Graded Carbon Nanotube Research Articles

Free vibration of doubly-curved panels reinforced by carbon nanotubes: New analytic solutions under non-Lévy-type boundary conditions

Existing analytic solutions for the free vibration of functionally graded carbon nanotube reinforced doubly-curved panels primarily address the cases with two parallel simply supported boundaries, known as Lévy-type boundary conditions (BCs). However, doubly-curved panels with non-Lévy-type BCs are more commonly encountered in practical engineering applications, yet their analytic solutions are rarely available due to significant mathematical challenges. This gap motivates us to develop new analytic free vibration solutions under these more complex BCs. The nanocomposites’ material properties are first computed according to the rule of mixture. The Hamiltonian-system governing equation for the free vibration of doubly-curved panels is then formulated from the Donnell-Mushtari theory, and is solved by adopting the analytic symplectic superposition method. The obtained analytic solutions are derived without requiring predefined solution forms, and have been thoroughly validated by comparison with the results from the finite element method. By utilizing the accurate analytic solutions, the effects of aspect ratios, BCs, types of CNT distributions, and volume fractions of CNT on the free vibration behaviors are further analyzed. The present solution procedure and the resulting analytic solutions are expected to be useful for dynamic modeling of composite shell panels, supporting both future research and practical applications.

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.

Sound radiation and wave propagation of functionally graded carbon nanotube reinforced composite plates

Sound radiation and wave propagation of functionally graded carbon nanotube reinforced composite plates

Enhancing stability of curved beams with functionally graded carbon nanotubes: a multi-faceted approach

Enhancing stability of curved beams with functionally graded carbon nanotubes: a multi-faceted approach

Thermomechanical bending of functionally graded carbon nanotubes reinforced composite plate by meshless method

AbstractFunctionally graded (FG) carbon nanotubes (CNTs) reinforced composite possessing continuously varied material properties have received extensive concern in the area of aviation, automotive, military, and spaceflight. The nonlinear bending of FG CNT composite is performed by developing a novel meshless model on the base of Smoothed Particle Hydrodynamics (SPH) method. The equivalent temperature relative material properties of composite is established in the light of the extended rule of mixture model. Different gradient dispersions of CNTs are taken into account and modeled. The governing equations are explored using the Mindlin theory and discretized though a symmetric SPH method with high prediction ability. The developed model is verified by analyzed several examples and compared to the solution obtained in literature. Influences of CNT dispersion, content, plate aspect ratio, boundary conditions and lamination angle on the bending response are elaborated. The present study provides an alternative potential method to solve the mechanical performances of FG CNT composites based on the meshless SPH formulation.Highlights Thermo‐mechanics of FG‐CNTRC is firstly solved by SPH method. The model is verified by solving the benchmarks in literature. Nonlinear bending behaviors of FG‐CNTRC are demonstrated. The minimum deflection take place in FG‐X type composite. For laminate plate, the smallest deformation occurs in 0° CNTRC.

Analytical Study of Geometric Effects on Functionally Graded Carbon Nanotube Sandwich Shells

Analytical Study of Geometric Effects on Functionally Graded Carbon Nanotube Sandwich Shells

Nonlinear forced vibration and detached resonance curves of axially moving functionally graded carbon nanotube reinforced composite plates

Nonlinear forced vibration and detached resonance curves of axially moving functionally graded carbon nanotube reinforced composite plates

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.

Active vibration control of piezoelectric multilayers FG‐CNTRC and FG‐GPLRC plates

AbstractThis study investigates the active vibration control of multilayer Functionally Graded Carbon NanoTube Reinforced polymer Composite (FG‐CNTRC) plates and multilayer functionally graded graphene platelets reinforced polymer composite (FG‐GPLRC) plates covered with a piezoelectric layer (PZTG1195N) serve as both actuators and sensors. The plate's heart is supposed to be reinforced nonlinearly using both types of reinforcement. Active vibration control of plates under uniform load was achieved by utilizing a velocity feedback control algorithm with a specific gain value. The finite element method is employed to analyze both the static and dynamic behavior of a square plate discretized into 9‐node quadratic finite elements with 5 ° of freedom per node. The material properties are assumed to change across the thickness direction following a power law distribution, exhibiting various patterns: Uniform Distribution (UD), as well as Functionally Graded types X, O, and A (FG‐X, FG‐O, and FG‐A). The geometrical nonlinear equations are solved by the Newmark integration method. This work begins by validating the natural frequencies of a Functionally Graded (FG) composite plate reinforced with four different distributions of Nano‐filler and then aims to show the effect of the nonlinear distribution of nanofillers on the plate's stiffness. The Results obtained after the execution of a developed code highlight the significance of employing a nonlinear distribution of nanofillers to attenuate vibrations.Highlights FG composite and sandwich plates reinforced with GPLs and CNTs. Effect of nonlinear nanofillers distribution of reinforcement on free and forced vibration. The ability of the nonlinear distribution of nanofillers to attenuate forced vibrations.

