The uniform heating and bending of a two-layer plate

The problem of the uniform heating of a two-layer plate is solved. The transversely isotropic layer considered (a soft plate) is in ideal contact with a rigid isotropic thin elastically deformed layer. The ends of the plate are load-free. A boundary layer of the soft plate (a thin contact layer) is introduced, which enables the boundary conditions on the ends of the plate to be formulated in such a way that the problem has a bounded smooth solution [1]. The two-layer plate, generally speaking, is bounded along the axis perpendicular to the axes directed along the length and thickness of the plate. The resultant force and the resultant moment, applied to the end transverse sections, are equal to zero. The exact solution of the temperature problem is sought using the equations of the theory of elasticity. The plane problem of the bending of a two-layer plate acted upon by a uniformly distributed pressure applied to the side surface of an anisotropic layer is solved by a similar method. The ends of the rigid isotropic layer are clamped.     

The thermoelastic problem for a halfspace with a semiinfinite elastic thermal insulator

Numerical conditions are given in an infinite and semiinfinite plate (heat insulator), which is connected by a vertical two-sided connection only with an elastic halfspace, in the interior of which is a concentrated source of heat, which generates a stationary heat field. The problem is reduced to the solution of an integral-differential equation of the Wiener-Hopf type with respect to the Fourier transform of the contact stress. Its exact solution is constructed using the factorization method, and the final solution is represented by a series with respect to Chebyshev-Laguerre polynomials. Calculations of bending moments and transverse forces are given in an infinite plate, semiinfinite, and infinite beam-rolling plates.Translated from Dinamicheskie Sistemy, No. 7, pp. 114–123, 1988.     

The use of the method of boundary states to analyse an elastic medium with cavities and inclusions

The analytical method of boundary states is developed and theoretically substantiated. A corollary of the Weierstrass theorem is proved according to which a function that is harmonic in a bounded, simply connected domain can be approximated by a series of homogeneous harmonic polynomials. A basis of the space of functions that are harmonic outside any neighbourhood of a point is constructed. An algorithm is developed for filling the basis of the space of the states of a multicavity elastic body. The method is used to solve a series of problems of determining of the stress-strain state of an unbounded elastic medium containing spherical cavities or inclusions with different boundary conditions: the boundary of the cavity is free (the Southwell problem), constrained or under conditions of contact with a rigid core. The effect of the width of the intercavity layer on the stress concentration is analysed in a non-axisymmetric problem with two cavities. The form of the relation between the mean-square discrepancy in the boundary conditions of the solution obtained and the number of elements in the basis is indicative of the numerical convergence of the solution of this problem.     

Contact of shells with a sharp linear stamp of finite length

The action of a plane, absolutely rigid stamp on a transversely isotropic shell is investigated. The use of the equations of shells with finite shear stiffness enables the correct formulation of the problem of the action on a shell by a stamp of fixed length. The problem is reduced to an integral equation. Applying the Fourier transform, the kernel of the integral equation is represented in the form of an expansion with respect to Chebyshev polynomials. By the representation of the solution of the integral equation in the form of a product, of a series of Chebyshev polynomials and a function that takes into account the singularities of the solution at the boundary of the contact zone, the considered problem is reduced to the solving of an infinite system of linear algebraic equations, whose coefficients have been determined by the methods of numerical integration. As an example a problem for a cylindrical shell has been solved.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 59–63, 1989.     

Some integral relations of Hankel transform type and applications to elasticity theory

The dominant part of an integral equation arising in connection with boundary value problems for the circular disc is evaluated in terms of orthogonal polynomials. This relation leads to an efficient method for numerical solution of the complete integral equation even in the presence of a complicated bounded kernel. The static problem of a circular crack in an infinite elastic body under general loads is used to illustrate vector boundary conditions leading to two coupled integral equations, while the problem of a vibrating flexible circular plate in frictionless contact with an elastic half space is solved by use of the associated numerical method.     

The difference method in a diffraction problem: The technique of interior boundary condition

The diffraction problem for a plane wave on a half-plane covered by thin layer with an interface is solved by the difference method. The system of difference equations is derived from the variational principle. A boundary solution at infinity must be imposed; this is a radiation condition, which is used in the form of the limit absorption principle. The arising infinite system of difference equations is reduced to a finite part of the boundary (the interface) by using the technique of so-called interior boundary conditions in the sense of Ryaben’kii. The real conditions are found by the Fourier method with respect to one spatial variable in the form of Fourier or Laurent series in the corresponding variable, which converge either inside, outside, or on the unit circle. Above the upper boundary of the layer, all unknowns are eliminated by using the so-called grid Green function, that is, the resolving function for the half-plane satisfying the radiation condition at infinity. For the unknowns on the upper boundary of the layer, an equation in terms of a function of a complex variable of Wiener-Hopf type is obtained, which is solved by factorization. Factorization is performed numerically: the logarithm of the function is expanded in a bi-infinite series, which is replaced by a discrete Fourier series. The closing system in a neighborhood of the interface has order proportional to the number of points on the interface. Solving this system yields all of the required characteristics of the solution.     

