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.

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.

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.

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.

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.

横向非均匀温度场作用的FGM夹层梁热屈曲分析

基于经典几何非线性梁理论,推导了受横向非均匀升温作用的FGM夹层梁热屈曲控制方程。对于端部夹紧边界条件梁,随着温度升高,梁发生分支屈曲,考虑到分支屈曲临界状态变形较小,将非线性边值问题退化成线性特征值问题,通过求线性特征值问题,得到了有量纲临界屈曲温度差解析解。另外,运用打靶法计算非线性边值问题,得到了热过屈曲平衡路径和平衡构型。利用算例探究层厚比、梯度指数、温度场级数项数目对FGM夹层梁热过屈曲平衡路径和平衡构型的影响。结果表明:当梯度层相对厚度和梯度指数增加时,FGM夹层梁临界屈曲温度差均增加。

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.

Small scale effects on buckling and postbuckling behaviors of axially loaded FGM nanoshells based on nonlocal strain gradient elasticity theory

By means of a comprehensive theory of elasticity, namely, a nonlocal strain gradient continuum theory, size-dependent nonlinear axial instability characteristics of cylindrical nanoshells made of functionally graded material(FGM) are examined. To take small scale effects into consideration in a more accurate way, a nonlocal stress field parameter and an internal length scale parameter are incorporated simultaneously into an exponential shear deformation shell theory. The variation of material properties associated with FGM nanoshells is supposed along the shell thickness, and it is modeled based on the Mori-Tanaka homogenization scheme. With a boundary layer theory of shell buckling and a perturbation-based solving process, the nonlocal strain gradient load-deflection and load-shortening stability paths are derived explicitly. It is observed that the strain gradient size effect causes to the increases of both the critical axial buckling load and the width of snap-through phenomenon related to the postbuckling regime, while the nonlocal size dependency leads to the decreases of them. Moreover, the influence of the nonlocal type of small scale effect on the axial instability characteristics of FGM nanoshells is more than that of the strain gradient one.

Relationship Between Thermal Shock Behavior and Cutting Performance of a Functionally Gradient Ceramic Tool

Based on the deep understanding of the requirements of cutting conditions on ceramic tools, a design model for functionally gradient ceramic tool materials with symmetrical composition distribution was presented in this paper, according to which an Al 2O 3-TiC functionally gradient ceramic tool material FG-1 was synthesized by powder-laminating and uniaxially hot-pressing technique. The thermal shock resistance of the Al 2O 3-TiC functionally gradient ceramics FG-1 was evaluated by water quenching and subsequent three-point bending tests of flexural strength diminution. Comparisons were made with results from parallel experiments conducted using a homogeneous Al 2O 3-TiC ceramics. Functionally gradient ceramics exhibited higher retained strength under all thermal shock temperature differences compared to homogeneous ceramics, indicating the higher thermal shock resistance. The experimental results were supported by the calculation of transient thermal stress field. The cutting performance of the Al 2O 3-TiC functionally gradient ceramic tool FG-1 was also investigated in rough turning the cylindrical surface of exhaust valve of diesel engine in comparison with that of a common Al 2O 3-TiC ceramic tool LT55. The results indicated that the tool life of FG-1 increased by 50 percent over that of LT55. Tool life of LT55 was mainly controlled by thermal shock cracking which was accompanied by mechanical shock. While tool life of FG-1 was mainly controlled by mechanical fatigue crack extension rather than thermal shock cracking, revealing the less thermal shock susceptibility of functionally gradient ceramics than that of common ceramics.

Effects of the Component and Fiber Gradient Distributions on the Strength of Cement-based Composite Materials

The effects of the component gradient distribution at interface and the fiber gradient distribution on the strength of cement-based materials were studied. The results show that the flexural strength and compressive strength of the mortar and concrete with interface component and fiber gradient distributions are obviously improved. The strengthes of the fiber gradient distributed mortar and concrete (FGDM/C) are higher than those of fiber homogeneously distributed mortar and concrete (FHDM/C). To obtain the same strength, therefore, a smaller fiber volume content in FGDM/C is needed than that in FHDM/C. The results also show that the component gradient distribution of the concrete can be obtained by means of multi-layer vibrating formation.

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.

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.

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.

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. [