The exact solution of Stokes＇ second problem including start-up process with fractional element

The start-up process of Stokes＇ second problem of a viscoelastic material with fractional element is studied. The fluid above an infinite flat plane is set in motion by a sudden acceleration of the plate to steady oscillation. Exact solutions are obtained by using Laplace transform and Fourier transform. It is found that the relationship between the first peak value and the one of equal-amplitude oscillations depends on the distance from the plate. The amplitude decreases for increasing frequency and increasing distance.     

横观各向同性饱和地基上无限板的稳态振动

研究了横观各向同性饱和土地基上无限板的稳态振动问题. 基于直角坐标系下横观各向同性饱和介质Biot波动方程的一般解，采用双重Fourier积分变换技术，建立了饱和地基与无限矩形板相互作用的动力方程，利用数值方法求解该方程，得到任意谐振荷载作用下饱和半空间体上无限板稳态响应的一般解. 数值结果表明，横观各向同性饱和地基上无限板的振动与各向同性饱和地基上的无限板的振动特性存在明显差异.     

The exact solution of Stokes' second problem including start-up process with fractional element

The start-up process of Stokes' second problem ofa viscoelastic material with fractional element is studied. Thefluid above an infinite flat plane is set in motion by a suddenacceleration of the plate to steady oscillation. Exact solutionsare obtained by using Laplace transform and Fourier transform.It is found that the relationship between the first peakvalue and the one of equal-amplitude oscillations dependson the distance from the plate. The amplitude decreases forincreasing frequency and increasing...     

Time-dependent motion of a two-dimensional floating elastic plate

Three methods are presented to determine the motion of a two-dimensional finite elastic plate floating on the water surface, which is released from rest and allowed to evolve freely. The first method is based on a generalized eigenfunction expansion and it is valid for all water depths. The second method is based on an integral equation derived from the Fourier transform, and it is valid for all water depths, although computations are made only for water of infinite depth. These two methods are both based on the frequency-domain solution—however no obvious connection exists between the two methods. The third method is valid only for shallow water, and it expresses the solution as the sum over decaying modes. We present a new derivation of the integral equation for a floating plate based on the Fourier transform of the equations of motion in the time domain. The solution obtained by each method is compared in the appropriate regime, and excellent agreement is found, thereby providing benchmark solutions. We also investigate the regime of validity of the infinite and shallow-depth solutions, and show that both give good results for a quite wide range of depths.     

Deflection of transient thermoelastic circular plate by Marchi-Zgrablich and Laplace integral transform technique

This paper deals with the determination of temperature distribution and thermal deflection function of a thin circular plate with the stated conditions. The transient heat conduction equation is solved by using Marchi-Zgrablich transform and Laplace transform simultaneously and the results of temperature distribution and thermal deflection function are obtained in terms of infinite series of Bessel function and it is solved for special case by using Mathcad 2007 software and represented graphically by using Microsoft excel 2007.     

多圆荷载下刚性道面变形和应力的计算

本文建立多圆荷载作用下弹性半空间体上薄板的挠度与应力的计算式。荷载数量及分布任意,每个圆荷载密度与轮迹半径彼此相异。对计算式中的反常积分及级数的收敛性予以证明。对含振荡函数反常积分建议一种方便的算法。     

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.     

Three-dimensional elastostatic solutions for transversely isotropic functionally graded material plates containing elastic inclusion

Based on the generalized England-Spencer plate theory, the equilibrium of a transversely isotropic functionally graded plate containing an elastic inclusion is studied. The general solutions of the governing equations are expressed by four analytic functions α(ζ), β(ζ), φ(ζ), and ψ(ζ) when no transverse forces are acting on the surfaces of the plate. Axisymmetric problems of a functionally graded circular plate and an infinite func-tionally graded plate containing a circular hole subject to loads applied on the cylindrical boundaries of the plate are firstly investigated. On this basis, the three-dimensional (3D) elasticity solutions are then obtained for a functionally graded infinite plate containing an elastic circular inclusion. When the material is degenerated into the homogeneous one, the present elasticity solutions are exactly the same as the ones obtained based on the plane stress elasticity, thus validating the present analysis in a certain sense.     

