二维切口问题完备特征解的研究现状与进展

简要回顾和介绍了文献中常见的几种求解二维切口问题特征根的方法,评述这些方法 的特点及其求解范围．重点介绍了在求解二维切口问题完备特征根方面的一些进展,包括特征 根的求解方法、特征根的重根探讨以及相应的特征函数计算,这一问题基本上得到了解决．为 基于特征展开研究二维断裂问题打下了基础．      

Stress Distribution in a Rigidly Clamped Composite Plate with Locally Curved Structures under Forced Vibration

A problem on the forced vibrations of a rectangular composite plate with locally curved structures is formulated using the exact three-dimensional equations of continuum mechanics and continuum theory. A technique for numerical solution of the problem is developed based on the semianalytic finite-element method. Numerical results are given for the stress distribution in the plate under forced vibrations. The results obtained are analyzed to study the effect of the curvature in the structure of the plate on the distribution of stress amplitudes. It is shown that the curvatures change significantly the stress pattern under either static or dynamic loading     

固接圆环板在集中力作用下的求解

本文根据板的基本理论，在文献￣［１］的基础上对因接圆环板在集中力作用下进行了求解，进而提出计算固接圆环板在非对称载荷作用下应力与应变的方法，并编制了计算程序。本文中还进行了有限元分析，二者符合较好。本文的方法和程序可用于有关的工程计算设计。     

Interaction of a rigid punch and a circular plate with a fixed side and a stress-free face

The axisymmetric contact problem of a rigid punch indentation into an elastic circular plate with a fixed side and a stress-free face is considered. The problem is solved by a method developed for finite bodies which is based on the properties of a biorthogonal system of vector functions. The problem is reduced to a Volterra integral equation (IE) of the first kind for the contract pressure function and to a system of two Volterra IE of the first kind for functions describing the derivative of the displacement of the plate upper surface outside the punch and the normal (or tangential) stress on the plate lower fixed surface. The last two functions are sought as the sum of a trigonometric series and a power-law function with a root singularity. The obtained ill-conditioned systems of linear algebraic equations are regularized by introducing small parameters and have a stable solution. A method for solving the Volterra IE is given. The contact pressure functions, the normal and tangential stresses on the plate fixed surface, and the dimensionless indentation force are found. Several examples of a plane punch computation are given.     

An Energy Method Applied to Large Elastic Deflection of a Thin Plate of Bimodulus Material

Abstract

Some composite materials and high-polymers are known to behave differently in simple tension and compression under static loads. The present paper is concerned with a method of analysis of the bending of bimodulus elastic plates employing Ambartsumyan-Khachatryan's model for isotropic bimodulus materials. This problem may be reduced to the conventional problem of minimizing the potential energy of the plate as a whole. A simply supported thin square plate subjected to lateral load is analyzed numerically by a simplex method. Results of the calculation show that the effect of the difference between the tensile and compressive elastic moduli on the deformation of the plate may be substantial     

各向异性平板开孔动应力集中问题的研究

采用各向异性平板弯曲波动理论及摄动方法，对正交各向异性平板开孔弯曲波的散射及动应力集中问题进行了分析研究，得到了此种平板稳态弯曲波动问题的渐近形式的分析解。同时采用保角映射技术，为求解正交各向异性平板开孔弹性波的散射及动应力集中问题提供了一种统一规范的方法。     

Coupled hydrodynamic and structural analysis of compressible jet impact onto elastic panels

This paper is concerned with the initial stage of a compressible liquid jet impact onto an elastic plate. The fluid flow is governed by the linear wave equation, while the response of the plate is governed by the classical linear dynamical plate equation. The coupling between the fluid flow and the plate deflection is taken into account through the dynamic and kinematic conditions imposed on the wetted part of the plate. The problem is solved numerically by the normal mode method. The principal coordinates of the hydrodynamic pressure and plate deflections satisfy a system of ordinary differential and integral equations. A time stepping method based on the Runge–Kutta scheme is used for the numerical integration of the system. Calculations are performed for two-dimensional, axisymmetric and three-dimensional jet impacts onto an elastic plate. The effects of the impact conditions and the elastic properties of the plate on the magnitudes of the elastic deflections and bending stresses are analysed.     

