On the Generalized Thermoelasticity Problem for an Infinite Fibre-Reinforced Thick Plate under Initial Stress

In this paper, the generalized thermoelasticity problem for an infinite fiber-reinforced transversely-isotropic thick plate subjected to initial stress is solved. The lower surface of the plate rests on a rigid foundation and temperature while the upper surface is thermally insulated with prescribed surface loading. The normal mode analysis is used to obtain the analytical expressions for the displacements, stresses and temperature distributions. The problem has been solved analytically using the generalized thermoelasticity theory of dual-phase-lags. Effect of phase-lags, reinforcement and initial stress on the field quantities is shown graphically. The results due to the coupled thermoelasticity theory, Lord and Shulman's theory, and Green and Naghdi's theory have been derived as limiting cases. The graphs illustrated that the initial stress, the reinforcement and phase-lags have great effects on the distributions of the field quantities.     

Impulse response of an infinitely long thick strip plate

This paper is a study of the dynamic response of an infinitely long thick strip plate subjected to an impulsive load. The plate is simply supported along the edges and resting on an elastic foundation. The problem is studied on the basis of a plate theory in which the effects of rotatory inertias and shear deformations are retained. Governing equations are solved by applying the methods of the Laplace transform with respect to time and the Fourier transform with respect to a longitudinal space variable. Dynamic coefficients (maximum dynamic displacement/static displacement, maximum dynamic bending moment/static bending moment) are calculated numerically for plates subjected to a step line load and shown graphically for various values of the parameters included.     

Large amplitude vibration of an initially stressed cross ply laminated plates

In this paper, nonlinear equations of large amplitude vibration for a laminated plate in a general state of nonuniform initial stress are derived. The equations include the effects of transverse shear and rotary inertia. Using these derived governing equations, the large amplitude vibration behaviour of an initially stressed cross-ply laminated plate is studied. The initial stress is taken to be a combination of pure bending stress plus an extensional stress in the plane of the plate. The Galerkin method is used to reduce the governing nonlinear partial differential equations to ordinary nonlinear differential equations and the Runge-Kutta method is used to obtain the nonlinear to linear frequencies. The frequency responses of nonlinear vibration are sensitive of the vibration amplitude, aspect ratio, thickness ratio, modulus ratio, stack sequence, layer number and state of initial stresses. The effects of various parameters on the large amplitude free vibrations are presented.     

Large amplitude vibration of an initially stressed moderately thick plate

Non-linear equations of motion for a transversely isotropic moderately thick plate in a general state of non-uniform initial stress where the effects of transverse shear and rotary inertia are included are derived. The large amplitude flexural vibration of a simply supported rectangular moderately thick plate subjected to initial stress is investigated. The initial stress is taken to be a combination of a pure bending stress plus an extensional stress in the plane of the plate. These equations are used to solve the vibrations problem by the Galerkin method. The effects of various parameters on the non-linear vibration frequencies are studied.     

一边平夾另一边受均布力矩作用的环形薄板大挠度问题

一.引言 作者曾在文献中指出,用摄动法来处理在均布边缘力矩作用之下的弹性圆薄板大挠度问题,不但远较用数字积分法来得简便,而且还提供了最大挠度以及应力的设计公式,使它们能直接应用到工程设计问题上去。本文则引申之,同样成功地处理了另一些具有实用价值但一直还未被研究过的问题,即外(内)边平夹,而沿内(外)边有均布边缘力矩作用的环形薄板大挠度问题。     

VIBRATION AND BUCKLING OF INITIALLY STRESSED CURVED BEAMS

Equations of motion for curved beams in a general state of non-uniform initial stresses are derived using the principle of virtual work. The equations are adjusted to a generic expression by using appropriate transformations. The free vibration behaviours of the curved beams subjected to a combination of uniform initial tensile of compressive stresses and uniform initial bending stress are examined. The Galerkin method is employed in obtaining accurate values of free frequencies and initial buckling stresses. The curved beam is assumed to be vibrating in its plane. Natural frequencies and initial buckling stresses for hinged supported curved beams are presented for validation. Effects of arc segment angles, elastic foundation, and initial stresses on the natural frequencies are investigated. Effects of arc segment angles, elastic foundation, and initial bending stresses on the initial buckling stresses are explored in this paper.     

