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.

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.

Pure bending of simply supported circular plate of transversely isotropic functionally graded material

This paper considers the pure bending problem of simply supported transversely isotropic circular plates with elastic compliance coefficients being arbitrary functions of the thickness coordinate. First, the partial differential equation, which is satisfied by the stress functions for the axisymmetric deformation problem is derived. Then, stress functions are obtained by proper manipulation. The analytical expressions of axial force, bending moment and displacements are then deduced through integration. And then, stress functions are employed to solve problems of transversely isotropic functionally graded circular plate, with the integral constants completely determined from boundary conditions. An elasticity solution for pure bending problem, which coincides with the available solution when degenerated into the elasticity solutions for homogenous circular plate, is thus obtained. A numerical example is finally presented to show the effect of material inhomogeneity on the elastic field in a simply supported circular plate of transversely isotropic functionally graded material (FGM).

Several three-dimensional solutions for transversely isotropic functionally graded material plate welded with circular inclusion

The problem of a transversely isotropic functionally graded material （FGM） plate welded with a circular inclusion is considered. The analysis starts with the general- ized England-Spencer plate theory for transversely isotropic FGM plates, which expresses a three-dimensional （3D） general solution in terms of four analytic functions. Several analytical solutions are then obtained for an infinite FGM plate welded with a circular inclusion and subjected to the loads at infinity. Three different cases are considered, i.e., a rigid circular inclusion fixed in the space, a rigid circular inclusion rotating about the x-, y-, and z-axes, and an elastic circular inclusion with different material constants from the plate itself. The static responses of the plate and/or the inclusion are investigated through numerical examples.

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

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.

Nonlinear oscillations of sigmoid functionally graded material plates moving in longitudinal direction

Geometrically nonlinear oscillations are investigated on sigmoid functionally graded material （S-FGM） plates with a longitudinal speed. The material properties of the plates obey a sigmoid distribution rule along the thickness direction. Based on the D＇Alembert＇s principle, a nonlinear equation of motion is derived for the moving S-FGM plates, where the von K^rm^n nonlinear plate theory is adopted. Utilizing the Galerkin method, the equation of motion is discretized and solved via the method of harmonic bal- ance. The approximate analytical solutions are validated through the adaptive step-size fourth-order Runge-Kutta method. Besides, the stability of the steady-state solutions is examined. The results reveal that the mode interaction behavior can happen between the first two modes of the moving S-FGM plates, leading to a complex nonlinear frequency response. It is further found that the power-law index, the longitudinal speed, the exci- tation amplitude, and the in-plane pretension force can significantly affect the nonlinear frequency-response characteristics of longitudinally traveling S-FGM plates.

Higher-order crack tip fields for functionally graded material plate with transverse shear deformation

The crack tip fields are investigated for a cracked functionally graded material （FGM） plate by Reissner＇s linear plate theory with the consideration of the transverse shear deformation generated by bending. The elastic modulus and Poisson＇s ratio of the functionally graded plates are assumed to vary continuously through the coordinate y, according to a linear law and a constant, respectively. The governing equations, i.e., the 6th-order partial differential equations with variable coefficients, are derived in the polar coordinate system based on Reissner＇s plate theory. Furthermore, the generalized displacements are treated in a separation-of-variable form, and the higher-order crack tip fields of the cracked FGM plate are obtained by the eigen-expansion method. It is found that the analytic solutions degenerate to the corresponding fields of the isotropic homogeneous plate with Reissner＇s effect when the in-homogeneity parameter approaches zero.

Free vibration of functionally graded sandwich plates using four-variable refined plate theory

This paper uses the four-variable refined plate theory （RPT） for the free vibration analysis of functionally graded material （FGM） sandwich rectangular plates. Unlike other theories, there are only four unknown functions involved, as compared to five in other shear deformation theories. The theory presented is variationally consistent and strongly similar to the classical plate theory in many aspects. It does not require the shear correction factor, and gives rise to the transverse shear stress variation so that the transverse shear stresses vary parabolically across the thickness to satisfy free surface conditions for the shear stress. Two common types of FGM sandwich plates are considered, namely, the sandwich with the FGM facesheet and the homogeneous core and the sandwich with the homogeneous facesheet and the FGM core. The equation of motion for the FGM sandwich plates is obtained based on Hamilton＇s principle. The closed form solutions are obtained by using the Navier technique. The fundamental frequencies are found by solving the eigenvalue problems. The validity of the theory is shown by comparing the present results with those of the classical, the first-order, and the other higher-order theories. The proposed theory is accurate and simple in solving the free vibration behavior of the FGM sandwich plates.

Accurate Free Vibration of Functionally Graded Skew Plates

The present study aims to analyze free vibration of thin skew plates made of functionally graded material(FGM)by using the weak form quadrature element method.The material properties vary continuously through the thickness according to a power-law form.A novel FGM skew plate element is formulated according to the neutral surface based plate theory and with the help of the differential quadrature rule.For verifications,Numerical results are compared with available data in literature.Results reveal that the non-dimensional frequency parameters of the FGM skew plates are independent of the power-law exponent and always proportional to those of homogeneous isotropic ones when the coupling and rotary inertias are neglected.In addition,employing the physical neutral surface based plate theory is equivalent to using the middle plane based plate theory with the reduced flexural modulus matrix.

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。

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.

