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 generalized 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.     

THERMAL BUCKLING ANALYSIS OF CIRCULAR PLATE EMBEDDED IN AN INFINITE ELASTIC PLATE 1)

分析了嵌入无限大弹性板中的圆板在变温时的热屈曲问题。由于圆板的热膨胀系数与无限大弹性板的热膨胀系数不同,温度变化时圆板中会产生压应力。当压应力达到其临界值时,圆板会发生热屈曲。首先,基于弹性力学平面应力问题的基本理论,得到圆板和无限大弹性板的应力和位移;然后建立圆板热屈曲的控制微分方程,求得临界屈曲温度的解析解和数值解,着重讨论圆板和无限大弹性板的材料物性参数的关系对圆板临界屈曲温度的影响。     

Elasticity solutions for a transversely isotropic functionally graded circular plate subject to an axisymmetric transverse load qrk

This paper considers the bending of transversely isotropic circular plates with elastic compliance coefficients being arbitrary functions of the thickness coordinate, subject to a transverse load in the form of qr^k (k is zero or a finite even number). The differential equations satisfied by stress functions for the particular problem are derived. An elaborate analysis procedure is then presented to derive these stress functions, from which the analytical expressions for the axial force, bending moment and displacements are obtained through integration. The method is then applied to the problem of transversely isotropic functionally graded circular plate subject to a uniform load, illustrating the procedure to determine the integral constants from the boundary conditions. Analytical elasticity solutions are presented for simply-supported and clamped plates, and, when degenerated, they coincide with the available solutions for an isotropic homogenous plate. Two numerical examples are finally presented to show the effect of material inhomogeneity on the elastic field in FGM plates.     

Non-axisymmetric thermal stress of a functionally graded coated circular inclusion in an infinite matrix

This paper is to study the non-axisymmetric two-dimensional problem of thermal stresses in an infinite matrix with a functionally graded coated circular inclusion based on complex variable method. With using the method of piece-wise homogeneous layers, the general solution for the functionally graded coating having radial arbitrary elastic properties is derived when the matrix is subjected to uniform heat flux at infinity, and then numerical results are presented for several special examples. It is found that the existence of the functionally graded coating can change interfacial thermal stresses, and choosing proper change ways of the radial elastic properties in the coating can obviously reduce the thermal stresses.     

功能梯度中厚圆/环板轴对称弯曲问题的解析解

基于一阶剪切变形板理论，导出了热/机载荷作用下，位移形式的功能梯度 中厚圆/环板轴对称弯曲问题的控制方程，获得了问题的位移和内力的一般解析解. 作为特 例，分别研究了边缘径向固定和可动的夹紧和简支的4种实心功能梯度圆板，给出了它们的 解，并分析了热/机载荷作用下解的形态，讨论了横向剪切变形、材料梯度常数和边界条件， 对板的轴对称弯曲行为的影响.     

Strain gradient solution for a finite-domain Eshelby-type anti-plane strain inclusion problem

A solution for the finite-domain Eshelby-type inclusion problem of a finite elastic body containing an anti-plane strain inclusion of arbitrary cross-sectional shape prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived in a general form using a simplified strain gradient elasticity theory (SSGET). The formulation is facilitated by an extended Betti’s reciprocal theorem and an extended Somigliana’s identity based on the SSGET and suitable for anti-plane strain problems. The disturbed displacement field is obtained in terms of the SSGET-based Green’s function for an infinite anti-plane strain elastic body. The solution reduces to that of the infinite-domain anti-plane strain inclusion problem when the boundary effect is not considered. The problem of a circular cylindrical inclusion embedded concentrically in a finite cylindrical elastic matrix undergoing anti-plane strain deformations is analytically solved by applying the general solution, with the Eshelby tensor and its average over the circular cross section of the inclusion obtained in closed forms. This Eshelby tensor, being dependent on the position, inclusion size, matrix size, and a material length scale parameter, captures the inclusion size and boundary effects, unlike existing ones. It reduces to the classical linear elasticity-based Eshelby tensor for the circular cylindrical inclusion in an infinite matrix if both the strain gradient and boundary effects are suppressed. Numerical results quantitatively show that the inclusion size effect can be quite large when the inclusion is small and that the boundary effect can dominate when the inclusion volume fraction is high. However, the inclusion size effect is diminishing with the increase of the inclusion size, and the boundary effect is vanishing as the inclusion volume fraction becomes sufficiently low.     

Axisymmetric bending analysis of thick functionally graded circular plates using fourth-order shear deformation theory

In the present article, axisymmetric bending and stretching of functionally graded (FG) circular plates subjected to uniform transverse loading based on fourth-order shear deformation plate theory (FOST) have been studied. Using a fourth-order shear deformation theory, the solutions for deflection and rotation functions of FG plates are presented in terms of the corresponding quantities for a homogeneous plate using the classical plate theory (CPT), from which solutions one can easily obtain the FOST solutions for axisymmetric bending of FG circular plates. It is assumed that the effective mechanical properties of the functionally graded plates through the thickness are continuous functions of the volume fractions of the constituent parts which are themselves defined by a power-law function. Numerical results for maximum deflection and shear stress are presented for various percentages of ceramic–metal volume fractions. These results are also compared with those obtained from the first-order shear deformation plate theory of Mindlin (FST), the third-order shear deformation plate theory of Reddy (TST) as well as the exact three-dimensional elasticity solution. It is found that although the maximum deflections obtained using FOST and TST are close to each other, the through-thickness shear stress is predicted more accurately by the FOST formulation than by the TST.     