Dynamic response of a prestressed,orthotropic thick plate strip to a moving line load

This paper is a study of the steady state response of an orthotropic plate strip to a moving line load. The plate is of infinite length and subjected to initial in-plane stresses parallel and perpendicular to the edges. The solution is obtained on the basis of a thick plate theory which takes into account the effects of shear deformation and rotatory inertia. The critical speed of the load which brings about a resonance effect in the system is determined. Further, the bending moment in the plate is calculated for several values of the load speed and the initial stress parameters and shown graphically as a function of the space variable moving with the load.     

Dynamics of a rectangular plate resting on an elastic foundation and partially in contact with a quiescent fluid

In this study, a method of analysis is presented for investigating the effects of elastic foundation and fluid on the dynamic response characteristics (natural frequencies and associated mode shapes) of rectangular Kirchhoff plates. For the interaction of the Kirchhoff plate–Pasternak foundation, a mixed-type finite element formulation is employed by using the Gâteaux differential. The plate finite element adopted in this study is quadrilateral and isoparametric having four corner nodes, and at each node four degrees of freedom are present (one transverse displacement, two bending moments and one torsional moment). Therefore, a total number of 16 degrees-of-freedom are assigned to each element. A consistent mass formulation is used for the eigenvalue solution in the mixed finite element analysis. The plate structure considered is assumed clamped or simply supported along its edges and resting on a Pasternak foundation. Furthermore, the plate is fully or partially in contact with fresh water on its one side. For the calculation of the fluid–structure interaction effects (generalized fluid–structure interaction forces), a boundary element method is adopted together with the method of images in order to impose an appropriate boundary condition on the fluid's free surface. It is assumed that the fluid is ideal, i.e., inviscid, incompressible, and its motion is irrotational. It is also assumed that the plate–elastic foundation system vibrates in its in vacuo eigenmodes when it is in contact with fluid, and that each mode gives rise to a corresponding surface pressure distribution on the wetted surface of the structure. At the fluid–structure interface, continuity considerations require that the normal velocity of the fluid is equal to that of the structure. The normal velocities on the wetted surface of the structure are expressed in terms of the modal structural displacements, obtained from the finite element analysis. By using the boundary integral equation method the fluid pressure is eliminated from the problem, and the fluid–structure interaction forces are calculated in terms of the generalized hydrodynamic added mass coefficients (due to the inertial effect of fluid). To asses the influences of the elastic foundation and fluid on the dynamic behavior of the plate structure, the natural frequencies and associated mode shapes are presented. Furthermore, the influence of the submerging depth on the dynamic behavior is also investigated.     

Thickness vibrations of a piezoelectric plate under biasing fields

Thickness vibrations of a piezoelectric plate under uniform initial thermoelectromechanical fields are investigated in this article. An exact solution is obtained for materials with general anisotropy. A plate of langasite, a relatively new piezoelectric material with strong piezoelectric coupling, is analyzed with imposed uniform acceleration as the biasing field as an example. The results show that the biasing fields have remarkable effect on the free vibrations of piezoelectric plate.     

On the vibration of an elastic plate on an elastic foundation

The forced vibration of an elastic plate under a time harmonic point force is studied. The plate is infinite in extent and supported by an elastic foundation. This study is made on the basis of the improved (Timoshenko) plate theory. The mathematical problem is to seek a fundamental solution (the Green's function) of the time-reduced plate equation of the improved plate theory. Such a fundamental solution is constructed by the distributional Fourier transform method. From the explicit expressions of the fundamental solution, the behavior of the fundamental singularity as a function of the vibration frequency and the foundation stiffness is examined. Conditions under which plate resonance occurs are also determined.     

