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.     

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.     

Three-dimensional elastostatic solutions for transversely isotropic functionally graded material plates containing elastic inclusion

Based on the generalized England-Spencer plate theory, the equilibrium of a transversely isotropic functionally graded plate containing an elastic inclusion is studied. The general solutions of the governing equations are expressed by four analytic functions α(ζ), β(ζ), φ(ζ), and ψ(ζ) when no transverse forces are acting on the surfaces of the plate. Axisymmetric problems of a functionally graded circular plate and an infinite func-tionally graded plate containing a circular hole subject to loads applied on the cylindrical boundaries of the plate are firstly investigated. On this basis, the three-dimensional (3D) elasticity solutions are then obtained for a functionally graded infinite plate containing an elastic circular inclusion. When the material is degenerated into the homogeneous one, the present elasticity solutions are exactly the same as the ones obtained based on the plane stress elasticity, thus validating the present analysis in a certain sense.     

Dynamic interaction between elastic thick circular plate and transversely isotropic saturated soil ground

IntroductionThe vibration of the plate on the porous saturated building foundation is a complicateddynamic contact problem.Its consideration is very important in both earthquake and geo-technical engineering.Lots of research works have been done in recent…     

层状横观各向同性地基上异性矩形薄板的弯曲解析解

选用更具广泛性的层状横观各向同性弹性地基模型,来分析四边自由各向异性矩形地基板的弯曲解析解。先基于直角坐标下横观各向同性体的静力胡海昌通解,借助双重傅里叶变换及矩阵传递法,获得层状横观各向同性地基的静力位移场和应力场;然后将异性薄板的弯曲控制方程,与基于层状横观各向同性弹性地基的位移解建立的板与地基变形协调方程相结合,先按对称性分解,再用三角级数法,得出层状横观各向同性弹性地基上四边自由各向异性矩形薄板的弯曲解析解,包括地基反力、板的挠度及板的内力的解析表达式。克服了数值法的弊端,取消了对地基反力的假设,且避免了矩阵指数函数的计算;同时考虑了地基的层状性及板和地基的各向异性,从而得到板的内力及地基反力更切实际的分布规律。算例结果与文献的有限元结果吻合良好,证明本文方法是切实可行的。     

On the boundary conditions for transversely isotropic piezoelectric plates

By invoking the theorem of work reciprocity for piezoelectric media, necessary conditions, which the prescribed edge data of the plate must fulfill in order that it should generate a decaying state within the plate, are established through generalizing the method proposed by Gregory and Wan. These decaying state conditions for the case of axisymmetric deformation of a transversely isotropic piezoelectric circular plate when stress and electric displacement conditions are imposed on the plate edge are derived explicitly, which are then used for the formulation of boundary conditions for the plate theory solution (or the interior solution). Also an analytical solution of the axisymmetric decaying state of transversely isotropic piezoelectric circular plates is derived. Furthermore, the corresponding necessary conditions for the axisymmetric deformation of elastic circular plates are indeed reproduced directly.     

Dynamic geometrically nonlinear analysis of transversely compressible sandwich plates

In order to construct a plate theory for a thick transversely compressible sandwich plate with composite laminated face sheets, the authors make simplifying assumptions regarding distribution of transverse strain components in the thickness direction. The in-plane stresses and σ_yy (Fig. 1) are computed from the constitutive equations, and the improved values of transverse stress components and σ_zz need to be computed by integration of pointwise equations of motion in a post-process stage of the finite element analysis. The improved values of the transverse strains can also be computed in the post-process stage by substituting the improved transverse stresses into the constitutive relations. A problem of cylindrical bending of a simply supported plate under a uniform load on the upper surface is considered, and comparison is made between the displacements, the in-plane stress and the improved transverse stresses (obtained by integration of the pointwise equations of motion), computed from the plate theory, with the corresponding values of exact elasticity solutions. In this comparison, a good agreement of both solutions is achieved. In the finite element analysis of sandwich plates in cylindrical bending with small thickness-to-length ratios, the shear locking phenomenon does not occur. The model of a sandwich plate in cylindrical bending, presented in this paper, has a wider range of applicability than the models presented in literature so far: it can be applied to the sandwich plates with a wide range of ratios of thickness to the in-plane dimensions, with both thin and thick face sheets (as compared to the thickness of the core) and to the sandwich plates with both transversely rigid and transversely compressible face sheets and cores.     

