功能梯度材料亚表面缺陷的热波多重散射问题

基于热传导波动模型,采用波函数展开法,研究了半无限功能梯度材料亚表面球形缺陷的热波多重散射.给出了热波散射的一般解.温度波由调制光束在材料表面激发,球形缺陷表面的边界条件为绝热,非均匀参数为指数函数变化.分析了结构几何参数和物理参数对温度分布的影响,并给出了温度变化的数值结果.本研究可为功能梯度材料的分析研究、物理反问题和红外热波成像等提供理论基础和参考数据.     

Scattering of the harmonic anti-plane shear waves by a crack in functionally graded piezoelectric materials

In the present paper, the dynamic behavior of a Griffth crack in the functionally graded piezoelectric material (FGPM) is investigated. It is assumed that the elastic stiffness, piezoelectric constant, dielectric permittivity and mass density of the FGPM vary continuously as an exponential function, and that FGPM is under the anti-plane mechanical loading and in-plane electrical loading. By using the Fourier transform and defining the jumps of displacement and electric potential components across the crack surface as the unknown functions, two pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement and electric potential components across the crack surface are expanded in a series of Jacobi polynomial. Numerical examples are provided to show the effects of material properties on the stress and the electric displacement intensity factors.     

Characteristics of waves in a functionally graded cylinder

An analytical‐numerical method is presented for analysing characteristics of waves in a cylinder composed of functionally graded material (FGM). In this method, the FGM cylinder is divided into a number of annular elements with three‐nodal‐lines in the wall thickness. The elemental material properties are assumed to vary linearly in the thickness direction so as to better model the spatial variation of material properties of FGM. The Hamilton principle is used to develop the dispersion equations for the cylinder, and the frequency and the group velocity are established in terms of the Rayleigh quotient. The method is applied to analyse several FGM cylinders, and its efficiency is demonstrated. Numerical results demonstrate that the ratio of radius to thickness has a stronger influence on the frequency spectra in the circumferential wave than on that in the axial wave, that negative group velocity presents at a range of smaller wave numbers and that the range varies as the wave normal and the ratio of radius to thickness of FGM cylinders. Copyright © 2001 John Wiley & Sons, Ltd.     

Transient waves in plates of functionally graded materials

A numerical method is proposed for analysing transient waves in plates of functionally graded material (FGM) excited by impact loads. The material properties of the FGM plate have a gradient in the thickness direction and are anisotropic in the plane of the plate. In the present method, the FGM plate is divided into layer elements in the thickness direction. For an accurate modelling of the variation of the material property of FGM plates, it is expressed by second‐order polynomials in the thickness direction within an element. This can further reduce the number of elements to obtain more accurate results effectively. The principle of virtual work is used to develop approximate dynamic equilibrium equations. The displacement response is determined by employing the Fourier transformation and the modal analysis. As examples, the displacement response of FGM plates excited by line, point and distributed loads is calculated. The computations have shown the efficiency of the present method. Copyright © 2001 John Wiley & Sons, Ltd.     

Multiple Scattering of Thermal Waves from a Subsurface Cylindrical Inclusion in Semi-infinite Functionally Graded Materials Using Non-Fourier Model

In this study, a theoretical method is applied to investigate the multiple scattering of thermal waves and temperature field resulting from a subsurface cylindrical inclusion in a semi-infinite functionally graded material (FGM). The adiabatic boundary condition at the semi-infinite surface is considered. The thermal waves are excited at the surface of semi-infinite functionally graded materials by modulated optical beams. The model includes the multiple scattering effects of the cylindrical thermal wave generated by the line heat source. According to the wave equation of heat conduction, a general solution of scattered thermal waves is presented. Numerical calculations illustrate the effect of subsurface inclusion on the temperature and phase change at the sample surface under different physical and geometrical parameters. It is found that the temperature above the conducting cylindrical inclusion decreases because of the existence of the inclusion. The effect of the inclusion on the temperature and phase change at the surface is also related to the non-homogeneous parameter of FGMs, the wave frequency of thermal waves, and the distance between the inclusion and the semi-infinite surface. Finally, the effect of the relaxation time of buried inclusion on the temperature and phase change at the surface is examined.     

Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory

Dynamic stability of microbeams made of functionally graded materials (FGMs) is investigated in this paper based on the modified couple stress theory and Timoshenko beam theory. This non-classical Timoshenko beam model contains a material length scale parameter and can interpret the size effect. The material properties of FGM microbeams are assumed to vary in the thickness direction and are estimated though Mori–Tanaka homogenization technique. The higher-order governing equations and boundary conditions are derived by using the Hamilton’s principle. The differential quadrature (DQ) method is employed to convert the governing differential equations into a linear system of Mathieu–Hill equations from which the boundary points on the unstable regions are determined by Bolotin’s method. Free vibration and static buckling are also discussed as subset problems. A parametric study is conducted to investigate the influences of the length scale parameter, gradient index and length-to-thickness ratio on the dynamic stability characteristics of FGM microbeams with hinged–hinged and clamped–clamped end supports. Results show that the size effect on the dynamic stability characteristics is significant only when the thickness of beam has a similar value to the material length scale parameter.     

基于正弦剪切变形理论的功能梯度材料三明治微梁的静动态特性

基于修正的偶应力理论和正弦剪切变形梁理论,研究了功能梯度材料三明治微梁的静态弯曲和自由振动行为。考虑两种不同类型的功能梯度材料三明治微梁,根据哈密顿变分原理建立其静动态力学行为的控制方程,应用Navier解法,得到了简支边界条件下弯曲变形和振动频率的解析解,同时,给出了固支等边界条件时的里兹法求解过程。数值算例表明,功能梯度三明治微梁的静动态力学行为具有明显的尺度效应,微梁的无量纲厚度、功能梯度指数、长厚比和结构形式等因素对其静动态响应有很大影响,相关结果和规律对功能梯度材料三明治微梁的结构设计和性能优化等实际工程应用具有一定的指导意义。      

Dynamic stress intensity factors around two parallel cracks in a functionally graded layer bonded to dissimilar half-planes subjected to anti-plane incident harmonic stress waves

The time-harmonic problem of determining the stress field around two parallel cracks in functionally graded materials (FGMs) is studied. The Fourier transform technique is used to reduce the boundary conditions to four simultaneous integral equations which are then solved by expanding the differences of crack surface displacements in a series. The unknown coefficients in the series are obtained by the Schmidt method. Numerical calculations are carried out for dynamic stress intensity factors (DSIF) in FGMs.     

Nonlinear theories of axisymmetric bending of functionally graded circular plates with modified couple stress

A microstructure-dependent nonlinear theory for axisymmetric bending of circular plates, which accounts for through-thickness power-law variation of a two-constituent material, is developed using the principle of virtual displacements. The formulation is based on a modified couple stress theory, power-law variation of the material, temperature-dependent properties, and the von Kármán geometric nonlinearity. Classical and first-order shear deformation theories are considered in the study. The modified couple stress theory contains a material length scale parameter that can capture the size effect in a functionally graded material plate. The theories presented herein can be used to develop analytical solutions of bending, buckling, and free vibration for the linear case and finite-element models for the nonlinear case to determine the effect of the geometric nonlinearity, power-law index, and microstructure-dependent constitutive relations on linear and nonlinear response of axisymmetric analysis of circular plates.     

Free vibration and stability of functionally graded circular cylindrical shells according to a 2D higher-order deformation theory

A two-dimensional (2D) higher-order deformation theory is presented for vibration and buckling problems of circular cylindrical shells made of functionally graded materials (FGMs). The modulus of elasticity of functionally graded (FG) shells is assumed to vary according to a power law distribution in terms of the volume fractions of the constituents. By using the method of power series expansion of continuous displacement components, a set of fundamental governing equations which can take into account the effects of both transverse shear and normal deformations, and rotatory inertia is derived through Hamilton’s principle. Several sets of truncated Mth order approximate theories are applied to solve the eigenvalue problems of simply supported FG circular cylindrical shells. In order to assure the accuracy of the present theory, convergence properties of the fundamental natural frequency for the fundamental mode r=s=1 are examined in detail. A comparison of the present natural frequencies of isotropic and FG shells is also made with previously published results. Critical buckling stresses of simply supported FG circular cylindrical shells subjected to axial stress are also obtained and a relation between the buckling stress and natural frequency is presented. The internal and external works are calculated and compared to prove the numerical accuracy of solutions. Modal transverse shear and normal stresses are calculated by integrating the three-dimensional (3D) equations of motion in the thickness direction satisfying the stress boundary conditions at the outer and inner surfaces. The 2D higher-order deformation theory has an advantage in the analysis of vibration and buckling problems of FG circular cylindrical shells.     

