Functionally Graded Carbon Nanotube Reinforced Composite Research Articles

Hygrothermal mechanical behaviors of a porous FG-CRC annular plate with variable thickness considering aggregation of CNTs

Experiment in literature has shown the aggregation phenomenon in carbon nanotube (CNT) reinforced polymeric composites. In this paper, considering the aggregation effect of CNTs and porosity in both polymeric matrix and aggregated CNT clusters, a mechanical model to investigate the coupled conduction of temperature and moisture as well as the hygrothermal mechanical behaviors of a porous functionally graded CNT reinforced composite (FG-CRC) rotating annular plate with variable thickness is built. Numerical results of the temperature and moisture fields as well as the hygrothermal mechanical responses are achieved by combining the differential quadrature method (DQM), the Runge-Kutta method and the Newmark method. Several useful conclusions as follow are obtained through numerical examples and discussions. In the numerical examples, influences of the aggregation effect of CNTs, coupled effect of temperature and moisture, material property (graded index and porosity parameters), geometric parameters as well as the rotating conditions to the hygrothermal mechanical responses of the porous FG-CRC annular plate are studied in detail. The theoretical process and conclusions will be helpful to the design and manufacture of novel porous graded materials and structures applying in hygrothermal environment and conditions.

Nonlinear primary and super-harmonic resonances of functionally graded carbon nanotube reinforced composite beams

The nonlinear primary and super-harmonic resonances of functionally graded carbon nanotube reinforced composite (FG-CNTRC) beams under transverse harmonic excitation are studied. Taking into account four different distribution patterns of carbon nanotubes (CNTs), the material properties of FG-CNTRC beams are predicted through the extended rule of mixture. The equations of motion for the transverse, longitudinal and rotational motions are derived from Timoshenko beam theory and von Kármán type of geometric nonlinearity. These nonlinear partial differential equations are reduced into nonlinear ordinary differential equations by the Galerkin method. The discretized equations of motion are solved by using the incremental harmonic balance (IHB) method, resulting in the vibration responses of the nonlinear system. For all the three coupled motions, the frequency response curves of primary, 2 super-harmonic and 3 super-harmonic resonances are constructed. Comparison studies on the dimensionless natural frequencies, nonlinear to linear frequency ratios and numerical simulation of the frequency response are conducted to verify the correctness and accuracy of this study. The numerical results illustrate the influences of CNT distributions, CNT volume fraction, slenderness ratio, dimensionless transverse excitation and damping coefficient on the primary and super-harmonic resonant responses.

Vibration analysis of pressurized sandwich FG-CNTRC cylindrical shells based on the higher-order shear deformation theory

The present study deals with the vibration analysis of sandwich cylindrical shells with the functionally graded carbon nanotube reinforced composite (FG-CNTRC) face sheets resting on elastic medium under internal pressure. Two FG-CNTRC face sheets along with homogeneous core are considered as the sandwich cylindrical shell. The overall mechanical properties of CNT-reinforced composites are presented in accordance to the refined rule of mixture. Based on the higher-order shear deformation theory (HSDT) and in the context of the variational differential quadrature (VDQ) method, the discretized version of governing equations is provided. Validation of the proposed model is demonstrated. Several numerical results are also represented to study the impacts of various material and geometrical factors on the vibration analysis of sandwich cylindrical shells. The results reveal that in the case of constant total thickness, increasing the core-to-face sheet thickness ratio decreases the dimensionless frequencies.

NURBS-based postbuckling analysis of functionally graded carbon nanotube-reinforced composite shells

An investigation into the postbuckling and geometrically nonlinear behaviors of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) shells is carried out in this study. The discrete nonlinear equation system is established based on non-uniform rational B-Spline (NURBS) basis functions and the first-order shear deformation shell theory (FSDT). The nonlinearity of shells is formed in the Total Lagrangian approach considering the von Karman assumption. The incremental solutions are obtained by using a modified Riks method. In the present formulation, the rule of mixture is used to estimate the effective material properties of FG-CNTRC shells. Effects of CNTs distribution, volume fraction and CNTs orientation on the postbuckling behavior of FG-CNTRC shells are particularly investigated. Exact geometries of shells are modeled by using NURBS interpolation. Several verifications are given to show the high reliability of the proposed formulation. Especially, some complex postbuckling curves of FG-CNTRC panels and cylinders are first provided that could be useful for future references.

