Isogeometric analysis for free vibration of bidirectional functionally graded plates in the fluid medium

This paper for first time proposes an isogeometric analysis (IGA) for free vibration response of bi-directional functionally graded (BDFG) rectangular plates in the fluid medium. Material properties of the BDFG plate change in both the thickness and length directions via power-law distributions and Mori-Tanaka model. The governing equation of motion of BDFG plate in the fluid-plate system is formulated basing on Hamilton's principle and the refined quasi three-dimensional (3D) plate theory with improved function f(z). The fluid velocity potential is derived from the boundary conditions of the fluid-plate system and is used to determine the added mass. The discrete system of equations is derived from the Galerkin weak form and numerically analyzed by IGA. The accuracy and reliability of the proposed solutions are verified by comparing the obtained results with those published in the literature. Moreover, the effects of the various parameters such as the interaction boundary condition, geometric parameter, submerged depth of plate, fluid density, fluid level, and the material volume control coefficients on the free vibration behavior of BDFG plate in the fluid medium are investigated in detail. Some major findings regarding the numerical results are withdrawn in conclusions.

Elasticity solutions for functionally graded plates in cylindrical bending

The plate theory of functionally graded materials suggested by Mian and Spencer is extended to analyze the cylindrical bending problem of a functionally graded rectangular plate subject to uniform load. The expansion formula for displacements is adopted. While keeping the assumption that the material parameters can vary along the thickness direction in an arbitrary fashion, this paper considers orthotropic materials rather than isotropic materials. In addition, the traction-free condition on the top surface is replaced with the condition of uniform load applied on the top surface. The plate theory for the particular case Of cylindrical bending is presented by considering an infinite extent in the y-direction. Effects of boundary conditions and material inhomogeneity on the static response of functionally graded plates are investigated through a numerical example.

Exact solution for thermal vibration of multi-directional functionally graded porous plates submerged in fluid medium

An analytical method for analyzing the thermal vibration of multi-directional functionally graded porous rectangular plates in fluid media with novel porosity patterns is developed in this study.Mechanical properties of MFG porous plates change according to the length,width,and thickness directions for various materials and the porosity distribution which can be widely applied in many fields of engineering and defence technology.Especially,new porous rules that depend on spatial coordinates and grading indexes are proposed in the present work.Applying Hamilton's principle and the refined higher-order shear deformation plate theory,the governing equation of motion of an MFG porous rectangular plate in a fluid medium(the fluid-plate system)is obtained.The fluid velocity potential is derived from the boundary conditions of the fluid-plate system and is used to compute the extra mass.The GalerkinVlasov solution is used to solve and give natural frequencies of MFG porous plates with various boundary conditions in a fluid medium.The validity and reliability of the suggested method are confirmed by comparing numerical results of the present work with those from available works in the literature.The effects of different parameters on the thermal vibration response of MFG porous rectangular plates are studied in detail.These findings demonstrate that the behavior of the structure within a liquid medium differs significantly from that within a vacuum medium.Thereby,they offer appropriate operational approaches for the structure when employed in various mediums.

THREE-DIMENSIONAL THERMOELASTIC ANALYSIS OF FUNCTIONALLY GRADED PLATE

Three-dimensional thermoelastic analysis is presented for an orthotropic functionally graded rectangular plate, which is simply supported and isothermal on its four lateral edges. With the assumption that material properties have arbitrary dependence on the thickness-coordinate a Peano-Baker series solution is obtained for the thermoelastic fields of the functionally graded plate subjected to mechanical[ and thermal loads on its upper and lower surfaces by means of state space method. The correctness of the obtained series solution is validated through numerical examples. The influence of different material properties distributions on the structural response of the plate is also studied.

