Bending Solutions for Inhomogeneous and Laminated Elastic Plates

A procedure has been developed in previous papers for constructing exact solutions of the equations of linear elasticity in a thick plate 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 thin plate or classical laminate theory equations (which are two-dimensional theories) generates, by straightforward substitutions, a solution of the three-dimensional elasticity equations for the homogeneous material. Recently this theory has been formulated in terms of functions of a complex variable. It was shown that the displacement and stress fields in the inhomogeneous material could be expressed in terms of four complex potentials that are analytic functions of the complex variable ζ = x + iy in the mid-plane of the plate. However, the analysis performed so far applies only to the case of a plate with traction-free upper and lower faces. The present paper extends these solutions to the case where the plate is bent by a pressure distribution applied to a face.     

Revisiting displacement functions in three-dimensional elasticity of inhomogeneous media

The paper presents a three-dimensional solution to the equilibrium equations for linear elastic transversely isotropic inhomogeneous media. We assume that the material has constant Poisson’s ratios, and its Young’s and shear moduli have the same functional form of dependence on the co-ordinate normal to the plane of isotropy. We show, apparently for the first time, that stresses and displacements in such an inhomogeneous transversely isotropic elastic solid can be represented in terms of two displacement functions which satisfy the second- and fourth-order partial differential equations. We examine and discuss key aspects of the new representation; they include the relationship between the new displacement functions and Plevako’s solution for isotropic inhomogeneous material with constant Poisson’s ratio as well as the application of the new representation to some important classes of three-dimensional elasticity problems. As an example, the displacement function is derived that can be used to determine stresses and displacements in an inhomogeneous transversely isotropic half-space which is subjected to a concentrated force normal to a free surface and applied at the origin (Boussinesq’s problem).     

Formulation of Problems in the General Kirchhoff—Love Theory of Inhomogeneous Anisotropic Plates

In this paper we study the procedure of reducing the three-dimensional problem of elasticity theory for a thin inhomogeneous anisotropic plate to a two-dimensional problem in the median plane. The plate is in equilibrium under the action of volume and surface forces of general form. À notion of internal force factors is introduced. The equations for force factors (the equilibrium equations in the median plane) are obtained from the thickness-averaged three-dimensional equations of elasticity theory. In order to establish the relation between the internal force factors and the characteristics of the deformed middle surface, we use some prior assumptions on the distribution of displacements along the thickness of the plate. To arrange these assumptions in order, the displacements of plate points are expanded into Taylor series in the transverse coordinate with consideration of the physical hypotheses on the deformation of a material fiber being originally perpendicular to the median plane. The well-known Kirchhoff—Love hypothesis is considered in detail. À closed system of equations for the theory of inhomogeneous anisotropic plates is obtained on the basis of the Kirchhoff—Love hypothesis. The boundary conditions are formulated from the Lagrange variational principle.     

Two-dimensional model of a plate made of an anisotropic inhomogeneous material

We present an asymptotic derivation of the two-dimensional equations of equilibrium of a thin elastic inhomogeneous plate manufactured of an anisotropic material of general form with 21 moduli of elasticity. We also consider simplified models obtained under special assumptions on the moduli. We use test examples to illustrate the error estimate of the proposed model and discuss its scope. The model is compared with the classical Kirchhoff–Love and Timoshenko–Reissner models.     

Energy Scaling of Compressed Elastic Films –Three-Dimensional Elasticity¶and Reduced Theories

We derive an optimal scaling law for the energy of thin elastic films under isotropic compression, starting from three-dimensional nonlinear elasticity. As a consequence we show that any deformation with optimal energy scaling must exhibit fine-scale oscillations along the boundary, which coarsen in the interior. This agrees with experimental observations of folds which refine as they approach the boundary. We show that both for three-dimensional elasticity and for the geometrically nonlinear Föppl-von Kármán plate theory the energy of a compressed film scales quadratically in the film thickness. This is intermediate between the linear scaling of membrane theories which describe film stretching, and the cubic scaling of bending theories which describe unstretched plates, and indicates that the regime we are probing is characterized by the interplay of stretching and bending energies. Blistering of compressed thin films has previously been analyzed using the Föppl-von Kármán theory of plates linearized in the in-plane displacements, or with the scalar eikonal functional where in-plane displacements are completely neglected. The predictions of the linearized plate theory agree with our result, but the scalar approximation yields a different scaling.     

