In addition, there is a reverse competition between couple stress or surface energy effects on the thermal buckling responses of nanobeams. As the temperature rise will cause the material elastic moduli softening, the thermal buckling load–deflection curves of nanobeams with the temperature-independent case are much higher than those with the temperature-dependent cases.

Critical speed and frequency behavior of rotating joined FG-CNTRC conical-conical shells

This paper provides information on critical speed and frequency behavior of rotating joined functionally graded carbon nanotube reinforced (FG-CNTRC) conical-conical shells. Furthermore, frequency bifurcation as a consequence of rotational effects is investigated. The shell is assumed to be composed of an isotropic polymer matrix that is reinforced by uniformly or functionally distributions of carbon nanotubes (CNTs). The structure’s kinematics originates from the first order shear deformation theory (FSDT). To derive spin-based motion equations, Coriolis and centrifugal forces together with initial hoop tensions are propounded through the Hamilton’s principle. Moreover, matching and boundary conditions are ascertained to supplement the governing equations. To discretize the governing equations meridionally, the generalized differential quadrature (GDQ) technique is employed. After validity checking, some significant parameters like cones’ angles, rotation speed, edge supports, CNT volume fractions and dispersion patterns that play effective roles on the critical speed and vibration characteristics are examined. It is revealed that by enhancing spinning speed, frequencies of different FG patterns converge. In addition, difference between backward and forward frequencies decreases as the circumferential mode number increases.

Vibration reduction of FG-CNTR piezoelectric laminated composite cantilever plate under aerodynamic load using full-dimensional state observer

A robust control method is developed to suppress the vibrations of the functionally graded carbon nanotube-reinforced (FG-CNTR) piezoelectric laminated composite cantilever plate subjected to the aerodynamic force and thermal environment. The distributions of the carbon nanotubes (CNTs) in the entire plate thickness are classified as the functionally graded (FG) or uniform distributions (UD). The effective material properties are obtained using the rule of mixtures. The classic laminated composite plate theory and Hamilton principle are used to establish the governing equations of motion for the FG-CNTR laminated composite cantilever plate under combined the aerodynamic force and thermal environment. Galerkin method is used to obtain a two-degree-of-freedom ordinary differential control equation. To actively suppress the vibration, the piezoelectric patches are used as the actuators and sensors which are attached to the upper and lower surfaces of the FG-CNTR laminated composite cantilever plate. A full-dimensional state observer is introduced to design the robust controller. To verify the efficiency of the control strategy, a comparison between the robust controller and velocity feedback controller (VFC) indicates that the robust controller has better control efficiency than the VFC. The effects of the CNT distribution, CNT volume fraction, temperature and aspect ratio on the dynamic behaviors of the laminated composite plates are studied. The effectiveness and accuracy of the proposed robust controller are verified through numerical simulations under different cases.

A layer-wise formulation for transient response of composite sandwich panel in the exposure of thermal shock loading by considering novel linear and torsional elastic foundation

In order to present practical ways to decline the adverse effects of thermal shock loading from the perspective of improving structural design, authors of this study put their main focus on the analysis of transient coupled poro-thermo-elastic behavior of the sandwich cylindrical panel under multi-directional initially stresses with functionally graded carbon nanotube reinforced (FG-CNTRC) face-layers and polymeric core resting on novel elastic foundation. With the aid of compatibility conditions, the sandwich structure with three layers are modeled. To guarantee the accuracy of the calculations, authors define the governing equations of the system within the context of the exact three-dimensional form of elasticity theory. Additionally, the Navier approach is employed as the analytical solution to find the response of the simply-supported structure in the content of shear stresses and transverse deflection. Dubner and Abate method is utilized to inverse the Laplace transform in order to determine the time history of the system. Convergence of the stress and deflections of the system with respect to passing time is examined and verified. The parametric study reveals that utilizing each of the CNT scattering patterns through the radial direction of the face-layers is useful for protecting specific proportion of the thickness against the applied thermal shock. Moreover, by tracking the effect of increasing the damper coefficient of the surrounding foundation on the different components of the stress, it is revealed that increasing this factor leads to increase of the transverse shear stress and decrease of the normal shear stress.

