Asymmetric Green's functions for a functionally graded transversely isotropic tri-material

This paper considers the elastic analysis of a functionally graded transversely isotropic tri-material solid under the arbitrary distribution of applied static loads. Using two displacement potential functions, for three-dimensional point-load and patch-load configurations, Green's functions for displacement and stress components are generated in the form of infinite line-integrals. These solutions are shown to be analytically reducible to the special cases of exponentially graded bi-material, exponentially graded half-space and homogeneous tri-material Green's functions. It also encompasses a functionally graded finite layer on a rigid base with surface loading with two cases of interfacial conditions, rigid-bonded and rigid-frictionless. Finally, for the special case of a functionally graded layer sandwiched between two homogeneous layers, using several numerical displays, the effect of material inhomogeneity on the responses is studied and the accuracy of numerical scheme is verified.     

Semi-analytic modelling of transversely isotropic magneto-electro-elastic materials under frictional sliding contact

Functionally graded magneto-electro-elastic (FGMEE) materials has been increasingly used in engineering applications, particularly in smart material or intelligent structure systems. This paper proposes a semi-analytical approach for sliding frictional contact problem between a rigid insulating sphere and a transversely isotropic FGMEE film and half-space based on frequency response functions (FRFs). Multilayered approximation is used to model the functionally graded material (FGM), and the FRFs for each MEE layer are derived explicitly. The unknown coefficients in FRFs are formulated by two matrix equations, and their efficient solution process is proposed. Based on the obtained FRFs, a highly efficient semi-analytical model (SAM) is developed which is able to solve the three-dimensional frictional contact of FGMEE materials with arbitrary layer designs. The model is validated with finite element method and the literature. Furthermore, the pressure/stress distribution and electric/magnetic potential are studied in different FGM designs to investigate the influence of material layout.     

3D dynamic responses of a multi-layered transversely isotropic saturated half-space under concentrated forces and pore pressure

Dynamic Green's function plays an important role in the study of various wave radiation, scattering and soil-structure interaction problems. However, little research has been done on the response of transversely isotropic saturated layered media. In this paper, the 3D dynamic responses of a multi-layered transversely isotropic saturated half-space subjected to concentrated forces and pore pressure are investigated. First, utilizing Fourier expansion in circumferential direction accompanied by Hankel integral transform in radial direction, the wave equations for transversely isotropic saturated medium in cylindrical coordinate system are solved. Next, with the aid of the exact dynamic stiffness matrix for in-plane and out-of-plane motions, the solutions for multi-layered transversely isotropic saturated half-space under concentrated forces and pore pressure are obtained by direct stiffness method. A FORTRAN computer code is developed to achieve numerical evaluation of the proposed method, and its accuracy is validated through comparison with existing solutions that are special cases of the more general problems addressed. In addition, selected numerical results for a homogeneous and a layered material model are performed to illustrate the effects of material anisotropy, load frequency, drainage condition and layering on the dynamic responses. The presented solutions form a complete set of Green's functions for concentrated forces (including horizontal load in x(y)-direction, vertical load in z-direction) as well as pore pressure, which lays the foundation for further exploring wave propagation of complex local site in a layered transversely isotropic saturated half-space by using the BEMs.     

State vector solutions for nonaxisymmetric problem of multilayered half space piezoelectric medium

A state space formulation is established for the nonaxisymmetric space problem of transversely isotropic piezoelectric media in a system of cylindrical coordinate by introducing the state vector. Using the Hankel transform and the Fourier series, the state vector equations are transformed into ordinary differential equations. By the use of the matrix methods, the analytical solutions of a single piezoelectric layer are presented in the form of the product of initial state variables and transfer matrix. The applications of state vector solutions are discussed. An analytical solution for a semiinfinite piezoelectric medium subjected to the vertical point forceP _z, horizontal point forceP _x along x-direction and point electric charge Q at the origin of the surface is presented. According to the continuity conditions at the interfaces, the general solution formulation forN-layered transversely isotropic piezoelectric media is given. Project supported by the National Natural Science Foundation of China (Grant No. 59648001).     

