Forced vibration analysis of functionally graded carbon nanotube-reinforced composite plates using a numerical strategy

In this paper, the nonlinear forced vibration behavior of composite plates reinforced by carbon nanotubes is investigated by a numerical approach. The reinforcement is considered to be functionally graded (FG) in the thickness direction according to a micromechanical model. The first-order shear deformation theory and von Kármán-type kinematic relations are employed. The governing equations and the corresponding boundary conditions are derived with the use of Hamilton's principle. The generalized differential quadrature (GDQ) method is utilized to achieve a discretized set of nonlinear governing equations. A Galerkin-based scheme is then applied to obtain a time-varying set of ordinary differential equations of Duffing-type. Subsequently, a time periodic discretization is done and the frequency response of plates is determined via the pseudo-arc length continuation method. Selected numerical results are given for the effects of different parameters on the nonlinear forced vibration characteristics of uniformly distributed carbon nanotube- and FG carbon nanotube-reinforced composite plates. It is found that with the increase of CNT volume fraction, the flexural stiffness of plate increases; and hence its natural frequency gets larger. Moreover, it is observed that the distribution type of CNTs significantly affects the vibrational behavior of plate. The results also show that when the mid-plane of plate is CNT-rich, the natural frequency takes its minimum value and the hardening-type response of plate is intensified.     

Geometrically nonlinear free vibration of shear deformable piezoelectric carbon nanotube/fiber/polymer multiscale laminated composite plates

The nonlinear free vibration of carbon nanotubes/fiber/polymer composite (CNTFPC) multi-scale plates with surface-bonded piezoelectric actuators is studied in this paper. The governing equations of the piezoelectric nanotubes/fiber/polymer multiscale laminated composite plates are derived based on first-order shear deformation plate theory (FSDT) and von Kármán geometrical nonlinearity. Halpin–Tsai equations and fiber micromechanics are used in hierarchy to predict the bulk material properties of the multiscale composite. The carbon nanotubes are assumed to be uniformly distributed and randomly oriented through the epoxy resin matrix. A perturbation scheme of multiple time scales is employed to determine the nonlinear vibration response and the nonlinear natural frequencies of the plates with immovable simply supported boundary conditions. The effects of the applied constant voltage, plate geometry, volume fraction of fibers and weight percentage of single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs) on the linear and nonlinear natural frequencies of the piezoelectric nanotubes/fiber/polymer multiscale composite plate are investigated through a detailed parametric study.     

Nonlinear free vibration behavior of functionally graded plates

In this paper, an analytical solution is provided for the nonlinear free vibration behavior of plates made of functionally graded materials. The material properties of the functionally graded plates are assumed to vary continuously through the thickness, according to a power-law distribution of the volume fraction of the constituents. The fundamental equations for thin rectangular plates of functionally graded materials are obtained using the von Karman theory for large transverse deflection, and the solution is obtained in terms of mixed Fourier series. The effect of material properties, boundary conditions and thermal loading on the dynamic behavior of the plates is determined and discussed. The results reveal that nonlinear coupling effects play a major role in dictating the fundamental frequency of functionally graded plates.     

VIBRATION CHARACTERISTICS AND TRANSIENT RESPONSE OF SHEAR-DEFORMABLE FUNCTIONALLY GRADED PLATES IN THERMAL ENVIRONMENTS

Free and forced vibration analyses for initially stressed functionally graded plates in thermal environment are presented. Material properties are assumed to be temperature dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Theoretical formulations are based on Reddy's higher order shear deformation plate theory and include the thermal effects due to uniform temperature variation. The plate is assumed to be clamped on two opposite edges with the remaining two others either free, simply supported or clamped. One-dimensional differential quadrature technique, Galerkin approach, and the modal superposition method are used to determine the transient response of the plate subjected to lateral dynamic loads. Comprehensive numerical results for silicon nitride/stainless-steel rectangular plates are presented in dimensionless tabular and graphical forms. The roles played by the constituent volume fraction index, temperature rise, shape and duration of dynamic loads, initial membrane stresses as well as the character of boundary conditions are studied. The results reveal that, when thermal effects are considered, functionally graded plates with material properties intermediate to those of isotropic ones do not necessarily have intermediate natural frequencies and dynamic responses.     

