Functionally Graded Carbon Nanotube Reinforced Composite Research Articles

The free vibration of the MEE sandwich microplate with the FG-CNTRC core using the MSGT

This study deals with the free vibration of the sandwich microplate with the core made of functionally graded carbon nanotube-reinforced composites (FG-CNTRC) and magneto-electro-elastic (MEE) face sheets. The governing equation of the microplates is derived by using the refined plate theory (RPT) with two variables and the modified strain gradient theory (MSGT). The Non-Uniform Rational B-Splines (NURBS) basis function of the isogeometric approach (IGA) is used for the approximation of the displacement and electric and magnetic fields of the microplates. The paper studies the effect of the length scale parameters (LSPs), CNTs distributions, CNTs volume fraction, magnetic and electric loads and geometry on the vibrational frequency of the MEE sandwich microplate.

Thermal Buckling and Postbuckling Analysis of Cracked FG-GPL RC Plates Using a Phase-Field Crack Model

A phase-field crack model is developed for numerical analysis of thermal buckling and postbuckling behavior of a functionally graded (FG) graphene platelet-reinforced composite (FG-GPLRC) plate with a central crack. The inclined central crack is represented according to a stable, effective phase-field formulation (PFF) by introducing a virtual crack rotation. The problem is formulated using first-order shear deformation theory (SDT) incorporated with von Kármán geometric nonlinearity. And it is approximated by combining regular Laplace interpolation functions and crack-tip singular functions in the framework of the 2D extended natural element method (XNEM). Troublesome shear locking is suppressed by applying the concept of the MITC (mixed-interpolated tensorial components)3+ shell element to the present numerical method. The results demonstrate the effectiveness of this method in accurately predicting the critical buckling temperature rise (CBTR) and the thermal postbuckling path. In addition, the parametric results reveal that the CBTR and postbuckling path of the FG-GPLRC plate are distinct from those of the FG carbon nanotube-reinforced composite (FG-CNTRC) plate and remarkably affected by the parameters associated with the crack and graphene platelet (GPL).

Vibration analysis of functionally graded carbon nanotube‐reinforced composite open cylindrical shells with damping film embedded

AbstractIn the present study, a discrete layer model was established to explore the vibration performance of Functionally Graded Carbon Nanotube‐Reinforced Composite (FG‐CNTRC) open cylindrical shells with damping film embedded based on the first‐order shell theory. There are four configurations of stacking arrangements considered for FG‐CNTRC open cylindrical shells with damping film embedded. The equivalent structural parameters of the top and bottom FG‐CNTRC panels are generated by implementing the extended mixing rule. Governing equations are derived based on the Hamilton principle and solved with the Naiver solution. Subsequently, after verifying the validity of this paper's solution by comparing it with the published literature, a parametric elaborated investigation discloses the variation patterns of vibration performance of four FG‐CNTRC open cylindrical shells with damping film embedded. The conclusions of the study can be used as a useful guide about open cylindrical composite shell structures with the design of high strength and damping.Highlights Discrete layer vibration model was bulit based on first‐order shell theory. Vbration performance of FG‐CNTRC open sandwich shells was studied. Variation patterns of frequency and loss factor was disclosed.

Free vibration analysis of functionally graded composite plates reinforced with linearly and nonlinearly distributed carbon nanotubes in hygrothermal environments

This article presents a novel investigation of the free vibration behavior of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plates under hygrothermal environments. It explores both uniform and non-uniform (functionally graded) distributions of CNTs across the plate thickness for the first time. The effective material properties of the CNTRC, considering temperature and moisture dependence, are determined using the extended rule of mixture. First-order shear deformation theory (FSDT) is employed to derive the governing equations incorporating hygro-elastic and thermo-elastic relations. These equations are solved using the finite element method. A validation study verifies the accuracy of the employed approaches. Subsequently, a comprehensive parametric study investigates the influence of plate geometry (length-to-width and width-to-thickness ratios), CNTs volume fraction, boundary conditions, linear and non-linear CNTs distributions, and hygrothermal environments on the free vibration behavior of polymeric nanocomposite plates reinforced with CNTs fillers is conducted. The results reveal that the increase in temperature and moisture leads to a decrease in the effective stiffness of the FG-CNTRC plates.