A transversely isotropic medium containing a penny-shaped crack subjected to a non-uniform axisymmetric loading via an anchored smooth rigid disk

A smooth rigid circular anchor disk encapsulated by a penny-shaped crack is embedded in and unbounded transversely isotropic medium. The lamellar rigid disk exerts a nonuniform axisymmetric loading to the upper face of the crack. With the aid of an appropriate stress function and Hankel transform, the governing equations are converted to a set of triple integral equations which in turn are reduced to a Fredholm integral equation of the second kind. For some transversely isotropic materials the normalized stiffness of the system falls well outside of the envelope pertinent to isotropic media. It is shown that mode I stress intensity factor is independent of the material properties and solely depends on the ratio of the radius of the rigid disk to that of the crack; moreover, for the cases where this ratio is less than about 0.9 a simple explicit approximate expression for the mode I stress intensity factor is derived. In contrast, the normalized mode II stress intensity factor is independent of the mentioned geometrical parameters but depends on the elastic properties of the material; depending on the material properties, the normalized mode II stress intensity factor can vary between 0 to ∞ for transversely isotropic materials and between 0 to π/4 for isotropic materials.     

横观各向同性电磁弹性固体耦合方程的一般解

横观各向同性电磁弹性固体的耦合特征由5个关于弹性位移、电位和磁位的二阶偏微分方程控制.基于势函数理论,耦合的方程组被简化为5个非耦合的关于势函数的广义Laplace方程.弹性场和电磁场由势函数表示,这构成了横观各向同性电磁弹性固体的一般解.     

Dynamic analysis of multi-directional functionally graded annular plates

Dynamic analysis of multi-directional functionally graded annular plates is achieved in this paper using a semi-analytical numerical method entitled the state space-based differential quadrature method. Based on the three-dimensional elastic theory and assuming the material properties having an exponent-law variation along the thickness, radial direction or both directions, the frequency equations of free vibration of multi-directional functionally graded annular plates are derived under various boundary conditions. Numerical examples are presented to validate the approach and the superiority of this method is also demonstrated. Then free vibration of functionally graded annular plates is studied for different variations of material properties along the thickness, radial direction and both directions, respectively. And the influences of the material property graded variations on the dynamic behavior are also investigated. The multi-directional graded material can likely be designed according to the actual requirement and it is a potential alternative to the unidirectional functionally graded material.     

横观各向同性饱和地基的三维动力响应

首先引入位移函数,将直角坐标系下横观各向同性饱和土Biot波动方程转化为2个解耦的六阶和二阶控制方程;然后基于双重Fourier变换,求解了Biot波动方程,得到以土骨架位移和孔隙水压力为基本未知量的积分形式的一般解,并用一般解给出了饱和土总应力分量的表达式．在此基础上系统研究了横观各向同性饱和半空间体的稳态动力响应问题,考虑表面排水和不排水两种情况,得到了半空间体在任意分布的表面谐振荷载作用下,表面位移的稳态动力响应,文末给出了算例．     

Reflection of plane waves from a rigid wall and a free surface in a transverse isotropic medium

For the general solution of two-dimensional equations of dynamics of a transverse isotropic medium with the Carrier–Gassmann condition, we give a representation in terms of two resolvent functions satisfying two separate wave equations. The problem of reflection of plane waves from a rigid wall and a free surface is solved. The coefficients of reflection and transformation of the plane waves are found. These formulas yield a solution for isotropic media too. Some special cases are consideredwhere the shapes (amplitudes) of the reflectedwaves are not uniquely determined, but linearly related with the shape of the incident wave.     