焊接残余应力平板多个共面表面裂纹的应力强度因子

采用线弹簧模型求解含焊接残余应力平板多个共面任意分布表面裂纹的应力强度因子.利用边裂纹权函数给出了裂纹表面上沿厚度非线性分布的残余应力向线性分布的转化公式.基于Reissner板理论和连续分布位错思想,将含多个共面任意分布表面裂纹的无限平板问题归结为一组Cauchy型奇异积分方程,并采用Gauss-Chebyshev方法获得了奇异积分方程的数值解.以三共面表面裂纹为例,计算了表面裂纹的应力强度因子,并讨论了裂纹间距、裂纹几何形状等因素对应力强度因子的影响.     

Unsteady flow of an Oldroyd-B fluid generated by a constantly accelerating plate between two side walls perpendicular to the plate

The velocity field and the adequate tangential stresses corresponding to the unsteady flow of an Oldroyd-B fluid induced by a constantly accelerating plate between two side walls perpendicular to the plate are established by means of Fourier sine transforms. The solutions corresponding to Maxwell, second grade and Newtonian fluids, performing the same motion, appear as limiting cases of the solutions obtained here. In the absence of the side walls, namely when the distance between walls tends to infinity, all solutions that have been determined reduce to those corresponding to the flow over an infinite plate. Finally, for comparison, the velocity field at the middle of the channel as well as the shear stress on the bottom wall is plotted as a function of y for several values of t and of the material constants. The influence of the side walls on the motion of the fluid is also emphasized by graphical illustrations.     

Unsteady stagnation-point flow of a Newtonian fluid in the presence of a magnetic field

The unsteady stagnation-point flow of a viscous fluid impinging on an infinite plate in the presence of a transverse magnetic field is examined and solutions are obtained. It is assumed that the infinite plate at y=0 is making harmonic oscillations in its own plane. A finite difference technique is employed and solutions for small and large frequencies of the oscillations are obtained for various values of the Hartmann's number.     

Mode I stress intensity factor solutions for spot welds in lap-shear specimens

The analytical solutions of the mode I stress intensity factor for spot welds in lap-shear specimens are investigated based on the classical Kirchhoff plate theory for linear elastic materials. First, closed-form solutions for an infinite plate containing a rigid inclusion under counter bending conditions are derived. The development of the closed-form solutions is then used as a guide to develop approximate closed-form solutions for a finite square plate containing a rigid inclusion under counter bending conditions. Based on the J integral, the closed-form solutions are used to develop the analytical solutions of the mode I stress intensity factor for spot welds in lap-shear specimens of large and finite sizes. The analytical solutions of the mode I stress intensity factor based on the solutions for infinite and finite square plates with an inclusion are compared with the results of the three-dimensional finite element computations of lap-shear specimens with various ratios of the specimen half width to the nugget radius. The results indicate that the mode I stress intensity factor solution based on the finite square plate model with an inclusion agrees well with the computational results for lap-shear specimens for the ratio of the half specimen width to the nugget radius between 4 and 15. Finally, a set of the closed-form stress intensity factor solutions for lap-shear specimens at the critical locations are proposed for future applications.     

MHD free convection flow of visco-elastic fluid past an infinite vertical porous plate

 This work provides a comprehensive theoretical analysis of a two-dimensional unsteady free convection flow of an incompressible, visco-elastic fluid past an infinite vertical porous plate. Solutions for the zero order perturbation velocity profile, the first order perturbation velocity and temperature profiles in closed form are obtained with the help of Laplace transform technique. The numerical solutions are carried out for the Prandtl number 0.1, 0.72, 1.0, 1.5 and 2.0 which are appropriate for different types of liquid metals and for different values of magnetic field parameter, M. Received on 1 September 1999     