Vibration analysis of a composite sandwich plate with a viscoelastic auxetic core, FG-CNTRC interior, and MEE–FGP exterior face sheets

This research has been conducted to analyze the free vibration behavior of a five-layered composite plate with a viscoelastic auxetic core. The plate comprises a viscoelastic auxetic core layer, functionally graded carbon nanotube reinforced composite (FG-CNTRC) interior, and magneto-electro-elastic functionally graded porous (MEE–FGP) exterior skins, which is rested on Winkler–Pasternak foundation. According to the magnetic–electric boundary conditions and Maxwell equations, the electric and magnetic potentials of the plate are determined. The macro-mechanical properties of the auxetic core have been derived based on Gibson’s model; the three-parameter Zener model has also been utilized to describe its viscoelastic behavior. Additionally, the equations of motion of the plate are obtained and solved by using Reddy’s third-order shear deformation theory (TSDT) and the Galerkin method, respectively. Moreover, various aspects of the current research have been validated by comparing the numerical results with those reported in the literature section. Overall, in the numerical result section, the effects of different parameters and conditions such as geometrical parameters, position of the plate’s interfaces of layers (thickness of the core and face layers), various distribution patterns and volume fractions of both CNT and porosity, external work parameters of the MEE skins, boundary conditions, Winkler–Pasternak stiffness coefficients, the relaxation time, and other parameters of the viscoelastic core on the natural frequency and loss factor of the composite plate are represented.

An analytical modelling of free vibration in porous FG-CNTRC plate resting on elastic foundations

This paper presents an analytical investigation of the vibrations of porous functionally graded carbon nanotube reinforced composite (FG-CNTRC) plate resting on elastic foundations. The plates are reinforced with randomly oriented straight carbon nanotubes, featuring four distinct reinforcement distribution patterns along the thickness. Material properties of the porous FG-CNTRC are graded along the thickness, with both symmetric and asymmetric porosity distributions considered. Utilizing high-order shear deformation theory (HSDT), the equations of motion are derived from Hamilton’s energy principle, and Navier’s method is employed for the solutions. The simply supported plate is assumed to rest on Winkler/Pasternak elastic foundations. The study examines the effects of various parameters, including CNT volume fractions, CNT distribution patterns, thickness ratio (a/h), aspect ratio (a/b), porosity coefficient, and elastic foundation parameters, on the dimensionless frequency parameter.

Dynamic behaviour of an inclined FG-CNTRC sandwich beam under a moving mass

This paper studies dynamic behavior of an inclined sandwich beam with a homogeneous core and functionally graded carbon nanotube reinforced composite (FG-CNTRC) face sheets under a moving mass. The effective properties of the face sheets are estimated by the extended rule of mixture. Three types of carbon nanotube distribution, namely uniform distribution (UD), functionally graded ( ) and V (FG-V) distributions, are considered. Based on the first-order shear deformation theory, a finite element formulation is formulated by using hierarchical functions to interpolate the displacements and rotation. Using the derived formulation, dynamic response the sandwich beam is computed with the aid of the Newmark method. The obtained result reveals that the inclined angle has a significant influence on the response of the beam, and the dynamic magnification factor decreases for the beam associated with a larger inclined angle. The effects of various parameter, including the nanotube volume fraction, the type of carbon nanotube distribution, the layer thickness ratio and the moving mass velocity on dynamic behavior of the sandwich beam are examined and highlighted.

Static, buckling, and free vibration responses of functionally graded carbon nanotube-reinforced composite beams with elastic foundation in non-polynomial framework

In this work, the static, buckling, and free vibration analysis of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beam resting on a Pasternak elastic foundation are studied. The secant function-based shear deformation theory (SFSDT) is used for this analysis. This theory fulfills the traction-free boundary conditions at the top and bottom surfaces of the beam, hence there is no need for a shear correction factor. Hamilton’s principle is used to determine the governing differential equations and boundary conditions whereas Navier’s solution technique is used for determining the closed-form solution. The analytical approach is used to examine the deflection, stresses, critical buckling load, and natural frequencies of the FG-CNTRC beam resting on the Pasternak elastic foundation including a shear layer and Winkler springs. To determine the material characteristics of FG-CNTRC beams, the Rule of the mixture is used. Uniform distribution (UD-beam), FG-X beam, FG-O beam, and FG-V beam are the different forms of CNT reinforcement distribution that are used in this study. Considering different span thickness ratios, the volume fraction and distribution of CNT, the Winkler spring, and the shear layer constant factors, all the structural responses are predicted. It is also observed that the present theory predicts the structural responses of the FG-CNTRC beam accurately when compared to other existing theories. A few new results are also included as the benchmark solutions for the new research.

