Functionally Graded Carbon Nanotube-reinforced Composite Plates Research Articles

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.

Four-node quadrilateral C 0-element based on cell-based smoothed strains strategy and third-order shear deformation theory for functionally graded carbon nanotube reinforced composite plates

This study indicates the analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates using a four-node quadrilateral element related to the C0-type of Reddy’s third-order shear deformation theory (C0 HSDT) and cell-based smoothed strains (CS) strategy. Reddy’s theory is surely taking the advantages and desirable properties of the third-order shear deformation theory. Besides, FG-CNTRC plates with advanced material properties are changed from the bottom to top surface with four kinds of carbon nanotube (CNTs). Numerical results and comparison with other reference solutions suggest that the benefits of the present element are accuracy and efficiency in analysis of FG-CNTRC plates.

Nonlinear buckling analysis of higher-order shear deformable FG-CNTRC plates stiffened by oblique FG-CNTRC stiffeners

The nonlinear buckling analysis of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plates subjected to axial compression load is analytically examined in this paper. Assuming that the FG-CNTRC plates are stiffened by an oblique FG-CNTRC stiffener system. Reddy’s higher-order shear deformation plate theory (HSDPT) with the geometrical nonlinearities of von Kármán is applied to establish the basic formulations. Moreover, the smeared stiffener technique is successfully improved for the higher-order shear deformable anisotropic oblique stiffener system by using a homogeneous model of the anisotropic beam. Galerkin’s method is used to achieve the expressions of critical buckling loads and postbuckling load-deflection curves in explicit form. The numerical values display the influences of FG-CNTRC stiffeners, material, and geometrical properties on the nonlinear buckling response of plates.

An analytical solution for vibration response of CNT/GPL/fibre/polymer hybrid composite micro/nanoplates

In the present article, a closed-form solution is carried out for the vibration response of CNT/GPL/fiber/polymer hybrid composite macro/micro/nanoplates resting on elastic support. Multi-layered fiber/polymer hybrid plates with either functionally graded carbon nanotube reinforced composite (FG-CNTRC) material or functionally graded graphene platelets reinforced composite (FG-GPLRC) material are two different types of plate considered here. The distribution of reinforcing nanocomposites can be uniform or functionally graded through the plate layers. The governing vibration equations are developed in the framework of the nonlocal elasticity theory of Eringen and using the first-order shear deformation plate theory. In the analytical solution procedure, the partial differential equations of motion are simplified to a series of homogeneous ordinary differential equations using a displacement field for Levy-type of the boundary conditions. The general solution of the obtained equations is expressed in terms of exponential functions. Using that the displacements should satisfy the remaining boundary conditions, the vibrational frequencies are derived. Finally, the influences of different parameters including geometry and material properties such as nonlocal parameter, number and angle of layers, volume fraction, and type of distribution of reinforcing nanocomposites are investigated and discussed. Natural frequencies for various boundary conditions are illustrated. The numerical results illustrate that the effects of the nonlocal parameter on the frequency decrease when one of the boundaries is free, and these effects increase when one of the boundaries is clamped for both FG-CNTRC and FG-GPLRC plates.

Influence of cut-out shape on the dynamic behavior of carbon nanotubes reinforced functionally graded plates

This research investigates the effects of cut-out on the dynamic behaviour of functionally graded carbon nanotube reinforced composite (FGCNTRC) plates. Finite element method (FEM) is utilized and SHELL181 type element of the ANSYS Workbench 17.2 program is utilized. To understand the vibration characteristics of the FG plates, various shapes of cut-outs such as triangular, square, rectangular, hexagonal, circular etc are considered. For comparison, solid FGCNTRC plate is modeled and natural frequencies of plates integrated with cut-outs of different shapes are determined using free vibration study. The mode superposition method is used to address the harmonic response analysis problem keeping the damping ratio constant. The stress, phase changes and displacement responses for definite frequency ranges have been used to interpret the findings. To learn more about the effects of cut-outs on the dynamic response, the plates are designed considering various grading patterns. The findings show that the shape of cut-outs significantly affects the harmonic responses as well as the fundamental frequency of the FG plates and are also depending on the grading patterns of these plates.

Investigation of Mechanical Behaviors of Functionally Graded CNT-Reinforced Composite Plates.

In this paper, the mechanical behavior of a functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plate is numerically investigated. According to the concept of a hierarchical model, the displacement is decomposed into the in-field functions and the assumed thickness-wise monomial. The former is defined on the plate midsurface and is approximated by the 2-D meshfree natural element method (NEM). The FG-CNTRC plate is modeled as a homogenized orthotropic body, and its effective elastic properties are determined by referring to MD simulation and the linear rule of mixtures. Regarding the thickness-wise distribution of CNTs, one uniform and three functionally gradient distributions are taken. Through comparative numerical experiments, the reliability of the numerical method is justified with the maximum relative difference of 6.12%. The effects of the volume fraction and vertical distribution of CNTs, the plate width-thickness and aspect ratios, and the boundary conditions on the bending, free vibration, and buckling behaviors of FG-CNTRC plates are examined. It is found that the mechanical behavior of FG-CNTRC plates is significantly dependent of these major parameters.