A relook at Reissner’s theory of plates in bending

Shear deformation and higher order theories of plates in bending are (generally) based on plate element equilibrium equations derived either through variational principles or other methods. They involve coupling of flexure with torsion (torsion-type) problem and if applied vertical load is along one face of the plate, coupling even with extension problem. These coupled problems with reference to vertical deflection of plate in flexure result in artificial deflection due to torsion and increased deflection of faces of the plate due to extension. Coupling in the former case is eliminated earlier using an iterative method for analysis of thick plates in bending. The method is extended here for the analysis of associated stretching problem in flexure.     

Contact problem for a ring plate on an elastic half-space. Variational approach

A contact problem of an axisymmetrically loaded flexible ring plate lying frictionlessly on an elastic half-space is considered. The plate subsidences are represented as a power series with unknown coefficients, which are determined by the Rayleigh-Ritz method using the minimum condition for the total strain energy of the plate and the elastic foundation. The method of orthogonal polynomials is used implicitly. Belarussian State Polytechnical Academy, Minsk 220027. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 1, pp. 193–198, January–February, 1999.     

ELASTIC DYNAMIC ANALYSIS OF MODERATELY THICK PLATE USING MESHLESS LRPIM

A meshless local radial point interpolation method （LRPIM） for solving elastic dy-namic problems of moderately thick plates is presented in this paper. The discretized system equation of the plate is obtained using a locally weighted residual method. It uses a radial basis function （RBF） coupled with a polynomial basis function as a trial function,and uses the quartic spline function as a test function of the weighted residual method. The shape function has the properties of the Kronecker delta function,and no additional treatment is done to impose essen-tial boundary conditions. The Newmark method for solving the dynamic problem is adopted in computation. Effects of sizes of the quadrature sub-domain and influence domain on the dynamic properties are investigated. The numerical results show that the presented method can give quite accurate results for the elastic dynamic problem of the moderately thick plate.     

Scattering of rayleigh lamb waves from a 2d-cavity in an elastic plate

The T-matrix method is applied to the problem of scattering of Rayleigh-Lamb modes from a twodimensional cavity in an elastic plate. A formal solution is obtained which is valid also for non-planar surfaces. Explicit expressions and numerical results are given for a plate with plane surfaces.     

Multiple reciprocity method with two series of sequences of high-order fundamental solution for thin plate bending

IntroductionIt’swell_knownthatthecomplicatedfundamentalsolution[1,2 ]forHelmholtzequationΔu(x) +k2 u(x) =0　　(x∈Ω:boundedopenregioninR2 )isu (x,y) =-iH(2 )0 (k x-y ) 4,thusit’snotconvenientfornumericalcomputation .IfapplyingthesimplefundamentalsolutionofLaplaceequationu 0 (x ,y) =-ln｜x-y｜(2π) ,theexpressionforthesolutionofequationintheclosedregion Ωisc(y)u(y) + ∫Γu(x) u 0 (x,y) nx -u 0 (x ,y) u(x) n dsx =-k2∫Ωu(x)u 0 (x,y)dΩx.Astherightsideappearstheregionalintegrationinclu…     

Numerical method for optimum motion of undulatory swimming plate in fluid flow

A numerical method for the optimum motion of an undulatory swimming plate is presented. The optimum problem is stated as minimizing the power input under the condition of fixed thrust. The problem is singular for the invisible modes, and therefore the commonly used Lagrange multiplier method cannot predict an optimum solution but just a saddle point. To eliminate the singularity, an additional amplitude inequality constraint is added to the problem. A numerical optimization code with a sequential quadratic programming method is used to solve the problem. The method is applied to several cases of the motion of two-dimensional and three-dimensional undulatory plates, and the optimum results are obtained.     

Contact problems for a circular plate with sliding fixation at the end face

Two mixed elasticity problems of punch indentation into a circular plate placed without clearance in a rigid cylindrical holder with smooth walls are considered. In the first problem, the plate lies without friction on a rigid base, and in the second problem, the plate is rigidly fixed to the base. The problems are solved by a method that was developed for bodies of finite dimensions and is based on the properties of closed systems of orthogonal functions. Each of the problems is reduced to two integral equations, namely, a Volterra integral equation of the first kind for the contact pressure function and a Fredholm integral equation of the first kind for the derivatives of the displacement of the plate upper surface outside the punch. The displacement function is sought as the sum of a trigonometric series and a power function with a root singularity. After truncation, the obtained illposed system of linear algebraic equation has a stable solution. A method for solving Volterra integral equations is given. The contact pressure distribution function and the dimensionless indentation force are determined. Examples of calculation of the plate interaction with the plane punch are given. Contact problems were earlier studied for a rectangle and a circular plate with a stress-free end both without taking account of their fixation [1, 2] and with regard for their fixation [3, 4]. The solution method described here was used to study the interaction of elastic hollow cylinder of finite length with a rigid bandage and a rigid insert [5, 6]. Other papers dealing with contact problems for bodies of finite dimensions, in particular, for a circular plate, should also be mentioned. In these papers, the problems under study were solved by the method of homogeneous solutions [7, 8] and by the method of coupled series-equations [9].     