Three-dimensional elastostatic solutions for transversely isotropic functionally graded material plates containing elastic inclusion

Based on the generalized England-Spencer plate theory, the equilibrium of a transversely isotropic functionally graded plate containing an elastic inclusion is studied. The general solutions of the governing equations are expressed by four analytic functions α（ζ）, β（ζ）, φ（ζ）, and ψ（ζ） when no transverse forces are acting on the surfaces of the plate. Axisymmetric problems of a functionally graded circular plate and an infinite func- tionally graded plate containing a circular hole subject to loads applied on the cylindrical boundaries of the plate are firstly investigated. On this basis, the three-dimensional （3D） elasticity solutions are then obtained for a functionally graded infinite plate containing an elastic circular inclusion. When the material is degenerated into the homogeneous one, the present elasticity solutions are exactly the same as the ones obtained based on the plane stress elasticity, thus validating the present analysis in a certain sense.

Influence of porosity distribution on nonlinear free vibration and transient responses of porous functionally graded skew plates

This article deals with the investigation of the effects of porosity distributions on nonlinear free vibration and transient analysis of porous functionally graded skew(PFGS)plates.The effective material properties of the PFGS plates are obtained from the modified power-law equations in which gradation varies through the thickness of the PFGS plate.A nonlinear finite element(FE)formulation for the overall PFGS plate is derived by adopting first-order shear deformation theory(FSDT)in conjunction with von Karman’s nonlinear strain displacement relations.The governing equations of the PFGS plate are derived using the principle of virtual work.The direct iterative method and Newmark’s integration technique are espoused to solve nonlinear mathematical relations.The influences of the porosity distributions and porosity parameter indices on the nonlinear frequency responses of the PFGS plate for different skew angles are studied in various parameters.The effects of volume fraction grading index and skew angle on the plate’s nonlinear dynamic responses for various porosity distributions are illustrated in detail.

Analysis of steady heat conduction for 3D axisymmetric functionally graded circular plate

The thermal conduction behavior of the three-dimensional axisymmetric functionally graded circular plate was studied under thermal loads on its top and bottom surfaces. Material properties were taken to be arbitrary distribution functions of the thickness. A temperature function that satisfies thermal boundary conditions at the edges and the variable separation method were used to reduce equation governing the steady state heat conduction to an ordinary differential equation （ODE） in the thickness coordinate which was solved analytically. Next, resulting variable coefficients ODE due to arbitrary distribution of material properties along thickness coordinate was also solved by the Peano-Baker series. Some numerical examples were given to demonstrate the accuracy, efficiency of the present model, mad to investigate the influence of different distributions of material properties on the temperature field. The numerical results confirm that the influence of different material distributions, gradient indices and thickness of plate to temperature field in plate can not be ignored.

Finite element analysis of functionally graded sandwich plates with porosity via a new hyperbolic shear deformation theory

This study focusses on establishing the finite element model based on a new hyperbolic sheareformation theory to investigate the static bending,free vibration,and buckling of the functionally graded sandwich plates with porosity.The novel sandwich plate consists of one homogenous ceramic core and two different functionally graded face sheets which can be widely applied in many fields of engineering and defence technology.The discrete governing equations of motion are carried out via Hamilton’s principle and finite element method.The computation program is coded in MATLAB software and used to study the mechanical behavior of the functionally graded sandwich plate with porosity.The present finite element algorithm can be employed to study the plates with arbitrary shape and boundary conditions.The obtained results are compared with available results in the literature to confirm the reliability of the present algorithm.Also,a comprehensive investigation of the effects of several parameters on the bending,free vibration,and buckling response of functionally graded sandwich plates is presented.The numerical results shows that the distribution of porosity plays significant role on the mechanical behavior of the functionally graded sandwich plates。

Nonlocal thermal buckling and postbuckling of functionally graded graphene nanoplatelet reinforced piezoelectric micro-plate

