Static/dynamic Analysis of Functionally Graded and Layered Magneto-electro-elastic Plate/pipe under Hamiltonian System

The 3-dimensional couple equations of magneto-electro-elastic structures are derived under Hamiltonian system based on the Hamilton principle. The problem of single sort of variables is converted into the problem of double sorts of variables, and the Hamilton canonical equations are established. The 3-dimensional problem of magneto-electro-elastic structure which is investigated in Euclidean space commonly is converted into symplectic system. At the same time the Lagrange system is converted into Hamiltonian system. As an example, the dynamic characteristics of the simply supported functionally graded magneto-electro-elastic material （FGMM） plate and pipe are investigated. Finally, the problem is solved by symplectic algorithm. The results show that the physical quantities of displacement, electric potential and magnetic potential etc. change continuously at the interfaces between layers under the transverse pressure while some other physical quantities such as the stress, electric and magnetic displacement are not continuous. The dynamic stiffness is increased by the piezoelectric effect while decreased by the piezomagnetic effect.

Cracked elastic substrate strip with functionally graded coating under thermal-mechanical loading

This paper investigates the functionally graded coating bonded to an elastic strip with a crack under thermal- mechanical loading. Considering some new boundary conditions, it is assumed that the temperature drop across the crack surface is the result of the thermal conductivity index which controls heat conduction through the crack region. By the Fourier transforms, the thermal-elastic mixed boundary value problems are reduced to a system of singular integral equations which can be approximately solved by applying the Chebyshev polynomials. The numerical computation methods for the temperature, the displacement field and the thermal stress intensity factors （TSIFs） are presented. The normal temperature distributions （NTD） with different parameters along the crack surface are analyzed by numerical examples. The influence of the crack position and the thermal-elastic non- homogeneous parameters on the TSIFs of modes I and 11 at the crack tip is presented. Results show that the variation of the thickness of the graded coating has a significant effect on the temperature jump across the crack surfaces when keeping the thickness of the substrate constant, and the thickness of functionally graded material （FGM） coating has a significant effect on the crack in the substrate. The results can be expected to be used for the purpose of gaining better understanding of the thermal-mechanical behavior of graded coatings.

Elastoplastic response of functionally graded cemented carbides due to thermal loading

Finite dement formulations are used to simulate the evolution of the elastoplastic response of functionally graded cemented carbides （FGCC） due to thermal loading. The geometry of specimens is an axisymmetric solid cylinder with a two-dimensional gradient. The elastoplastic constitutive relationship is developed by constraint factors. Numerical results show that compressive stresses occur in the surface zone and tensile stresses in the cobalt rich zone when the temperature drops from the initial stress-free temperature of 800 to 0℃. The maximum value of the surface compressive stress is 254 MPa and the maximum value of the tensile stress is 252 MPa in the cobalt rich zones. When the cobalt concentration difference in the specimens is equal to or greater than 0.3, there is pronounced plastic flow in cobalt rich zone. When the temperature heats up from 0 to 800 ℃, the total plastic strain reaches 0.001 4. Plastic flow has a significant effect on the reduction of thermal stress concentration.

热敏灸治疗颈型颈椎病的Meta分析及GRADE证据等级评价

目的系统评价热敏灸及其综合疗法用于颈型颈椎病的临床效果,为热敏灸治疗颈型颈椎病提供更好的证据来源。方法收集国内外各大数据库热敏灸及其综合疗法用于颈型颈椎病的随机对照试验(RCT),通过Revman 5.3软件进行Meta分析,按照GRADE标准对结局指标进行证据质量评价。结果纳入9篇文献共855例患者,Meta分析结果提示:热敏灸及其综合疗法组在提高总有效率(RR=1.19,P<0.01)和治愈率(RR=1.73,P<0.01)、降低视觉模拟量表(VAS)评分(SMD=-0.73,P<0.01)、简化McGill疼痛问卷(SF-MPQ)评分(SMD=-1.28,P<0.05)以及NPQ颈痛量表评分(SMD=-0.81,P<0.05)方面都优于对照组;GRADE证据等级评价提示:前三者为低质量证据,后二者为极低质量证据。结论热敏灸及其综合疗法用于颈型颈椎病在提高临床效果、镇痛以及改善患者生活质量方面更具优势。

FREE VIBRATION OF FUNCTIONALLY GRADED, MAGNETO-ELECTRO-ELASTIC, AND MULTILAYERED PLATES

The state-space method is employed to evaluate the modal parameters of functionally graded, magneto-electro-elastic, and multilayered plates. Based on the assumption that the properties of the functionally graded material are exponential, the state equation of structural vibration which takes the displacement and stress of the structure as state variables is derived. The natural frequencies and modal shapes are calculated based on the general solutions of the state equation and boundary conditions given in this paper. The influence of the functionally graded exponential factor on the elastic displacement, electric, and magnetic fields of the structure are discussed by assuming a sandwich plate model with different stacking sequences.

