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.     

Characterization of the variation of the material properties in a freestanding inhomogeneous thin film

This Letter presents a new technique for measuring the variation of the material properties along the thickness in a freestanding inhomogeneous thin film. The analytical results reveal a simple relation between the material properties and the set of cut-off frequencies of Lamb waves. The influence of the graded properties on the variation of cut-off frequencies in three different kinds of models, including artificial FGM model, sub-surface damage model, and nano-porous thin film model, is discussed. These results provide theoretical guidance for characterizing the material property variations of MEMS/NEMS.     

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.     

Free vibration analysis of a rotating hub–functionally graded material beam system with the dynamic stiffening effect

A comprehensive dynamic model of a rotating hub–functionally graded material (FGM) beam system is developed based on a rigid–flexible coupled dynamics theory to study its free vibration characteristics. The rigid–flexible coupled dynamic equations of the system are derived using the method of assumed modes and Lagrange's equations of the second kind. The dynamic stiffening effect of the rotating hub–FGM beam system is captured by a second-order coupling term that represents longitudinal shrinking of the beam caused by the transverse displacement. The natural frequencies and mode shapes of the system with the chordwise bending and stretching (B–S) coupling effect are calculated and compared with those with the coupling effect neglected. When the B–S coupling effect is included, interesting frequency veering and mode shift phenomena are observed. A two-mode model is introduced to accurately predict the most obvious frequency veering behavior between two adjacent modes associated with a chordwise bending and a stretching mode. The critical veering angular velocities of the FGM beam that are analytically determined from the two-mode model are in excellent agreement with those from the comprehensive dynamic model. The effects of material inhomogeneity and graded properties of FGM beams on their dynamic characteristics are investigated. The comprehensive dynamic model developed here can be used in graded material design of FGM beams for achieving specified dynamic characteristics.     

Graded Index 532HR/1064HT Filter by Glancing Angle Deposition

Glancing angle deposition is a novel method to prepare graded index coatings. By using this method and physical vapour deposition, ZrO2 is used to engineer graded index filter on BK7 glass substrate. Controlling the deposition rate and the periodic oscillation of oblique angle of deposited material, a 10-period graded index ZrO2 filter with high reflection near 532nm and high transmittance at wavelength 1064nm is fabricated. The causes of difference between the theoretical and experimental results are discussed in detail. The material properties and electron gun nonlinearity are possibly the main origins of the difference, which result in the variations in both thickness control and deposition rate of the film material.     

聚变堆第一壁连续W/Cu 梯度材料的热工性能优化

针对不同体积分布指数p的W/Cu连续功能梯度材料的偏滤器第一壁结构,采用有限元软件计算了 8MW•m⁻²稳态运行热加载以及等离子体破裂条件下1GW•m⁻²热流冲击下的力学响应。相同稳态加载条件下,W/Cu 连续功能梯度材料的最优分布指数与分层梯度材料存在较大差异,其最优等效应力比分层梯度材料要小26%,表现出更优异的性能。在热冲击响应过程中,连续梯度W/Cu材料塑性损伤随p值不同也存在较大变化,其最优p值与其稳态运行时热应力最优p值存在一定差异,从第一壁应用条件考虑,应综合选取,最佳p值在1.2附近。综合来看,连续梯度W/Cu材料具有更连续变化的热物理属性及力学性能,在聚变堆第一壁结构设计中具有更大的应用潜力。     

�۱�ѵ�һ������W/Cu �ݶȲ��ϵ��ȹ������Ż�

针对不同体积分布指数p的W/Cu连续功能梯度材料的偏滤器第一壁结构,采用有限元软件计算了8MW.m?2稳态运行热加载以及等离子体破裂条件下1GW.m?2热流冲击下的力学响应。相同稳态加载条件下,W/Cu连续功能梯度材料的最优分布指数与分层梯度材料存在较大差异,其最优等效应力比分层梯度材料要小26%,表现出更优异的性能。在热冲击响应过程中,连续梯度W/Cu材料塑性损伤随p值不同也存在较大变化,其最优p值与其稳态运行时热应力最优p值存在一定差异,从第一壁应用条件考虑,应综合选取,最佳p值在1.2附近。综合来看,连续梯度W/Cu材料具有更连续变化的热物理属性及力学性能,在聚变堆第一壁结构设计中具有更大的应用潜力。     

A theoretical study of the propagation of Rayleigh waves in a functionally graded piezoelectric material (FGPM)