Bending and free vibration analysis of symmetric and unsymmetric functionally graded CNT reinforced sandwich beams containing softcore

In the present work, bending and free vibration analyses of functionally graded carbon nanotube-reinforced (FG-CNTR) sandwich beams are carried out using finite element-based higher-order zigzag theory. Face sheets are assumed to be made up of FG-CNTR composite, and the core is assumed to be made up of balsa wood (softcore). The present formulation also takes into account transverse normal stresses. The computational model incorporates transverse shear stress and transverse normal stress continuity condition at interfaces. Zero transverse shear stress condition at the bottom and top surfaces of the beam is also satisfied. The principle of minimum potential energy is employed for carrying out bending analysis, while Hamilton’s principle is adopted for free vibration analysis. The investigation is carried out for different gradation laws which govern the distribution of CNTs across the thickness of face sheets. The influence of the core’s thickness on stresses and displacements is also critically analyzed in the present work. It has been observed that the thickness of the core and CNT gradation law significantly affect the mechanical behavior of the sandwich FG-CNTRC beam.

Finite element framework for static analysis of temperature dependent IHSDT based functionally graded CNT reinforced plates

In this study, a finite element model is presented to investigate the effect of temperature dependent material properties of single-walled carbon nanotube (SWCNT) and PMPV on the static response of functionally graded carbon-nano tube reinforced (FG-CNTR) plate with four different boundary conditions. The governing equations are developed on the basis of inverse hyperbolic shear deformation theory (IHSDT) and Hamilton’s principle. The effective material properties are evaluated using the extended rule of mixture. The pattern of CNTs distribution in the plate is assumed to be uniformly distributed (UD) and functionally graded (FG-V, FG-Λ, FG-O, and FG-X) along the thickness direction. The governing equations are solved using a C0 isoparametric finite element method (FEM) and discretized with an isoparametric eight noded element with seven degrees of freedom per node. The authors have performed various parametric studies to observe the effect of plate span to thickness ratio, volume fraction, CNTs distribution on the static response of FG-CNTR plate subjected to sinusoidal (SSL) and uniformly distributed (UDL) loading conditions. The numerical solutions for the deflection and stresses are obtained and compared with known results in the literature. The study concludes that IHSDT accurately predicts the behavior of the FG-CNTR plate, also temperature dependent material properties have significantly influenced the static behavior of the FG-CNTR plate.

Free vibration and buckling analyses of CNT reinforced laminated non-rectangular plates by discrete singular convolution method

This paper presents the free vibration and buckling analyses of functionally graded carbon nanotube-reinforced (FG-CNTR) laminated non-rectangular plates, i.e., quadrilateral and skew plates, using a four-nodded straight-sided transformation method. At first, the related equations of motion and buckling of quadrilateral plate have been given, and then, these equations are transformed from the irregular physical domain into a square computational domain using the geometric transformation formulation via discrete singular convolution (DSC). The discretization of these equations is obtained via two-different regularized kernel, i.e., regularized Shannon’s delta (RSD) and Lagrange-delta sequence (LDS) kernels in conjunctions with the discrete singular convolution numerical integration. Convergence and accuracy of the present DSC transformation are verified via existing literature results for different cases. Detailed numerical solutions are performed, and obtained parametric results are presented to show the effects of carbon nanotube (CNT) volume fraction, CNT distribution pattern, geometry of skew and quadrilateral plate, lamination layup, skew and corner angle, thickness-to-length ratio on the vibration, and buckling analyses of FG-CNTR-laminated composite non-rectangular plates with different boundary conditions. Some detailed results related to critical buckling and frequency of FG-CNTR non-rectangular plates have been reported which can serve as benchmark solutions for future investigations.

Buckling analyses of carbon nanotube reinforced functionally graded composite cylindrical panels

This paper investigates the buckling analyses of carbon nanotube (CNT) reinforced composite cylindrical panels via a geometrically nonlinear finite element model with large rotations based on the first-order shear deformation (FOSD) hypothesis. Fully geometrically nonlinear strain-displacement relations and large rotation of shells are considered in the model. First, the proposed model is validated by a frequency analysis of a simply supported CNT reinforced composite cylindrical panel from an existing reference. Then, the model is applied to simulate the behaviors of carbon nanotube reinforced functionally graded (CNT-FG) composite cylindrical panels. The effects of curvature ratio, different buckling behaviors and four representative forms of CNT distributions are studied for their material performance comparatively.