拉压不同弹性模量薄板上圆孔的应力分析

采用弹性理论研究了拉压不同弹性模量薄板上圆孔的孔边应力集中问题．采用广义虎克定律推导出了拉压不同弹性模量薄板上圆孔边的应力平衡方程,并联合利用应力函数及边界条件得到了拉压不同弹性模量薄板上圆孔边的应力表达式．算例分析表明,当薄板材料的拉压弹性模量相差较大时,采用经典弹性理论研究薄板上圆孔的孔边应力是不合适的,当经典弹性理论与拉压不同弹性模量弹性理论的计算结果间的差别超过工程允许误差5%时,应该采用拉压不同弹性模量弹性理论进行计算．     

功能梯度矩形厚板的三维热弹性分析

直接从三维热弹性力学基本方程出发，通过引入两个位移函数和两个应力函数，导出了一个二阶的齐次状态方程和一个四阶的非齐次状态方程。分析中采用了层合近似模型，即将板划分成厚度足够小的若干薄层，从面可将每一层内的材料常数近似为常数。给出了任意厚度的四边简支横观各向同性功能梯度矩形板的热弹性分析，特别当板较薄时，与薄板理论进行了数值对比，发现两者结果吻合很好。最后研究了材料梯度指标对热弹性场的影响，结果显示梯度指标对热应力和位移都有着显著的作用：在不同的区间，梯度指标对它们有不同的影响；并且在同一区间，梯度指标对两者的影响程度也有所不同。     

Benchmark trigonometric and 3-D elasticity solutions for an exponentially graded thick rectangular plate

The bending problem of a transverse load acting on an isotropic inhomogeneous rectangular plate using both two-dimensional (2-D) trigonometric and three-dimensional (3-D) elasticity solutions is considered. In the present 2-D solution, trigonometric terms are used for the displacements in addition to the initial terms of a power series through the thickness. The effects due to transverse shear and normal deformations are both included. The form of the assumed 2-D displacements is simplified by enforcing traction-free boundary conditions at the faces of the plate. No transverse shear correction factors are needed because a correct representation of the transverse shearing strain is given. The plate material is exponentially graded, meaning that Lamé’s coefficients vary exponentially in a given fixed direction (the thickness direction). A wide variety of results for the displacements and stresses of an exponentially graded rectangular plate are presented. The validity of the present 2-D trigonometric solution is demonstrated by comparison with the 3-D elasticity solution. The influence of aspect ratio, side-to-thickness ratio and the exponentially graded parameter on the bending response are investigated.     

AN ACCURATE TWO-DIMENSIONAL THEORY OF VIBRATIONS OF ISOTROPIC,ELASTIC PLATES

An infinite system of two-dimensional equations of motion of isotropic elastic plates with edge and corner conditions are deduced from the three-dimensional equations of elasticity by expansion of displacements in a series of trigonometrical functions and a linear function of the thickness coordinate of the plate. The linear term in the expansion is to accommodate the in-plane displacements induced by the rotation of the plate normal in low-frequency flexural motions. A system of first-order equations of flexural motions and accompanying boundary conditions are extracted from the infinite system. It is shown that the present system of equations is equivalent to the Mindlin’s first-order equations, and the dispersion relation of straight-crested waves of the present theory is identical to that of the Mindlin’s without introducing any corrections. Reduction of present equations and boundary conditions to those of classical plate theories of flexural motions is also presented.     

THREE-DIMENSIONAL MODELING FOR THIN PLATE-LIKE STRUCTURES INCLUDING SURFACE EFFECTS BY USING STATE SPACE METHOD

<正>A three-dimensional(3-D)approach based on the state space method is proposed to study size-dependent mechanical properties of ultra-thin plate-like elastic structures considering surface effects.The structure is modeled as a laminate composed of a bulk bounded with upper and bottom surface layers,which are allowed to have different material properties from the bulk layer.State equations,including the surface properties of the structure,can be established on the basis of 3-D fundamental elasticity to analyze the size-dependent static characteristics of the thin plate-like structure.Compared with two-dimensional plate theories based size-dependent models for thin film structures in literature,the present 3-D approach is exact,which can provide benchmark results to assess the accuracy of 2-D plate theories and various numerical approaches. To show the feasibility of the proposed approach,a 3-D analytical solution for a simply supported plate-like thin structure including surface layers is derived.An algorithm is proposed for the calculation of the state equations obtained to ensure that the numerical results can reveal the surface effects clearly even for extremely thin surface layers.Numerical examples are carried out to exhibit the surface effects and some discussions are provided based on the results obtained.     

Modified equations of finite-size layered plates made of orthotropic material. Comparison of the results of numerical calculations with analytical solutions

This paper describes the modified bending equations of layered orthotropic plates in the first approximation. The approximation of the solution of the equation of the three-dimensional theory of elasticity by the Legendre polynomial segments is used to obtain differential equations of the elastic layer. For the approximation of equilibrium equations and boundary conditions of three-dimensional theory of elasticity, several approximations of each desired function (stresses and displacements) are used. The stresses at the internal points of the plate are determined from the defining equations for the orthotropic material, averaged with respect to the plate thickness. The construction of the bending equations of layered plates for each layer is carried out with the help of the elastic layer equations and the conjugation conditions on the boundaries between layers, which are conditions for the continuity of normal stresses and displacements. The numerical solution of the problem of bending of the rectangular layered plate obtained with the help of modified equations is compared with an analytical solution. It is determined that the maximum error in determining the stresses does not exceed 3 %.     