Buckling and post-buckling of anisogrid lattice-core sandwich plates with nanocomposite skins

This research offers an in-depth examination of the buckling and post-buckling responses observed in sandwich plates distinguished by the presence of an anisogrid lattice core and face layers composed of functionally graded carbon nanotube-reinforced composites (FG-CNTRC). To model this structure, a global continuous model for the lattice core is employed, while the displacement field is approximated using the First-Order Shear Deformation Theory (FSDT). Geometrically nonlinear strain relations based on the von-Kármán assumptions are incorporated to accurately capture the structural response. The homogenization of the FG-CNTRC skins is achieved through the modified rule of mixture, enabling estimations of effective material properties. To separate pre-buckling and post-buckling phases, we implement the adjacent-equilibrium criterion. Solving the linear (pertaining to buckling) and nonlinear (related to post-buckling) equations is based on the Generalized Differential Quadrature (GDQ) technique alongside an iterative method grounded in the displacement control strategy. Our analysis explores the influence of core and skin characteristics on the stability of these sandwich plates. By examining critical buckling thresholds, post-buckling equilibrium paths, and the underlying mechanisms driving these responses, we contribute insights into the structural stability and performance of such composite structures. This study not only enhances our understanding of buckling behavior in anisogrid lattice core sandwich plates but also holds promise for diverse engineering applications where structural stability is paramount.

Natural frequencies and modal shapes of folded sandwich plates made of porous core and FG-CNTRC coating layers resting on two parameters elastic foundation

Folded structures hold significant importance in aerospace technology due to their distinct design, effectively reconciling strength with a lightweight framework. Within aerospace applications, these structures fulfill diverse roles, serving as pivotal elements in aircraft wings, fuselage sections, and essential components of spacecraft and satellites. This study delves into the free vibration behavior of folded sandwich plates composed of a porous core and functionally graded carbon nanotube-reinforced composite (FG-CNTRC) coatings, resting on a two-parameter elastic foundation. Leveraging first-order shear deformation plate theory (FSDT), elasto-dynamic equations are established for each individual plate segment. Utilizing the two-dimensional generalized differential quadrature method (2D-GDQM), these equations are discretized. Subsequently, by applying boundary conditions to all six edges, including traditional clamped, simply supported, and free configurations, and incorporating continuity conditions for displacements, forces, and moments at the joint borders, natural frequencies and modal shapes are extracted for the folded plate structure. The validity of the proposed formulation and results is demonstrated by comparing them with numerical examples for both composite flat and folded plates. This comparison provides strong support for the accuracy and reliability of the present study. The study comprehensively examines the influence of various parameters on natural frequencies, including modal shape visualizations under different boundary configurations, plate thickness, crank angle, porosity parameter, core thickness, foundation stiffness, and CNTs distribution type.

1/2 and 1/3 Subharmonic Resonance Behaviors of a Rotating FG-CNTRC Beam with Geometric Imperfections

This paper aims at investigating 1/2 and 1/3 subharmonic resonances of a rotating functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beam in the presence of geometric imperfections. A general function is introduced to denote three types of imperfections containing sine, global, and local modes. At first, based on the von-Kármán geometric nonlinearity assumption, the forced vibration equation is set up. Then, the Galerkin method and multiple scale method are utilized to study subharmonic resonance behaviors. Finally, coupled effects of FG-CNTRCs, rotating motion, and geometric imperfections on subharmonic resonance responses are investigated. It is found that a specific range of the excitation amplitude and frequency motivates the subharmonic resonance behavior. The coupling of geometric imperfections and geometric nonlinearities generates the 1/2 resonance phenomena, and geometric nonlinearities result in the 1/3 resonance behavior. Both the imperfection mode and amplitude can affect the resonance response and resonance regions. Moreover, in contrast with the 1/3 subharmonic resonance, geometric imperfections produce greater influence on the 1/2 subharmonic resonance.

Vibration analysis of functionally graded carbon nanotubes reinforced composite nanoplates

This work presents the analytical analysis for free linear vibration behavior of functionally graded-carbon nanotubes reinforced composite (FG-CNTRC) nanoplates in the framework of nonlocal strain gradient theory (NSGT) and the first-order shear deformation plate theory (FSDPT). The nanoplate is considered made of a mixture of an isotropic polymer matrix and reinforced carbon nanotubes (CNTs). Four different distributions of CNTs are examined including uniformly distributed and FG reinforcements (FG-O, FG-X, and FG-V). The governing equations of motion are established based on the Hamilton’s principle. The closed-form analytical solution for the natural frequency of FG-CNTRC nanoplates with simply supported all edges is carried out by using the Navier-type solution. The impact of some key parameters on the natural frequencies of FG-CNTRC nanoplates is also studied and discussed. The result shows that FG-CNTRC nanoplates reveal the softening- or hardening-stiffness effects depending on the relationship between the nonlocal parameter and the material length scale parameter. The aspect ratios of FG-CNTRC nanoplates, the volume fraction, and the distribution pattern of CNTs have also an important impact on the vibration behavior of FG-CNTRC nanoplates.