Vibrational behavior of doubly curved smart sandwich shells with FG-CNTRC face sheets and FG porous core

As a first endeavor, the free flexural vibration behavior of doubly curved complete and incomplete sandwich shells with functionally graded (FG) porous core, FG carbon nanotube reinforced composite (FG-CNTRC) face sheets and integrated piezoelectric layers is investigated. The variable radii shells with the three most common types of geometries, i.e., elliptical, cycloid and parabolic, are considered. The system equations are derived based on the general higher-order shear deformation theory and Maxwell's equation. The generalized differential quadrature (GDQ) method is employed to discretize the governing partial differential equations subjected to different boundary conditions. The accuracy and reliability of the approach are verified by comparing the results with the existing solutions in open literature. The effects of porosity parameter and porosity distribution through the thickness direction, carbon nanotube (CNT) volume fraction, different boundary conditions and various shell geometrical parameters on the flexural vibrational behavior of the smart sandwich shell structures are investigated and useful results are presented.

Geometrically non-linear analysis of FG-CNTRC shell structures with surface-bonded piezoelectric layers

This work makes a first attempt to conduct a geometrically non-linear analysis of functionally graded carbon nanotube reinforced composites (FG-CNTRC) structures, with surface-bonded active layers, based on Kirchhoff shell theory. Uniform and three different distribution types of functionally graded nanocomposites are presented. These distributions are assumed to be uniaxially aligned in the axial direction and functionally graded in the shell thickness direction, while the electric potential through the thickness of the active layers is assumed to be linear. A comparison study is carried out in order to show the applicability of the current formulation applied to various shapes of shell structures. The effects of various parameters as volume fraction, distribution of nanotubes, geometrical characteristics as well as load boundary on non-linear behavior of the smart FG-CNTRC structures are investigated.

Thermal effects on the mechanical analysis of temperature-dependent FG-CNTRC annular plates

The present study deals with the vibration and buckling of functionally graded carbon-nanotube-reinforced composite (FG-CNTRC) annular plates subjected to thermal loading. Thermo-mechanical characteristics of the materials are temperature-dependent (TD). Based on Mindlin’s plate theory and using Hamilton’s principle, the governing equations are derived. The generalized differential quadrature (GDQ) approach and periodic differential operators are employed to numerically solve the problem. The proposed approach is verified through different comparative studies. Additionally, to study the impacts of involved factors on the instability and vibration characteristics of CNT-reinforced composites annular plates, a wide range of results is presented. The results reveal that considering the TD material properties considerably affect the mechanical characteristics of thermally-induced CNT-reinforced composite plates.

Higher-Order Thermo-Elastic Analysis of FG-CNTRC Cylindrical Vessels Surrounded by a Pasternak Foundation

This study analyses the two-dimensional thermo-elastic response of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) cylindrical pressure vessels, by applying the third-order shear deformation theory (TSDT). The effective properties of FG-CNTRC cylindrical pressure vessels are computed for different patterns of reinforcement, according to the rule of mixture. The governing equations of the problem are derived from the principle of virtual works and are solved as a classical eigenproblem under the assumption of clamped supported boundary conditions. A large parametric investigation aims at showing the influence of some meaningful parameters on the thermo-elastic response, such as the type of pattern, the volume fraction of CNTs, and the Pasternak coefficients related to the elastic foundation.