An exact approach is used to investigate Rayleigh waves in a functionally graded piezoelectric material (FGPM) layer bonded to a semi infinite homogenous solid. The piezoelectric material is polarized when the six fold symmetry axis is put along the propagation direction x₁. The FGPM character imposes that the material properties change gradually with the thickness of the layer. Contrary to the analytical approach, the adopted numerical methods, including the ordinary differential equation (ODE) and the stiffness matrix method (SMM), treat separately the electrical and mechanical gradients. The influences of graded variations applied to FGPM film coefficients on the dispersion curves of Rayleigh waves are discussed. The effects of gradient coefficients on electromechanical coupling factor, displacement fields, stress distributions and electrical potential, are reported. The obtained deviations in comparison with the ungraded homogenous film are plotted with respect to the dimensionless wavenumber. Opposite effects are observed on the coupling factor when graded variations are applied separately. A particular attention has been devoted to the maximum of the coupling factor and it dependence on the stratification rate and the gradient coefficient. This work provides with a theoretical foundation for the design and practical applications of SAW devices with high performance.     

Wave field localization in a prestressed functionally graded layer

Characteristic features of wave field formation caused by a surface source of harmonic vibration in a prestressed functionally graded layer are investigated. It is assumed that the elastic moduli and the density of the material vary with depth according to arbitrary laws. The initial material of the medium is represented by a model hyperelastic material with third-order elastic moduli. The boundary-value problem for a set of Lamè equations is reduced to a set of Cauchy problems with initial conditions, which is solved by the Runge–Kutta–Merson method modified to fit the specific problem under study. Considering shear vibrations of a functionally graded layer as an example, effects of the type of its inhomogeneity, variations in its properties, and nature of its initial stressed state on the displacement distribution in depth are investigated. Special attention is paid to characteristic features of displacement localization in a layer with an interface-type inclusion near critical frequencies. A direct relation between the inhomogeneous layer structure and the type of displacement localization in depth is demonstrated. It is found that the role of initial stresses and variations in material parameters considerably increases in the vicinities of critical frequencies.     

Free Vibration Analysis of Nanocomposite Plates Reinforced by Graded Carbon Nanotubes Based on First-Order Shear Deformation Plate Theory

Based on the Mindlin's first-order shear deformation plate theory this paper focuses on the free vibration behavior of functionally graded nanocomposite plates reinforced by aligned and straight single-walled carbon nanotubes (SWCNTs). The material properties of simply supported functionally graded carbon nanotube-reinforced (FGCNTR) plates are assumed to be graded in the thickness direction. The effective material properties at a point are estimated by either the Eshelby-Mori-Tanaka approach or the extended rule of mixture. Two types of symmetric carbon nanotubes (CNTs) volume fraction profiles are presented in this paper. The equations of motion and related boundary conditions are derived using the Hamilton's principle. A semi-analytical solution composed of generalized differential quadrature (GDQ) method, as an efficient and accurate numerical method, and series solution is adopted to solve the equations of motions. The primary contribution of the present work is to provide a comparative study of the natural frequencies obtained by extended rule of mixture and Eshelby-Mori-Tanaka method. The detailed parametric studies are carried out to study the influences various types of the CNTs volume fraction profiles, geometrical parameters and CNTs volume fraction on the free vibration characteristics of FGCNTR plates. The results reveal that the prediction methods of effective material properties have an insignificant influence of the variation of the frequency parameters with the plate aspect ratio and the CNTs volume fraction.     

A new approach for free vibration of axially functionally graded beams with non-uniform cross-section

This paper studies free vibration of axially functionally graded beams with non-uniform cross-section. A novel and simple approach is presented to solve natural frequencies of free vibration of beams with variable flexural rigidity and mass density. For various end supports including simply supported, clamped, and free ends, we transform the governing equation with varying coefficients to Fredholm integral equations. Natural frequencies can be determined by requiring that the resulting Fredholm integral equation has a non-trivial solution. Our method has fast convergence and obtained numerical results have high accuracy. The effectiveness of the method is confirmed by comparing numerical results with those available for tapered beams of linearly variable width or depth and graded beams of special polynomial non-homogeneity. Moreover, fundamental frequencies of a graded beam combined of aluminum and zirconia as two constituent phases under typical end supports are evaluated for axially varying material properties. The effects of the geometrical and gradient parameters are elucidated. The present results are of benefit to optimum design of non-homogeneous tapered beam structures.     