Vibrations of carbon nanotube-reinforced composites

This work deals with a study of the vibrational properties of carbon nanotube-reinforced composites by employing an equivalent continuum model based on the Eshelby-Mori-Tanaka approach. The theory allows the calculation of the effective constitutive law of the elastic isotropic medium (matrix) with dispersed elastic inhomogeneities (carbon nanotubes). The devised computational approach is shown to yield predictions in good agreement with the experimentally obtained elastic moduli of composites reinforced with uniformly aligned single-walled carbon nanotubes (CNTs). The primary contribution of the present work deals with the global elastic modal properties of nano-structured composite plates. The investigated composite plates are made of a purely isotropic elastic hosting matrix of three different types (epoxy, rubber, and concrete) with embedded single-walled CNTs. The computations are carried out via a finite element (FE) discretization of the composite plates. The effects of the CNT alignment and volume fraction are studied in depth to assess how the modal properties are influenced both globally and locally. As a major outcome, the lowest natural frequencies of CNT-reinforced rubber composites are shown to increase up to 500 percent.     

Three-dimensional analysis for transient coupled thermoelastic response of a functionally graded rectangular plate

The generalized coupled thermoelasticity based on the Lord-Shulman theory is considered to study the transient thermoelastic response of functionally graded rectangular plates. The state equations of functionally graded rectangular plate subjected to time-dependent thermal loads were established by using of state space approach, in which three displacement components, three stress components, the temperature and the heat flux were chosen as state variables. By giving simply supported boundary conditions and assuming that the material properties of the plate have an exponential law distribution along the thickness-coordinate, the equations were solved by the numerical Laplace transformations and shooting methods for transient thermal responses of a three dimensional functionally graded rectangular plate due to a thermal shock on its top surface. Effects of the volume fraction distributions of material constituents on the thermal responses, including the temperature change, the displacement and the stresses distributions were investigated.     

Nonlinear dynamic response and active vibration control for piezoelectric functionally graded plate

The nonlinear dynamic response and active vibration control of the piezoelectric functionally graded plate are analyzed in this paper. Based on higher-order shear plate theory and elastic piezoelectric theory, the nonlinear geometric and constitutive relations of the piezoelectric functionally graded plate are established, and then the nonlinear motion equations of the piezoelectric functionally graded plate are obtained through Hamilton's variational principle. The nonlinear active vibration control of the structure is carried out with adoption of the negative velocity feedback control algorithm. By applying finite difference method, the whole problem is solved by using iterative method synthetically. In numerical examples, the effects of mechanical load, electric load, the volume fraction and the geometric parameters on the dynamic response and vibration control of the piezoelectric FGM plate are investigated.     

Wave propagation of functionally graded material plates in thermal environments

The wave propagation of an infinite functionally graded plate in thermal environments is studied using the higher-order shear deformation plate theory. The thermal effects and temperature-dependent material properties are both taken into account. The temperature field considered is assumed to be a uniform distribution over the plate surface and varied in the thickness direction only. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Considering the effects of transverse shear deformation and rotary inertia, the governing equations of the wave propagation in the functionally graded plate are derived by using the Hamilton’s principle. The analytic dispersion relation of the functionally graded plate is obtained by solving an eigenvalue problem. Numerical examples show that the characteristics of wave propagation in the functionally graded plate are relates to the volume fraction index and thermal environment of the functionally graded plate. The influences of the volume fraction distributions and temperature on wave propagation of functionally graded plate are discussed in detail. The results carried out can be used in the ultrasonic inspection techniques and structural health monitoring.     

Multi-scale bending,buckling and vibration analyses of carbon fiber/carbon nanotube-reinforced polymer nanocomposite plates with various shapes

Using a finite element-based multi-scale modeling approach, the bending, buckling and free vibration of hybrid polymer matrix composites reinforced by carbon fibers and carbon nanotubes (CF/CNT-RP) are analyzed herein. Thick composite plates with rectangular, circular, annular and elliptical shapes are considered. First, the equivalent material properties of CF/CNT-RP are calculated for different volume fractions of CF and CNT. To accomplish this aim, a two-step procedure is presented through which the coupled effects of nano- and micro-scale are taken into account. In the first step, modeling of dispersion of CNTs into the polymer matrix is done with considering interphase formed by their chemical interaction with the matrix, and the equivalent properties of resulting composite material are determined accordingly. CFs are then dispersed into CNT-RP which is considered a homogenous material in this step. Both distributions of CNTs and CFs are assumed to be random. After computing the equivalent properties of CF/CNT-RP for different volume fractions of its constituents, the bending, buckling and free vibration analyses of plates with different shapes are performed. It is shown that the reinforcement of the polymer matrix with both CF and CNT significantly affects the bending, buckling and free vibration characteristics of plates.     