半无限功能梯度材料结构中圆孔对弹性波的多重散射

基于弹性波多体散射理论,采用波函数展开法,研究了半无限指数梯度材料中圆孔对弹性波的多重散射和动应力集中,得到了问题的解析解,给出了圆孔动应力集中系数的数值解,分析了圆孔与边界的距离、入射波波数以及材料的非均匀参数对圆孔周围动应力集中系数的影响。分析表明:梯度材料的非均匀参数小于零时对最大动应力影响较小,但是对动应力在圆周的分布有较大影响;大于零时对最大动应力和动应力在圆周的分布影响都很大,特别是在圆孔与边界的距离较小时影响更大。     

Transient thermal stress analysis of an edge crack in a functionally graded material

An edge crack in a strip of a functionally graded material (FGM) is studied under transient thermal loading conditions. The FGM is assumed having constant Young's modulus and Poisson's ratio, but the thermal properties of the material vary along the thickness direction of the strip. Thus the material is elastically homogeneous but thermally nonhomogeneous. This kind of FGMs include some ceramic/ceramic FGMs such as TiC/SiC, MoSi₂/Al₂O₃ and MoSi₂/SiC, and also some ceramic/metal FGMs such as zirconia/nickel and zirconia/steel. A multi-layered material model is used to solve the temperature field. By using the Laplace transform and an asymptotic analysis, an analytical first order temperature solution for short times is obtained. Thermal stress intensity factors (TSIFs) are calculated for a TiC/SiC FGM with various volume fraction profiles of the constituent materials. It is found that the TSIF could be reduced if the thermally shocked cracked edge of the FGM strip is pure TiC, whereas the TSIF is increased if the thermally shocked edge is pure SiC.     

Investigation of anti-plane shear behavior of a Griffith permeable crack in functionally graded piezoelectric materials by use of the non-local theory

In this paper, the non-local theory of elasticity is applied to obtain the behavior of a Griffith crack in functionally graded piezoelectric materials under the anti-plane shear loading for the permeable electric boundary conditions. To make the analysis tractable, it is assumed that the material properties vary exponentially with coordinate vertical to the crack. By means of the Fourier transform, the problem can be solved with the help of a pair of dual-integral equations that the unknown variable is the jump of the displacement across the crack surfaces. These equations are solved by use of the Schmidt method. Numerical examples are provided. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularities are present near the crack tips. The non-local elastic solutions yield a finite hoop stress at the crack tips, thus allows us to using the maximum stress as a fracture criterion. The finite hoop stresses at the crack tips depend on the crack length, the functionally graded parameter and the lattice parameter of the materials, respectively.     

Influence of microstructure on fracture toughness distribution in ceramic-metal functionally graded materials

This paper deals with the influence of microstructure on fracture toughness distribution in functionally graded materials (FGMs) consisting of partially stabilized zirconia (PSZ) and austenitic stainless steel SUS 304. FGMs and non-graded composites (non-FGMs) with fine and coarse microstructures are fabricated by powder metallurgy using PSZ and two kinds of SUS 304 powders. The fracture toughness is determined by conventional tests for several non-FGMs with each material composition and by a method utilizing stable crack growth in FGMs. The obtained results on the fracture toughness are as follows: (1) The fracture toughness increases with an increase in a content of SUS 304 on both FGMs and non-FGMs. (2) On the fracture toughness of the non-FGMs, the influence of microstructure is negligible. (3) On the FGMs, the fracture toughness is higher in the FGM with fine microstructure than in the FGM with coarse microstructure. (4) The fracture toughness of the FGMs is higher than that of the non-FGMs especially in the case of fine microstructure. Finally, the residual stress in the FGMs created in a fabrication process is estimated from the difference in fracture toughness between the FGMs and non-FGMs.     

梯度材料中任意形孔对反平面剪切波散射与动应力

该文基于弹性动力学理论,采用复变函数与保角映射方法,研究了指数梯度材料中任意形孔洞对弹性波的散射与动应力集中,给出了问题的解析解.并以求解椭圆孔动应力集中系数为例,分析了入射波数和材料非均匀参数等对椭圆孔动应力分布的影响.     