Semi-Analytical Solution for Functionally Graded Solid Circular and Annular Plates Resting on Elastic Foundations Subjected to Axisymmetric Transverse Loading

In this paper, the static analysis of functionally graded (FG) circular plates resting on linear elastic foundation with various edge conditions is carried out by using a semi-analytical approach. The governing differential equations are derived based on the three dimensional theory of elasticity and assuming that the mechanical properties of the material vary exponentially along the thickness direction and Poisson's ratio remains constant. The solution is obtained by employing the state space method (SSM) to express exactly the plate behavior along the graded direction and the one dimensional differential quadrature method (DQM) to approximate the radial variations of the parameters. The effects of different parameters (e.g., material property gradient index, elastic foundation coefficients, the surfaces conditions (hard or soft surface of the plate on foundation), plate geometric parameters and edges condition) on the deformation and stress distributions of the FG circular plates are investigated.     

Effect of side walls on the motion of a viscous fluid induced by an infinite plate that applies an oscillating shear stress to the fluid

The velocity field corresponding to the unsteady motion of a viscous fluid between two side walls perpendicular to a plate is determined by means of the Fourier transforms. The motion of the fluid is produced by the plate which after the time t = 0, applies an oscillating shear stress to the fluid. The solutions that have been obtained, presented as a sum of the steady-state and transient solutions satisfy the governing equation and all imposed initial and boundary conditions. In the absence of the side walls they are reduced to the similar solutions corresponding to the motion over an infinite plate. Finally, the influence of the side walls on the fluid motion, the required time to reach the steady-state, as well as the distance between the walls for which the velocity of the fluid in the middle of the channel is unaffected by their presence, are established by means of graphical illustrations.     

Interaction of counterpropagating electromagnetic waves in an absorbing plate inside a waveguide

The interaction of two counterpropagating coherent transverse electric and transverse magnetic electromagnetic waves in an absorbing plate that is placed in a waveguide with an arbitrary transverse cross section is analyzed. It is assumed that the waves with different initial phases are incident on the plate boundaries from two sides. An analytical expression for the interference transmission coefficient with respect to power is derived. Several physical features of the tunneling interference in the plate are revealed. It is demonstrated that a scenario in which an electromagnetic energy flux exists on the left-hand side of the plate and vanishes on the right-hand side or vice versa is possible at certain relations of the initial phases and amplitudes of the counterpropagating waves.     

Investigation of area mobility of a finite thin plate subject to a uniform velocity excitation

Methods for theoretically calculating area mobility of a finite thin plate are derived for a typical excitation pattern,uniform velocity excitation,over a contact area.In the derivation,the contact area is discretised into sub-regions and the excitation and response continuously distributed over these sub-regions are simplified as acting at the centers of the sub-regions.Therefore,according to the concept of effective point mobility,the theoretical formula for area mobility of a finite thin plate subject to a uniform velocity excitation is obtained.According to the relation between complex power and area mobility,the measurement method of a finite plate is also obtained by means of transfer point mobility.When comparing results generated by theoretical calculation to patterns measured in experiment,consistent patterns are found,suggesting that the proposed methods are practical.The results of calculation show that the accuracy of theoretical calculation increases with the number of sub-regions and is influenced by the moment caused by the experiment device,especially when the excitation frequency is high.     

Nonlocal elasticity theory for thermal buckling of nanoplates lying on Winkler–Pasternak elastic substrate medium

In the present work, thermal buckling of single-layered graphene sheets lying on an elastic medium is analyzed. For this purpose, the sinusoidal shear deformation plate theory in tandem with the nonlocal continuum theory, which takes the small scale effects into account, is employed. The non-linear stability equations, which contain the reaction of Winkler–Pasternak elastic substrate medium, are derived and then solved analytically for a plate with various boundary conditions and based on various plate theories. Closed form solutions are formulated for three types of thermal loadings as uniform, linear and nonlinear temperature rise through the thickness of the plate. A number of examples are presented to illustrate the numerical results concerned with the buckling temperature response of nanoplates resting on two-parameter elastic foundations. The influences played by transversal shear deformation, plate aspect ratio, side-to-thickness ratio, nonlocal parameter, and elastic foundation parameters are all investigated.     