Nonlinear bending analysis of FG-CNTRC plate resting on elastic foundation by natural element method

A nonlinear numerical analysis is addressed for a functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plate on elastic foundation. The large deflection bending problem is formulated according to the von Kármán nonlinear theory and the (1,1,0)* hierarchical model. And, it is numerically solved by 2-D (two-dimensional) natural element method (NEM) which is characterized by a high accuracy even for coarse grid. The derived nonlinear matrix equations are iteratively calculated by combining the load increment scheme and the Newton-Raphson iteration method. The present method was verified through the benchmark test, where the present method shows excellent agreement with the reference solutions. The nonlinear behavior of FG-CNTRC plate on elastic foundation is evaluated in terms of the non-dimensional deflection and bending moment. These non-dimensional quantities are examined with respect to the foundation stiffness, the volume fraction and gradient pattern of CNT, the aspect and width-thickness ratios of plate and the boundary condition. From the parametric experiments, it is found that both non-dimensional quantities are significantly influenced by such parameters.

Active Vibration Control of Functionally Graded Carbon Nanotube Reinforced Composite Plate with Coupled Electromechanical Actuation

Piezoelectric materials possess excellent electromechanical coupling characteristics, which are functional and suitable in structural vibration control. This study investigates the active control of free and forced vibration for piezoelectric-integrated functionally graded carbon nanotube reinforced composite (FG-CNTRC) plate using the finite element method (FEM). Based on the first-order shear deformation theory (FSDT), the governing equations of the motion of a piezoelectric-integrated FG-CNTRC plate are derived by Hamilton’s principle. The convergence and accuracy of the numerical method is verified through the results of natural frequencies. The influences of CNT volume fraction, CNT distribution type, piezoelectric layer thickness-to-plate thickness ratio, and boundary condition on the natural frequencies are investigated. A constant gain velocity feedback algorithm is used to achieve the dynamic response control of the piezoelectric-integrated FG-CNTRC plate. In addition, the effects of dynamic load, feedback control gain, and boundary condition on the dynamic response of the plate are studied. Numerical results indicate that active control is promising for practical applications in civil and mechanical engineering.

Thermal stress analysis of perforated unsymmetric FG-CNTRC plate using a general analytical solution

In this study, based on the development of the complex variable method, a general solution is presented to calculate thermal stresses in unsymmetric functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plates weakened by a hole under uniform heat flux. The extracted equations can also be used for thermal analysis of perforated FGM plates and unsymmetric composites laminates. Due to the fact that size and temperature affect the mechanical properties of carbon nanotubes (CNT), the material properties of the unsymmetric FG-CNTRC plates are approximated based on gradual changes in the direction of thickness and extended rule of the mixture. The present analysis considers the effect of important parameters such as CNT volume fraction, CNT angle, hole aspect ratio, hole orientation, and heat flux angle on the distribution of thermal stresses and moments around elliptical and circular holes. The present comprehensive parametric study shows that the aforementioned variables have a significant effect on the thermal stresses in perforated FG-CNTRC plates. Therefore, by properly selecting their values in the design of such plates, thermal stresses can be significantly reduced.

Thermally Induced Bistable Functionally Graded Nanocomposite Plate

This paper explores the feasibility of using functionally graded carbon nanotube-reinforced composite (FG-CNTRC) in designing bistable plates. Single-walled carbon nanotubes (SWCNTs) in this nanocomposite are assumed to have a functionally graded distribution across the thickness direction following a power law. By varying the distribution type, volume fraction, and volume fraction exponent of SWCNTs, laminates with multiple thermal properties can be generated. A higher-order Rayleigh–Ritz model is presented to investigate the bistability and buckling behaviours of the proposed bistable plates. Their out-of-plane displacements and snap-through forces are predicted using this model. In the buckling analysis, transverse concentrated forces are applied to the panel corners to induce snap-through. To trace the load-displacement path during the loading process, the arc-length method is used to solve derived nonlinear equilibrium equations. The analytical results are compared using nonlinear finite element (FE) analysis for validation. A comprehensive parametric study is conducted to analyse the effects of curing temperature, distribution type, volume fractions, and volume fraction exponents of SWCNTs on the bistable behaviours of FG-CNTRC plates. It is found that, by varying these factors, bistable FG-CNTRC plates with multiple stable shapes and a wide range of snap-through forces can be generated. FG- CNTRC thus offers a rich design space for producing bistable plates.

