Dynamic stress from a subsurface cylindrical inclusion in a functionally graded material layer under anti-plane shear waves

The multiple scattering of shear waves and dynamic stress resulting from a subsurface cylindrical inclusion in a functionally graded material (FGM) layer bonded to homogeneous materials are investigated, and the analytical solution of this problem is derived. Image method is used to satisfy the traction free boundary condition of the FGM layer. The analytical solutions of wave fields around the actual and image inclusions are expressed by employing wave functions expansion method, and the expanded mode coefficients are determined by satisfying the continuous boundary conditions around the inclusions. Through the numerical solutions of dynamic stress concentration factors (DSCFs) around the inclusion, the effects of the position of the inclusion in the material layer, the properties of the inclusion, and the properties of the two phases of composites on the DSCFs are analyzed. Analyses show that when the cylindrical inclusion is stiffer than the two phases of FGMs, the dynamic stress around the inclusion increases greatly. When the distance between the surface of the structure and the inclusion is smaller, the effect of the properties of the inclusion becomes greater. When the cylindrical inclusion is softer than the two phases of FGMs, the maximum dynamic stress shows little difference; however, the variation of the distribution of the dynamic stress around the inclusion is greater.     

Dynamic stress from a cylindrical inclusion buried in a functionally graded piezoelectric material layer under electro-elastic waves

This paper presents a theoretical method to investigate the multiple scattering of electro-elastic waves and dynamic stress around a subsurface cylindrical inclusion in a functionally graded piezoelectric material layer bonded to homogeneous piezoelectric materials. The analytical solutions of wave fields are expressed by employing wave function expansion method, and the expanded mode coefficients are determined by satisfying the boundary conditions around the inclusion. The image method is used to satisfy the mechanical and electrically short conditions at the free surface of the structure. Through the numerical solutions of dynamic stress concentration factors around the inclusion, it is found that when the cylindrical inclusion possesses higher rigidity and greater piezoelectric constant than the two phases of functionally graded materials, the dynamic stress around the inclusion increases greatly. When the distance between the surface of the structure and the inclusion is smaller, the effect of the properties of the inclusion becomes greater. When the cylindrical inclusion possesses lower rigidity and smaller piezoelectric constant than the two phases of functionally graded materials, the maximum dynamic stress shows little difference; however, the variation of the distribution of the dynamic stress around the inclusion is greater. The effect of the properties of the inclusion on the dynamic stress around the inclusion is greater than that on the electric field. The effects of wave frequency on the dynamic stress and electric field are also examined.     

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。     

Multiple scattering of shear waves and dynamic stress from a circular cavity buried in a semi-infinite slab of functionally graded materials

Based on the theory of elastodynamics and employing image method, the multiple scattering and dynamic stress in a semi-infinite slab of functionally graded materials with a circular cavity are investigated. The analytical solution of this problem is derived, and the numerical solutions of the dynamic stress concentration factor around the cavity are also presented. The effects of the distance between the cavity and the boundaries of the semi-infinite slab, the incident wave number and the non-homogeneity parameter of materials on the dynamic stress concentration factors are analyzed. Analyses show that the dynamic stress around the cavity increases with increasing non-homogeneity parameter of materials and incident wave number. The boundaries of the semi-infinite slab have great effect on both the maximum dynamic stress and the distribution of dynamic stress around the circular cavity, and the effect increases with increasing incident wave number.     

覆盖层为功能梯度材料弹性半平面中的Love波

对均匀各向同性弹性半平面上覆盖一层功能梯度材料中存在的Love波的频散问题进行了研究,给出了Love波频散方程的一般形式。利用WKBJ近似理论,给出了功能梯度材料层的位移、应力近似解析解,导出了Love波WKBJ近似频散方程的一般形式。该文以功能梯度材料层的剪切弹性模量和质量密度沿厚度方向均为指数函数变化为例,进行了实例计算和分析,给出了频散曲线,讨论了Love波在功能梯度材料覆盖层弹性半平面中传播的一般性质。这些结论对无损检测和反问题分析方法的改进提供理论依据。     