Large-amplitude nonlinear free vibrational behavior of the rotating rectangular plate reinforced by functionally graded carbon nanotubes

The present study discusses the nonlinear free vibrational behavior of the rotating rectangular plate reinforced by functionally graded carbon nanotubes (FG-CNTs). Mechanical properties of the plate have been continuously changed along the thickness of the plate by considering four different distribution patterns of CNTs. Based on the first-order shear deformation theory (FSDT) and von Ḱarḿan’s geometrical nonlinearity, the governing equations of the plate are derived using Hamilton’s principle. Galerkin discretization technique is employed to convert the partial differential equations of motion to ordinary differential equations. Afterward, the nonlinear time-dependent equations of the plate were analytically solved using the multiple time scales method. Eventually, the effect of CNT parameters, the geometrical ratios and rotational speed on the nonlinear frequency of the plate have been studied. For verification purposes, the present numerical results have been compared with those available in the literature with evident agreement. The present research may give benchmark solutions and some guidelines for designing FG-CNTs rotating plates.

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.

A nonlocal higher-order shear deformation approach for nonline ar static analysis of magneto-electro-elastic sandwich Micro/Nano-plates with FG-CNT core in hygrothermal environment

In this study, the nonlocal static analysis of Magneto-Electro-Elastic (MEE) sandwich micro/nano-plates with Functionally Graded Carbon Nanotubes (FG-CNTs) core in a hygrothermal environment is studied. The nonlocal static formulation is developed based on Reddy's High-order Shear Deformation theory (HSDT). The von Kármán nonlinear strains are used and the governing equations of the HSDT are derived accounting for Eringen's nonlocal stress-gradient model. The governing equations are solved using the Galerkin method, which involves expanding the displacement field based on orthogonal functions. The variation of the volume fractions through the thickness of FG-CNTs core layer has been computed using two different homogenization techniques: the rule of mixtures and the Mori–Tanaka scheme. A detailed parametric study to show the characteristics of the FG-CNTs core, nonlocal parameters, thermal and moisture changes, types of CNT reinforcement and volume fraction, electric and magnetic potentials, and geometric parameters. The results are compared with the first-order shear deformation theory (FSDT) and classical plate theory to show the accuracy of nonlocal High-order Shear Deformation theory(NHSDT) formulation.

Behavior of functionally graded carbon nanotube reinforced composite sandwich beams with pultruded GFRP core under bending effect

A novel generation of composite sandwich beams with laminated carbon fiber-reinforced polymer skins and pultruded glass fiber-reinforced polymer core materials was examined for their flexural behavior. The strength and failure mechanisms of the composite sandwich beams in flatwise and edgewise configurations were investigated using three-point static bending tests. These sophisticated composite structures must be designed and used in a variety of sectors, and our research provides vital insights into their performance and failure patterns. In comparison to the reference specimens (FGM-1), the carbon nanotube-reinforced specimens’ bending capacity was affected and ranged from −2.5% to 7.75%. The amount of the carbon nanotube addition had a substantial impact on the beams’ application level and load-carrying capacity. Particularly, the application of 0.5 wt% additive in the outermost fiber region of the beams, such as in FGM-4, led to an increase in the bending capacity. However, the stiffness values at the maximum load were decreased by 0.3%–18.6% compared to FGM-1, with the minimum level of the decrease in FGM-4. The experimental results were compared with the theoretical calculations based on the high-order shear deformation theory, which yielded an approximation between 11.99% and 12.98% by applying the Navier’s solution.

A local–global optimization approach for maximizing the multiphysics frequency response of laminated functionally graded CNTs reinforced magneto-electro-elastic plates

Smart materials have garnered significant attention due to their distinct properties and potential applications in various engineering fields. In line with this, this study presents a local–global optimization approach aimed at maximizing the multiphysics frequency of laminated functionally graded carbon nanotube reinforced magneto-electro-elastic (FG-CNTMEE) plates. The laminated FG-CNTMEE plate is composed of layers of functionally graded carbon nanotubes embedded in an electro-elastic matrix, rendering it capable of exhibiting exceptional properties in response to magnetic and electric fields. To optimize its performance, a Balance Enhanced Symbiotic Organisms Search (BESOS) algorithm with new modifications of phases is proposed for global optimization, and integrated with a multiphysical isogeometric analysis using refined zigzag theory (MIARZ) for precise local–global analysis of laminated FG-CNTMEE plates within a Computer-Aided Design (CAD) environment. The effectiveness, robustness, and reliability of the proposed approach are demonstrated through numerical examples that highlight its applicability in achieving optimal designs for smart laminated FG-CNTMEE plates.

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.