Free vibration of functionally graded carbon nanotube-reinforced composite beam

The purpose of this paper is to investigate the free vibration behavior of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beam. The carbon nanotubes (CNTs) are aligned and distributed in polymeric matrix with different patterns of reinforcement. The material properties of the FG-CNTRC beam are estimated by using the rule of mixture. The trigonometric shear deformation beam theory is employed to deal with the problem. The mathematical models provided in this paper are numerically validated by comparison with some available results. New results of free vibration analyses of FG-CNTRC beam are presented and discussed in detail. The effects of the length to thickness ratio and CNT volume fraction are taken into investigation.

A NURBS-based IGA using zig-zag plate theory for nonlinear passive/semi-active damping analysis of laminated FG-CNTRC plates

This study introduces an efficient and versatile numerical approach for simulating passive/semi-active damping treatments on laminated structures composed of functionally graded carbon nanotube-reinforced composites (FG-CNTRC) plates. This novel approach integrates an isogeometric finite element formulation that leverages basic functions generated from Non-Uniform Rational Basis Spline (NURBS) with the Murakami zig-zag theory, Von Kármán’s nonlinear strain–displacement model, and the electro-mechanical coupling properties of the constraining layers. The proposed isogeometric formulation can be transformed into the Laplace domain using the Golla–Hughes–McTavish model to successfully simulate the time-dependent passive/semi-active damping mechanism of viscoelastic materials. Subsequently, results obtained in the Laplace domain are reconverted into the time domain using inverse techniques. The validation of the proposed model through two numerical examples showcases its high reliability and demonstrates the feasibility of simulating and controlling FG-CNTRC plates with various geometries within a Computer-Aided Design-Computer-Aided Engineering (CAD-CAE) analysis framework.

A New Analytical Approach for Nonlinear Global Buckling of Axially Compressed and Tensiled Sandwich Toroidal Shell Segments with CNTRC Coatings and Corrugated Core in Thermal Environment

In this paper, the nonlinear global buckling and postbuckling responses of axially compressed and tensiled sandwich toroidal shell segments with functionally graded carbon nanotube-reinforced composite (FG-CNTRC) coatings and corrugated core in the thermal environment are investigated. Three distributed types of FG-CNTRC coatings with two geometrical types of corrugated core, and three shell types including convex, concave, and cylindrical shells are considered. By adding the thermal forces to the governing equations, a homogenization model for corrugated structures is developed to simulate the thermal and mechanical behavior of the trapezoidal and round corrugated core. The potential energy expression of sandwich shells is established according to the Donnell shell theory with the nonlinearities of deflection, an algorithm to solve the governing equation is developed using the Ritz energy method, and three equilibrium equations are obtained respecting three amplitudes of deflection. The axial critical buckling loads and postbuckling curves of shells can be achieved after some calculations. The numerical examples prove the beneficial effects of corrugated core and CNTRC coatings on nonlinear global buckling responses of sandwich toroidal shell segments significantly.

Stochastic free vibration analysis of FG-CNTRC plates based on a new stochastic computational scheme

Stochastic analysis can provide more accurate results compared to deterministic analysis. In this study, the spatial variability of material parameters is introduced into the free vibration analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates to achieve a higher degree of accuracy, and a new stochastic computational scheme is proposed to handle the low-level uncertainties. First-order shear deformation theory (FSDT) is employed to establish the displacement field of the plates. The Young's moduli of carbon nanotube (CNT) and matrix are regarded as one-dimensional (1D) and two-dimensional (2D) random fields, respectively, which is discretized by Karhunen-Loève (K-L) expansion. The obtained random variables are substituted into modified perturbation stochastic method (MPSM) and radial point interpolation method (RPIM) to calculate the first two order estimates of the stochastic non-dimensional natural frequencies ω¯. The sensitivity of the first to fourth ω¯ to the random fields is analyzed, and the corresponding stochastic bands are plotted. The results indicate that the presented approach can accurately solve the generalized eigenvalue problem in stochastic free vibration; ω¯ is sensitive to random fields E11CNT and Em, and insensitive to random field E22CNT; the CNT distribution mode also affects the sensitivity.