Stokes' First Problem for an MHD Burgers Fluid

This work is related to the flow of a magnetohydrodynamic Burgers fluid.The flow of an incompressible conducting Burgers fluid in the presence of a uniform transverse magnetic field over a plate that is moved suddenly is considered.By the application of the Laplace and Fourier sine transforms techniques,the exact analytical expressions for the velocity field and associated shear stress are determined in simple forms.They are written as a sum of steady-state and transient solutions.The graphical results are plotted for different values of indispensable parameters and some interesting results are concluded.The corresponding solutions for the hydrodynamic Burgers fluid appear as the limiting cases of the obtained solutions.     

Free vibrations of an infinite plate of parabolically varying thickness on elastic foundation

A simple model is presented for use by research workers in seismology, soil mechanics and earthquake vibrations in problems where the resistance offered by the soil foundation is very close to that of Winkler's assumption. In earth vibrations one earth layer can be treated as an infinite plate of variable thickness while layers below it may approximately serve as an elastic foundation. Here, the transverse displacement of a plate of parabolically varying thickness on an elastic foundation is expressed as a power series and the frequencies, deflections and moments corresponding to the first two modes of vibration are computed for various values of foundation modulus and taper constant, and for two combinations of boundary conditions. A problem discussed in previous work [1] can be considered as a particular case of the present problem if the foundation modulus is taken to be zero.     

基于波像差函数建立大口径施密特校正板方程

施密特光学系统由施密特校正板和球面反射镜组成,校正板设置在球面反射镜的球心处,系统的焦点不一定和反射镜的焦点重合。为了得到精确的校正板面形初始结构参数,基于波像差函数建立了带有离焦量的大口径施密特校正板的数学模型,同时校正了系统的三级和五级球差。利用光学设计软件对校正板口径为1000mm、主镜的曲率半径为2000mm、F数为1的系统进行了设计和分析,来验证校正板面形数学模型的正确性。结果显示此校正板的数学模型与优化后结果吻合得较好,校正板面形初始结构参数的精度得到了极大的提高,这为大口径、大相对孔径的施密特光学系统的设计提供了理论基础。     

Some laws of polarization and dispersion of a drop in an electrostatic field

The laws of distribution among contributions in various interactions to the total polarization energy of a conductor in a uniform electrostatic field was analyzed. It is shown that in a closed system, spontaneous shape variations of a liquid conductor with a free surface in an external magnetic field are possible only if they are accompanied by an increase in the conductor dipole moment. Variations of the intrinsic energy of a conductor are studied by the example of a conductive liquid drop in the case where a drop affected by a polarization charge becomes unstable. Analytical expressions defining the sizes and charges of the droplets ejected out of the initial drop under the conditions of instability are derived.     

Bending and springback theory of metal-polymer sandwich laminates

In this report an analysis is made of the behavior of sandwich beams in which the core polymer is laminated on both sides with surface metal sheets, each of which has a different thickness and mechanical properties when they are loaded with a uniform bending moment which is then released resulting in springback of the bent sandwich beam. It is assumed that the polymer behaves elastically because the bending strain in the core is small and its elastic limit is much larger than that of metals. Sandwich beams have various elastoplstic stess distributions when bent depending on the mutual relationships between their dimensions, the mechanical moduli, and the applied bending moment. Further, residual curvatures, shifting position of neutral axis, and residual stress distributions in sandwich beams variously elastoplastically stressed initially on the decrease of applied bending moment are analyzed.