Nonlinear flutter analysis of quadrilateral plates consisting of functionally graded carbon nanotubes reinforced composites using Isogeometric Analysis

The current investigation deals with the aeroelastic behavior of quadrilateral plates composed of Functionally Graded Carbon Nanotubes Reinforced Composites (FG-CNTRC) with different distribution models in supersonic airflow using Isogeometric Analysis (IGA). To fulfill this aim, the aeroelastic equations of the mentioned plates are constituted utilizing extended Hamilton's Principle. Hence, First-order Shear Deformation Theory (FSDT) and von Karman's nonlinear strains are exploited to extract structural dynamics equations. The first-order piston theory is utilized to include aerodynamic forces in aeroelastic equations. Carbon nanotube distribution is assumed to be across the thickness either uniformly or non-uniformly and estimated by using the extended rule of mixture. The Isogeometric approach is operated to discretize the obtained relations. The natural frequencies and the flutter boundaries are then attained by applying eigenvalue analysis to the linear aeroelastic model. The flutter behavior of the plates is studied by applying the well-known Newmark method to the mentioned equation. The post-flutter analysis is also conducted by simultaneously applying the Newton–Raphson and Newmark methods to the nonlinear aeroelastic equation. The impacts of nanotube distribution and its volume fraction, flow yaw angle, and geometry on the plates’ behavior are investigated. The results are also validated by comparing them with well-known references presented in the literature.

Free vibration analysis of functionally graded carbon nanotubes reinforced double plates

The specific objective of this study is to analyze the dynamics of functionally graded carbon nanotubes (FGCNT) reinforced double plates. Connected via an elastic layer, the plates have simply supported boundary conditions. In the current study, three carbon nanotubes functionally graded patterns, varying in the thickness direction are considered, including uniformly distributed, functionally graded O-pattern, and functionally graded X-pattern. Following the development of the coupled equations of motion using the Hamilton principle while considering the influences of the elastic layer, the equations are subsequently solved utilizing a two-spatial-variable modal decomposition method. For verification purposes, the equations developed are compared to simplified configurations provided in the existing studies. The solution methodology is verified through comparing against numerical results of simplified configurations of plates obtained from the development of the finite element method and existing studies. Both verifications have shown very good agreement. Influences of plates’ dimensions, carbon nanotubes reinforcement, and the stiffness of elastic layer are analyzed and provided in this study. The transverse-motion natural frequencies of the double plates are also identified, and they follow a decreasing trend as the aspect ratio increases for all the cases. The fundamental lateral-motion and axial-motion natural frequency also follows a similar trend as the aspect ratio increases. The reinforcement effect of carbon nanotubes on the transverse-motion natural frequencies is less obvious for thinner plates. An increase in the elastic layer stiffness increases the second series transverse-motion natural frequencies of the double-plate system. Among the considered functionally graded patterns, the functionally graded X-pattern reinforcement provides the largest increase in the transverse-motion natural frequencies.

Low speed impact on sandwich beam with flexible core and face sheets reinforced with FG-CNTs

Purpose The purpose of this study is to model the theory of the low-velocity impact (LVI) process on sandwich beams consisting of flexible cores and face sheets reinforced with functionally graded carbon nanotubes (CNTs). Design/methodology/approach A series of parameters derived from molecular dynamics are used to consider the size scale in the mixture rule for the combination of CNTs and resin. A procedure involving the use of the first-order shear deformation theory of the beam is used to provide the displacement field of the sandwich beam. The energy method and subsequently the generalized Lagrange method are used to derive the motion equations. Due to the use of Hertz’s nonlinear theory to calculate the contact force, the equations of motion are nonlinear. Validation of the problem is carried out by comparing natural frequencies with other papers. Findings The influence of a series of parameters such as CNTs distributions pattern in the face sheets, the influence of the CNTs volume fraction and the influence of the core thickness to the face sheets thickness ratio in the issue of LVI on sandwich beams with clamped-clamped boundary conditions is investigated. The result shows that the type of CNTs pattern in the face sheet and the CNTs volume fraction have a very important effect on the answer to the problem, which is caused by the change in the value of the Young’s modulus of the beam at the contact surface. Changes in the core thickness to the face sheets thickness ratio has little effect on the impact response. Originality/value Considering the important application of sandwich structures in vehicles, aviation and ships, in this research, sandwich beams consisting of flexible core and CNTs-reinforced face sheets are investigated under LVI.

Efficient analysis on buckling of FG-CNT reinforced composite joined conical–cylindrical laminated shells based on GDQ method under multiple loading conditions

This work presents an efficient analysis of buckling for functionally graded carbon nanotube reinforced composite (FG-CNTRC) joined conical–cylindrical laminated shells with lateral pressure and axial compressive load considering thermal effect. The governing equilibrium equation of the joined conical–cylindrical shell (CO-CYL) is determined using the first-order shear deformation theory (FSDT), Donnell kinematic assumptions, von-Kármán geometrical nonlinearity, and considering thermal effect. A semi-analytical solution, generalized differential quadrature (GDQ) method, and artificial springs are applied during the solution procedure. The comparison studies have demonstrated a remarkable 97% reduction in running time compared to the finite element method (FEM).