A refined theory of elastic thick plates for extensional deformation

Based on elasticity theory, various two-dimensional (2D) equations and solutions for extensional deformation have been deduced systematically and directly from the three-dimensional (3D) theory of thick rectangular plates by using the Papkovich–Neuber solution and the Lur’e method without ad hoc assumptions. These equations and solutions can be used to construct a refined theory of thick plates for extensional deformation. It is shown that the displacements and stresses of the plate can be represented by the displacements and transverse normal strain of the midplane. In the case of homogeneous boundary conditions, the exact solutions for the plate are derived, and the exact equations consist of three governing differential equations: the biharmonic equation, the shear equation, and the transcendental equation. With the present theory a solution of these can satisfy all the fundamental equations of 3D elasticity. Moreover, the refined theory of thick plate for bending deformation constructed by Cheng is improved, and some physical or mathematical explanations and proof are provided to support our justification. It is important to note that the refined theory is consistent with the decomposition theorem by Gregory. In the case of nonhomogeneous boundary conditions, the approximate governing differential equations and solutions for the plate are accurate up to the second-order terms with respect to plate thickness. The correctness of the stress assumptions in the classic plane-stress problems is revised. In an example it is shown that the exact or accurate solutions may be obtained by applying the refined theory deduced herein.     

Love solutions in the linear inhomogeneous transversely isotropic theory of elasticity

A general Love solution for the inhomogeneous transversely isotropic theory of elasticity with the elastic constants dependent on the coordinate z is proposed. This result may be considered as a generalization of the Love solutions we recently derived for the inhomogeneous isotropic theory of elasticity. The key steps of deriving the Love solution for the classical linear homogeneous transversely isotropic theory of elasticity are described for further use of the derivation procedure, which is then generalized to the inhomogeneous transversely isotropic case. Some particular cases of inhomogeneity traditionally used in the theory of elasticity are also examined. The significance of the derived solutions and their importance for the modeling of functionally graded materials are briefly discussed     

An exact analytical approach for in-plane and out-of-plane free vibration analysis of thick laminated transversely isotropic plates

In this article, the governing equations of motion of thick laminated transversely isotropic plates are derived based on Reddy’s third-order shear deformation theory. These equations are exactly converted to four uncoupled equations to study the in-plane and out-of-plane free vibrations of thick laminated plates without any usage of approximate methods. Based on the present analytical approach, exact Levy-type solutions are obtained for thick laminated transversely isotropic plates and, for some boundary conditions, the exact characteristic equations hitherto not reported in the literature are given. Also, the in-plane and out-of-plane deformed mode shapes are plotted for different boundary conditions. The present solutions can accurately predict both the in-plane and out-of-plane natural frequencies and mode shapes of thick laminated transversely isotropic plates.     

Three-dimensional elasticity solution for bending of transversely isotropic functionally graded plates

This paper presents a three-dimensional elasticity solution for a simply supported, transversely isotropic functionally graded plate subjected to transverse loading, with Young’s moduli and the shear modulus varying exponentially through the thickness and Poisson’s ratios being constant. The approach makes use of the recently developed displacement functions for inhomogeneous transversely isotropic media. Dependence of stress and displacement fields in the plate on the inhomogeneity ratio, geometry and degree of anisotropy is examined and discussed. The developed three-dimensional solution for transversely isotropic functionally graded plate is validated through comparison with the available three-dimensional solutions for isotropic functionally graded plates, as well as the classical and higher-order plate theories.     

薄板理论的正交关系及其变分原理

利用平面弹性与板弯曲的相似性理论，将弹性力学新正交关系中构造对偶向量的思路推广到 各向同性薄板弹性弯曲问题，由混合变量求解法直接得到对偶微分方程并推导了对应的变分 原理. 所导出的对偶微分矩阵具有主对角子矩阵为零矩阵的特点. 发现了两个独立的、对称 的正交关系，利用薄板弹性弯曲理论的积分形式证明了这种正交关系的成立. 在恰当选择对 偶向量后，弹性力学的新正交关系可以推广到各向同性薄板弹性弯曲理论.     

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.     

