Unsteady motion of a source in a fluid under a floating plate

This paper studies the three-dimensional unsteady problem of the hydroelastic behavior of a floating infinite plate under the impact of waves generated by horizontal straight motion of a point source of mass in a fluid of infinite depth. The solution is carried out using known integral and asymptotic methods. The formulas obtained are used to numerically analyze the effect of plate thickness, depth of submergence of the source, and its acceleration, deceleration, and velocity of uniform motion on the deflection of the floating plate.     

Effect of the Conductance and Thickness of a Conducting Plate on the Signal from a Material–Velocity Inductive Transducer

The perturbation problem of the magnetic field of a constant–current turn located above a conducting plate set into motion by a plane shock wave with a rectangular profile is considered. It is shown that not only the velocity of the plate but also its dynamic conductivity can be determined on the basis of the electromotive force of induction recorded by means of the turn. For the case where the conductance of the plate is known for both the conducting half–space and for a plate whose thickness is comparable with the skin–layer thickness, approximatecalculated dependences for the velocity of the plate are obtained. A comparison with experimental data and the clarification of the calculated dependences allows one to conclude that the approaches proposed can be used for determining the conductance of metals in shock–wave processes.     

Unsteady vibration of a growing circular plate

Forced vibration of a thickening circular plate is studied within the framework of the small deformation theory. The plate material is assumed to be elastic and isotropic and the plate thickness to be continually increasing due to influx of material from outside. It is also assumed that the plate thickness varies with time but is independent of the space coordinates. Moreover, in the process of growth, the midsurface position does not change, which suggests that the plate grows symmetrically on both its surfaces.     

Motion of a slender body in a fluid under a floating plate

This paper studies the three-dimensional unsteady problem of the hydroelastic behavior of a floating infinite plate under the impact of waves generated by horizontal rectilinear motion of a slender solid in a fluid of infinite depth. An analytic solution of the problem is found based on the known solutions for the unsteady motion of a point source of mass in a fluid of infinite depth under a floating plate. Asymptotic formulas are obtained which model the motion of a solid slender body in a fluid by replacing the body with a source-sink system. These formulas are used to numerically analyze the effect of plate thickness, depth of the body, its dimensions and the velocity of rectilinear motion on the amplitude of deflection of the floating plate. The motion of a submarine under a nonbreakable plate was modeled experimentally. Theoretical and experimental data are in good agreement.     

Active flow control applications with a jet and vortex actuator in a laminar cross flow

In this study the effect of an oscillatory, zero-net-mass flux device called a Jet and Vortex Actuator (JaVA) on the laminar boundary layer is investigated. The JaVA can be utilized to energize the boundary layer over a flat plate by creating jets and vortices thus it can delay or prevent boundary layer separation if it is used properly.It appears to produce qualitatively different flow regimes depending on its actuation parameters, e.g. frequency and amplitude or subtle changes in geometry such as the position of the actuator plate with respect to the JaVA-cavity. This latter effect was only discovered recently. Since little is known about the underlying fluid dynamics and because of a complete lack of unsteady data, a device is designed and built for experiments in water. Unsteady flow fields have been recorded for visualization and furthermore quantitative evaluation by means of Particle Image Velocimetry (PIV) has been carried out in addition to numerical simulations. Results show that the JaVA-induced vortices ejected into the flat plate boundary layer significantly enhance the velocity profiles and its characteristics such as the displacement thickness and the momentum thickness if the plate is oscillating at high frequencies as it is flush-mounted or inside the cavity. But if the plate is extracted out of the cavity then there is no improvement in the flow fields hence separation can be delayed or prevented for long downstream distances only if the actuation parameters and plate positions are selected properly.     

Derivation of Plate and Rod Equations for a Piezoelectric Body from a Mixed Three-Dimensional Variational Principle

We use a mixed 3-dimensional variational principle to derive 2-dimensional equations for an anisotropic plate-like piezoelectric body and one-dimensional equations for an anisotropic beam-like piezoelectric body. The formulation accounts for double forces without moments which may change the thickness of the plate and deform the cross-section of the rod. The dependence of the bending rigidities of a transversely isotropic plate upon the angle between the normal to the midsurface and the direction of transverse isotropy is exhibited. The plate equations are used to study the cylindrical deformations of a transversely isotropic plate due to equal and opposite charges applied to its top and bottom surfaces. It is also found that a piezoelectric circular rod with axis of transverse isotropy not coincident with its centroidal axis and subjected to electric charges at the end faces is deformed into a non-prismatic body.     