Sound transmission loss analysis of double-walled sandwich functionally graded carbon nanotube-reinforced composite magneto-electro-elastic plates under thermal environment

Sandwich structures with functionally graded (FG) cores have gained interest in vibroacoustic applications. This article considers the vibroacoustic properties of double-walled sandwich magneto-electro-elastic (MEE) plates with a functionally graded carbon nanotube-reinforced composite (FG-CNTRC) core layer in a thermal environment. A coupled multiphysics model is developed based on third-order shear deformation theory (TSDT) and acoustic-structure interaction. Special attention is paid to the transmission loss of this arrangement for simply supported and clamped boundaries for four different kinds of CNT distributions, namely, UD, FG-V, FG-O, and FG-X. Sound velocity potential, normal velocity continuity conditions, and Hamilton's principle are used to generate the coupled vibroacoustic equations, which are then solved using the Galerkin method. The effects of boundary conditions, CNT distributions, cavity depth, multiphysics coupling fields, and temperature on the STL are comprehensively studied. The results provide guidelines for tailoring the dynamic response of these materials to achieve enhanced acoustic insulation performance. It enhances fundamental understanding and enables the engineering design of multilayered composite panels with optimized vibration and noise control capabilities.

Isogeometric vibration of the magneto-electro-elastic sandwich plate with functionally graded carbon nanotube reinforced composite core

This paper investigates the free vibration analysis of the magneto-electro-elastic (MEE) sandwich plates with functionally graded carbon nanotube reinforced composite (FG-CNTRC) core using isogeometric approach (IGA). The sandwich plate is composed of the homogeneous MEE face sheets and FG-CNTRC core with four types of carbon nanotubes (CNTs) distribution, including CNT-UD, CNT-O, CNT-V and CNT-X. The external electric voltage and magnetic potential are applied in the top and bottom layers of the MEE sandwich plate. Employing the refined plate theory (RPT) and Hamilton principle, the governing equation for free vibration of the MEE sandwich plate is derived. The IGA employs Non-Uniform Rational B-Splines (NURBS) basic functions to approximate the displacement fields and the magnetic and electric potentials in the RPT model. The study examines and discusses the impact of different factors on the frequency of the MEE sandwich plate, including parameters like CNTs distributions, CNTs volume fraction, external electric voltage and magnetic potential, and the geometrical parameter of the plate. This research has revealed several important discoveries regarding the fabrication of MEE sandwich structures.

A new analytical approach for nonlinear buckling and postbuckling of torsion-loaded FG-CNTRC sandwich toroidal shell segments with corrugated core in thermal environments

The design of the torsion-loaded toroidal shell segment with two functionally graded carbon nanotube-reinforced composite (FG-CNTRC) coatings and the corrugated core is presented. Round and trapezoidal shapes of the corrugated core are considered. A homogenization model for the corrugated core is applied to predict the stiffnesses of the corrugated core. The total potential energy equation, according to the Donnell shell theory, is established. The Ritz energy method is applied to determine the expressions of the buckling and postbuckling responses. The numerical investigations present the significantly beneficial effects of FG-CNTRC coatings and corrugated core on the buckling responses of the shells.

Stability and nonlinear vibration of carbon nanotubes-reinforced composite pipes conveying fluid

This paper focuses on the stability and nonlinear vibration characteristics of carbon nanotubes-reinforced composite pipes conveying fluid, and the mechanism of the combined effects of multiple factors. Based on the assumption of Euler beam and Von-karman nonlinear stress-strain relationship, the dynamic governing equations and generalized boundary conditions of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) pipes are established by Hamiltonian variational principle. Firstly, the modal function suitable for generalized boundary conditions is derived by Galerkin method. The stability of FG-CNTRC pipes system and the relationship between natural frequency and flow velocity are analyzed in this paper. The effect of flow velocity on the natural frequency of pipelines is examined under various carbon nanomaterial distribution patterns and wall thicknesses in this research. Then, the nonlinear vibration characteristics of FG-CNTRC pipes are analyzed by multi-scale method. The method of deducing nonlinear frequency through high-order partial differential equations is also extended in nonlinear analysis. The results show that, in the case of uniform distribution of carbon nanotubes, increasing the wall thickness can improve the stability of the pipe system. The influence of amplitude on the nonlinear frequency is more obvious at the second-order frequency.