THEORY OF ELASTICITY OF AN ANISOTROPIC BODY FOR THE BENDING OF BEAMS

A theory of elasticity for the bending of orthogonal anisotropic beams has been developed by analogy with the special case, which can be obtained by applying the theory of elasticity for bending of transversely isotropic plates to the problems of two deminsions. In this paper, we present a method to solve the problems of bending of orthogonal anisotropic beams and a new theory of the deep-beam whose ratio of depth to length is larger. It is pointed out that Reissner's theory to account for the effect of transverse shear deformation is not very approximate in the components of stress,     

复合材料层合厚板锯齿理论及有限元分析

对于较厚复合材料弯曲问题,已有锯齿型厚板理论最大误差超过35%。为了合理地分析较厚复合材料弯曲问题,发展了准确高效的锯齿型厚板理论。此理论位移变量个数独立于层合板层数,其面内位移不含有横向位移一阶导数,构造有限元时仅需C0插值函数,故称此理论为C0型锯齿厚板理论。基于发展的锯齿理论,构造了六节点三角形单元并推导了复合材料层合/夹层板弯曲问题有限元列式。为验证C0型锯齿厚板理论性能,分析了复合材料层合/夹层厚板弯曲问题,并与已有C1型锯齿理论对比。结果表明,本文的C0型锯齿厚板理论最大误差15%,比已有锯齿型厚板理论准确高效。     

Representation of the solution of refined bending theory for isotropic plates

A solution of the bending problem for isotropic plates in a refined statement based on the system of six-order differential equations is proposed. A procedure for determining the general solutions of the corresponding biharmonic and metaharmonic equations is suggested. A method for satisfying the boundary conditions is given. The results of numerical studies of the stress state of an infinite plate with an elliptic cavity are given.     

3D thermally induced analysis of annular plates of functionally graded materials

Within the framework of three-dimensional elasticity theory, this paper investigates the thermal response of functionally graded annular plates in which the material can be transversely isotropic and vary along the thickness direction in an arbitrary manner. The generalized Mian and Spencer method is utilized to obtain the analytical solutions of annular plates under a through-thickness steady temperature field. The present analytical solutions are validated through comparisons against those available in open literature. A parametric study is conducted to examine the effects of gradient distribution, different temperature fields, different diameter ratio and boundary conditions on the deformation and stress fields of the plate. The results show that these factors can have obvious effects on the thermo-elastic behavior of functionally gradient materials(FGM)annular plates.     

The simply supported rectangular plate: An exact,three dimensional,linear elasticity solution

An exact three dimensional solution for the problem of a transversely loaded, simply supported rectangular plate of arbitrary thickness is presented within the linear theory of elastostatics. The solution, obtained in a semi-inverse fashion, satisfies all the boundary conditions of the problem in a pointwise manner and is in the form of a double Fourier sine series. The classical Navier solution for the problem is shown to be the limit of the present solution as the plate thickness aspect ratio approaches zero. It is noted that the solution presented provides a benchmark against which approximate theories of transversely loaded plates may be measured. The new elasticity solution also provides a heuristic basis for a novel theory of thick plates of arbitrary planform and edge support recently given by the author.     

A new trigonometric shear deformation theory for isotropic,laminated composite and sandwich plates

A new trigonometric shear deformation theory for isotropic and composite laminated and sandwich plates, is developed. The new displacement field depends on a parameter “m”, whose value is determined so as to give results closest to the 3D elasticity bending solutions. The theory accounts for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surface, thus a shear correction factor is not required. Plate governing equations and boundary conditions are derived by employing the principle of virtual work. The Navier-type exact solutions for static bending analysis are presented for sinusoidally and uniformly distributed loads. The accuracy of the present theory is ascertained by comparing it with various available results in the literature. The results show that the present model performs as good as the Reddy’s and Touratier’s shear deformation theories for analyzing the static behavior of isotropic and composite laminated and sandwich plates.     

Static analysis of simply supported functionally graded and layered magneto-electro-elastic plates

In this article, static analysis of functionally graded, anisotropic and linear magneto-electro-elastic plates have been carried out by semi-analytical finite element method. A series solution is assumed in the plane of the plate and finite element procedure is adopted across the thickness of the plate such a way that the three-dimensional character of the solution is preserved. The finite element model is derived based on constitutive equation of piezomagnetic material accounting for coupling between elasticity, electric and magnetic effect. The present finite element is modeled with displacement components, electric potential and magnetic potential as nodal degree of freedom. The other fields are calculated by post-computation through constitutive equation. The functionally graded material is assumed to be exponential in the thickness direction. The numerical results obtained by the present model are in good agreement with available functionally graded three-dimensional exact benchmark solutions given by Pan and Han [Pan, E., Han, F., in press. Green’s function for transversely isotropic piezoelectric functionally graded multilayered half spaces. Int. J. Solids Struct.]. Numerical study includes the influence of the different exponential factor, magneto-electro-elastic properties and effect of mechanical and electric type of loading on induced magneto-electro-elastic fields. In addition further study has been carried out on non-homogeneous transversely isotropic FGM magneto-electro-elastic plate available in the literature [Chen, W.Q., Lee, K.Y., Ding, H.J., 2005. On free vibration of non-homogeneous transversely isotropic magneto-electro-elastic plates].     