Combined effects of surface energy and couple stress on the nonlinear bending of FG-CNTR nanobeams

This paper investigates the size-dependent nonlinear bending of functionally graded carbon nanotube-reinforced (FG-CNTR) nanobeams. Chen–Yao’s surface elasticity and modified couple stress theories are adopted to describe surface effects and couple stress effects, respectively. These nanobeams, in which the carbon nanotube (CNT)-reinforced phases are assumed to be distributed in a gradient along the thickness, are subjected to a uniform pressure and rest on a nonlinear elastic foundation. In accordance with the Euler–Lagrange variational principle, the governing equations and boundary conditions for the FG-CNTR nanobeams, which involve geometric nonlinearity due to the von Kármán strain relations, are obtained. Then, with the assistance of the two-step perturbation technique, the load-deflection relationship is determined for nanobeams subjected to simply supported (SS) and clamped–clamped (CC) boundary conditions. Finally, the impacts of various factors, including surface properties, characteristic material length, elastic foundation, geometric factors, layout type and volume fraction of CNTs, on the mechanical behaviors of CNT-based nanobeams are examined. The numerical results reveal that the combination of surface effects and couple stress helps to enhanceq the stiffness of the nanobeams. Furthermore, the size-dependent nonlinear bending of the FG-CNTR nanobeam is markedly affected by the content and layout type of the reinforcements.

Effect of CNT reinforcements on the flutter boundaries of cantilever trapezoidal plates under yawed supersonic fluid flow

A comprehensive study is presented on the supersonic flutter analysis of functionally graded carbon nanotube reinforced (FG-CNTRC) cantilever trapezoidal plates. First-order shear deformation theory (FSDT) is employed to model the structure, effective mechanical properties are calculated based on the extended rule of mixture, aerodynamic pressure is estimated according to the piston theory and the set of governing equations and boundary conditions are derived using Hamilton’s principle. A numerical solution is done using generalized differential quadrature method (GDQM) and natural frequencies, corresponding mode shapes, critical speed and flutter frequency are calculated. Convergence and accuracy of the presented solution are confirmed and effect of various parameters on the flutter boundaries are investigated including geometrical parameters, yaw angle, volume fraction, and distribution of carbon nanotubes (CNTs). In the near future, results of this article can be considered as a useful tool in design and analysis of aeronautic vehicles flying at supersonic speeds.

Nonlinear Bending Analysis of Functionally Graded CNT-Reinforced Shallow Arches Placed on Elastic Foundations

This research explores the nonlinear bending behaviors of functionally graded carbon nanotube-reinforced (FG-CNTR) shallow arches with unmovable simply supported ends and clamped–clamped ends; these arches are subjected to a uniform radial pressure and rest on a nonlinear elastic foundation. The temperature-dependent material properties of the arches are considered. Within the framework of Reddy shear deformation theory possessing von Karman nonlinearity, the motion equations and boundary conditions for the FG-CNTR arches are determined by the Euler–Lagrange variational principle. Then, a two-step perturbation technique is adopted to determine the load–deflection relationship analytically. To verify the validity of the developed model and related perturbation solutions, a numerical investigation is conducted for shallow arches with five distribution patterns of carbon nanotube (CNT) reinforcements uniaxially aligned in the axial direction. Finally, the influences of various factors, including the elastic foundation, layout type, and volume fraction of CNTs and geometric factors, on the nonlinear behaviors of FG-CNTR shallow arches are examined. The obtained results show that the load–deflection curves exhibit less snap-through instability as the CNT volume fraction increases. The transverse shear stress versus the thickness of FG-CNTR shallow arches is markedly affected by the layout type and content of reinforcements.