Nonlinear buckling and post-buckling of functionally graded CNTs reinforced composite truncated conical shells subjected to axial load

This study deals with the nonlinear static analysis of functionally graded carbon nanotubes reinforced composite (FG-CNTRC) truncated conical shells subjected to axial load based on the classical shell theory. Detailed studies for both nonlinear buckling and post-buckling behavior of truncated conical shells. The truncated conical shells are reinforced by single-walled carbon nanotubes which alter according to linear functions of the shell thickness. The nonlinear equations are solved by both the Airy stress function and Galerkin method based on the classical shell theory. In numerical results, the influences of various types of distribution and volume fractions of carbon nanotubes, geometrical parameters, elastic foundations on the nonlinear buckling and post-buckling behavior of FG-CNTRC truncated conical shells are presented. The proposed results are validated by comparing with other authors.

A comprehensive study on the free vibration of arbitrary shaped thick functionally graded CNT-reinforced composite plates

The main objective of this paper is to analyze the free vibration of arbitrary shaped thick functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plates based on the higher-order shear deformation theory (HSDT) using a variational differential quadrature approach. By means of the generalized differential quadrature (GDQ) numerical operators and Hamilton’s principle, the discretized equations of motion are obtained. In order to use the GDQ differential and integral operators appropriately, the coordinate transformation is considered through the conventional finite element approach for transforming the irregular domain of the plate into the regular computational one. Employing a unified numerical approach to analyze different shapes of thick FG-CNTRC plates based on HSDT is the main novel aspect of the present study. To imply the accuracy of the present model, a wide range of comparison studies are presented. The results indicate the efficiency of the developed numerical methodology to study the vibration of arbitrary shaped thick FG-CNTRC plates. Several results are also given to investigate the impacts of geometrical parameters and material properties on the vibrational behavior of FG-CNTRC plates.

Nonlinear bending analysis of FG-CNTRC annular plates with variable thickness on elastic foundation

As a first endeavor, the nonlinear bending behavior of functionally graded carbon nanotube reinforced composite (FG-CNTRC) annular plates with variable thickness on an elastic foundation is scrutinized in this research. The nonlinear governing equations are established according to the third-order shear deformation theory (TSDT) in conjunction with the nonlinear von Karman strain field. A Pasternak elastic foundation is assumed to be in contact with the annular plate during deformation. To discretize the nonlinear governing equations and boundary conditions, the generalized differential quadrature method (GDQM) is adopted and the nonlinear system of equations is solved via the Newton-Raphson iterative method. Through the numerical results, effects of the elastic foundation, geometrical parameters, boundary conditions, CNT dispersion and volume fraction, thickness-to-radius ratio and thickness profile are investigated in detail. It is shown that CNT volume fraction and dispersion have a remarkable effect on improving bending behavior of the annular plate. It is found that thickness profile plays an essential role in the bending response of the annular plate.

Nonlinear free and forced vibration analysis of FG‐CNTRC annular sector plates

This article deals with the large amplitude free and forced vibration analysis of functionally graded carbon nanotube‐reinforced composite (FG‐CNTRC) annular sector plates based on a numerical approach. The modified rule of mixture is used to estimate the material properties. The equations of motion are developed based on the first‐order shear deformation theory (FSDT) and the von Kármán geometric nonlinearity. First, the discretized form of energy functional of structure is given with the aid of variational differential quadrature method. Then, a time periodic discretization is performed and the frequency response of the nanocomposite plate is determined using the pseudo‐arc length continuation method. After verifying the correctness of the proposed approach, a comprehensive parametric study is presented to investigate the effects of important factors on the nonlinear vibration characteristics of the FG‐CNTRC annular sector plates. The results imply that the volume fraction and distribution type of nanotubes have considerable effects on the fundamental frequency as well as nonlinear frequency response curves. POLYM. COMPOS., 40:E1364–E1377, 2019. © 2018 Society of Plastics Engineers