Stability of a cantilevered skew inhomogeneous plate in supersonic gas flow

This paper considers the vibrations of a skew inhomogeneous plate in gas flow. The plate is clamped in a certain section of one of its sides. Interaction of the flow with the plate is described using piston theory. The problem solution is based on the Hamilton’s variational principle and finite element method. The calculation results are compared with known data of theoretical studies and experiments. For the inhomogeneous plate, similarity parameters were established for the problem, which, in practically important cases, appears to be self-similar for one of the similarity parameters. This allows one to reduce the solution of this problem to the solution of an algebraic eigenvalue problem.     

A simple and accurate mixed FE-DQ formulation for free vibration of rectangular and skew Mindlin plates with general boundary conditions

A simple and accurate mixed finite element-differential quadrature formulation is proposed to study the free vibration of rectangular and skew Mindlin plates with general boundary conditions. In this technique, the original plate problem is reduced to two simple bar (or beam) problems. One bar problem is discretized by the finite element method (FEM) while the other by the differential quadrature method (DQM). The mixed method, in general, combines the geometry flexibility of the FEM and high accuracy and efficiency of the DQM and its implementation is more easier and simpler than the case where the FEM or DQM is fully applied to the problem. Moreover, the proposed formulation is free of the shear locking phenomenon that may be encountered in the conventional shear deformable finite elements. A simple scheme is also presented to exactly implement the mixed natural boundary conditions of the plate problem. The versatility, accuracy and efficiency of the proposed method for free vibration analysis of rectangular and skew Mindlin plates are tested against other solution procedures. It is revealed that the proposed method can produce highly accurate solutions for the natural frequencies of rectangular and skew Mindlin plates with general boundary conditions.     

Indentation of a Rigid Body into an Elastic Plate

The problem of a punch shaped like an elliptic paraboloid pressed into an elastic plate is studied under the assumption that the contact region is small. The action of the punch on the plate is modeled by point forces and moments. The method of joined asymptotic expansions is used to formulate the problem of one–sided contact for the internal asymptotic representation; the problem is solved with the use of the results obtained by L. A. Galin. The coordinates of the center of the elliptic contact region, its dimensions, and the angle of rotation are determined. The moments which ensure translational indentation of the punch are calculated and an equation that relates displacements of the punch to the force acting on it is given.     

Scattering of water waves by inclined thin plate submerged in finite-depth water

Summary  The problem of water wave scattering by an inclined thin plate submerged in water of uniform finite depth is investigated here under the assumption of irrotational motion and linear theory. A hypersingular integral equation formulation of the problem is obtained by an appropriate use of Green's integral theorem followed by utilization of the boundary condition on the plate. This hypersingular integral equation involves the discontinuity in the potential function across the plate, which is approximated by a finite series involving Chebyshev polynomials. The coefficients of this finite series are obtained numerically by collocation method. The quantities of physical interest, namely the reflection and transmission coefficients, force and moment acting on the plate per unit width, are then obtained numerically for different values of various parameters, and are depicted graphically against the wavenumber. Effects of finite-depth water, angle of inclination of the plate with the vertical over the deep water and vertical plate results for these quantities are shown. It is observed that the deep-water results effectively hold good if the depth of the mid-point of the submerged plate below the free surface is of the order of one-tenth of the depth of the bottom. Received 30 November 2000; accepted for publication 26 June 2001     

加强板的弯矩函数列式

本文首先谇薄板弯曲问题矩函数的物理意义,据此,将弯矩函数列式推广到具有加强条的薄板弯曲问题,给出了与平面弹性问题完全对应的余能原理。     

Computation of the stress resultants of a floating Mindlin plate in response to linear wave forces

In this paper, we focus on the computation of stress resultants of a floating elastic plate using the Mindlin plate theory. The proposed method makes use of the linear wave theory and shallow-draft assumption. However, the usual Kirchhoff theory is replaced by the Mindlin theory for the plate. For a single frequency, the coupled water-plate problem is solved using a higher-order-coupled finite element–boundary element method. The solutions for the stress-resultants computed using the proposed method are more satisfactory than these based on the Kirchhoff plate theory.