Random dynamic response and reliability of a crack in a functionally graded material layer between two dissimilar elastic half-planes

An analytical approach is presented for the random dynamic analysis of a functionally graded material (FGM) layer between two dissimilar elastic half-planes. This FGM layer contains a crack and its material properties vary randomly in the thickness direction, while their mean values are exponential functions of field position. The transient loadings applied on the crack faces are assumed to be stochastic processes of time. In order to obtain the solution, the FGM layer is divided into several sub-layers, and the material properties of each layer are reduced to random variables by an average method. A fundamental problem is constructed for the solution. Based on the use of Laplace and Fourier transforms, the boundary conditions are reduced to a set of singular integral equations, which can be solved by the Chebyshev polynomial expansions. Both stress intensity factor history with its statistics and dynamic reliability are analytically derived. Numerical calculations are provided to show the effects of related parameters.     

On a moving dielectric crack in a piezoelectric interface with spatially varying properties

This paper provides a comprehensive theoretical analysis of a finite crack propagating with constant speed along an interface between two dissimilar piezoelectric media under inplane electromechanical loading. The interface is modeled as a graded piezoelectric layer with spatially varying properties (functionally graded piezoelectric materials, i.e., FGPMs). The analytical formulations are developed using Fourier transforms and the resulting singular integral equations are solved with Chebyshev polynomials. Using a dielectric crack model with deformation-dependent electric boundary condition, the dynamic stress intensity factors, electric displacement intensity factor, crack opening displacement (COD) intensity factor, and energy release rate are derived to fully understand this inherent mixed mode dynamic fracture problem. Numerical simulations are made to show the effects of the material mismatch, the thickness of the interfacial layer, the crack position, and the crack speed upon the dynamic fracture behavior. A critical state for the electromechanical loading applied to the medium is identified, which determines whether the traditional impermeable (or permeable) crack model serves as the upper or lower bound for the dielectric model considering the effect of dielectric medium crack filling.     

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.     

3D analytical solution for a functionally graded transversely isotropic piezoelectric circular plate under tension and bending

A three-dimensional (3D) analysis of a functionally graded piezoelectric circular plate under tension and bending is carried out. A direct displacement method is developed, with analytical solutions obtained for plate with either free or simply-supported edge conditions. The material properties of the plate can vary arbitrarily along the thickness except that the strain-energy function should be positive definite as required for stable materials and certain integrable conditions are assumed valid during the derivation. The validity of the present solutions is discussed both analytically and numerically. Numerical analyses are made for a specific functionally graded material to show the influence of material heterogeneity on the piezoelastic field.     

Determination of dynamic effective properties in functionally graded materials

Summary This work is dedicated to the investigation of the dynamic effective properties in functionally graded materials resulting from an anti-plane shear wave. A micromechanics-based elastodynamic model is developed to predict the dynamic behavior of two-phase functionally graded materials, and the distribution of dynamic effective properties in the gradation direction is presented. Generally speaking, in functionally graded materials there exist two microstructurally distinct zones: a fiber-matrix zone and a transition zone. In the fiber-matrix zone, the dispersion relation for the effective wave number is derived using the effective medium method, and the dynamic effective properties for any macroscopic material points are determined in the corresponding microstructural representative volume element (RVE). In the transition zone, a transition function is introduced to make the wave fields continuous and differentiable. Numerical examples of the dynamic effective properties in the gradation direction under different parameters are presented graphically. The obtained results reveal that the distribution of dynamic effective properties in the gradation direction is dependent on the material properties of each phase, the incident frequency, and the gradation parameter of the materials. Comparisons between numerical solutions and experimental data are also made. At last, the results are discussed in detail.     

Analytical solution of a cantilever functionally graded beam

The problem of a cantilever functionally graded beam subjected to different loads is studied. In terms of Airy stress function a general two-dimensional solution is presented for a cantilever functionally graded beam, assuming that all the elastic moduli of the material have the same variations along the beam-thickness direction. Explicit expressions of analytical solutions to some specific examples under different boundary conditions are obtained to demonstrate the usefulness of the proposed general solution technique. This solution will be useful in analyzing functionally graded beam with arbitrary variations of material properties and it can serve as a basis for establishing simplified functionally graded beam theories.     