Thermal postbuckling and vibration analyses of functionally graded plates

Thermal postbuckling and vibration behaviors of the functionally graded (FG) plate are investigated. The material properties of the FG plate are assumed to vary continuously through the thickness of the plate and as temperature with the nonlinearity. The nonlinear finite element equations based on the first-order shear deformation plate theory are formulated for the FG plate. The von Karman nonlinear strain–displacement relationship is used to account for the large deflection of the plate. The incremental form considering the initial displacement and initial stress is adopted for the nonlinear temperature-dependent material properties of the functionally graded material. The numerical result shows the characteristics of the thermal postbuckling and vibration of the FG plate in the pre- and post-buckled regions.     

A New Higher Order Shear Deformation Model for Static Behavior of Functionally Graded Plates

In this paper, a new displacement based high-order shear deformation theory is introduced for the static response of functionally graded plate. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. The theory presented is variationally consistent, has strong similarity with classical plate theory in many aspects, does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. The mechanical properties of the plate are assumed to vary continuously in the thickness direction by a simple power-law distribution in terms of the volume fractions of the constituents. Numerical illustrations concerned flexural behavior of FG plates with Metal-Ceramic composition. Parametric studies are performed for varying ceramic volume fraction, volume fraction profiles, aspect ratios and length to thickness ratios. The validity of the present theory is investigated by comparing some of the present results with those of the classical, the first-order and the other higher-order theories. It can be concluded that the proposed theory is accurate and simple in solving the static behavior of functionally graded plates.     

Free vibration analysis of functionally graded plates using the element-free kp-Ritz method

A free vibration analysis of metal and ceramic functionally graded plates that uses the element-free kp-Ritz method is presented. The material properties of the plates are assumed to vary continuously through their thickness according to a power-law distribution of the volume fractions of the plate constituents. The first-order shear deformation plate theory is employed to account for the transverse shear strain and rotary inertia, and mesh-free kernel particle functions are used to approximate the two-dimensional displacement fields. The eigen-equation is obtained by applying the Ritz procedure to the energy functional of the system. Convergence studies are performed to examine the stability of the proposed method, and comparisons of the solutions derived with those reported in the literature are provided to verify its accuracy. Four types of functionally graded rectangular and skew plates—Al/Al₂O₃, Al/ZrO₂, Ti–6Al–4V/Aluminum oxide, and SUS304/Si₃N₄—are included in the study, and the effects of the volume fraction, boundary conditions, and length-to-thickness ratio on their frequency characteristics are discussed in detail.     

Interfacial stress analysis of functionally graded beams strengthened with a bonded hygrothermal aged composite plate

This paper reports an improved analytical solution for analysis the problem of interface stresses in functionally graded beam (FGB) strengthened with bonded hygrothermal aged composite plates. The material properties of the functionally graded beam are supposed to vary according to power law distribution of the volume fraction of the constituents through the beam thickness. The obtained results are compared with the existing solutions in the literature to verify the validity of the new analytical approach. It is found that the inhomogeneities play an important role in reducing the stress concentrations along bi-material interfaces. Finally, a parametric study was carried out to show the effects of the fiber volume fraction, the hygrothermal effect, and some design variables, e.g. thickness of adhesive layer and FRP plate on the magnitude of maximum shear and normal stress.     

Semi-Analytical Solution for Functionally Graded Solid Circular and Annular Plates Resting on Elastic Foundations Subjected to Axisymmetric Transverse Loading

In this paper, the static analysis of functionally graded (FG) circular plates resting on linear elastic foundation with various edge conditions is carried out by using a semi-analytical approach. The governing differential equations are derived based on the three dimensional theory of elasticity and assuming that the mechanical properties of the material vary exponentially along the thickness direction and Poisson's ratio remains constant. The solution is obtained by employing the state space method (SSM) to express exactly the plate behavior along the graded direction and the one dimensional differential quadrature method (DQM) to approximate the radial variations of the parameters. The effects of different parameters (e.g., material property gradient index, elastic foundation coefficients, the surfaces conditions (hard or soft surface of the plate on foundation), plate geometric parameters and edges condition) on the deformation and stress distributions of the FG circular plates are investigated.     

Three-dimensional free vibration of thick functionally graded annular plates in thermal environment

The free vibration analysis of functionally graded (FG) thick annular plates subjected to thermal environment is studied based on the 3D elasticity theory. The material properties are assumed to be temperature dependent and graded in the thickness direction. Considering the thermal environment effects and using Hamilton's principle, the equations of motion are derived. The effects of the initial thermal stresses are considered accurately by obtaining them from the 3D thermoelastic equilibrium equations. The differential quadrature method (DQM) as an efficient and accurate numerical tool is used to solve both the thermoelastic equilibrium and free vibration equations. Very fast rate of convergence of the method is demonstrated. Also, the formulation is validated by comparing the results with those obtained based on the first-order shear deformation theory and also with those available in the literature for the limit cases, i.e. annular plates without thermal effects. The effects of temperature rise, material and geometrical parameters on the natural frequencies are investigated. The new results can be used as benchmark solutions for future researches.     