Elastic and viscoelastic solutions to rotating functionally graded hollow and solid cylinders

Analytical solutions to rotating functionally graded hollow and solid long cylinders are developed. Young＇s modulus and material density of the cylinder are assumed to vary exponentially in the radial direction, and Poisson＇s ratio is assumed to be constant. A unified governing equation is derived from the equilibrium equations, compatibility equation, deformation theory of elasticity and the stress-strain relationship. The governing second-order differential equation is solved in terms of a hypergeometric function for the elastic deformation of rotating functionally graded cylinders. Dependence of stresses in the cylinder on the inhomogeneous parameters, geometry and boundary conditions is examined and discussed. The proposed solution is validated by comparing the results for rotating functionally graded hollow and solid cylinders with the results for rotating homogeneous isotropic cylinders. In addition, a viscoelastic solution to the rotating viscoelastic cylinder is presented, and dependence of stresses in hollow and solid cylinders on the time parameter is examined.

Electromagnetoelastic behaviors of functionally graded piezoelectric solid cylinder and sphere

Analytical studies on electromagnetoelastic behaviors are presented for the functionally graded piezoelectric material （FGPM） solid cylinder and sphere placed in a uniform magnetic field and subjected to the external pressure and electric loading. When the mechanical, electric and magnetic properties of the material obey an identical power law in the radial direction, the exact displacements, stresses, electric potentials and perturbations of magnetic field vector in the FGPM solid cylinder and sphere are obtained by using the infinitesimal theory of electromagnetoelasticity. Numerical examples also show the significant influence of material inhomogeneity. It is interesting to note that selecting a specific value of inhomogeneity parameter β can optimize the electromagnetoelastic responses, which will be of particular importance in modern engineering designs.

Three-dimensional free vibration analysis of multi-directional functionally graded piezoelectric annular plates on elastic foundations via state space based differential quadrature method

The three-dimensional free vibration analysis of a multi-directional func- tionally graded piezoelectric （FGP） annular plate resting on two parameter （Pasternak） elastic foundations is investigated under different boundary conditions. The material properties are assumed to vary continuously along the radial and thickness directions and have exponent-law distribution. A semi-analytical approach named the state space based differential quadrature method （SSDQM） is used to provide an analytical solution along the thickness using the state space method （SSM） and an approximate solution along the radial direction using the one-dimensional differential quadrature method （DQM）. The influence of the Winkler and shear stiffness of the foundation~ the material property graded variations, and the circumferential wave number on the nomdimensional natural frequency of multi-directional FGP annular plates is studied.

Generalized thermoelastic interaction in functional graded material with fractional order three-phase lag heat transfer

The present work is concerned with the solution of a problem on thermoelastic interactions in a functional graded material due to thermal shock in the context of the fractional order three-phase lag model. The governing equations of fractional order generalized thermoelasticity with three-phase lag model for functionally graded materials(FGM)(i.e., material with spatially varying material properties) are established. The analytical solution in the transform domain is obtained by using the eigenvalue approach.The inversion of Laplace transform is done numerically. The graphical results indicate that the fractional parameter has significant effects on all the physical quantities. Thus, we can consider the theory of fractional order generalized thermoelasticity an improvement on studying elastic materials.

Vibration analysis of axisymmetric functionally graded viscothermoelastic spheres

The present study focuses on the analysis of free vibrations of axisymmetric functionally graded hollow spheres. The material is assumed to be graded in radial di- rection with a simple power law. Matrix Frrbenious method of extended power series is employed to derive the analytical solutions for displacement, temperature, and stresses. The dispersion relations for the existence of various types of pos- sible modes of vibrations in the considered hollow sphere are derived in a compact form. In order to explore the character- istics of vibrations, the secular equations are further solved by using fixed point iteration numerical technique with the help of MATLAB software. The numerical results have been presented graphically for polymethyl methecrylate materials in respect of natural frequencies, frequency shift, inverse quality factor, displacement, temperature change, and radial stress.