Thermal buckling analysis of temperature-dependent FG-CNTRC quadrilateral plates

The main objective of the present study is to analyze the thermal buckling of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) quadrilateral plates. Functionally graded patterns are introduced for the distribution of the carbon nanotubes (CNTs) through the thickness direction of the plate. The effective material properties of nanocomposite plate reinforced by CNTs are considered to be temperature-dependent (TD) and estimated using the micromechanical model. By the use of minimum total potential energy principle and based on the first-order shear deformation theory of plates, the stability equations are obtained. In order to use the generalized differential quadrature (GDQ) method and solve the stability equations, the irregular domain of quadrilateral plate is transformed into regular computational domain employing the mapping technique. The efficiency and accuracy of the proposed approach are first validated. Then, a comprehensive parametric study is presented to examine the effects of model parameters on the thermal buckling of FG-CNTRC quadrilateral plates. The results indicate that considering temperature dependency of the material properties plays an important role in the stability of the FG-CNTRC quadrilateral plates subjected to thermal loading.

Dynamic Stability of Rotating FG-CNTRC Cylindrical Shells under Combined Static and Periodic Axial Loads

The dynamic stability behavior of rotating functionally graded carbon nanotube reinforced composite (FG-CNTRC) cylindrical shells under combined static and periodic axial forces is investigated. The governing equations are derived based on the first-order shear deformation theory (FSDT) of shells. The initial mechanical stresses due to the steady state rotation of the shell are evaluated by solving the dynamic equilibrium equations. The equations of motion under different boundary conditions are discretized in the spatial domain and transformed into a system of Mathieu–Hill type equations using the differential quadrature method (DQM) together with the trigonometric series. The influences of both the initial mechanical stresses and Coriolis acceleration are considered. Then, the parametric resonance is analyzed and the dynamic instability regions are determined by employing the Bolotin’s first approximation. After validating the approach, the effects of rotational speed, Coriolis acceleration, carbon nanotubes (CNTs) distribution in the thickness direction, CNTs volume fraction, length and thickness-to-mean radius ratios on the principal dynamic instability regions are examined in detail.

Geometrically nonlinear dynamic analysis of FG-CNTRC plates subjected to blast loads using the weak form quadrature element method

Geometrically nonlinear dynamic analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) rectangular plates subjected to blast loads is conducted based on Reddy’s higher-order shear deformation theory using the weak form quadrature element method. The von-Kármán strain terms are introduced to consider the geometrically nonlinear effects. The polymer composite plate is reinforced by single-walled carbon nanotubes (SWCNTs) with the uniform and functionally graded distribution in the plate thickness direction. The effective material properties of the FG-CNTRC plates are estimated with the extended rule of mixture. The Newmark-β time integration scheme and the Newton–Raphson iteration technique are adopted to solve the nonlinear incremental dynamic equilibrium equation in temporal domain. Comparative and convergence studies are carried out to validate the accuracy, efficiency and numerical stability of the presented weak form quadrature element formulation. The effects of the carbon nanotube distribution and volume fraction, plate width-to-thickness ratio, plate aspect ratio, load type and boundary condition on the dynamic response of the FG-CNTRC plates under blast loads are systematically investigated through the parametric studies.

A comprehensive analytical study on functionally graded carbon nanotube-reinforced composite plates

This paper presents a model which is effective and simple based upon Second Order Shear Deformation Theory (SSDT) to comprehensive study of Functionally Graded Carbon Nanotube Reinforced Composite (FG-CNTRC) plates including size effects for the first time. Also it is the first time that the size-dependent static, stability and dynamic response of FG-CNTRC plates considering Winkler–Pasternak elastic foundation effects are investigated. It is assumed that the material properties of FG-CNTRC are varied through the thickness direction using four different distributions of carbon nanotubes (CNTs). In order to include the size effects in our modeling, the nonlocal elasticity theory presented by Eringen is utilized. The influences of different parameters such as volume fraction of carbon nanotubes, nonlocality and elastic foundation effects on the response of FG-CNTRC plates are studied. Numerical results prove high accuracy and reliability of the present method in comparison with other available numerical or analytical methods.