Oscillations of a plate cascade in a transonic gas flow

The transonic unsteady flow of a gas through a cascade of thin, slightly curved plates is quite complex and has received little study. The main difficulties are associated with the nonlinear dependence of the aerodynamic characteristics on the plate thickness. In [1] it is shown that, for a single thin plate performing high-frequency oscillations in a transonic gas stream, the variation of the unsteady aerodynamic characteristics with plate thickness may be neglected. For a plate cascade, the flow pattern is complicated by the aerodynamic interference between the plates, which may depend significantly on their shape. A solution of the problem of transonic flow past a cascade without account for the plate thickness has been obtained by Hamamoto [2].The objective of the present study is the clarification of the dependence of the aerodynamic characteristics of a plate cascade on plate thickness in transonic unsteady flow regimes. The nonlinear equation for the velocity potential is linearized under the assumption that the motionless plate causes significantly greater disturbances in the stream than those due to the oscillations. A similar linearization was carried out for a single plate in [3]. The aerodynamic interference between the plates is determined by the method presented in [4]. As an example, the aerodynamic forces acting on a plate oscillating in a duct and in a free jet are calculated.     

基于高阶剪切变形理论的四边形求积元板单元及其应用

近年来由各类新型复合材料或功能梯度材料构成的板结构在工程领域得到了广泛应用,其显著特点是材料性能沿板厚变化.为合理考虑横向剪切应变,许多学者基于Reddy高阶剪切变形理论,构建了不同的有限元单元对该类板结构进行分析,但其中满足$C^{1}$连续条件的单元相对较少.本文基于Reddy高阶剪切变形理论,采用求积元方法,建立了$C^{1}$连续的四边形板单元.利用该单元对均质材料、复合材料、功能梯度材料构成的等厚度矩形板、变厚度矩形板及等厚度斜板的线弹性弯曲和自由振动问题进行了计算分析,并与现有文献中的相应计算结果进行了对比.研究表明:基于高阶剪切变形理论的四边形求积元板单元具有较高的计算效率和良好的适应性,文中各类材料构成的等变厚度矩形板及等厚度斜板均只需1个单元即可得到理想的计算结果.对于等/变厚度矩形板,可仅使用9$\times$9个积分点,而对于等厚度斜板,随着斜角的增大,所需积分点的数目逐渐增多至15$\times $15.该四边形求积元板单元可进一步用于新型复合材料板的非线性分析.      

Stiffness and buckling optimization of thin plates with BEM

The thickness optimization is used to maximize the stiffness or the buckling load of a Kirchhoff plate having constant volume. The shape of the plate is arbitrary and it is subjected to any type of admissible boundary conditions. The optimization consists in establishing the thickness variation law, for which either the stiffness of the plate or the buckling load is maximized. Beside the equality constraint of constant volume, the thickness variation is subjected also to inequality constraints resulting from serviceability requirements (upper and lower thickness bounds) as well as from the condition that the Kirchhoff plate theory remains valid. The latter constraint is new and it is derived herein by approximating the plate with a three-dimensional prismatic elastic body having curved upper and lower surface. The optimization problem is solved using the sequential quadratic programming algorithm. The bending and the plane stress problem of a plate with variable thickness, required for the evaluation of the objective function, are solved using the analog equation method. The thickness is approximated using integrated radial basis functions that approximate accurately not only the thickness function but also its first and second derivatives involved in the plate equation and in the constraints. Several plate optimization problems have been studied giving realistic and meaningful optimum designs without violating the validity of the thin plate theory.     

Photothermoelastic study of stress concentrations in a plate with internal heating

This paper describes a photothermoelastic method for simulating, in a three-dimensional model, the temperature gradients that occur in structural parts subjected to internal heating such as is frequently encountered in certain areas of nuclear-reactor design. The method is applied to a plate which has a step change in thickness and sustains a nonlinear temperature gradient through its thickness. The shapes of the gradients simulate internal heating of the plate material. The values for the highest stresses on the free surfaces of the plate, within the thickness of the plate, and at the root of the step are presented in graphical form for a range of internal heat-generated conditions. Thermal-stress-concentration factors are presented for a step change in the thickness of a plate under this type of heating. Its design significance is discussed. The same stress and stress-concentration values are shown to also apply to nonnuclear problems. During shut-down in conventional thermal plants, when the walls sustain linear steady-state temperature drops across their thicknesses, temperature profiles exactly analogous to those presented in this paper occur. The stresses can then be computed from the values presented here.     