Nonlinear free vibration of functionally graded CNT-reinforced composite plates

This paper is concerned with the development of a nonlinear numerical method for solving the nonlinear vibration problem and the parametric investigation of nonlinear free vibration characteristics of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plates. The von Kármán nonlinear vibration problem is numerically formulated in the framework of 2-D natural element method (NEM) and the (1,1,0)* hierarchical model which is equivalent to the first-order shear deformation theory. The selectively reduced integration technique is employed to suppress the shear locking, and the resulting nonlinear modal equation is solved by a three-step direct iterative scheme. The nonlinearity in free vibration is evaluated in terms of the nonlinear to linear fundamental frequency ratio with respect to the relative vibration amplitude. Through the comparison with the other reference methods, the reliability of present nonlinear method is verified with the maximum relative difference less than 8.0%. The nonlinear free vibration characteristics FG-CNTRC plates are parametrically investigated with respect to the gradient pattern and volume fraction of CNTs and the relative thickness and aspect ratio of plates. The gradient pattern gives rise to the significant affect and the relative thickness and the aspect ratio shows the bounded effect, while the variation to the volume fraction of CNTs is found to be insignificant.

A refined plate theory for functionally graded carbon nanotube-reinforced plates with piezoelectric actuator

With the advantage of outstanding mechanical characteristics, carbon nanotube (CNT) is nowadays considered as one of the most ideal reinforcements for composite materials. Mechanical properties of functionally graded carbon nanotube reinforced composite (FG-CNTRC) structures have been extensively investigated in the published literature. However, analytical results on transverse shear stresses of smart FG-CNTRC plates subjected to electro-mechanical loadings are scarce in literature. To fill this gap, a refined plate theory is proposed for bending analysis of laminated FG-CNTRC plates integrated with a piezoelectric fiber reinforced composite (PFRC) actuator. The kinematic assumptions of the proposed plate theory are obtained by superimposing an efficient zigzag function on the first-order shear deformation theory. Laminated FG-CNTRC plates with four distribution types of CNT through the thickness are considered. The governing equations are obtained by using the principle of minimum potential energy and the Navier's method is used to solve the boundary value problem. The present results are compared with the three-dimensional (3D) elasticity solutions and the results computed by using the chosen models. Numerical results show that the proposed zigzag function can effectively improve the accuracy of proposed plate model in predicting the displacements and stresses. In order to understand clearly the static behaviors of smart FG-CNTRC plates, the influences of electro-mechanical loading, CNT distribution through the thickness direction, CNT volume fraction and span-to-thickness ratio are thoroughly investigated.

Dynamic Analysis of Multi-Stepped Functionally Graded Carbon Nanotube Reinforced Composite Plate with General Boundary Condition

This study presents the multi-stepped functionally graded carbon nanotube reinforced composite (FG-CNTRC) plate model for the first time, and its free and forced vibration is analyzed by employing the domain decomposition method. The segmentation technique is employed to discretize the structure along the length direction. The artificial spring technique is applied to the structural boundary and piecewise interface for satisfying the boundary conditions and the combined conditions between subplates. Based on this, the boundary conditions of subdomains could be considered as a free boundary constraint, reducing the difficulty in constructing the allowable displacement function. Since all the structures of subdomains are identical, the allowable displacement functions of them can be uniformly constructed using the two-dimensional ultraspherical polynomial expansion. The potential energy function of the plate is derived from the first-order shear deformation theory (FSDT). The allowable displacement function is substituted into the potential energy function, and then the natural frequencies and mode shapes of the multi-stepped FG-CNTRC plate are decided by using the Rayleigh–Ritz method. The accuracy and reliability of the proposed method are confirmed by the results of the previous literature and finite element method (FEM). On this basis, the influences of the geometric and material parameters on free and forced vibration of the multi-stepped FG-CNTRC plate are also studied.

A comparative study of different dynamic condensation techniques applied to multi-damage identification of FGM and FG-CNTRC plates

The dynamic condensation method has been recognized as an effective alternative for structural damage identification using spatially-incomplete modal measurements. However, comparative studies of different dynamic condensation techniques applied to the subject of structural damage identification have been scarcely found, especially for composite structures. In this regard, we conduct a comparative study of six typical dynamic condensation techniques utilized for addressing damage identification problems of composite plates made of functionally graded materials (FGM) and functionally graded carbon nanotube-reinforced composite (FG-CNTRC) materials. Firstly, the six techniques consisting of Guyan’s method, Kidder’s method, Neumann series expansion-based second-order model reduction (NSEMR-II) method, improved reduced system (IRS) method, iterated IRS (IIRS) method, and iterative order reduction (IOR) method are reviewed. Then, their performance for reduced Eigen and optimization-damage identification problems are evaluated by studying two numerical examples of FGM plate and FG-CNTRC plate. For solving the optimization-damage identification problem of plate structures, the article proposes to use a hybrid global–local algorithm, Manta Ray Foraging Optimization—Sequential Quadratic Programming (MRFO-SQP), where the MRFO algorithm is utilized for global exploration and the SQP algorithm is used for the local searching process. The comparative study indicates that the IOR technique is the best dynamic condensation technique and is effective for addressing the structural damage identification problems when comparing with the other five techniques. It is also found that the damage identification approach based on the hybrid MRFO–SQP algorithm combined with the IOR technique can archive the high accuracy and low computational cost for damage localization and quantification.