SHEAR-HORIZONTAL WAVES IN A ROTATED Y-CUT QUARTZ PLATE WITH AN ISOTROPIC ELASTIC LAYER OF FINITE THICKNESS

We study shear-horizontal (SH) waves in a rotated Y-cut quartz plate carrying an isotropic elastic layer of finite thickness.The three-dimensional theories of anisotropic elasticity and isotropic elast...     

Elastic stability of inhomogeneous thin plates on an elastic foundation

We study the elastic stability of infinite inhomogeneous thin plates on an elastic foundation under in-plane compression. The elastic stiffness constants depend on the coordinate variable in the thickness direction of the plate. The elastic foundation is represented as a Winkler-type model characterized by linear and nonlinear spring constants. First we derive the Föppl–von Kármán equations by taking variations of the elastic strain energy. Next we develop the linear stability analysis of the plate under uniform in-plane compression and explicitly derive the critical loads and wave numbers for particular three cases. The effects of the material inhomogeneity, material orthotropy and loading orthotropy on the critical states are examined independently. Finally, we perform a weakly nonlinear analysis of the plate at the onset of the buckling instability. With the multiple scales method, the amplitude equations for the unstable modes that provide insight into the mode type and its amplitude are derived and then the effect of the material inhomogeneity on buckling modes are evaluated qualitatively.     

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.     

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.     

Solution of a Riemann problem for elasticity

This paper describes a numerical algorithm for the Riemann solution for nonlinear elasticity. We assume that the material is hyperelastic, which means that the stress-strain relations are given by the specific internal energy. Our results become more explicit under further assumptions: that the material is isotropic and that the Riemann problem is uniaxial. We assume that any umbilical points lie outside the region of physical relevance. Our main conclusion is that the Riemann solution can be obtained by the iterative solution of functional equations (Godunov iterations) each defined in one- or two-dimensional spaces.Supported in part by AFOSR-88-0025.     

Elastic and viscoelastic solutions to rotating functionally graded hollow and solid cylinders

Analytical solutions to rotating functionally graded hollow and solid long cylinders are developed. Young＇s modulus and material density of the cylinder are assumed to vary exponentially in the radial direction, and Poisson＇s ratio is assumed to be constant. A unified governing equation is derived from the equilibrium equations, compatibility equation, deformation theory of elasticity and the stress-strain relationship. The governing second-order differential equation is solved in terms of a hypergeometric function for the elastic deformation of rotating functionally graded cylinders. Dependence of stresses in the cylinder on the inhomogeneous parameters, geometry and boundary conditions is examined and discussed. The proposed solution is validated by comparing the results for rotating functionally graded hollow and solid cylinders with the results for rotating homogeneous isotropic cylinders. In addition, a viscoelastic solution to the rotating viscoelastic cylinder is presented, and dependence of stresses in hollow and solid cylinders on the time parameter is examined.     

End Effects in Anti-plane Shear for an Inhomogeneous Isotropic Linearly Elastic Semi-infinite Strip

The purpose of this research is to further investigate the effects of material inhomogeneity on the decay of Saint-Venant end effects in linear isotropic elasticity. This is carried out within the context of anti-plane shear deformations of an inhomogeneous isotropic elastic solid. The mathematical issues involve the effects of spatial inhomogeneity on the decay rates of solutions to Dirichlet or Neumann boundary-value problems for a second-order linear elliptic partial differential equation with variable coefficients on a semi-infinite strip. In previous work [1], the elastic coefficients were assumed to be smooth functions of the transverse coordinate so that the material was inhomogeneous in the lateral direction only. Here we develop a new technique, based on a change of variable, to study generally inhomogeneous isotropic materials. The governing partial differential equation is transformed to a Helmholtz equation with a variable coefficient, which facilitates analysis of the influence of material inhomogeneity on the diffusion of end effects. For certain classes of inhomogeneous materials, an explicit optimal decay estimate is established. The results of this paper are applicable to continuously inhomogeneous materials and, in particular, to functionally graded materials. This revised version was published online in August 2006 with corrections to the Cover Date.     

An asymptotic strain gradient Reissner-Mindlin plate model

In this paper we derive a strain gradient plate model from the three-dimensional equations of strain gradient linearized elasticity. The deduction is based on the asymptotic analysis with respect of a small real parameter being the thickness of the elastic body we consider. The body is constituted by a second gradient isotropic linearly elastic material. The obtained model is recognized as a strain gradient Reissner-Mindlin plate model. We also provide a mathematical justification of the obtained plate model by means of a variational weak convergence result.