This paper analyzes the nonlocal thermal buckling and postbuckling behaviors of a multi-layered graphene nanoplatelet(GPL)reinforced piezoelectric micro-plate.The GPLs are supposed to disperse as a gradient pattern in the composite micro-plate along its thickness.The effective material properties are calculated by the Halpin-Tsai parallel model and mixture rule for the functionally graded GPL reinforced piezoelectric(FG-GRP)micro-plate.Governing equations for the nonlocal thermal buckling and postbuckling behaviors of the FG-GRP micro-plate are obtained by the first-order shear deformation theory,the von Kármán nonlinear theory,and the minimum potential energy principle.The differential quadrature(DQ)method and iterative method are introduced to numerically analyze the effects of the external electric voltage,the distribution pattern and characteristic of GPLs,and the nonlocal parameter on the critical buckling behaviors and postbuckling equilibrium path of the FG-GRP micro-plate in thermal environment.

Static and free vibration analyses of functionally graded porous variable-thickness plates using an edge-based smoothed finite element method

The main purpose of this paper is to present numerical results of static bending and free vibration of functionally graded porous(FGP) variable-thickness plates by using an edge-based smoothed finite element method(ES-FEM) associate with the mixed interpolation of tensorial components technique for the three-node triangular element(MITC3), so-called ES-MITC3. This ES-MITC3 element is performed to eliminate the shear locking problem and to enhance the accuracy of the existing MITC3 element. In the ES-MITC3 element, the stiffness matrices are obtained by using the strain smoothing technique over the smoothing domains formed by two adjacent MITC3 triangular elements sharing an edge. Materials of the plate are FGP with a power-law index(k) and maximum porosity distributions(U) in the forms of cosine functions. The influences of some geometric parameters, material properties on static bending, and natural frequency of the FGP variable-thickness plates are examined in detail.

Buckling and large deflection behaviors of radially functionally graded ring-stiffened circular plates with various boundary conditions

The buckling and large deflection behaviors of axis-symmetric radially functionally graded （RFG） ring-stiffened circular plates are investigated by the dynamic relaxation （DR） method combined with the finite difference discretization technique. The material properties of the constituent components of the RFG plate are assumed to vary continuously according to the Mori-Tanalka distribution along the radial direction. The nonlinear governing equations are obtained in the incremental form based on the firstorder shear deformation plate theory （FSDT） and the von Karman relations for large deflection. In the buckling analysis, an external in-plane load is applied to the plate in- crementally so that, in each load-step, the incremental form of the governing equations can be solved by a numerical code prepared based on the DR method. After converging the DR code in the first increment, the latter load-step is added to the previous one, and the program is repeated again. The critical buckling load is determined from the compressive load-displacement curve obtained by solving the incremental form of the governing equations. Based on the present incremental form of formulation, a bending analysis can also be conducted if the whole load is applied simultaneously. Finally, a detailed parametric study is carried out to investigate the influences of various boundary conditions, grading indices, thickness-to-radius ratios, stiffener＇s positions and depths on the critical buckling load, and displacements and stresses resulted from the bending analysis. It is observed that the effect of the stiffener on the results is much greater in the functionally graded plate with higher material grading indices. The results also reveal that, by increasing the depth of the stiffer, the values of ascending the critical buckling load are approximately identical for both simply supported and clamped boundary conditions.

Nonlinear oscillations of sigmoid functionally graded material plates moving in longitudinal direction

Geometrically nonlinear oscillations are investigated on sigmoid functionally graded material （S-FGM） plates with a longitudinal speed. The material properties of the plates obey a sigmoid distribution rule along the thickness direction. Based on the D＇Alembert＇s principle, a nonlinear equation of motion is derived for the moving S-FGM plates, where the von K^rm^n nonlinear plate theory is adopted. Utilizing the Galerkin method, the equation of motion is discretized and solved via the method of harmonic bal- ance. The approximate analytical solutions are validated through the adaptive step-size fourth-order Runge-Kutta method. Besides, the stability of the steady-state solutions is examined. The results reveal that the mode interaction behavior can happen between the first two modes of the moving S-FGM plates, leading to a complex nonlinear frequency response. It is further found that the power-law index, the longitudinal speed, the exci- tation amplitude, and the in-plane pretension force can significantly affect the nonlinear frequency-response characteristics of longitudinally traveling S-FGM plates.