Dynamic stability of functionally graded cantilever cylindrical shells under distributed axial follower forces

In this paper, flutter of functionally graded material (FGM) cylindrical shells under distributed axial follower forces is addressed. The first-order shear deformation theory is used to model the shell, and the material properties are assumed to be graded in the thickness direction according to a power law distribution using the properties of two base material phases. The solution is obtained by using the extended Galerkin's method, which accounts for the natural boundary conditions that are not satisfied by the assumed displacement functions. The effect of changing the concentrated (Beck's) follower force into the uniform (Leipholz's) and linear (Hauger's) distributed follower loads on the critical circumferential mode number and the minimum flutter load is investigated. As expected, the flutter load increases as the follower force changes from the so-called Beck's load into the so-called Leipholz's and Hauger's loadings. The increased flutter load was calculated for homogeneous shell with different mechanical properties, and it was found that the difference in elasticity moduli bears the most significant effect on the flutter load increase in short, thick shells. Also, for an FGM shell, the increase in the flutter load was calculated directly, and it was found that it can be derived from the simple power law when the corresponding increase for the two base phases are known.     

Analysis of Functionally Graded Piezoelectric Cylinders in a Hygrothermal Environment

This paper presents an analytical solution for the interaction of electric potentials, electric displacement, elastic deformations, and describes hygrothermal effect responses in hollow and solid cylinders, subjected to mechanical load and electric potential. Exact solutions for displacement, stresses and electric potentials in functionally graded piezoelectric material are determined using the infinitesimal theory. The material properties coefficients of the present cylinder are assumed to be graded in the radial direction by a power law distribution. Numerical examples display the significant of influence of material inhomogeneity. It is interesting to note that selecting a specific value of inhomogeneity parameter can optimize the piezoelectric hollow and solid cylinders responses, which will be of particular importance in modern engineering designs.     

Two-dimensional thermoelasticity solution for functionally graded thick beams

Two-dimensional thermoelasticity analysis of functionally graded thick beams is presented using the state space method coupled with the technique of differential quadrature. Material properties vary continuously and smoothly through the beam thickness, leading to variable coefficients in the state equation derived from the elasticity equations. Approximate laminate model is employed to translate the state equation into the one with constant coefficients in each layer. To avoid numerical instability, joint coupling matrices are introduced according to the continuity conditions at interfaces in the approximate model. The differential quadrature procedure is applied to discretizing the beam in the axial direction to make easy the treatment of arbitrary end conditions. A simply-supported beam with exponentially varying material properties is considered to validate the present method. Numerical examples are performed to investigate the influences of relative parameters.     

Free vibration of FGM Timoshenko beams with through-width delamination

Free vibration of functionally graded beams with a through-width delamination is investigated.It is assumed that the material property is varied in the thickness direction as power law functions and a single through-width delamination is located parallel to the beam axis.The beam is subdivided into three regions and four elements.Governing equations of the beam segments are derived based on the Timoshenko beam theory and the assumption of‘constrained mode’.By using the differential quadrature element method to solve the eigenvalue problem of ordinary differential equations governing the free vibration,numerical results for the natural frequencies of the beam are obtained.Natural frequencies of delaminated FGM beam with clamped ends are presented.Effects of parameters of the material gradients,the size and location of delamination on the natural frequency are examined in detail.     

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.     

Semi-analytical solution for three-dimensional transient response of functionally graded annular plate on a two parameter viscoelastic foundation

The three-dimensional transient analysis of functionally graded annular plates with arbitrary boundary conditions is carried out in this paper. The material properties of the FGM plate are assumed to vary smoothly in an exponential law along the thickness direction. The plate is assumed to rest on a two parameter viscoelastic foundation. A semi-analytical method, which integrates the state space method (SSM), Laplace transform and its inversion, as well as the one-dimensional differential quadrature method (DQM), is proposed to obtain the transient response of the plate. The state space method is used to obtain the analytical solution in the thickness direction. The differential quadrature method is employed to approximate the solution in the radial direction. The Laplace transform and the numerical inversion are used to obtain the solution in time domain. Numerical results show a good agreement between the response histories obtained by the present method and finite element method. The effects of the boundary conditions at the edges, the material graded index, the Winkler and shearing layer elastic coefficients, and the damping coefficient are studied. Numerical examples show that the peak response decreases as the material graded index, the Winkler and shearing layer elastic coefficients, and the damping coefficient increase. The results obtained in this paper can serve as benchmark data in further research.     

Non-linear free vibration of a simply-supported beam by programming techniques

An account is given of a study of free vibrations of a simply-supported beam with non-linear material properties. The material is of the Ramberg-Osgood type. Non-linear programming technique was used to find the response of the system. The variation of frequency with amplitude has been obtained for different values of material properties. The results indicate that the beam behaves like a soft spring for the type of non-linearity introduced by the material. This method can be used for all modes directly without reference to the lower modes.