Transfer matrix method for free and forced vibrations of multi-level functionally graded material stepped beams with different boundary conditions

Functionally graded materials(FGMs)are a novel class of composite materials that have attracted significant attention in the field of engineering due to their unique mechanical properties.This study aims to explore the dynamic behaviors of an FGM stepped beam with different boundary conditions based on an efficient solving method.Under the assumptions of the Euler-Bernoulli beam theory,the governing differential equations of an individual FGM beam are derived with Hamilton’s principle and decoupled via the separation-of-variable approach.Then,the free and forced vibrations of the FGM stepped beam are solved with the transfer matrix method(TMM).Two models,i.e.,a three-level FGM stepped beam and a five-level FGM stepped beam,are considered,and their natural frequencies and mode shapes are presented.To demonstrate the validity of the method in this paper,the simulation results by ABAQUS are also given.On this basis,the detailed parametric analyses on the frequencies and dynamic responses of the three-level FGM stepped beam are carried out.The results show the accuracy and efficiency of the TMM.

Complete solutions for elastic fields induced by point load vector in functionally graded material model with transverse isotropy

The paper develops and examines the complete solutions for the elastic field induced by the point load vector in a general functionally graded material(FGM)model with transverse isotropy.The FGMs are approximated with n-layered materials.Each of the n-layered materials is homogeneous and transversely isotropic.The complete solutions of the displacement and stress fields are explicitly expressed in the forms of fifteen classical Hankel transform integrals with ten kernel functions.The ten kernel functions are explicitly expressed in the forms of backward transfer matrices and have clear mathematical properties.The singular terms of the complete solutions are analytically isolated and expressed in exact closed forms in terms of elementary harmonic functions.Numerical results show that the computation of the complete solutions can be achieved with high accuracy and efficiency.

DESIGN AND PROPERTIES OF A FUNCTIONALLY GRADIENT CERAMIC TOOL MATERIAL

Based on the analyses of the severity of cutting process as well as the failure mechanisms of ceramic tools, a model for designing functionally gradient ceramic tool materials with symmetrical distribution is presented, by which a Al 2O 3/(W,Ti)C ceramic tool material FG 2 was developed. Multi objective optimization method was employed in designing the compositional distribution of this ceramic tool material. The results of both continuous and intermittent cutting tests are indicative of the much better cutting behavior of the functionally gradient ceramic tool FG 2 than that of the common ceramic tool SG 4.

NVESTIGATION OF PRODUCTOPN OF FUNCTIONALLY GRADIENT MATERIAL BY ELECTROLESS PLATING

Thispaper proposed a new methodof producing Ceramic/ Metalfunctionally gradient mate rialby electroless platingtechnique. The experimentof producing SiC/ Ni PFGM wascar ried out with self made electroless plating facilities. The results show that the thickness of FGMcoating andthegradientdistribution ofcompositioncanbecontrolled byoptimizingelec trolessplating technology and changing the parameters such as plating time, the additionspeed and concentration of SiCparticles. Analysisdemonstratesthatthereisastrongrelation ship amongthe SiCcontent,the microstructureandthe mechanicalproperty ofthe FGM.

Crack tip field in functionally gradient material with exponential variation of elastic constants in two directions

This paper presents an exact solution of the crack tip field in functionally gradient material with exponential variation of elastic constants. The dimensionless Poisson＇s ratios v0 of the engineering materials （iron, glass …… ） are far less than one; therefore, neglecting them, one can simplify the basic equation and the exact solution is easy to obtain. Although the exact solution for the case v0 ≠ 0 is also obtained, it is very complicated and the main result is the same with the case v0 = 0 （it will be dealt with in Appendix VII）. It has been found that the exponential term exp（ax ＋ by） in the constitutive equations becomes exp（ ax /2 ＋ by/2- kr /2 ） in the exact solution.

Synthesis,characterization and biological activity in vitro of FeCrAl(f)/HA asymmetrical biological functionally gradient materials

FeCrAl(f)/HA biological functionally gradient materials(FGMs) were successfully fabricated by the hot pressing technique.Scanning electron microscope(SEM),energy dispersive spectrometer(EDS) and bending strength test machine were utilized to characterize the microstructure,component,mechanical properties and the formation of the Ca-deficient apatite on the surface of these materials.The results indicate that an asymmetrical FeCrAl(f)/HA FGM,consolidating powders prepared by mixing HA with 3%–15%(volume fraction) is successfully prepared.Both of the matrix and FeCrAl fiber are integrated very tightly and bite into each other very deeply.And counter diffusion takes place to some extent in two phase interfaces.The elemental compositions of the FeCrAl(f)/HA FGM change progressively.Ca and P contents increase gradually with immersion time increasing,and thereafter approach equilibrium.The bone-like apatite layer forms on the materials surface,which possesses benign bioactivity,and the favorable biocompatibility can provide potential firm fixation between FeCrAl(f)/HA asymmetrical FGM implants and human bone.

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.