任意形状弹性薄板弯曲的精确解*

本文提供了一个求解任意形状弹性薄板弯曲的新方法,在求得了极坐标系中弹性薄板弯曲微分方程的精确解后,将解代入薄板的边界条件,利用Fourier级数将边界方程展开,可确定出各待定常数,所得结果是精确的。     

Effect of hydrostatic pressure on vibrating functionally graded circular plate coupled with bounded fluid

This study investigates free vibration of a thick FG circular plate in contact with an inviscid, and incompressible fluid. Analysis of plate is based on First-order Shear Deformation Reissner–Mindlin Theory (FSDT) with consideration of rotational inertial effects and transverse shear stresses. Potential theory together Bernouli's equation are utilized to obtain the fluid pressure on the free surface of the plate. The governing equation of the oscillatory behavior of the fluid is obtained by solving Laplace equation and satisfying its boundary conditions. The natural frequencies and mode shapes of the plate are determined using Chebyshev polynomials. The effects of the geometrical parameters such as plate thickness to its radius ratio, boundary conditions, fluid density, volume fraction index, and height of the fluid on natural frequencies and mode shapes are investigated. Comparison of analytically outcome of this study is made with similar publications in the literature.     

Analytical and experimental evaluation of stresses in elastic annuli subjected to three-point loading on the outer surface

An analytical solution based on the implementation of the Fourier series is presented for the full-field analysis of stresses in an annular plate subjected to external force loadings uniformly distributed over three small segments of the outer periphery when two of those segments are symmetrical about the diameter passing through the center of the third segment. This problem refers to an annular plate which rests upon two equidistant platens when compressed by a vertical locally-distributed force. The obtained theoretical results are compared with a series of photoelastic experiments for different distances between the loaded segments of the rim.  The feasibility of the mathematical model for the mixed-type boundary conditions is analyzed by comparison of the experimental results, the theoretical analysis, and FEM numerical simulation.     

Formation of cracks on compressing an unbounded brittle body with a circular opening

A simplified model of a brittle body /1/ is used as a basis for investigating the appearance of cracks originating at the boundary of a circular cavity in a body in a state of plane deformation caused by uniaxial compression at infinity. The singular integral equation of the problem is reduced to an integral Fredholm equation with a degenerate kernel. The solution is obtained in the form of a Fourier series in terms of Legendre polynomials.     

Reissner厚板弹性弯曲的一般解析解

针对大型工程建设中的Reisner厚板弹性弯曲问题,本文采用复级数方法求解相应的常系数偏微分方程组的边值问题,并首次得到了任意边界条件下的一般解析解.该解形式简单,计算方便、可靠.以四边简支和三边固支一边自由两种支撑条件下厚板承受均布载荷为例进行了分析验算,与已有的计算结果相比,计算结果相当满意.同时本文还着重对解的收敛速度、正确性(合理性)及边界满足情况进行了考察.     

变厚度圆薄板非对称非线性弯曲问题

首先将直角坐标系中的横向变厚度薄板的大挠度方程,转化到极坐标系中的变厚度圆薄板的非对称大挠度方程·　此方程和极坐标系中径向、切向两个平衡方程联立求解·　将物理方程和中面应变非线性变形方程,代入3个平衡方程,可得用3个变形位移表示的3个非对称非线性方程·　用Fourier级数表示的解代入基本方程,获得相应的基本方程·　在周边夹紧边界条件下,用修正迭代法求解·　作为算例,研究了余弦形式载荷作用下的问题,还给出了载荷与挠度的特征曲线,曲线依据变厚度参数变化而变化,其结果和物理概念完全吻合·     

An analytical spectral stiffness method for buckling of rectangular plates on Winkler foundation subject to general boundary conditions

An analytical spectral stiffness method is proposed for the efficient and accurate buckling analysis of rectangular plates on Winkler foundation subject to general boundary conditions (BCs). The method combines the advantages of superposition method, stiffness-based method and the Wittrick–Williams algorithm. First, exact general solutions of the governing differential equation (GDE) of plate buckling considering both elastic foundation and biaxial loading is derived by using a modified Fourier series. The superposition of such general solutions satisfy the GDE exactly and BCs approximately, which guarantees the rapid convergence and high accuracy. Then, based on the exact general solution, the spectral stiffness matrix which relates the coefficients of plate generalized displacement BCs and force BCs is symbolically developed. As a result, arbitrary BCs can be prescribed straightforwardly in the stiffness-based model. As an efficient and reliable solution technique, the Wittrick–Williams algorithm with the J₀ problem resolved is applied to obtain the critical buckling solutions. The accuracy and efficiency of the method are verified by comparing with other methods. Benchmark buckling solutions are provided for plates with all possible boundary conditions. Also, dependence of various factors such as foundation stiffness, load combinations and aspect ratio on the buckling behaviors are investigated.     