Fundamental solutions of the thermoelastic equations for transversely isotropic plates

The problem of thermoelasticity for transversely isotropic plates acted upon by concentrated heat sources is solved. The {1, 2}-order equations of thermoelasticity that incorporate the transverse shear and normal stresses are used. A bending heat source with symmetric heat transfer is considered. The dependence of thermal stress components on the thermal and thermomechanical parameters of transversely isotropic plates is studied     

An exact solution for transversely isotropic,simply supported and layered rectangular plates

Levinson's solution for the problem of a simply supported rectangular plate of arbitrary thickness by normal surface loads is extended to the transversely isotropic and layered case. The exact closed form solution is obtained by using the propagator matrix method in a system of vector functions. As a special case of the layered medium, the normal displacement or deflection of a homogeneous plate of arbitrary thickness by normal surface loads is also given. It is shown that it approaches the classical solution for the transversely isotropic thin plate as the thickness approaches zero on the one hand, and on the other hand reduces to the thick plate expression as given by Levinson when the medium is isotropic.     

Experimental and theoretical analysis of the deformation of transversely isotropic plates under creep conditions

This paper describes the results of calculations and experiments on the torsion of plates made of isotropic and transversely isotropic VT-20 and 1163T alloys with low resistance to creep strain in the direction perpendicular to the median surface. The numerical simulation results for plates of different thicknesses related to the class of rigid and flexible plates are compared using the pure bending theory and the finite element method. It is found that the curvature values are smaller in the case of deformation of a plate made of anisotropic material into a sign-variable saddle surface than in the case of a plate of isotropic material. The calculation in the assumption of pure bending provides an upper bound of the curvature difference in the deformation of plates made of transversely isotropic and isotropic materials.     

Static analysis of a long and thick orthotropic circular cylindrical shell of revolution

Summary An elasticity solution has been obtained for a long, thick transversely isotropic circular cylindrical shell subjected to distributed pinch load using a set of three displacement functions. Numerical results are presented for different materials and thickness to mean radius ratios. The results obtained from this analysis have been compared with classical and first-order shear deformation shell theories of Flugge, Sanders, Love and Donnell.     

A Note on the Iterative Solution of Stress Problems for Plates and Shallow Shells

A technique based on a refined iterative theory and the numerical method of local variations is developed and used to determine the stress–strain state of transversely isotropic shallow shells and plates. All the components of the stress–strain state and boundary-layer effects are taken into account. The solutions are analyzed for accuracy and convergence.     

横观各向同性热弹性梁的精化理论

本文从横观各向同性梁的二维问题出发,研究了横观各向同性热弹性梁的精化理论。首先,在不作任何预先假设的条件下,利用横观各向同性热弹性理论和Lur’e算子函数,获得了由梁中线上的物理量表示的位移场和应力场。对热弹性梁上下表面承受非齐次边界条件的情况,推导出梁的近似控制微分方程。再舍去温度项,则横观各向同性热弹性梁的精化理论退化为横观各向同性梁的精化理论。     

Exact Solutions for a Thick Elastic Plate with a Thin Elastic Surface Layer

A procedure has been developed in previous papers for constructing exact solutions of the equations of linear elasticity in a plate (not necessarily thin) of inhomogeneous isotropic linearly elastic material in which the elastic moduli depend in any specified manner on a coordinate normal to the plane of the plate. The essential idea is that any solution of the classical equations for a hypothetical thin plate or laminate (which are two-dimensional theories) generates, by straightforward substitutions, a solution of the three-dimensional elasticity equations for the inhomogeneous material. In this paper we consider a thick plate of isotropic elastic material with a thin surface layer of different isotropic elastic material. It is shown that the interface tractions and in-plane stress discontinuities are determined only by the initial two-dimensional solution, without recourse to the three-dimensional elasticity theory. Two illustrative examples are described.