The effect of initial geometric imperfection on the nonlinear resonance of functionally graded carbon nanotube-reinforced composite rectangular plates

The purpose of the present study is to examine the impact of initial geometric imperfection on the nonlinear dynamical characteristics of functionally graded carbon nanotube-reinforced composite(FG-CNTRC) rectangular plates under a harmonic excitation transverse load. The considered plate is assumed to be made of matrix and single-walled carbon nanotubes(SWCNTs). The rule of mixture is employed to calculate the effective material properties of the plate. Within the framework of the parabolic shear deformation plate theory with taking the influence of transverse shear deformation and rotary inertia into account, Hamilton’s principle is utilized to derive the geometrically nonlinear mathematical formulation including the governing equations and corresponding boundary conditions of initially imperfect FG-CNTRC plates. Afterwards, with the aid of an efficient multistep numerical solution methodology, the frequency-amplitude and forcing-amplitude curves of initially imperfect FG-CNTRC rectangular plates with various edge conditions are provided, demonstrating the influence of initial imperfection,geometrical parameters, and edge conditions. It is displayed that an increase in the initial geometric imperfection intensifies the softening-type behavior of system, while no softening behavior can be found in the frequency-amplitude curve of a perfect plate.

Vibration and sound transmission loss characteristics of porous foam functionally graded sandwich panels in thermal environment

This study investigates the vibration and acoustic properties of porous foam functionally graded(FG)plates under the influence of the temperature field.The dynamics equations of the system are established based on Hamilton's principle by using the higher-order shear deformation theory under the linear displacement-strain assumption.The displacement shape function is assumed according to the four-sided simply-supported(SSSS)boundary condition,and the characteristic equations of the system are derived by combining the motion control equations.The theoretical model of vibro-acoustic coupling is established by using the acoustic theory and fluid-structure coupling solution method under the simple harmonic acoustic wave.The system's natural frequency and sound transmission loss(STL)are obtained through programming calculations and compared with the literature and COMSOL simulation to verify the validity and reliability of the theoretical model.The effects of various factors,such as temperature,porosity coefficients,gradient index,core thickness,width-to-thickness ratio on the vibration,and STL characteristics of the system,are discussed.The results provide a theoretical basis for the application of porous foam FG plates in engineering to optimize vibration and sound transmission properties.

Thermal buckling and postbuckling of FGM circular plates with in-plane elastic restraints

Based on von Karman＇s plate theory, the axisymmetric thermal buckling and post-buckling of the functionally graded material （FGM） circular plates with in- plane elastic restraints under transversely non-uniform temperature rise are studied. The properties of the FGM media are varied through the thickness based on a simple power law. The governing equations are numerically solved by a shooting method. The results of the critical buckling temperature, post-buckling equilibrium paths, and configurations for the in-plane elastically restrained plates are presented. The effects of the in-plane elastic restraints, material property gradient, and temperature variation on the responses of thermal buckling and post-buckling are examined in detail.

3D thermoelasticity solutions for functionally graded thick plates

Thermal-mechanical behavior of functionally graded thick plates, with one pair of opposite edges simply supported, is investigated based on 3D thermoelasticity. As for the arbitrary boundary conditions, a semi-analytical solution is presented via a hybrid approach combining the state space method and the technique of differential quadrature. The temperature field in the plate is determined according to the steady-state 3D thermal conduction. The Mori-Tanaka method with a power-law volume fraction profile is used to predict the effective material properties including the bulk and shear moduli, while the effective coefficient of thermal expansion and the thermal conductivity are estimated using other micromechanics-based models. To facilitate the im-plementation of state space analysis through the thickness direction, the approximate laminate model is employed to reduce the inhomogeneous plate into a homogeneous laminate that delivers a state equation with constant coefficients. The present solutions are validated by comparisons with the exact ones for both thin and thick plates. Effects of gradient indices, volume fraction of ceramics, and boundary conditions on the thermomechanical behavior of functionally graded plates are discussed.