Three-dimensional magneto-thermo-elastic analysis of functionally graded cylindrical shell

The present paper presents the three-dimensional magneto-thermo-elastic analysis of the functionally graded cylindrical shell immersed in applied thermal and magnetic fields under non-uniform internal pressure. The inhomogeneity of the shell is assumed to vary along the radial direction according to a power law function, whereas Poisson＇s ratio is supposed to be constant through the thickness. The existing equations in terms of the displacement components, temperature, and magnetic parameters are derived, and then the effective differential quadrature method （DQM） is used to acquire the analytical solution. Based on the DQM, the governing heat differential equations and edge boundary conditions are transformed into algebraic equations, and discretized in the series form. The effects of the gradient index and rapid temperature on the displacement, stress components, temperature, and induced magnetic field are graphically illustrated. The fast convergence of the method is demonstrated and compared with the results obtained by the finite element method （FEM）.

Generalized thermoelastic functionally graded spherically isotropic solid containing a spherical cavity under thermal shock

This paper is concerned with the determination of thermoelastic displacement, stress and temperature in a functionally graded spherically isotropic infinite elastic medium having a spherical cavity, in the context of the linear theory of generalized thermoelasticity with two relaxation time parameters （Green and Lindsay theory）. The surface of cavity is stress-free and is subjected to a time-dependent thermal shock. The basic equations have been written in the form of a vector-matrix differential equation in the Laplace transform domain, which is then solved by an eigenvalue approach. Numerical inversion of the transforms is carried out using the Bellman method. Displacement, stress and temperature are computed and presented graphically. It is found that variation in the thermo-physical properties of a material strongly influences the response to loading. A comparative study with a corresponding homogeneous material is also made.

Buckling analysis of functionally graded material(FGM) sandwich truncated conical shells reinforced by FGM stiffeners filled inside by elastic foundations

An analytical solution for buckling of an eccentrically stiffened sandwich truncated conical shell is investigated. The shell consists of two functionally graded material （FGM） coating layers and a core layer which are metal or ceramic subjected to an axial compressive load and an external uniform pressure. Shells are reinforced by stringers and rings, in which the material properties of shells and stiffeners are graded in the thickness direction following a general sigmoid law distribution. Two models of coated shell-stiffener arrangements are investigated. The change of the spacing between stringers in the meridional direction is taken into account. A couple set of three-variable- coefficient partial differential equations in terms of displacement components are solved by the Galerkin method. A closed-form expression for determining the buckling load is obtained. The numerical examples are presented and compared with previous works.

Buckling analysis of functionally graded plates partially resting on elastic foundation using the differential quadrature element method

We extend the differential quadrature element method (DQEM) to the buckling analysis of uniformly in-plane loaded functionally graded (FG) plates fully or partially resting on the Pasternak model of elastic support. Material properties of the FG plate are graded in the thickness direction and assumed to obey a power law distribution of the volume fraction of the constituents. To set up the global eigenvalue equation, the plate is divided into sub-domains or elements and the generalized differential quadrature procedure is applied to discretize the governing, boundary and compatibility equations. By assembling discrete equations at all nodal points, the weighting coefficient and force matrices are derived. To validate the accuracy of this method, the results are compared with those of the exact solution and the finite element method. At the end, the effects of different variables and local elastic support arrangements on the buckling load factor are investigated.

THERMAL AND THERMO-ELASTIC-PLASTIC RESPONSE OF CERAMIC-METAL FUNCTIONALLY GRADED MATERIALS—THERMAL SHOCK PROBLEM

The thermal and thermo-elastic-plastic response of newly developed ceramic-metal functionally graded materials under a thermal shock load is studied. The materials are heated at the ceramic surface with a sudden high-intensity heat flux input, and cooled at the metal surface with a flowing liquid nitrogen. Emphasis is placed on two aspects: (1) the influence of the graded composition of the materials on the temperature and stress response; and (2) the optimum design of the graded composition from a unified viewpoint of the heat insulation property and stress relaxation property. Moreover, a comparison between the thermoelastic stress and the thermo-elastic-plastic stress is also made to indicate the plasticity effect.