The derivation of a thick and thin plate formulation without ad hoc assumptions

For the plate formulation considered in this paper, appropriate three-dimensional elasticity solution representations for isotropic materials are constructed. No a priori assumptions for stress or displacement distributions over the thickness of the plate are made. The strategy used in the derivation is to separate functions of the thickness variable z from functions of the coordinates x and y lying in the midplane of the plate. Real and complex 3-dimensional elasticity solution representations are used to obtain three types of functions of the coordinates x, y and the corresponding differential equations. The separation of the functions of the thickness coordinate can be done by separately considering homogeneous and nonhomogeneous boundary conditions on the upper and lower faces of the plate. One type of the plate solutions derived involves polynomials of the thickness coordinate z. The other two solution forms contain trigonometric and hyperbolic functions of z, respectively. Both bending and stretching (or in-plane) solutions are included in the derivation.     

Winkler地基上变厚度圆板的轴对称弯曲

本文提出了Winkler地基上变厚度圆板轴对称弯曲的传递矩阵算法。首先,根据贝塞尔函数理论获得了等厚度圆板和环板单元在任意荷载作用下轴对称弯曲的解析解,这些解均由通解和特解两部分组成。基于这些解析解,导出了等厚度圆板和环板单元的传递矩阵。然后沿径向将变厚度圆板划分成一个等厚度圆板单元和一系列等厚度环板单元,应用传递矩阵算法原理获得了变厚度圆板的整体传递矩阵。引入圆板的边界条件,给出了该板每条节线上的挠度、径向转角、径向弯矩和径向剪力。最后,讨论了受均布荷载作用的简支线性变厚度圆板的弯曲,将本文数值解与解析解进行比较,证实了本文方法的有效性,并简要地讨论了地基参数对板挠度和径向弯矩的影响。     

Fracture of metal plates on normal impact by a rigid conical body

Punching through aluminum plates on normal impact by a rigid conical body is considered. Plates of D16AT and AMTsM were used. The ballistic limit is determined depending on the plate thickness, its material, and the cone angle of the punching body. The fracture mechanisms are determined for various combinations of these parameters.     

Photoelastic analysis of an asymmetrically reinforced circular cut-out in a flat plate subjected to uniform unidirectional stress

Three-dimensional photoelastic studies of stresses around an asymmetrically reinforced circular cut-out in a flat plate under uniform unidirectional stress are reported. The frozen-stress technique, with Hysol 4290 material, was used to determine the stress distribution through four critical points on the boundary of the reinforced hole. Included were models with different cross sections of reinforcement, with various interface fillet radii and with different plate widths. For the majority of models, the ratio of volume of reinforcement to volume of hole was unity. It is concluded that, for reducing the stress concentration, there is a limit on the effectiveness of increasing the fillet radius beyond half the plate thickness. It was found that a reinforcement having a thickness of approximately 40 percent of the plate thickness was optimum and that the stress concentration decreases with volume of reinforcement. The authors believe that, with judgment, some of the conclusions reached may be applied, for design purposes, beyond the specific dimensional ranges and loading conditions of the tests.     

Nonlinear analysis of a thin circular functionally graded plate and large deflection effects on the forces and moments

The dynamic von Karman equations are used for nonlinear analysis of a thin circular plate made of a functionally graded material. The thickness of the plate is constant and the properties of the functionally graded material depend on temperature and vary throughout the thickness. It is assumed that the plate oscillates with large amplitudes. The forces and moments in the plate are determined in solving the equations for harmonic vibrations. Relevant results are obtained in the case of stead-state free vibrations. These results indicate that the volume fraction has a strong effect on the forces, moments, and material properties Published in Prikladnaya Mekhanika, Vol. 44, No. 6, pp. 134–144, June 2008. An erratum to this article can be found at     

Three Dimensional Vibration Transmission of a Plate on a Layer of Porous Medium Due to a Moving Load

Three-dimensional (3D) transmission of vibration in an infinite elastic thin plate on a layer of poroviscoelastic medium, due to a harmonic, rectangular moving load, is investigated theoretically based on Biot’s theory. The material of the medium is idealized as a uniform, fully saturated poroviscoelastic layer on bedrock. By introducing four scalar potential functions and Helmholtz decomposition theorem, analytical solutions of stress, displacement, and pore pressure with and without thin plate are derived using Fourier transform technique. Numerical results are obtained with the help of inverse Fourier transform and are used to analyze the influence of load velocity, porosity, permeability, relative stiffness of plate versus ground, and the thickness of plate on the vibration. Furthermore, the results are compared with the available dynamic response results of a non-moving load on a layer of viscoelastic material.     