Bending, free vibration and buckling of functionally graded carbon nanotube-reinforced sandwich plates, using the extended Refined Zigzag Theory

The paper presents an application of the extended Refined Zigzag Theory (eRZT) in conjunction with the Ritz method to the analysis of bending, free vibration and buckling of functionally graded carbon nanotube-reinforced (FG-CNTR) sandwich plates. Two stacking sequences are taken into consideration: sandwich panels with a homogeneous core and functionally graded face-sheets and sandwich panels with homogeneous face-sheets and a functionally graded core.After validating the convergence characteristics and the numerical accuracy of the developed approach using orthogonal and non-orthogonal admissible functions, a detailed parametric numerical investigation is carried out. Bending under bi-sinusoidal and uniform transverse pressure, free vibration and buckling loads under uniform in-plane uniaxial, biaxial and shearing loadings of FG-CNTR sandwich plates are studied. Numerical results for square and rectangular FG-CNTR sandwich plates under various combinations of geometry (core-to-face sheet thickness ratio and side to thickness ratio), different set of boundary conditions, CNTs volume fraction and grading laws are presented and discussed in detail. It is concluded that the eRZT predicts the response for static, stability and free vibration problems more accurately than first-order (FSDT) and third-order (TSDT) shear deformation theories, also for FG-CNTR sandwich plates.

Dynamic Response of FG-CNT Composite Plate Resting on an Elastic Foundation Based on Higher-Order Shear Deformation Theory

AbstractIn this paper, the dynamic response of a functionally graded carbon nanotube–reinforced (FG-CNTRC) composite plate is obtained based on higher-order quasi-three-dimensional (3D) shear defor...

Dynamic stability of the rotating carbon nanotube-reinforced adaptive sandwich beams with magnetorheological elastomer core

This paper reports on the numerical results that are performed with a freely vibrating magnetorheological elastomer sandwich beams reinforced by carbon nanotubes, exposed to an angular velocity and a magnetic field. The shear deformable beam model of the Timoshenko is employed to model the top and bottom functionally graded carbon nanotube reinforced (FGCNTR) layers, whereas a linear viscoelastic behavior is considered for the magnetorheological elastomer core layer. Using the Hamilton principle as well as a differential quadrature method, the natural frequencies and corresponding loss factors are derived. Convergence study and comparison results with the literature show a high stability and accuracy of the present approach. Enhancement of angular velocity causes a significant increasing and decreasing trends for the natural frequencies and loss factors, respectively. Observations reveal that there is an optimum value for the magnetic field intensity in which the loss factor is the maximum. Also, the effects of carbon nanotube distributions and geometric parameters of the magnetorheological elastomer and FGCNTR layers on the modal parameters are examined.

Nonlinear vibration and modal analysis of FG nanocomposite sandwich beams reinforced by aggregated CNTs

In the present work, by considering the aggregation effect of single‐walled carbon nanotubes (SWCNT), the nonlinear vibration of functionally graded (FG) nanocomposite sandwich Timoshenko beams resting on Pasternak foundation are presented. The material properties of the FG nanocomposite sandwich beam are estimated using the Eshelby–Mori–Tanaka approach and differential quadrature method (DQM) is used to obtain natural frequency. The nonlinear governing equations and boundary conditions are derived using the Hamilton principle and von Kármán geometric nonlinearity. The higher order nonlinear governing equations and boundary conditions are calculated using the Hamilton principle. A direct iterative method is employed to determine the nonlinear frequencies and mode shapes of the beams. It is shown that the mechanical properties and therefore vibration of functionally graded carbon nanotube reinforced (FG‐CNTR) sandwich beams are severely affected by CNTs aggregation. A detailed parametric study is carried out to investigate the influences of Winkler foundation modulus, shear elastic foundation modulus, length to span ratio, thicknesses of face sheets on the nonlinear vibration of the structure. POLYM. ENG. SCI., 59:1362–1370 2019. © 2019 Society of Plastics Engineers

Aeroelastic analysis of CNT reinforced functionally graded laminated composite plates with damage under subsonic regime

This research paper presents the dynamic and aeroelastic analysis of functionally graded carbon nanotube reinforced (FG-CNTRC) laminated composite plates with damage in the subsonic regime. First order shear deformation theory (FSDT) is applied to develop finite element code in MATLAB environment for structural analysis. The anisotropic damage formulation based on the concept of stiffness reduction is introduced into the finite element formulation to include the damage in the plate. To implement aeroelastic analysis through PK-method, the required aerodynamic force matrix coefficients, are generated in MSC.Nastran by supplying modal parameters (obtained from structural analysis) through direct matrix abstract program (DMAP) using doublet-lattice-method (DLM). The accuracy of the present model has been analyzed by tackling the various numerical illustrations and validated with some of earlier published literature. Further, the effect of various parameters like CNT fiber volume fraction, lamination sequence, damage location and orientation on flutter characteristic are studied.