A Hierarchical Finite Element for Geometrically Non-linear Vibration of Thick Plates

A p-version, hierarchical finite element for moderately thick isotropic plates is derived and free vibrations are studied. The effects of the rotatory inertia, transverse shear and geometrical non-linearity, due to moderately large displacements, are taken into account. The time domain free vibration equations of motion are obtained by applying the principle of virtual work, and are mapped into the frequency domain by the harmonic balance method. The ensuing frequency domain equations are solved by a predictor–corrector method. The convergence properties of the element, the influence of the plate's thickness and of the width to length ratio on the backbone curves and on the non-linear mode shapes are investigated. The first and higher order modes are analysed and results are compared with published results.     

A Theory of Chiral Cosserat Elastic Plates

The mechanical behaviour of chiral materials is of interest for the investigation of carbon nanotubes, honeycomb structures, auxetic materials and bones. This paper is concerned with a theory of chiral Cosserat elastic plates. In this theory, in contrast with the case of achiral plates, the stretching and flexure cannot be treated independently of each other. First, we derive the basic equations which characterize the deformation of chiral plates. Then we establish a uniqueness result in the dynamical theory. In the equilibrium theory we establish conditions under which the Neumann problem admits solutions. Finally, the deformation of an infinite plate with a circular hole is studied. It is shown that, in contrast with the theory of Cosserat achiral plates a uniform pressure acting on the boundary of the hole produces a microrotation of the material particles.     

Existence and Integral Representations of Weak Solutions for Elastic Plates with Cracks

The existence and continuous dependence on the data are investigated in Sobolev spaces for the problem of bending of a Reissner-Mindlin-type plate weakened by a crack when the displacements or the moments and force are prescribed along the two sides of the crack. The cases of both an infinite and a finite plate are considered, and representations are sought for the solutions in terms of single layer and double layer potentials with distributional densities. This revised version was published online in August 2006 with corrections to the Cover Date.     

The Bending-Gradient Theory for Thick Plates: Existence and Uniqueness Results

This paper is devoted to the mathematical justification of the Bending-Gradient theory which is considered as the extension of the Reissner-Mindlin theory (or the First Order Shear Deformation Theory) to heterogeneous plates. In order to rigorously assess the well-posedness of the Bending-Gradient problems, we first assume that the compliance tensor related to the generalized shear force is positive definite. We define the functional spaces to which the variables of the theory belong, then state and prove the existence and uniqueness theorems of solutions of the Bending-Gradient problems for clamped and free plates, as well as for simply supported plates. The obtained results are afterward extended to the general case, i.e., when the compliance tensor related to generalized shear forces is not definite.     

Integration of an Equilibrium System in an Enhanced Theory of Bending of Elastic Plates

The complete integral of the system of partial differential equations governing the equilibrium bending of elastic plates with transverse shear deformation and transverse normal strain is constructed by means of complex variable methods. The process helps to elucidate the physical meaning of certain analytic constraints imposed on the asymptotic behavior of the solutions and shows that in the case of an infinite plate, any analytic solution has finite energy if and only if the bending and twisting moments, the transverse shear force, the displacements in vertical planes, and two other characteristic quantities vanish at infinity. An example is discussed to illustrate the theory.     

弹性薄板的后屈曲特性

文章用三维弹性理论的非线性应变公式研究了弹性薄板的后屈曲特性，旨在探讨冯·卡门平板大挠度方程的可靠性范围。计算结果表明，当板中薄膜力较大时，用冯·卡门方程描述后屈曲问题会产生较大的误差。     

Thermal Stresses in Elastic: Plates*

A method is proposed for the solution of elastic quadrilateral plates under thermal loads. The procedure employs eigenfunctions for angular regions, together with a least squares technique. An electronic computer is a necessary adjunct to the proposed method of solution. Numerical results are given     

弹性薄圆板的后屈曲分析

本文对薄圆板的后屈曲进行了研究。采用Galerkin法,试函数选为Legendre多项式,控制方程是Von-karman大挠度方程。考虑了简支,夹支两种边界条件。计算结果与有关文献[1]进行了比较,表明以Legendre多项式为试函数收敛快,精度高,且计算工作量较文献[1]为小。     

损伤弹性薄板的弯曲

通过损伤弹性薄板的变分方法,推导了损伤弹性薄板弯曲的运动控制方程．选取满足边界条件的挠度函数,采用Ritz法和 Galerkin法,将原问题转化为线性方程组的求解．通过算例分析,得到y=b/2处挠度和损伤随x的变化曲线,结果表明损伤薄板中任一点的位移总是大于无损薄板中的位移．     

Some Theorems in the Theory of Microstretch Thermomagnetoelectroelasticity

The linear dynamic theory of microstretch thermomagnetoelectroelasticity is studied in this paper.First,a reciprocity relation which involves two processes at different instants is established to form the basis of a uniqueness result and a reciprocal theorem.The proof of the reciprocal theorem avoids both using the Laplace transform and incorporating the initial conditions into the equations of motion.The uniqueness theorem is derived with no definiteness assumption on the elastic constitutive coefficients.Then the continuous dependence theorem is discussed upon two external data systems.Finally,the variational principle of Hamilton type which fully characterizes the solution of the mixed boundary-initial-value problem(mixed problem) is obtained.These theorems lay a solid foundation for further theoretical and numerical studies on microstretch thermomagnetoelectroelastic materials.     