A deep feed-forward neural network for damage detection in functionally graded carbon nanotube-reinforced composite plates using modal kinetic energy

This paper proposes a new Deep Feed-forward Neural Network(DFNN)approach for damage detection in functionally graded carbon nanotube-reinforced composite(FG-CNTRC)plates.In the proposed approach,the DFNN model is developed based on a data set containing 20000 samples of damage scenarios,obtained via finite element(FE)simulation,of the FG-CNTRC plates.The elemental modal kinetic energy(MKE)values,calculated from natural frequencies and translational nodal displacements of the structures,are utilized as input of the DFNN model while the damage locations and corresponding severities are considered as output.The state-of-the art Exponential Linear Units(ELU)activation function and the Adamax algorithm are employed to train the DFNN model.Additionally,in order to enhance the performance of the DFNN model,the mini-batch and early-stopping techniques are applied to the training process.A trial-and-error procedure is implemented to determine suitable parameters of the network such as the number of hidden layers and the number of neurons in each layer.The accuracy and capability of the proposed DFNN model are illustrated through two distinct configurations of the CNT-fibers constituting the FG-CNTRC plates including uniform distribution(UD)and functionally graded-V distribution(FG-VD).Furthermore,the performance and stability of the DFNN model with the consideration of noise effects on the input data are also investigated.Obtained results indicate that the proposed DFNN model is able to give sufficiently accurate damage detection outcomes for the FG-CNTRC plates for both cases of noise-free and noise-influenced data.

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.

A unified solution method for free vibration analysis of functionally graded rotating type plates with elastic boundary condition

In this paper,a unified solution method for analyzing the free vibration characteristics of functionally graded rotating type plate(FGRTP)of which the distribution of material is defined by four parameters is presented.It is assumed that the considered functionally graded materials(FGM)are distributed in the thickness direction according to four parameters.The displacement fields of any point on the FGRTP are determined by the first order shear deformation theory(FSDT),and all displacement functions are extended by ultraspherical polynomial.By applying the Ritz method to the energy function of the whole system,the constitutive equation of FGRTP is obtained and the natural frequencies are obtained by solving the eigenvalue problem.The boundary conditions are generalized to arbitrary boundary conditions by artificial elastic technique.The accuracy of the proposed method is verified by comparing with the previous literatures.The effects of different parameters on the free vibration characteristics of FGRTP are studied through some numerical examples.

A strategy resisting wrinkling of sandwich structures reinforced using functionally-graded carbon nanotubes

Sandwich structures have been widely applied in the wing and the horizontal tail of the aircraft,so face sheets of such structure might occur wrinkling deformation in the process of service,which will largely decrease capability of sustaining loads.As a result,this paper aims at proposing a reasonable strategy resisting wrinkling deformation of sandwich structures.To this end,an enhanced higher-order model has been proposed for wrinkling analysis of sandwich structures.Buckling behaviors of a five-layer sandwich plate are firstly analyzed,which is utilized to assess performance of the proposed model.Subsequently,wrinkling behaviors of four sandwich plates are further investigated by utilizing present model,which have been evaluated by using quasi threedimensional(3D)elasticity solutions,3D Finite Element Method(3D-FEM)results and experimental datum.Finally,the present model is utilized to study the buckling and the wrinkling behaviors of sandwich plates reinforced by Carbon Nano Tubes(CNTs).In addition,influence of distribution profile of CNTs on wrinkling behaviors has been analyzed,and a typical distribution profile of CNTs has been chosen to resist wrinkling deformation.Without increase of additional weight,the present strategy can effectively resist wrinkling deformation of sandwich plates,which is rarely reported in published literature.