Finite element evaluation of mixed mode stress intensity factors in functionally graded materials

This paper is directed towards finite element computation of fracture parameters in functionally graded material (FGM) assemblages of arbitrary geometry with stationary cracks. Graded finite elements are developed where the elastic moduli are smooth functions of spatial co‐ordinates which are integrated into the element stiffness matrix. In particular, stress intensity factors for mode I and mixed‐mode two‐dimensional problems are evaluated and compared through three different approaches tailored for FGMs: path‐independent J^*_k‐integral, modified crack‐closure integral method, and displacement correlation technique. The accuracy of these methods is discussed based on comparison with available theoretical, experimental or numerical solutions. Copyright © 2001 John Wiley & Sons, Ltd.     

Effect of a graded interlayer on the stress intensity factor of cracks in a joint under thermal loading

In a bimaterial joint with and without a graded interlayer, the stress intensity factor of cracks perpendicular to the interface was calculated for a thermal loading by a homogeneous change in the temperature. In joints without an interlayer, the stress intensity factor increases to infinity as the crack approaches the interface for the case of the Young’s moduli E₁/E₂>1 (crack in material 1). Introducing a graded interlayer with a continuous transition in the material properties between the two joined materials leads to a continuous change in the stress intensity factor if the crack propagates from material 1 into material 2. Results are presented for different transition functions of the material properties and for different thickness ratios of the layers. The possible beneficial effect of a graded interlayer is discussed.     

Analysis of a transiently propagating crack in functionally graded materials under mode I and II

Crack tip stress and displacement fields for a transiently propagating crack along gradient in functionally graded materials (FGMs) with a linear variation of shear modulus are developed. The higher order terms of the transient stress and displacement fields at crack tip were obtained by transforming the general partial differential equations of the dynamic equilibrium into Laplace’s equations whose solutions have harmonic functions. Thus, the fields can be expressed very simply. Using these stress components, isochromatics and the first invariant at crack tip are generated.The results show that the isochromatics (constant maximum shear stress) for mode I crack tilt backward around the crack tip with an increase of crack tip acceleration , and tilt forward around the crack tip with an increase of rate of change of dynamic mode I stress intensity factor . The isochromatics for mixed mode crack move to upper direction with an increases of and , and lower direction with an increase of . Contours of the first stress invariant for mode I crack enlarge around the crack tip with an increase of , and decrease around the crack tip with an increase of . As decreases at crack initiation, the predicted kinking angles increase. As increases, the predicted kinking angles also increase.     

Consideration of spatial variation of the length scale parameter in static and dynamic analyses of functionally graded annular and circular micro-plates

This article introduces new methods for static and free vibration analyses of functionally graded annular and circular micro-plates, which can take into account spatial variation of the length scale parameter. The underlying higher order continuum theory behind the proposed approaches is the modified couple stress theory. A unified way of expressing the displacement field is adopted so as to produce numerical results for three different plate theories, which are Kirchhoff plate theory (KPT), Mindlin plate theory (MPT), and third-order shear deformation theory (TSDT). Governing partial differential equations and corresponding boundary conditions are obtained following the variational approach and the Hamilton's principle. Derived systems of differential equations are solved numerically by utilizing the differential quadrature method (DQM). Comparisons to the results available in the literature demonstrate the high level of accuracy of the numerical results generated through the developed methods. Extensive analyses are presented in order to illustrate the influences of various geometric and material parameters upon static deformation profiles, stresses, and natural vibration frequencies. In particular, the length scale parameter ratio -which defines the length scale parameter variation profile-is shown to possess a profound impact on both static and dynamic behaviors of functionally graded annular and circular micro-plates.     

Dynamic stress field for torsional impact of a penny-shaped crack in a transversely isotropic functional graded strip

The torsional impact response of a penny-shaped crack in a transversely isotropic strip is considered. The shear moduli are assumed to be functionally graded such that the mathematics is tractable. Laplace and Hankel transforms are used to reduce the problem to solving a Fredholm integral equation. The crack tip stress field is obtained by considering the asymptotic behavior of Bessel function. Investigated are the effects of material nonhomogeneity and orthotropy and strip’s highness on the dynamic stress intensity factor. The peak of the dynamic stress intensity factor can be suppressed by increasing the shear moduli’s gradient and/or increasing the shear modulus in a direction perpendicular to the crack surface. The dynamic behavior varies little with the increasing of the strip’s highness.