Harmonic and pulse excitations of multiply connected cylindrical bodies

The three-dimensional problem of the theory of elasticity of the harmonic oscillations of cylindrical bodies (a layer with several tunnel cavities on a cylinder of finite length) is considered for uniform mixed boundary conditions on its bases. Using the Φ-solutions constructed, the boundary-value problems are reduced to a system of well-known one-dimensional singular integral equations. The solution of the problem of the pulse excitation of a layer on the surface of a cavity is “assembled” from a packet of corresponding harmonic oscillations using an integral Fourier transformation with respect to time. The results of calculations of the dynamic stress concentration in a layer (a plate) weakened by one and two openings of different configuration are given, as well as the amplitude-frequency characteristics for a cylinder of finite length with a transverse cross section in the form of a square with rounded corners, and data of calculations for a trapeziform pulse, acting on the surface of a circular cavity, are presented.     

Approximate analytic solution of heat conduction problems with a mismatch between initial and boundary conditions

We consider a heat conduction problem for an infinite plate with a mismatch between initial and boundary conditions. Using the method of integral relations, we obtain an approximate analytic solution to this problem by determining the temperature perturbation front. The solution has a simple form of an algebraic polynomial without special functions. It allows us to determine the temperature state of the plate in the full range of the Fourier numbers (0≤F<∞) and is especially effective for very small time intervals.     

A mixed problem for an inhomogeneous wave equation with a summable potential

A mixed problem for an inhomogeneous wave equation with fixed ends in the case of a summable potential is studied. Using the Krylov method for acceleration of the convergence of Fourier series, a classical solution under minimal conditions on the initial data and a generalized solution in the case of quadratic summable initial data and perturbing function are obtained.     

Solution of the Falkner–Skan equation for wedge by Adomian Decomposition Method

The Adomian Decomposition Method is employed in the solution of the two dimensional laminar boundary layer of Falkner–Skan equation for wedge. This work aims at the solution of momentum equation in the case of accelerated flow and decelerated flow with separation. The Adomian Decomposition Method is provided an analytical solution in the form of an infinite power series. The effect of Adomian polynomials terms is considered on accuracy of the results. The velocity profiles in boundary layer are obtained. Results show a good accuracy compared to the exact solution.     

Analysis of the structure of the boundary layer in the problem of the torsion of a laminated spherical shell

The structures of the boundary layer in the problem of the torsion of a radially stratified spherical segment (shell) with an arbitrary number of alternating hard and soft layers are investigated. It is shown that weakly attenuating boundary-layer solutions exist. Despite the fact that a stress state, self-balanced in the section, corresponds to these elementary solutions, they may penetrate fairly deeply and considerably change the stress–strain state pattern far from the ends. Using an asymptotic analysis of the problem, an applied theory of torsion is proposed which takes into account weakly attenuating boundary-layer solutions.     

Mixed problem for a first-order partial differential equation with involution and periodic boundary conditions

The Fourier method is used to find a classical solution of the mixed problem for a first-order differential equation with involution and periodic boundary conditions. The application of the Fourier method is substantiated using refined asymptotic formulas obtained for the eigenvalues and eigenfunctions of the corresponding spectral problem. The Fourier series representing the formal solution is transformed using certain techniques, and the possibility of its term-by-term differentiation is proved. Minimal requirements are imposed on the initial data of the problem.     

The Love's problem for hyperbolic thermoelasticity

In our paper we investigated the initial-boundary value problem for elastic layer situated on half space of another elastic medium. In this medium the thermomechanical interactions were taken into consideration. The system of equations with initial-boundary conditions describes the phenomenon of wave propagation with finite speed. In our problem there are two surfaces ie. free surface and contact surface between layer and half space. On the free surface are setting boundary conditions for normal and tangent surface force. We consider two types of contact between layer and half-space: rigid contact and slip contact. The initial-boundary value problem was solved by using integral transformations and Cagniard-de Hoope methods. From the solution of this problem follows that in layer and half space exist some kind of thermoelastic waves. We investigated moreover the conditions which should be fullfiled for propagation of Rayleigh and Love's type waves on the contact surface between layers and half space. The results obtained in our investigation were used in technical applications especially engineering design and diagnostics of roads and airfields. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)