压电材料空间轴对称问题的通解及其应用

本文根据横观各向同性压电材料空间轴对称问题场方程的结构特点,利用逐次引进势函数的方法,最后得到将位移分量和电势函数用满足特定偏微分方程的单一势函数表示的所谓通解,推导过程表明这种形式的通解是完备的,作为应用举例,文中用通解求解了压电材料半无限体表面受集中力的问题,得到位移、应力、电位移分量及电势函数的解析表达式,本文所提供的通解可作为分析含空腔、夹杂或币形裂纹等缺陷的压电材料的机-电耦合行为的工具,算例所得结果可直接用于求解压电体相互间或压电体与普通弹性体间的接触问题。     

横观各向同性含液饱和多孔介质中应力波传播的特征分析

根据广义特征理论,对横观各向同性含液饱和多孔介质中应力波传播特性进行了特征分析．给出了特征曲面的微分方程以及沿次特征线的相容条件,得到了波阵面的解析表达式．详细地讨论了应力波在横观各向同性含液饱和多孔介质中传播时,其速度曲面和波阵面的形状及性质．分析结果亦表明,纯固体中应力波传播的特征方程,是含液饱和多孔介质中应力波特征方程的特例．     

Elasticity Solutions for Cylindrical Bending of Functionally Graded Piezoelectric Material Plates北大核心CSCD

功能梯度压电材料（FGPM）同时兼具功能梯度材料和压电材料特性,可为多功能或智能化轻质结构设计提供支撑,在诸多领域有着广泛的应用前景.将Mian和Spencer功能梯度板理论由功能梯度弹性材料推广到功能梯度压电材料,解析研究了FGPM板的柱面弯曲问题,其中,材料弹性常数、压电和介电参数沿板厚方向可以任意连续变化.最终,给出了FGPM板受横向均布荷载作用下柱面弯曲问题的弹性力学解.通过算例分析,重点讨论了压电效应对FGPM板静力响应的影响.     

Two collinear antiplane cracks in functionally graded magnetoelectroelastic composite materials

An analysis of two collinear antiplane cracks in inhomogeneous (functionally graded) anisotropic magnetoelectroelastic materials is presented. In designing components involving functionally graded materials, an important aspect of the problem is the fracture failure. The problem is formulated for transversely isotropic functionally graded magnetoelectroelastic materials. An integral transform is employed to reduce the problem to a singular integral equation that can be solved.     

Higher-Order Asymptotic Approximations for Surface Love Waves of SH Type in Tranversely Isotropic Elastic Media

A uniform asymptotics of the surface Love modes for a special case of anisotropy (tranverse isotropy) of an elastic media is obtained. In constructing the asymptotics of surface waves, the space-time (ST) ray method is employed. The wave field of each Love mode is represented as the sum of the ST caustic expansion involving the Airy functions with a real eikonal and two correction terms that are ST ray solutions, which in fact are inhomogeneous waves with complex eikonals. The eikonals and coefficients of the caustic and ray series are sought in the form of expansions in powers of two variables. The first variable is the distance from the surface, whereas the other characterizes the proximity of the caustic of a ray field to the boundary surface. Thanks to the specific structure of the elasticity tensor for a transversely isotropic medium, the boundary surface is necessarily a plane. A recursion process of computation of higher terms of the asymptotic expansion allows one to trace the conversion of the formulas obtained to the known ray solutions for isotropic elastic media. Relations between the elasticity parameters of a medium are obtained that ensure the existence of SH Love waves in a transversely isotropic medium and that are consistent with the conditions of the positiveness of the elastic energy of deformation. Bibliography: 6 titles.     

含椭圆孔或裂纹压电介质平面问题的基本解

应用复变函数的方法，并基于精确的电边界条件，导出了含一椭圆孔或裂纹的横观各向同性压电体在任意集中力和集中电荷作用下的复变函数解，即Ｃｒｅｎ函数解·叠加该解，得到了裂纹表面作用任意集中载荷或分布载荷时的一般解·这些解不但澄清了从前文献中一些不合理的结果，同时也为应用边界元法求解更复杂的压电介质断裂力学问题提供了基本解·     

二维各向异性压电介质机电耦合场的基本解

本文研究各向异性压电介质的机电耦台问题．应用平面波分解法和留数定理，首次得到了线力和线电荷作用下一般二维各向异性压电介质机电耦合场的基本解．本文的解适用于平面问题、反平面问题以及平面和反平面相互耦合问题．作为特例，文中给出了横观各向同性压电介质的基本解．     