Analytical solution of axisymmetric contact problem about indentation of a circular indenter into a soft functionally graded elastic layer

The paper addresses a contact problem of the theory of elasticity,i.e.,the penetration of a circular indenter with a flat base into a soft functionally graded elastic layer.The elastic properties of a functionally graded layer arbitrarily vary with depth,and the foundation is assumed to be elastic,yet much harder than a layer.Approximated analytical solution is constructed,and it is shown that the solutions are asymptotically exact both for large and small values of characteristic dimensionless geometrical parameter of the problem.Numerical examples are analyzed for the cases of monotonic and nonmonotonic variations of elastic properties.Numerical results for the case of homogeneous layer are compared with the results for nondeformable foundation.

Thermoelastic vibration analysis of micro-scale functionally graded material fluid-conveying pipes in elastic medium

Micro-scale functionally graded material(FGM)pipes conveying fluid have many significant applications in engineering fields.In this work,the thermoelastic vibration of FGM fluid-conveying tubes in elastic medium is studied.Based on modified couple stress theory and Hamilton’s principle,the governing equation and boundary conditions are obtained.The differential quadrature method(DQM)is applied to investigating the thermoelastic vibration of the FGM pipes.The effect of temperature variation,scale effect of the microtubule,micro-fluid effect,material properties,elastic coefficient of elastic medium and outer radius on thermoelastic vibration of the FGM pipes conveying fluid are studied.The results show that in the condition of considering the scale effect and micro-fluid of the microtubule,the critical dimensionless velocity of the system is higher than that of the system which calculated using classical macroscopic model.The results also show that the variations of temperature,material properties,elastic coefficient and outer radius have significant influences on the first-order dimensionless natural frequency.

Thermoelastic Analysis of Non-uniform Pressurized Functionally Graded Cylinder with Variable Thickness Using First Order Shear Deformation Theory(FSDT) and Perturbation Method

Recently application of functionally graded materials（FGMs） have attracted a great deal of interest. These materials are composed of various materials with different micro-structures which can vary spatially in FGMs. Such composites with varying thickness and non-uniform pressure can be used in the aerospace engineering. Therefore, analysis of such composite is of high importance in engineering problems. Thermoelastic analysis of functionally graded cylinder with variable thickness under non-uniform pressure is considered. First order shear deformation theory and total potential energy approach is applied to obtain the governing equations of non-homogeneous cylinder. Considering the inner and outer solutions, perturbation series are applied to solve the governing equations. Outer solution for out of boundaries and more sensitive variable in inner solution at the boundaries are considered. Combining of inner and outer solution for near and far points from boundaries leads to high accurate displacement field distribution. The main aim of this paper is to show the capability of matched asymptotic solution for different non-homogeneous cylinders with different shapes and different non-uniform pressures. The results can be used to design the optimum thickness of the cylinder and also some properties such as high temperature residence by applying non-homogeneous material.

Revisiting the elastic solution for an inner-pressured functionally graded thick-walled tube within a uniform magnetic field

In this paper, the mechanical responses of a thick-walled functionally graded hollow cylinder subject to a uniform magnetic field and inner-pressurized loads are studied. Rather than directly assume the material constants as some specific function forms displayed in pre-studies, we firstly give the volume fractions of different constituents of the functionally graded material(FGM) cylinder and then determine the expressions of the material constants. With the use of the Voigt method, the corresponding analytical solutions of displacements in the radial direction, the strain and stress components, and the perturbation magnetic field vector are derived. In the numerical part, the effects of the volume fraction on the displacement, strain and stress components, and the magnetic perturbation field vector are investigated. Moreover, by some appropriate choices of the material constants, we find that the obtained results in this paper can reduce to some special cases given in the previous studies.

Effect of partial elastic foundation on free vibration of fluid-filled functionally graded cylindrical shells

The free vibration characteristics of fluid-filled functionally graded cylindrical shells buried partially in elas- tic foundations are investigated by an analytical method. The elastic foundation of partial axial and angular dimen- sions is represented by the Pasternak model. The motion of the shells is represented by the first-order shear defor- mation theory to account for rotary inertia and transverse shear strains. The functionally graded cylindrical shells are composed of stainless steel and silicon nitride. Material prop- erties vary continuously through the thickness according to a power law distribution in terms of the volume fraction of the constituents. The governing equation is obtained using the Rayleigh-Ritz method and a variation approach. The fluid is described by the classical potential flow theory. Numerical examples are presented and compared with existing available results to validate the present method.