The Linear and Nonlinear Bending Analyses of Functionally Graded Carbon Nanotube-Reinforced Composite Plates Based on the Novel Four-Node Quadrilateral Element

This paper presents the linear and nonlinear analyses of functionally graded carbon nanotube- reinforced composite (FG-CNTRC) plates using a four-node quadrilateral element based on the C0-type of Shi’s third-order shear deformation theory (C0-STSDT). Shi’s theory is taking the advantages and desirable properties of the third-order shear deformation theory. Besides, material properties of FG-CNTRC plates are changed from the bottom to top surface and based on the rule of mixture. Numerical results and comparison with other reference solutions suggest that the advantages of present element are accuracy and efficiency in analysis of FG-CNTRC plates. Some nonlinear numerical results of FG-CNTRC plates are also given in this paper and this contributes to providing additional data for future research work.

Vibration and nonlinear dynamic analysis of sandwich FG-CNTRC plate with porous core layer

This paper focuses on the influence of CNTs, porosity, mechanical and thermal loading on the vibration and dynamic response of the sandwich functionally graded carbon nanotube-reinforced composite (FG-CNTRC) composite plate. The plate is made by three layers in which the core layer is porous FGM materials, bottom and top surfaces are FG-CNTRC. The motion equations are given based on Hamilton’s principle, TSDT, Galerkin method and the fourth-order Runge–Kutta method. The numerical illustration is shown to examine the influence of various parameters such as porosity distribution, CNTs volume fraction, geometrical parameters, elastic foundations, temperature, mechanical loads on the dynamic behaviors of the plate. Highlights The FG-CNTRC plates with porous core layer The vibration and nonlinear dynamic analysis Elastic foundation and temperature Analytical solutions, Hamilton’s principle with the high-order shear deflection theory is used Galerkin method and the fourth-order Runge–Kutta method are applied

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

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

Combined analytical and numerical approach for auxetic FG-CNTRC plate subjected to a sudden load

In the current work, the dynamic behavior of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plate with negative Poisson’s ratio (NPR) is investigated by combining higher-order shear deformation theory and large deflection theory. First, explicit solutions are proposed to predict the effective Poisson’s ratio (EPR) of the laminates. Taking carbon nanotube-reinforced composite (CNTRC) as an example, the maximum NPR is obtained for $$\left( { \pm \theta } \right)_{{3{\text{T}}}}$$ laminate as well. Results show that the EPR ( $$v_{13}^{\text{e}}$$ , $$v_{23}^{\text{e}}$$ ) can range from a positive value of 0.311 to a negative value of 0.63. For the dynamic response problem, the asymptotic solutions with a two-step perturbation approach are derived for FG-CNTRC plates to capture the relationship between the center deflection and time. Several key factors such as functionally graded distribution, variations in the elastic foundation, and thermal stress produced by changing the temperature field are considered in the subsequent analysis. Numerical simulations are carried out to examine the corresponding dynamic behavior of FG-CNTRC plates when these factors are taken into account.

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

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

Isogeometric analysis of FG-CNTRC plates in combination with hybrid type higher-order shear deformation theory

Abstract Bezier extraction based isogeometric finite element analysis method coupled with a new hybrid type higher-order shear deformation theory (HSDT) is presented for the static bending and buckling analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates. By mapping non-uniform rational B-spline basis functions to Bezier element in the forms of Bernstein polynomial basis via the Bezier extraction operator, the isogeometric analysis (IGA) can be implemented in the conventional finite element method (FEM) framework. This study focuses on the introduction of the Bezier extraction based IGA combined with the HSDT for the static analysis through the formulation and implementation of the second-order derivatives of the Bezier element basis functions, which has hardly been attempted in the open literature. Compared to the conventional FEM based on first-order shear deformation theory, the present IGA method based on the HSDT surmounts the shear locking without exploiting the shear correction factor. The proposed isogeometric approach is thus believed to be more accurate and effective and achieves higher convergence as the polynomial order increases. Performance accuracy of the Bezier extraction based isogeometric approach is first evidenced by comparing the computed results with the reference solutions, followed by the illustrative examples to further explore the flexural and buckling behavior of FG-CNTRC plates with various reinforcement patterns.