The concentration of stress and strain in finite thickness elastic plate containing a circular hole

The elastic stress and strain fields of finite thickness large plate containing a hole are systematically investigated using 3D finite element method. It is found that the stress and strain concentration factors of the finite thickness plate are different even if the plate is in elasticity state except at notch root of plate surface. The maximum stress and strain do not always occur on the mid plane of plate. They occur on the mid plane only in thin plate. The maximum stress and strain concentration factors are not on mid plane and the locations of maximum stress and strain concentration factors are different in thick plate. The maximum stress and strain concentration factors of notch root increase from their plane stress value to their peak values, then decrease gradually with increasing thickness and tend to each constant related to Poisson’s ratio of plate, respectively. The stress and strain concentration factors at notch root of plate surface are the same and are the monotonic descent functions of thickness. Their values decrease rapidly and tend to each constant related to Poisson’s ratio with plate thickness increasing. The difference between maximum and surface value of stress concentration factor is a monotonic ascent function of thickness. The thicker the plate is or the larger the Poisson’s ratio is, the larger the difference is. The corresponding difference of strain concentration factor is similar to the one of stress concentration factor. But the difference magnitude of stress concentration factor is larger than that of strain concentration factor in same plate.     

Study of thermal gradient effect on vibrations of a non-homogeneous orthotropic rectangular plate having bi-direction linearly thickness variations

Here, thermal gradient effect on vibration of non-homogeneous orthotropic rectangular plate having bi-direction linearly thickness variations is studied. The non-homogeneity is assumed to arise due to the variation in the density of the plate material. Rayleigh Ritz method is used to evaluate the fundamental frequencies and deflection function. The two-dimensional thickness variation is taken as the Cartesian product of linear variations along the two concurrent edges of the plate. Yet the method used for the solution of present problem provides an approximate solution but it is quite convenient and authentic one. A two-term deflection function is used. Frequency corresponding to first mode of vibration is calculated for clamped plate for different values of temperature gradient, taper constants and non-homogeneity constant. In special cases, comparisons have been made with results that are available in the literature.     

Extended meshfree analysis of transverse and inplane loading of a laminated anisotropic plate of general planform geometry

An extended meshfree method is presented for the analysis of a laminated anisotropic plate under elastostatic loading. The plate may be of any planform shape with its thickness profile composed of perfectly bonded uniform thickness layers of distinct anisotropic materials. Both transverse and inplane loads are considered using a first order shear deformation theory for flexural behavior and generalized plane stress for the membrane behavior. In this extended meshfree method, a rectangular domain is initially considered with the plate of arbitrary geometry inscribed within it. A particular solution in the form of an analytic generalized Navier solution (a compound double Fourier series) is used to capture the response due to the loading within the rectangular domain. Then, a homogeneous solution by meshfree analysis is added to treat the augmented boundary conditions on the actual contour of the plate. These augmented conditions are composed of the prescribed values and that of the particular solution evaluated around the plate’s contour.Concentrated transverse and inplane loads in the form of uniform loads over a very small patch are considered with this generalized Navier solution representation. When a meshfree portion is added to account for the boundary conditions, such solutions constitute the Green’s functions for the plate. The viability of these double Fourier series representations is shown by the convergence rates for the kinematic and force/moment fields. An additional example of a two layer ±30° angleply circular plate is given to illustrate the capability of this extended meshfree method.     

Dynamics of thin periodic plates resting on a periodically inhomogeneous Winkler foundation

Summary The aim of the paper is to investigate the dynamic response of thin elastic plates having periodic substructure in planes parallel to the plates midplane and interacting with a Winkler foundation. The main goal of the analysis is to describe the effect of substructure size on the plates dynamics. For this purpose, the method proposed in [4, 5] is used. Two special cases are analysed: a plate band with a constant thickness interacting with a periodically inhomogeneous Winkler foundation, and a plate band with a periodically variable thickness interacting with a homogeneous Winkler foundation. The physical correctness conditions of the model are also discussed. Received 14 July 1998; accepted for publication 7 January 1999