Integral Solution of a Problem with Non-Standard Boundary Conditions in an Enhanced Theory of Bending of Elastic Plates

A modified boundary integral equation method is used to solve a specific type of mixed boundary value problem in an enhanced theory of bending of elastic plates in which the effects of transverse shear deformation and transverse normal strain are taken into account. The problem considered is characterized by the fact that a combination of transverse displacement and bending and twisting moments is prescribed on the curve which bounds the middle surface of the plate. Both interior and exterior problems are formulated and the corresponding existence and uniqueness results derived.     

Design of Inhomogeneous Plates on an Elastic Half-Space

An applied model is developed for designing an inhomogeneous plate on an elastic half-space. The plate is discretized by the finite-element method and the half-space by the boundary-element method. The model allows us to evaluate the stress–strain state of the plate immediately under a die, whose dimensions are comparable with the plate's cross section. The plate may be inhomogeneous across the thickness and in plan. Separation of the plate from the elastic base is possible.     

弹性地基上各向异性板的静力分析

根据弹性地基上各向异性矩形板弯曲挠度的微分方程精确的求得了适用于各种载荷的非齐次解和各类齐次解。其中由三角函数和双曲线函数组成的齐次解能满足四个边为任意边界条件的问题;由代数多项式和双正弦级数组成的齐次解能满足四个角为任意边界条件的问题。通过适当选取建立了满足任意边界条件和任意载荷作用的一般解。解中的积分常数完全由边界条件来决定。以四边简支承受均布载荷和局部分布载荷的对称迭层复合材料方板为例进行了计算和分析。其结果与已有文献结果是一致的。由于集中载荷不能求得作用点的弯矩,故在例题中改用局部分布载荷因而求得了最大弯矩。     

各向异性弹性体中矩形孔边缘应力分布

推导了各向异性平面弹性体中矩形孔洞边缘的应力分布公式并给出了数值计算结果。工程中会遇到矩形孔洞的问题，而矩形孔导致的应力集中增加了工程问题的复杂性。目前尚没有简单实用的公式计算矩形孔在各向异性体中应力的分布。本文采用复变函数保角变换的方法推导了矩形孔边缘的应力公式。在利用本文给出的方法，可以计算矩形孔在不同性质的各向异性材料中，承受任意的双轴荷载下孔边缘的应力分布情况。     

Boundary Integral Equations in Dynamic Problems for Elastic Plates

Initial-boundary value problems with Dirichlet and Neumann conditions arising in the theory of bending of plates with transverse shear deformation are reduced to time-dependent boundary integral equations by means of layer potentials. The solvability of these equations is then investigated in Sobolev-type spaces. This revised version was published online in June 2006 with corrections to the Cover Date.     

Elastic Wave Scattering and Dynamic Stress Concentrations in Stretching Thick Plates with Two Cutouts by Using the Refined Dynamic Theory

Based on the refined dynamic equation of stretching plates, the elastic tensioncompression wave scattering and dynamic stress concentrations in the thick plate with two cutouts are studied. In view of the problem that the shear stress is automatically satisfied under the free boundary condition, the generalized stress of the first-order vanishing moment of shear stress is considered. The numerical results indicate that, as the cutout is thick, the maximum value of the dynamic stress factor obtained using the refined dynamic theory is 19% higher than that from the solution of plane stress problems of elastic dynamics.     

Equilibrium of a Prestressed Strip Reinforced with Elastic Plates

The contact problem for a prestressed elastic strip reinforced with equally spaced elastic plates is considered. The Fourier integral transform is used to construct an influence function of a unit concentrated force acting on the infinite elastic strip with one edge constrained. The transmission of forces from the thin elastic plates to the prestressed strip is analyzed. On the assumption that the beam bending model and the uniaxial stress model are valid for an elastic plate subjected to both vertical and horizontal forces, the problem is mathematically formulated as a system of integro-differential equations for unknown contact stresses. This system is reduced to an infinite system of algebraic equations solved by the reduction method. The effect of the initial stresses on the distribution of contact forces in the strip under tension and compression is studied     

圆板非线性振动有限元分析的一种迭代方法

同时考虑横向振动和板平面内的运动,用3节点有限元研究均匀圆板的轴对称大振幅非线性振动,构造了一个避免发散加速收敛的平均迭代法,并将计算结果与文献的已有结果做了比较。