Three-dimensional analysis of a thick FGM rectangular plate in thermal environment

The thermal behavior of a thick transversely isotropic FGM rectangular plate was investigated within the scope of three-dimensional elasticity. Noticing many FGMs may have temperature-dependent properties, the material constants were further considered as functions of temperature. A solution method based on state-space formulations with a laminate approximate model was proposed. For a thin plate, the method was clarified by comparison with the thin plate theory. The influences of material inhomogeneity and temperature-dependent characteristics were finally discussed through numerical examples.

Free axisymmetric vibration of FGM circular plates

Based on the three-dimensional theory, this work presents a direct displacement method to investigate the free axisymmetric vibration of transversely isotropic circular plates, whose material is functionally graded and properties obey the exponential law along the thickness direction of the plate. Under two boundary conditions, the solution satisfies all basic equations and the Corresponding boundary condition at every point. Thus, it is three-dimensional exact. Numerical examples are presented and compared with previous works. The present method can also be extended to the case of arbitrary distribution of the material properties along the thickness direction of the plate.

Thermal buckling and postbuckling of FGM circular plates with in-plane elastic restraints

Based on von Karman＇s plate theory, the axisymmetric thermal buckling and post-buckling of the functionally graded material （FGM） circular plates with in- plane elastic restraints under transversely non-uniform temperature rise are studied. The properties of the FGM media are varied through the thickness based on a simple power law. The governing equations are numerically solved by a shooting method. The results of the critical buckling temperature, post-buckling equilibrium paths, and configurations for the in-plane elastically restrained plates are presented. The effects of the in-plane elastic restraints, material property gradient, and temperature variation on the responses of thermal buckling and post-buckling are examined in detail.

An analysis of surface acoustic wave propagation in a plate of functionally graded materials with a layered model

In a homogeneous plate, Rayleigh waves will have a symmetric and anti-symmetric mode regarding to the mid-plane with different phase velocities. If plate properties vary along the thickness, or the plate is of functionally graded material (FGM), the symmetry of modes and frequency behavior will be modified, thus producing dif-ferent features for engineering applications such as amplifying or reducing the velocity and deformation. This kind of effect can also be easily realized by utilizing a layered structure with desired material properties that can produce these effects in terms of velocity and displacements, since Rayleigh waves in a solid with gen-eral material property grading schemes are difficult to analyze with known methods. Solutions from layered structures with exponential and polynomial property grad-ing schemes are obtained from the layered model and comparisons with known analytical results are made to validate the method and examine possible applica-tions of such structures in engineering.

FGM圆环板面内自由振动的DQM求解

基于二维线弹性理论,假定材料物性沿圆环板的径向按照幂律梯度分布,建立了FGM薄圆环板面内自由振动的运动微分方程,采用微分求积法数值研究了FGM圆环板面内自由振动的量纲一频率特性,并与各向同性材料圆环板面内自由振动的量纲一频率进行了比较,说明本文的分析方法有效.结果表明,不同边界条件,FGM梯度指标以及FGM圆环板内、外半径比对量纲一频率均有影响,其计算结果和分析方法可供设计参考.

HA／316L粉非对称生物FGM的微观组织与增韧机制

采用热压工艺制备了HA/316L粉非对称生物功能梯度材料(FGM)。HA/316L粉非对称生物FGM在宏观上呈现明显的梯度,微观上则表现出成分连续变化,且各成分分布均匀、弥散。在各梯度层内部及界面都没有裂纹及大孔洞出现,界面结合紧密。随着316L粉的含量增加,韧窝的数量逐渐增加,韧窝形貌由浅变深,边缘由尖锐逐渐变得圆滑,表明材料由脆性断裂向韧性断裂转化。纯HA梯度层为典型的脆性断裂,HA80/316L和HA60/316L梯度层表现为典型的晶间断裂,HA40/316L和HA20/316L梯度层断裂性质为晶间断裂中掺杂有韧性断裂,而316L梯度层则表现为典型的韧性断裂。316L粉的加入改变了HA/316L粉生物FGM各梯度层的断裂方式,从而提高了材料的力学性能。从整体上而言,HA/316L粉生物FGM主要增韧机制包括层间裂纹偏转增韧与裂纹偏转增韧。

FGM中厚圆板轴对称自由振动的打靶法求解

研究了由陶瓷和金属两种材料组成的功能梯度材料(FGM)中厚圆板的自由振动问题。基于考虑横向剪切变形中厚板的几何方程、物理方程及平衡方程,建立了以中面转角和横向位移为基本未知量的功能梯度中厚圆板轴对称自由振动问题的控制方程;假定功能梯度中厚圆板的材料性质方向按照幂函数连续变化规律;采用打靶法数值求解所得非线性两点边值问题出,获得了多种边界下功能梯度中厚圆板的无量纲自然频率以及振动模态。讨论了材料梯度指数、板的厚度以及边界条件对自然频率的影响。

温度影响下FGM圆环板的面内自由振动分析

基于二维弹性理论和Hamilton原理,假设材料物理性质随温度变化且沿圆环板径向按照幂律梯度分布,导出了温度影响下FGM薄圆环板面内自由振动的运动微分方程。用微分求积法(DQM)计算了温度影响下FGM圆环板面内自由振动的无量纲频率,并与各向同性材料圆环板面内自由振动的无量纲频率进行了比较,说明该分析方法的有效性。同时考虑了沿圆环板径向均匀升温和非均匀升温两种情况下,几何参数、材料性质和温度变化对面内自由振动频率的影响。