Analytical solution for a functionally graded beam with arbitrary graded material properties

The plane stress problem of an orthotropic functionally graded beam with arbitrary graded material properties along the thickness direction is investigated by the displacement function approach for the first time. A general two-dimensional solution is obtained for a functionally graded beam subjected to normal and shear tractions of arbitrary form on the top and bottom surfaces and under various end boundary conditions. For isotropic case explicit solutions are given to some specific through-the-thickness variations of Young’s modulus such as exponential model, linear model and reciprocal model. The influence of different grade models on the stress and displacement fields are illustrated in numerical examples. These analytical solutions can serve as a basis for establishing simplified theories and evaluating numerical solutions of functionally graded beams.     

Effect of an interphase layer on the dynamic stress concentration in a Mg-matrix surrounding a SiC-particle

The potential of using an interphase layer to reduce stress concentrations under a dynamic loading in a Mg-matrix surrounding a SiC-particle is investigated in this study. An interphase layer was applied between the particle and the matrix and the contact between them was assumed to be perfect. Both constant-property materials and functionally graded materials were considered for the interphase. A constant-property interphase was modelled as a single layer while a functionally graded interphase was divided into a number of sublayers and each sublayer was treated as having constant material properties. Numerical results reveal that the interphase layer made of a constant-property material shows better stress concentration reduction than that made of functionally graded materials. An interphase layer with low values of both shear modulus and Poisson’s ratio is necessary for a significant stress concentration reduction. Studies were focused on determining the maximum stress concentration that occurs over a range of frequencies. This investigation has revealed that a stress concentration reduction of up to 44% could be realized.     

Analysis of functionally graded plates by meshless method: A purely analytical formulation

Functionally graded plates under static and dynamic loads are investigated by the local integral equation method (LIEM) in this paper. Plate bending problem is described by the Reissner moderate thick plate theory. The governing equations for the functionally graded material with respect to the neutral plane are presented in the Laplace transform domain and therefore the in-plane and bending problems are uncoupled. Both isotropic and orthotropic material properties are considered. The local integral equation method is developed with the locally supported radial basis function (RBF) interpolation. As the closed forms of the local boundary integrals are obtained, there are no domain or boundary integrals to be calculated numerically in this approach. The solutions of the nodal values for the entire plate are obtained by solving a set of linear algebraic equation system with certain boundary conditions. Details of numerical procedures are presented and the accuracy and convergence characteristics of the method are examined. Several examples are presented for the functionally graded plates under static and dynamic loads and the accuracy for proposed method has been observed compared with 3D analytical solutions.     

Exact solutions for characterization of electro-elastically graded materials

The present paper deals with a class of functionally graded materials (FGM), called active FGM that has electro-elastically graded material phases. An active FGM system leads to minimization of stress concentration that arises due to mismatch in the electrical and elastic properties of the constituent phases. This work focuses on the characterization of the through thickness stresses of an active FGM subjected to electrical excitation. The structure is comprised of a substrate, an electro-elastically graded layer and an active layer. A formulation for exact solutions of the system based on Euler–Bernoulli theory is presented. Power-law variation of the composition of the two phases in the graded layer is considered. Performance of linearly gradient FGM for a range of stiffness and electrical property ratios of the active and substrate materials have been studied. It is observed that the electrical strain component and the compositional gradation significantly influence the stress characteristics of the active FGM.     

Thermomechanical postbuckling analysis of functionally graded plates and shallow cylindrical shells

Summary. In this paper, an analytic solution is provided for the postbuckling behavior of plates and shallow cylindrical shells made of functionally graded materials under edge compressive loads and a temperature field. The material properties of the functionally graded shells are assumed to vary continuously through the thickness of the shell according to a power law distribution of the volume fraction of the constituents. The fundamental equations for thin rectangular shallow shells of FGM are obtained using the von Karman theory for large transverse deflection, and the solution is obtained in terms of mixed Fourier series. The effect of material properties, boundary conditions and thermomechanical loading on the buckling behavior and stress field are determined and discussed. The results reveal that thermomechanical coupling effects and the boundary conditions play a major role in dictating the response of the functionally graded plates and shells under the action of edge compressive loads.     