Influence of elastic properties on the stress state of a nonthin transversely isotropic plate with a circular hole

The paper outlines a method for constructing the general analytic solution to the system of equilibrium equations of nonthin transversely isotropic plates. The method uses the Fourier–Legendre series expansion of the unknown functions with respect to the thickness coordinate. The stress state near a circular hole in a nonthin plate subject to tensile and shear stresses at infinity is analyzed     

Tension of a nonthin transversely isotropic plate with a noncircular cylindrical cavity

The three-dimensional stress state of a transversely isotropic plate with a nearly circular cylindrical cavity is examined. The cavity surface is subject to normal and tangential stresses and the plate is subject at infinity to tensile and shear forces. The problem is solved by expanding unknown functions into Fourier-Legendre series in the thickness coordinate and using the boundary-shape perturbation method. The equations and recurrence formulas needed to solve the problem in an arbitrary approximation are presented __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 11, pp. 101–113, November 2006.     

Solving the equilibrium equations for a transversely isotropic piezoceramic plate

The system of equilibrium equations for nonthin transversely isotropic piezoceramic plates subject to antisymmetric (about the mid-plane) deformation is set up by expanding the unknown functions into Fourier-Legendre series in the thickness coordinate. To find the general solution of the system of equations, a method is proposed, which is then used to solve the stress problem for a plate with a circular hole under mechanical loading __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 10, pp. 96–106, October 2006.     

EXACT SOLUTION TO THE PROBLEM OF A HALF-PLANE CRACK IN A TRANSVERSELY ISOTROPIC PIEZOELECTRIC BODY SUBJECTED TO ANTISYMMETRIC TANGENTIAL POINT FORCES

An exact and complete solution of the problem of a half-plane crack in an infinite transversely isotropic piezoelectric body is presented. The upper and lower crack faces are assumed to be loaded antisymmetrically by a couple of tangential point forces in opposite directions. The solution is derived through a limiting procedure from that of a penny-shaped crack. The expressions for the electroelastic field are given in terms of elementary functions. Finally, the numerical results of the second and third mode stress intensity factorsk ₂ andk ₃ of piezoelectric materials and elastic materials are compared in figures. Project supported by the National Natural Science Foundation of China (No. 19872060 and 69982009) and the Postdoctoral Foundation of China.     

Boundary Conditions for Plate Bending in One-dimensional Hexagonal Quasicrystals

For plate bending in one-dimensional (1D) hexagonal quasicrystals (QCs), the reciprocal theorem and the general solution for QCs media are applied in a novel way to obtain the appropriate stress and mixed boundary conditions accurate to all order for plates of general edge geometry and loadings. Through generalizing the method developed by Gregory and Wan, a set of necessary conditions on the edge-data for the existence of a rapidly decaying solution is established. The prescribed data must satisfy these conditions in order that they should generate a decaying state. When a set of stress edge-data or mixed edge-data is imposed on the plate edge, these decaying state conditions for the case of axisymmetric deformation of 1D hexagonal QC plates are derived explicitly. They are then used for the correct formulation of boundary conditions for the plate theory solution (or the interior solution). Furthermore, in the absence of phonon–phason fields coupling effect, corresponding necessary conditions for the case of transversely isotropic elastic plates are derived subsequently, and their isotropic elastic counterparts are also obtained.      

横观各向同性弹性半空间地基上正交异性矩形中厚板弯曲解析解

对横观各向同性体通解进行双重傅里叶变换,获得了直角坐标系下横观各向同性弹性半空间地基受任意竖向荷载作用下的位移积分变换解;在此基础上建立了板与地基的变形协调方程,并与三个广义位移变量描述的弹性地基上四边自由正交各向异性矩形中厚板的弯曲控制方程相结合,用三角级数法,得出横观各向同性弹性半空间地基上四边自由正交异性矩形中厚板受任意竖向荷载作用的弯曲解析解。相关算例分析表明,本文方法是有效的。     

横观各向同性饱和地基上中厚圆板的非轴对称振动

研究横观各向同性饱和土地基上中厚弹性圆板的非轴对称振动问题。基于横观各向同性饱和介质Biot波动方程的一般解,按混合边值问题建立了饱和地基与弹性中厚圆板非轴对称动力相互作用的对偶积分方程,并将对偶积分方程转化为易于计算的第二类Fredholm积分方程;采用数值方法求解该积分方程。数值算例结果表明,当h/a>0.05时,饱和半空间体上中厚度圆板在不同频率下的振动特性与相应频率下的刚性板的振动特性基本相同,当h/a<0.05时,板中心的位移将随h/a的减小而增大。     