Three-dimensional nonlinear bending analysis of FG-CNTs reinforced composite plates using the element-free Galerkin method based on the S-R decomposition theorem

A three-dimensional element-free Galerkin method (EFG) based on the Strain-Rotation (S-R) decomposition theorem, named 3D-SR-EFG, is developed to investigate the nonlinear bending behavior of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates. Due to its overcoming the deficiencies of classic finite deformation theories, S-R decomposition theorem can provide a reliable theoretical support for the geometrically nonlinear simulation. The incremental variational formulation based on the S-R decomposition theorem for three-dimensional static large deformation problems is derived from the updated co-moving coordinate formulation and principle of potential energy rate. Global weak-form EFG is adopted to obtain the discrete form of the formulation. Convergence and comparison studies are conducted to validate the numerical stability and accuracy of the proposed 3D-SR-EFG. The influences of volume fraction and distributions of CNTs, plate’s aspect ratio and width-to-thickness ratio, boundary conditions on the nonlinear bending response of the CNTRC plates are numerically analyzed and discussed in parametric studies. Results demonstrate that the 3D-SR-EFG can effectively predict the nonlinear bending behavior of the CNTRC plates. This work also further extends the applications of the S-R decomposition theorem.

Thermal and mechanical buckling analysis of FG carbon nanotube reinforced composite plates using modified couple stress theory and isogeometric approach

Abstract This paper investigates the thermal and mechanical buckling analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates by using isogeometric analysis (IGA) based on modified couple stress theory (MCST). A refined hyperbolic shear deformation theory is used for buckling analysis, which satisfies free transverse shear stress conditions on the top and bottom surfaces of plate without a need for shear correction factor. The material properties of carbon nanotube reinforced composite plates are assumed to be temperature-dependent. For numerical analysis the IGA method using B-Spline or Non-Uniform Rational B-Spline (NURBS) functions is employed. The obtained results are compared with those available in the literature. Also, the influence of different parameters on mechanical and thermal buckling analysis is investigated. These parameters include material length scale parameter, boundary conditions, aspect and length-to-thickness ratios of plate, different types of FG-CNTRC distribution, volume fraction of CNTs and temperature dependency.

Size-dependent analysis of FG-CNTRC microplates based on modified strain gradient elasticity theory

We present in this study a size-dependent computational approach based on the modified strain gradient theory (MSGT) and higher-order shear deformation theory for static bending and free vibration analyses of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) microplates. Three material length scale parameters (MLSPs) are taken into account in MSGT to capture size effects of microplate behavior. The effective material properties of FG-CNTRC microplates are obtained by an extended rule of mixture. Four types of carbon nanotube distributions, which are either uniform or functionally graded (FG) through the plate thickness, are considered. The governing equations are derived from the principle of virtual work and are then solved by isogeometric analysis (IGA). The IGA is suitable for a numerical implementation of the size-dependent models since it requires higher-order gradients in the weak form. The inclusion of geometrical parameters, boundary conditions, distributed types of carbon nanotube and material length scale parameters are studied to evaluate the displacement and natural frequency of FG-CNTRC microplates. In addition, the present size-dependent model can be retrieved into the modified couple stress model or classical model when a few MLSPs are ignored.

Numerical study on the thermal buckling analysis of CNT-reinforced composite plates with different shapes based on the higher-order shear deformation theory

In the present study, based on the higher-order shear deformation plate theory, the unified numerical formulation is developed in variational framework to investigate the thermal buckling of different shapes of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates. Since the thermal environment has considerable effects on the material properties of carbon nanotubes (CNTs), the temperature-dependent (TD) thermo-mechanical material properties are taken into account. In order to present the governing equations, the quadratic form of the energy functional of the plate structure is derived and its discretized counterparts are presented employing the variational differential quadrature (VDQ) approach. The discretized equations of motion are finally obtained based on Hamilton's principle. In order to convenient application of differential quadrature numerical operators in irregular physical domain, the mapping procedure is considered in accordance to the conventional finite element formulation. Some comparison and convergence studies are performed to show validity and efficiency of the proposed approach. A wide range of numerical results are also reported to analyze the thermal buckling behavior of different shaped FG-CNTRC plates.