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.

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.

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.

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.

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.

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.

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.

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.

Optimization design and residual thermal stress analysis of PDC functionally graded materials

The distribution of thermal stresses in functionally graded polycrystalline diamond compact (PDC) and in single coating of PDC are analyzed respectively by thermo-mechanical finite element analysis (FEA). It is shown that they each have a remarkable stress concentration at the edge of the interfaces. The diamond coatings usually suffer premature failure because of spallation, distortion or defects such as cracks near the interface due to these excessive residual stresses. Results showed that the axial tensile stress in FGM coating is reduced from 840 MPa to 229 MPa compared with single coating, and that the shear stress is reduced from 671 MPa to 471 MPa. Therefore, the single coating is more prone to spallation and cracking than the FGM coating. The effects of the volume compositional distribution factor (n) and the number of the graded layers (L) on the thermal stresses in FGM coating are also discussed respectively. Modelling results showed that the optimum value of the compositional distribution factor is 1.2, and that the best number of the graded layers is 6.

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.

Electrical, Thermal, and Mechanical Properties of Cu/Ti3AlC2 Functional Gradient Materials Prepared by Low-temperature Spark Plasma Sintering

Cu/Ti3AlC2 composite and functional-gradient materials with excellent electrical conductivity and thermal conductivity as well as good flexural properties were prepared by low-temperature spark plasma sintering of Cu and Ti3AlC2 powder mixtures. The phase compositions of the materials were analyzed by X-ray diffraction, and their microstructure was characterized by scanning electron microscopy and energy dispersive X-ray spectroscopy. Further, the electrical conductivity, thermal conductivity, and flexural properties of the materials were tested. Results show that, for the composite materials, the resistivity rises from 0.75 × 10^-7 Ω·m only to 1.32 × 10^-7 Ω·m and the thermal diffusivity reduces from 82.5 mm^2/s simply to 39.8 mm^2/s, while the flexural strength improves from 412.9 MPa to 471.3 MPa, as the content of Ti3AlC2 is increased from 5 wt%to 25 wt%. Additionally, the functional-gradient materials sintered without interface between the layers exhibit good designability, and their overall electrical conductivity, thermal conductivity, and flexural strength are all higher than those of the corresponding uniform composite material.

Preparation of Ti-Mo system functionally graded materials by co-sedimentation

A new kind of method, co sedimentation method, was used to fabricate functional gradient material(FGM) in order to eliminate the interfaces in gradient materials. The deposit bodies obtained from the sedimentation of Ti and Mo particles were densified at 1 673 K under a pressure of 20 MPa in flowing argon for 1 h. Finally, Ti Mo system FGM with continuous change of composition was successfully prepared. The results reveal that the sedimentation method is an effective way to manufacture FGM with continuous change of composition. Moreover, the results also show that the compositional gradient of FGM can be adjusted in a wide range through both particle size distribution and the ratio of powders. [

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

Preparation of Al/Si functionally graded materials using ultrasonic separation method

Functionally graded materials （FGM） have been widely used in many industries such as aerospace, energy and electronics. In this experimental study of fabricating FGM, an approach was developed to prepare AI/Si FGM using power ultrasonic separation method. Material sample with continuously changing composition and performance/properties was successfully produced. Results showed that the microstructure of the FGM sample transited, from its top to bottom, from the hypereutectic structure with a large quantity of primary Si gradually to the eutectic, and finally to the hypoeutectic with numerous primary AI dendrites. The distribution of primary Si and microhardness of the FGM sample also presented graded characteristics, resulting that the wear resistance of the FGM sample decreased from top to bottom. Preliminary discussion was made on the mechanism of the formation of AI/Si FGM.