Analytical elastic solutions for pressurized hollow cylinders with internal functionally graded coatings

The main objective of this work is to obtain analytical solutions for thick-walled cylinders subjected to internal and external pressure in which the entire wall is made of functionally graded material or of only a thin functionally graded coating present on the internal homogeneous wall. We assume that the materials are isotropic with constant Poisson’s ratio; as far as the Young modulus is concerned, we consider a power and an exponential. The proposed analytical solutions show the effects of the different profiles describing the graded properties of the materials on the stress and displacement fields; in addition, comparisons between graded coating and conventional homogeneous coating highlight the advantage of the graded material on the interface stress reduction. Furthermore, we show how even a thin graded coating can be useful to satisfy the requirements of a specific application without having to make an entire wall with graded properties. This investigation permits us to optimize the elastic response of cylinders under pressure by tailoring the thickness variation of the elastic properties and to reduce manufacturing costs given by the technological limitations that occur to produce entire functionally graded walls.     

Theoretical analysis of heat conduction problems of nonhomogeneous functionally graded materials for a layer sandwiched between two half-planes

Functionally graded materials are the materials whose material properties are smoothly varying along one axis, and they are used as buffer layers to connect two dissimilar materials. By choosing proper functionally graded parameters, the material properties at the interface can be identical to prevent the interfacial fracture problem. This study analyzes the heat conduction problem of nonhomogeneous functionally graded materials for a layer sandwiched between two half-planes. From the Fourier transform method, the full-field solutions of temperature and heat flux are obtained in explicit forms. Numerical calculations based on the analytical solutions are performed and are discussed in detail. The continuous characteristics of the temperature and heat flux along the interface are emphasized, and some interesting phenomena are presented in this study. It is noted that the temperature and heat flux fields along the interface for nonhomogeneous functionally graded materials are continuous if the conductivities are identical at the interface. Furthermore, the temperature and heat flux q _y have the identical contour slopes across the interface.     

Dynamic behavior of a crack in a functionally graded piezoelectric strip bonded to two dissimilar half piezoelectric material planes

Summary. The dynamic behavior of a crack in a functionally graded piezoelectric material (FGPM) strip bonded to two half dissimilar piezoelectric material planes subjected to combined harmonic anti-plane shear wave and in-plane electrical loading was studied under the limited permeable and permeable electric boundary conditions. It was assumed that the elastic stiffness, piezoelectric constant and dielectric permittivity of the functionally graded piezoelectric layer vary continuously along the thickness of the strip. By using the Fourier transform, the problem can be solved with a set of dual integral equations in which the unknown variables are the jumps of the displacements and the electric potentials across the crack surfaces. In solving the dual integral equations, the jumps of the displacements and the electric potentials across the crack surfaces were expanded in a series of Jacobi polynomials. Numerical results illustrate the effects of the gradient parameter of FGPM, electric loading, wave number, thickness of FGPM strip and electric boundary conditions on the dynamic stress intensity factors (SIFs).     

功能梯度曲梁弯曲问题的解析解

该文采用弹性力学逆解法,求得了功能梯度曲梁在端部受弯矩作用的解析解。假设弹性模量E=E₀rn沿径向呈幂函数的梯度分布。根据弹性力学平面问题的基本方程,在极坐标系下,引入应力函数,得到了弯曲问题的解析解。进而将功能梯度曲梁问题进行扩展,求得了整环或厚壁圆筒以及向错问题的解析解。将所得到的解退化到均匀弹性情况,与经典的理论解一致。最后对梯度函数按幂函数变化的算例进行了分析,结果显示梯度因子n对应力及位移的分布产生了巨大的影响。该文所得到的结论可以作为功能梯度曲梁构件优化设计的理论基础。