A new analytical-numerical method for calculating interacting stresses of a multi-hole problem under both remote and arbitrary surface stresses

Based on the elementary solutions and new integral equations, a new analytical-numerical method is proposed to calculate the interacting stresses of multiple circular holes in an infinite elastic plate under both remote stresses and arbitrarily distributed stresses applied to the circular boundaries. The validity of this new analytical-numerical method is verified by the analytical solution of the bi-harmonic stress function method, the numerical solution of the finite element method, and the an...     

ANALYSIS OF TWO DISSIMILAR FUNCTIONALLY GRADED STRIPS CONTAINING INTERFACE CRACK UNDER PLANE DEFORMATION

In this paper the plane elasticity problem of two bonded dissimilar functionally graded strips containing an interface crack is studied.The governing equation in terms of Airy stress function is formulated and exact solutions are obtained for several special variations of material properties in Fourier transformation domain.The mixed boundary problem is reduced to a system of singular integral equations that are solved numerically.Numerical results show that fracture toughness of materials can be greatly improved by graded variation of elastic modulus and the influence of the specific form of elastic modulus on the fracture behavior of FGM is limited.     

On asymmetric bending of functionally graded solid circular plates

Based on the three-dimensional elasticity equations, this paper studies the elastic bending response of a transversely isotropic functionally graded solid circular plate subject to transverse biharmonic forces applied on its top surface. The material properties can continuously and arbitrarily vary along the thickness direction. By virtue of the generalized England’s method, the problem can be solved by determining the expressions of four analytic functions. Expanding the transverse load in Fourier series along the circumferential direction eases the theoretical construction of the four analytic functions for a series of important biharmonic loads. Certain boundary conditions are then used to determine the unknown constants that are involved in the four constructed analytic functions. Numerical examples are presented to validate the proposed method. Then, we scrutinize the asymmetric bending behavior of a transversely isotropic functionally graded solid circular plate with different geometric and material parameters.     

Mode I stress intensity factor solutions for spot welds in lap-shear specimens

The analytical solutions of the mode I stress intensity factor for spot welds in lap-shear specimens are investigated based on the classical Kirchhoff plate theory for linear elastic materials. First, closed-form solutions for an infinite plate containing a rigid inclusion under counter bending conditions are derived. The development of the closed-form solutions is then used as a guide to develop approximate closed-form solutions for a finite square plate containing a rigid inclusion under counter bending conditions. Based on the J integral, the closed-form solutions are used to develop the analytical solutions of the mode I stress intensity factor for spot welds in lap-shear specimens of large and finite sizes. The analytical solutions of the mode I stress intensity factor based on the solutions for infinite and finite square plates with an inclusion are compared with the results of the three-dimensional finite element computations of lap-shear specimens with various ratios of the specimen half width to the nugget radius. The results indicate that the mode I stress intensity factor solution based on the finite square plate model with an inclusion agrees well with the computational results for lap-shear specimens for the ratio of the half specimen width to the nugget radius between 4 and 15. Finally, a set of the closed-form stress intensity factor solutions for lap-shear specimens at the critical locations are proposed for future applications.     

Analytical solution of axisymmetric contact problem about indentation of a circular indenter into a soft functionally graded elastic layer

The paper addresses a contact problem of the theory of elasticity,i.e.,the penetration of a circular indenter with a flat base into a soft functionally graded elastic layer.The elastic properties of a functionally graded layer arbitrarily vary with depth,and the foundation is assumed to be elastic,yet much harder than a layer.Approximated analytical solution is constructed,and it is shown that the solutions are asymptotically exact both for large and small values of characteristic dimensionless geometrical parameter of the problem.Numerical examples are analyzed for the cases of monotonic and nonmonotonic variations of elastic properties.Numerical results for the case of homogeneous layer are compared with the results for nondeformable foundation.     

Three-dimensional stress and free vibration analyses of functionally graded plates with circular holes by the use of the graded finite element method

Static and free vibration analyses of plates with circular holes are performed based on the three-dimensional theory of elasticity. The plates are made of a functionally graded material (FGM), and the volume fractions of the constituent materials vary continuously across the plate. The effective properties of the FGM plate are estimated by using the Mori–Tanaka homogenization method. A graded finite element method based on the Rayleigh–Ritz energy formulation is used to solve the problem. Effects of different volume fractions of the materials and hole sizes on the behavior of FGM plates under uniaxial tension are investigated. Natural frequencies of a fully clamped FGM plate with a circular cutout are derived. The results obtained are compared with available experimental data.     