ANTI-PLANE SHEAR CRACK IN A FUNCTIONALLY GRADIENT PIEZOELECTRIC MATERIAL

The main objective of this paper is to study the singular natureof the crack-tip stress and electric displacement field in afunctionally gradient piezoelectric medium having materialcoefficients with a discontinuous derivative. The problem isconsidered for the simplest possible loading and geometry, namely,the anti-plane shear stress and electric displacement in -plane oftwo bonded half spaces in which the crack is parallel to theinterface.

Genetic Algorithm for the Thermal Stresses Optimum Design ofFunctionally Gradient Material Plate

Based on the thermal stress distribution for functionally gradient material (FGM) plates, a Genetic Algorithm (GA) method for the thermal stresses optimum design of FGM plate with computer technologies is given. The minimum thermal stresses combination distribution for FGM is obtained.

Nonlinear stability of functionally graded material(FGM)sandwich cylindrical shells reinforced by FGM stiffeners in thermal environment

In this paper, Donnell＇s shell theory and smeared stiffeners technique are improved to analyze the postbuckling and buckling behaviors of circular cylindrical shells of stiffened thin functionally graded material （FGM） sandwich under an axial loading on elastic foundations, and the shells are considered in a thermal environment. The shells are stiffened by FGM rings and stringers. A general sigmoid law and a general power law are proposed. Thermal elements of the shells and reinforcement stiffeners are considered. Explicit expressions to find critical loads and postbuckling load-deflection curves are obtained by applying the Galerkin method and choosing the three-term approximate solution of deflection. Numerical results show various effects of temperature, elastic foundation, stiffeners, material and geometrical properties, and the ratio between face sheet thickness and total thickness on the nonlinear behavior of shells.

Microstructure and Mechanical Properties of ZrO_2-Ni Functionally Gradient Material

The fabrication. microstructure and mechanical properties of ZrO2-Ni functionally gradient materials (FGM ) have been studied. FGM as well as non-FG M of ZrO2-Ni system was developed by powder metallurgical process. X-ray diffractometer (XRD ). electron probe microanalyzer (EPMA), scanning electron microscope (SEM ) and optical microscope were employed to investigate the crystalline phases. chemical composition and microstructure Experimental results demonstrate that the composition and microstructure of ZrO2-Ni FGM have the expected gradient distribution. There are no distinct interfaces in the FGM due to the gradient change of components. that is, the constituents are continuous in microstructure everywhere. Moreover, Vickers hardness and flexural strength were measured for the common composites as a function of composition. It is made clear that the mechanical properties of the FGM vary corresponding to the constitutional changes as well

Ti/Al_2O_3 Functionally Gradient Material Prepared by the Explosive Compaction/SHS Process

Ti1Al2O3 Functionally Gradient Material (FGM) was prepared by an explosive compaction/SHS process. Ten sheets of the compounding powder were laminated and pressed to get a green body of FGM. It was then compacted explosively By burying the explosive compaction body into a stoichiometric Al/TiO2 mixture and igniting the combustion of the stoichiometric Al/TiO2 mixture, the SHS reaction of the explosive compaction body was initiated by the heat released from the combustion of the stoichiometric Al/TiO2 mixture. In this way, Ti/Al2O3 FGM was synthesized. The adiabatic temperatures of each gradient layer were calculated when the preheating temperatures were 298 K and 1173 K, respectively The microstructure, composition and properties of Ti/Al2O3 FGM and the reaction mechanism of each gradient layer were studied. It was found that Ti/Al2O3 FGM prepared by the explosive compaction/SHS process had a high density and a high microhardness. Its structure, composition and properties showed apparent gradient distribution. The structure of the standard stoichiometric ratio gradient layer of FGM was a network structure. Its reaction mode could be described as follows: Al powder melted first, then the molten Al penetrated into the TiO2 zone and reacted with TiO2, and big pores were left in the original positions of Al powder. The reaction of gradient layers with the addition of Al3O3 as diluents was similar to that of the standard stoichiometric ratio gradient layer, so were their structure and composition. However, the reaction of gradient layers with the addition of Ti as diluents was more complex and the composition deviated slightly from the designed one

THERMAL STRESSES ANALYSIS OF CERAMIC/METAL FUNCTIONALLY GRADIENT MATERIAL CYLINDER

In this paper, the composite cylinder system is made of three layers: metal, functionally gradient material (FGM) and ceramics is studied. The formulas of the steady_state temperature distribution and the associate thermal stress distribution in the cylinder are obtained. For ZrO 2/Ti_6Al_4V system, the distribution of steady_state temperature and thermal stress are calculated and discussed.

ANTI-PLANE FRACTURE ANALYSIS OF FUNCTIONALLY GRADIENT MATERIAL INFINITE STRIP WITH FINITE WIDTH

The special case of a crack under mode Ⅲ conditions was treated, lying parallel to the edges of an infinite strip with finite width and with the shear modulus varying exponentially perpendicular to the edges. By using Fourier transforms the problem was formulated in terms of a singular integral equation. It was numerically solved by representing the unknown dislocation density by a truncated series of Chebyshev polynomials leading to a linear system of equations. The stress intensity factor （SIF） results were discussed with respect to the influences of different geometric parameters and the strength of the non-homogeneity. It was indicated that the SIF increases with the increase of the crack length and decreases with the increase of the rigidity of the material in the vicinity of crack. The SIF of narrow strip is very sensitive to the change of the non-homogeneity parameter and its variation is complicated. With the increase of the non-homogeneity parameter, the stress intensity factor may increase, decrease or keep constant, which is mainly determined by the strip width and the relative crack location. If the crack is located at the midline of the strip or if the strip is wide, the stress intensity factor is not sensitive to the material non-homogeneity parameter.