Three-dimensional elastic solutions for functionally graded circular plates

Three-dimensional elastic solutions are obtained for a functionally graded thick circular plate subject to axisymmetric conditions. We consider a isotropic material where the Young modulus depends exponentially on the position along the thickness, while the Poisson ratio is constant. The solution method utilises a Plevako’s representation form which reduces the problem to the construction of a potential function satisfying a linear fourth-order partial differential equation. We write this potential function in terms of Bessel functions and we pointwise assign mixed boundary conditions. The analytic solution is obtained in a general form and explicitly presented by assuming transversal load on the upper face and zero displacements on the mantle; this is done by superposing the solutions of problems with suitably imposed radial displacement. We validate the solution by means of a finite element approach; in this way, we highlight the effects of the material inhomogeneity and the limits of the employed numerical method near the mantle, where the solution shows a large sensitivity to the boundary conditions.     

Free axisymmetric vibration of transversely isotropic piezoelectric circular plates

Based on three-dimensional elastic theory of piezoelectric materials, the axisymmetric state space formulation of piezoelectric laminated circular plates is derived. Finite Hankel transforms are used and the boundary variables in free terms are replaced, for two kinds of boundary conditions, to obtain ordinary differential equations with constant coefficients. Regarding the axisymmetric free vibration problem, two exact solutions for two different boundary conditions are found. Discarding piezoelectric effect, the exact solutions for transversely isotropic circular laminates are also obtained through the same procedure. Numerical examples are given and compared with those of Finite Element Method (FEM) .     

Winkler地基上四边自由正交各向异性矩形薄板弯曲问题的辛叠加解

研究Winkler地基上正交各向异性矩形薄板弯曲方程所对应的Hamilton正则方程, 计算出其对边滑支条件下相应Hamilton算子的本征值和本征函数系, 证明该本征函数系的辛正交性以及在Cauchy主值意义下的完备性, 进而给出对边滑支边界条件下Hamilton正则方程的通解, 之后利用辛叠加方法求出Winkler地基上四边自由正交各向异性矩形薄板弯曲问题的解析解. 最后通过两个具体算例验证了所得解析解的正确性.     

BENDING OF SECTORIAL LEVINSON PLATES: RELATIONSHIPS WITH CLASSICAL SOLUTIONS*

ABSTRACT

Relationships between bending solutions (deflections and stress resultants) of the Levinson third-order plate theory are presented in terms of the solutions of the classical plate theory for sectorial plates. These relationships are algebraic in nature, and can be readily used to determine the shear deformation theory (based on the Levinson plate theory) solutions of sectorial plates without going through the process of solving the governing equations. The sectorial plates considered have simply supported radial edges while the circular curved edges may be either simply supported, clamped, or free. Numerical results are presented to illustrate the usefulness of the developed relationships.

     

Green’s functions of a surface-stiffened transversely isotropic half-space

Green’s functions of a transversely isotropic half-space overlaid by a thin coating layer are analytically obtained. The surface coating is modeled by a Kirchhoff thin plate perfectly bonded to the half-space. With the aid of superposition technique and employing appropriate displacement potential functions, the Green’s functions are expressed in two parts; (i) a closed-form part corresponding to the transversely isotropic half-space with surface kinematic constraints, and (ii) a numerically evaluated part reflecting the interaction between the half-space and the plate in the form of semi-infinite integrals. Some limiting cases of the problem such as surface-stiffened isotropic half-space, Boussinesq and Cerruti loadings, and extremely flexible and rigid plates are also studied. For the classical Cerruti problem in transversely isotropic materials, the effects of incompressibility are highlighted. Numerical results are provided to show the effects of material anisotropy, relative stiffness factor, and load buried depth. The obtained Green’s functions play a key role in treating further mixed-boundary-value problems in surface stiffened transversely isotropic half-spaces.     

Elasticity solutions for functionally graded annular plates subject to biharmonic loads

Based on England’s expansion formula for displacements, the elastic field in a transversely isotropic functionally graded annular plate subjected to biharmonic transverse forces on its top surface is investigated using the complex variables method. The material parameters are assumed to vary along the thickness direction in an arbitrary fashion. The problem is converted to determine the expressions of four analytic functions α (ζ), β (ζ), ? (ζ) and ψ (ζ) under certain boundary conditions. A series of simple and practical biharmonic loads are presented. The four analytic functions are constructed carefully in a biconnected annular region corresponding to the presented loads, which guarantee the single-valuedness of the mid-plane displacements of the plate. The unknown constants contained in the analytic functions can be determined from the boundary conditions that are similar to those in the plane elasticity as well as those in the classical plate theory. Numerical examples show that the material gradient index and boundary conditions have a significant influence on the elastic field.