Buckling of 2D-FG Cylindrical Shells under Combined External Pressure and Axial Compression

This paper presents the stability of two-dimensional functionally graded (2D-FG) cylindrical shells subjected to combined external pressure and axial compression loads, based on classical shell theory. The material properties of functionally graded cylindrical shell are graded in two directional (radial and axial) and determined by the rule of mixture. The Euler's equation is employed to derive the stability equations, which are solved by GDQ method to obtain the critical mechanical buckling loads of the 2D-FG cylindrical shells. The effects of shell geometry, the mechanical properties distribution in radial and axial direction on the critical buckling load are studied and compared with a cylindrical shell made of 1D-FGM. The numerical results reveal that the 2D-FGM has a significant effect on the critical buckling load.     

Dispersion of waves and characteristic wave surfaces in functionally graded piezoelectric plates

An inhomogeneous layer element method is presented to analyze the dispersion of waves and characteristic wave surfaces in plates of functionally graded piezoelectric material (FGPM). In this method, the FGPM plate is divided into a number of layered elements. The elemental elastic and electric properties are assumed as linear functions of the thickness to adopt the variety of the material property of FGPM. The Hamilton principle is applied to determine the governing equations. The phase velocity surface, phase slowness surface, phase wave surface, group velocity surface, group slowness surface, and group wave surface for FGPM plate are formulated using Rayleigh quotient and the orthogonality condition of the eigenvectors. These six surfaces are then used to illustrate the characteristics of waves in FGPM plates. Numerical examples are presented using the present formulations to analyze dispersions and characteristics of waves in FGPM plates.     

Vibration analysis of functionally graded annular plates with mixed boundary conditions in thermal environment

The free vibration analysis of functionally graded annular plates with mixed boundary conditions in thermal environment is carried out by the 3D elasticity theory and the Chebyshev–Ritz method. The material properties are assumed to be temperature dependent and graded in the thickness direction. The mixed boundary conditions which include upper and lower surfaces partially fixed, inner side partially fixed and outer side partially fixed are considered, respectively. The accuracy of the present approach for solving the free vibration of the plates with different boundary conditions is validated by comparing the present numerical results with the results available. The effects of the different mixed boundary conditions, the temperature rise, the material graded index and the geometrical parameters on the eigen-frequencies are studied.     

Complex permittivity and microwave absorption properties of carbon nanotubes/polymer composite: A numerical study

Based on an equivalent resistance-capacitance (RC) network, we investigate theoretically the complex permittivity and microwave absorption properties of carbon nanotubes (CNTs)/polymer composite in the frequency range of 50 MHz-3 GHz using the logarithmic mixing rule. Both the real and imaginary parts of the permittivities of CNTs and polymer are considered in detail. The simulated results show that the real and imaginary permittivities of the composite increase explicitly with increasing volume fraction of CNTs, and the latter is more sensitive. The calculated complex permittivity spectra of the composite are in good agreement with the available experimental data. In addition, a good linear relationship between microwave absorbance and frequency is found.     

Vibro-acoustic response and sound transmission loss analysis of functionally graded plates

This paper presents analytical studies on the vibro-acoustic and sound transmission loss characteristics of functionally graded material (FGM) plates using a simple first-order shear deformation theory. The material properties of the plate are assumed to vary according to power law distribution of the constituent materials in terms of volume fraction. The sound radiation due to sinusoidally varying point load, uniformly distributed load and obliquely incident sound wave is computed by solving the Rayleigh integral with a primitive numerical scheme. Displacement, velocity, acceleration, radiated sound power level, radiated sound pressure level and radiation efficiency of FGM plate for varying power law index are examined. The sound transmission loss of the FGM plate for several incidence angles and varying power law index is studied in detail. It has been found that, for the plate being considered, the sound power level increases monotonically with increase in power law index at lower frequency range (0–500 Hz) and a non-monotonic trend is appeared towards higher frequencies for both point and distributed force excitations. Increased vibration and acoustic response is observed for ceramic-rich FGM plate at higher frequency band; whereas a similar trend is seen for metal-rich FGM plate at lower frequency band. The dBA values are found to be decreasing with increase in power law index. The radiation efficiency of ceramic-rich FGM plate is noticed to be higher than that of metal and metal-rich FGM plates. The transmission loss below the first resonance frequency is high for ceramic-rich FGM plate and low for metal-rich FGM plate and further depends on the specific material property. The study has found that increased transmission loss can be achieved at higher frequencies with metal-rich FGM plates.