Method of lines for functionally gradient materials

The method of lines(MOL) for solving the problems of functionally gradient materials(FGMs) was studied. Navier’s equations for FGMs were derived, and were semi-discretized into a system of ordinary differential (equations(ODEs)) defined on discrete lines with the finite difference. By solving the system of ODEs, the solutions to the problem can be obtained. An example of three-point bending was given to demonstrate the application of MOL for a crack problem in the FGM. The computational results show that the more accurate results can be obtained with less computational time and resources. The obvious difficulties of numerical method for crack problems in FGMs, such as the effect of material nonhomogeneity and the existence of high gradient stress and strain near a crack tip, can be overcome without additional consideration if this method is adopted.

Wave propagation of a functionally graded plate via integral variables with a hyperbolic arcsine function

Several studies on functionally graded materials(FGMs)have been done by researchers,but few studies have dealt with the impact of the modification of the properties of materials with regard to the functional propagation of the waves in plates.This work aims to explore the effects of changing compositional characteristics and the volume fraction of the constituent of plate materials regarding the wave propagation response of thick plates of FGM.This model is based on a higher-order theory and a new displacement field with four unknowns that introduce indeterminate integral variables with a hyperbolic arcsine function.The FGM plate is assumed to consist of a mixture of metal and ceramic,and its properties change depending on the power functions of the thickness of the plate,such as linear,quadratic,cubic,and inverse quadratic.By utilizing Hamilton’s principle,general formulae of the wave propagation were obtained to establish wave modes and phase velocity curves of the wave propagation in a functionally graded plate,including the effects of changing compositional characteristics of materials.

Bending and wave propagation analysis of axially functionally graded beams based on a reformulated strain gradient elasticity theory

A new size-dependent axially functionally graded(AFG) micro-beam model is established with the application of a reformulated strain gradient elasticity theory(RSGET). The new micro-beam model incorporates the strain gradient, velocity gradient,and couple stress effects, and accounts for the material variation along the axial direction of the two-component functionally graded beam. The governing equations and complete boundary conditions of the AFG beam are derived based on Hamilton's principle. The correctness of the current model is verified by comparing the static behavior results of the current model and the finite element model(FEM) at the micro-scale. The influence of material inhomogeneity and size effect on the static and dynamic responses of the AFG beam is studied. The numerical results show that the static and vibration responses predicted by the newly developed model are different from those based on the classical model at the micro-scale. The new model can be applied not only in the optimization of micro acoustic wave devices but also in the design of AFG micro-sensors and micro-actuators.

Free vibration analysis of functionally graded material beams based on Levinson beam theory

Free vibration response of functionally graded material （FGM） beams is studied based on the Levinson beam theory （LBT）. Equations of motion of an FGM beam are derived by directly integrating the stress-form equations of elasticity along the beam depth with the inertial resultant forces related to the included coupling and higherorder shear strain. Assuming harmonic response, governing equations of the free vibration of the FGM beam are reduced to a standard system of second-order ordinary differential equations associated with boundary conditions in terms of shape functions related to axial and transverse displacements and the rotational angle. By a shooting method to solve the two-point boundary value problem of the three coupled ordinary differential equations, free vibration response of thick FGM beams is obtained numerically. Particularly, for a beam with simply supported edges, the natural frequency of an FGM Levinson beam is analytically derived in terms of the natural frequency of a corresponding homogenous Euler-Bernoulli beam. As the material properties are assumed to vary through the depth according to the power-law functions, the numerical results of frequencies are presented to examine the effects of the material gradient parameter, the length-to-depth ratio, and the boundary conditions on the vibration response.

Thermo-magnetic analysis of thick-walled spherical pressure vessels made of functionally graded materials

This study presents an analytical solution of thermal and mechanical displacements, strains, and stresses for a thick-walled rotating spherical pressure vessel made of functionally graded materials (FGMs). The pressure vessel is subject to axisymmetric mechanical and thermal loadings within a uniform magnetic field. The material properties of the FGM are considered as the power-law distribution along the thickness. Navier’s equation, which is a second-order ordinary differential equation, is derived from the mechanical equilibrium equation with the consideration of the thermal stresses and the Lorentz force resulting from the magnetic field. The distributions of the displacement, strains, and stresses are determined by the exact solution to Navier’s equation. Numerical results clarify the influence of the thermal loading, magnetic field, non-homogeneity constant, internal pressure, and angular velocity on the magneto-thermo-elastic response of the functionally graded spherical vessel. It is observed that these parameters have remarkable effects on the distributions of radial displacement, radial and circumferential strains, and radial and circumferential stresses.