Nonlinear finite element analysis of functionally graded plates integrated with patches of piezoelectric fiber reinforced composite

In this paper, a nonlinear static finite element analysis of simply supported smart functionally graded (FG) plates in the presence/absence of the thermal environment has been presented. The substrate FG plate is integrated with the patches of piezoelectric fiber reinforced composite (PFRC) material which act as the distributed actuators of the plate. The material properties of the FG substrate plate are assumed to be temperature dependent and graded along the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The derivation of this nonlinear thermo-electro-mechanical coupled finite element model is based on the first order shear deformation theory and the Von Karman type geometric nonlinearity. The numerical solutions of the nonlinear equations of the finite element model are obtained by employing the direct iteration method. The numerical illustrations suggest the potential use of the distributed actuator made of the PFRC material for active control of nonlinear deformations of smart FG structures. The effects of volume fraction index of the FG material of the substrate plates and the locations of the PFRC patches on the control authority of the patches are investigated. Emphasis has also been placed on investigating the effect of variation of piezoelectric fiber orientation angle in the PFRC patches on their actuation capability for counteracting the large deflections of FG plates.     

A general solution for three dimensional steady-state transversely isotropic hygro-thermo-magneto-piezoelectric media

In this article, we extract the general solution of three dimensional (3D) equations using potential theory method (PTM) for steady-state, transversely isotropic, hygro-thermo-magneto-piezoelectric media (HTMPM). The governing equations are simplified by introducing the displacement functions. A general solution is completely determined by advantage of the superposition principle and operator theory, which is connected in terms of two functions, fulfilling a second-order and twelfth-order homogeneous partial differential equation (PDE), separately. With the help of Almansi’s theorem, the general solution can be further shortened, which is stated by seven harmonic functions only. The acquired general solutions are straightforward structure and helpful in boundary value problems of HTMPM. Further, we apply the 3D fundamental solutions inside an infinite and on the surface of semi-infinite of a steady point heat source united with a steady point moisture source transversely isotropic HTMPM. Comprehensive and exact solutions are given in the form of elementary functions, which appear as a standard for various types of approximate solutions and numerical codes. Some numerical simulation is conducted based on the obtained general solutions.     

An efficient and simple refined theory for buckling analysis of functionally graded plates

In this paper, an efficient and simple refined theory is presented for buckling analysis of functionally graded plates. The theory, which has strong similarity with classical plate theory in many aspects, accounts for a quadratic variation of the transverse shear strains across the thickness and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. The mechanical properties of functionally graded material are assumed to vary according to a power law distribution of the volume fraction of the constituents. Governing equations are derived from the principle of minimum total potential energy. The closed-form solutions of rectangular plates are obtained. Comparison studies are performed to verify the validity of present results. The effects of loading conditions and variations of power of functionally graded material, modulus ratio, aspect ratio, and thickness ratio on the critical buckling load of functionally graded plates are investigated and discussed.     

Thermoelastic vibration and buckling analysis of functionally graded piezoelectric cylindrical shells

This paper presents the report of an investigation into thermoelastic vibration and buckling characteristics of the functionally graded piezoelectric cylindrical, where the functionally graded piezoelectric cylindrical shell is made from a piezoelectric material having gradient change along the thickness, such as piezoelectricity and dielectric coefficient et al. Here, utilizing Hamilton’s principle and the Maxwell equation with a quadratic variation of the electric potential along the thickness direction of the cylindrical shells and the first-order shear deformation theory, and taking into account both the direct piezoelectric effect and the converse piezoelectric effect, the thermoelastic vibration and buckling characteristics of functionally graded piezoelectric cylindrical shells composed of BaTiO₃/PZT − 4, BaTiO₃/PZT − 5A and BaTiO₃/PVDF are, respectively, calculated. The effects of material composition (volume fraction exponent), thermal loading, external voltage applied and shell geometry parameters on the free vibration characteristics are described, and the axial critical load, critical temperature and critical control voltage are obtained.