Thermal buckling of functionally graded circular plates based on higher order shear deformation plate theory

In this research, thermal buckling of circular plates compose of functionally graded material (FGM) is considered. Equilibrium and stability equations of a FGM circular plate under thermal loads are derived, based on the higher order shear deformation plate theory (3rd order plate theory). Assuming that the material properties vary as a power form of the thickness coordinate variable z and using the variational method, the system of fundamental partial differential equations is established. A buckling analysis of a functionally graded circular plate (FGCP) under various types of thermal loads is carried out and the result are given in closed-form solutions. The results are compared with the critical buckling temperature obtained for FGCP based on first order (1st order plate theory) and classical plate theory (0 order plate theory) given in the literature. The study concludes that higher order shear deformation theory accurately predicts the behavior of FGCP, whereas the first order and classical plate theory overestimates buckling temperature.     

Stress Concentration Factor in Functionally Graded Plates with Circular Holes Subjected to Anti-plane Shear Loading

Stress concentration factors due to the presence of geometrical discontinuities (circular holes) in functionally graded plates are derived. The material property inhomogeneity is assumed to be in the radial direction originating at the center of the plate. Variable separable closed-form solutions are obtained for the stresses and displacements in functionally graded plates (without and with holes) subjected to anti-plane shear loading. The stresses in functionally graded plates without a hole are not homogeneous as it is in the case of homogeneous plates. Either a stress concentration (more than the applied stress) or dilution (less than the applied stress) occurs depending on whether the modulus increases (hardening graded material) or decreases (softening graded material) away from the center of the graded plate without a hole. A novel definition of the stress concentration factor due to the geometrical discontinuity in functionally graded plates is derived. The effect of the circular hole in functionally graded plates is to magnify (compared to homogeneous plates) the stress concentration when the modulus decreases away from the center of the hole (softening material). Beneficial reduction of the stress concentration factor is achieved in hardening functionally graded materials.     

Inclusions in a finite elastic body

Within the framework of 2D or 3D linear elasticity, a general approach based on the superposition principle is proposed to study the problem of a finite elastic body with an arbitrarily shaped and located inclusion. The proposed approach consists in decomposing the initial inclusion problem into the problem of the inclusion embedded in the corresponding infinite body and the auxiliary problem of the finite body subjected to the appropriate boundary loading provided by solving the former problem. Thus, our approach renders it possible to circumvent the difficulty due to the unavailability of the relevant Green function, use various existing solutions for the problem of an inclusion inside an unbounded body and clearly makes appear the finite boundary effects. The general approach is applied and specified in the context of 2D isotropic elasticity. The complex potentials for the problem of an inclusion in an infinite body are given as two boundary integrals, and the boundary integral equation governing the complex potentials for the auxiliary problem is provided. In the important particular situation where a finite body with an arbitrarily shaped and located inclusion is circular, the exact explicit expressions for the complex potentials are derived, leading to those for the strain, stress and Eshelby’s tensor fields inside and outside the inclusion. These results are analytically detailed and numerically illustrated for the cases of a square inclusion placed concentrically, and a circular inclusion located eccentrically, inside a circular body.     

Dynamic fracture mechanics analysis for composite material with material non-homogeneity in thickness direction

The problem considered here is the response of a non-homogeneous composite material containing some cracks subjected to dynamic loading. It is assumed that the composite material is orthotropic and all the material properties depend only on the coordinatey (along the thickness direction). In the analysis, the elastic region is divided into a number of plies of infinite length. The material properties are taken to be constants for each ply. By utilizing the Laplace transform and Fourier transform technique, the general solutions for plies are derived. The singular integral equations of the entire elastic region are obtained and solved by the virtual displacement principle. Attention is focused on the time-dependent full field solutions of stress intensity factor(SIF) and strain energy release rate. As a numerical illustration, the dynamic stress intensity factor of a substrate/functionally graded film structure with two cracks under suddenly applied forces on cracks face are presented for various material non-homogeneity parameters.     

功能梯度压电圆板自由振动问题的三维精确分析

本文对周边为广义刚性滑动和广义简支两种边界条件下的功能梯度压电材料圆板自由振动问题进行分析。根据轴对称横观各向同性压电材料基本方程,并利用有限Hankel变换得到了功能梯度压电材料圆板的状态空间方程。假设材料的机械和电学性质均沿板厚方向按统一的指数函数形式梯度分布,从而获得了周边为广义刚性滑动和广义弹性简支两种边界条件下功能梯度压电圆板自由振动问题的三维精确频率方程,该方程是一个关于自由振动频率的超越方程,通过求解该超越方程可得到在不同板厚以及不同的材料性质梯度变化情况下的圆板自由振动频率值,结果表明在相同的材料性质梯度变化情况下频率均随着板厚增加而增大,而在相同的板厚情况下频率则随材料性质梯度变化指数的增大而减小的结论。     

多圆荷载下刚性道面变形和应力的计算

本文建立多圆荷载作用下弹性半空间体上薄板的挠度与应力的计算式。荷载数量及分布任意,每个圆荷载密度与轮迹半径彼此相异。对计算式中的反常积分及级数的收敛性予以证明。对含振荡函数反常积分建议一种方便的算法。