A SEMI-ANALYTICAL METHOD FOR THE VIBRATION OF AND SOUND RADIATION FROM A TWO-DIMENTIONAL BEAM-STIFFENED PLATE

A semi-analytical method based on space harmonics to investigate the vibration of and sound radiation from an infinite,fluid-loaded plate is presented.The plate is reinforced with two sets of orthogonally and equally spaced beam stiffeners,which are assumed to be line forces.The response of the stiffened plate to a convected harmonic pressure in the wave-number space is obtained by adopting the Green’s function and Fourier transform methods.Using the boundary conditions and space harmonic method,we establish the relationship between the stiffener forces and the vibration displacement of the plate.In this paper,the stiffener forces are expressed in terms of harmonic amplitudes of the plate displacement,which are calculated by using a numerical reduction technique.Finally,the Fourier inverse transform is employed to find expressions of the vibration and sound radiation in physical space.Agreements with existing results prove the validity of this approach and more numerical results are presented to show that this method converges rapidly.     

Dynamic stress intensity factors around two rectangular cracks in an infinite elastic plate under impact load

A dynamic problem for two equal rectangular cracks in an infinite elastic plate is considered. The two cracks are placed perpendicular to the plane surfaces of the plate. An incoming shock tensile stress is returned by the cracks. In the Laplace transform domain, the boundary conditions at the two sides of the plate are satisfied using the Fourier transform technique. The mixed boundary conditions are reduced to dual integral equations. Crack displacement is expanded in a series of functions which are zero outside of the cracks. The unknown coefficients in the series are determined by the Schmidt method. The stress intensity factors are defined in the Laplace transform domain and these are inverted using a numerical method.     

The Rayleigh-Stokes-Problem for a fluid of Maxwellian type

The velocity fields corresponding to an incompressible fluid of Maxwellian type subjected to a linear flow on an infinite flat plate and within an infinite edge are determined by means of the Fourier sine transforms. They are in close proximity of those of a second grade fluid. The well known solutions for a Navier-Stokes fluid appear as a limiting case of our solutions.     

Contact problems for a finitely deformed incompressible elastic halfspace

This paper examines the class of problems related to the interaction between a finitely deformed incompressible elastic halfspace and contacting elements that include smooth, flat rigid indenters with elliptical and circular shapes and a thick plate of infinite extent. The contact between the finitely deformed elastic halfspace and the contacting elements is assumed to be bilateral. The interaction between both the rigid circular indenter and the finitely deformed halfspace is induced by a Mindlin force that acts at the interior of the halfspace regions and by exterior loads. Similar considerations apply for the contact between the flexible plate of infinite extent and the finitely deformed elastic halfspace. The theory of small deformations superposed on large deformations proposed by Green et al. (Proc R Soc Ser A 211:128–155, 1952) is used as the basis for the formulation of the problem, and results of potential theory and integral transform techniques are used to develop the analytical results. In particular, explicit results are presented for the displacement of the rigid elliptical indenter and the maximum deflection of the flexible plate induced by the Mindlin forces, when the finitely deformed halfspace region has a strain energy function of the Mooney–Rivlin form.     

Exact solutions for unsteady unidirectional flows of a generalized second-order fluid through a rectangular conduit

The exact solutions are obtained for unsteady unidirectional flows of a generalized second-order fluid through a rectangular conduit. The fractional calculus in the constitutive relationship of a non-Newtonian fluid is introduced. We construct the solutions by means of Fourier transform and the discrete Laplace transform of the sequential derivatives and the double finite Fourier transform. The solutions for Newtonian fluid between two infinite parallel plates appear as limiting cases of our solutions.     

无穷大周期加筋微穿孔板结构振动响应及吸声特性

研究在平面声波斜入射情况下,无穷大双周期加筋的微穿孔薄板结构的振动响应及吸声性能.首先在马大猷和Takahashi微穿孔板声学理论基础上,建立了微穿孔加筋薄板结构振动的半解析模型;然后应用傅里叶变换及空间波数分析方法,将周期加筋微穿孔薄板的振动位移及辐射声压表示为频域内波数的分量迭加形式;最后通过对波数分量进行求解,并利用傅里叶逆变换得到双周期加筋的微穿孔薄板的振动响应及吸声系数表达式.计算结果表明,薄板的弯曲振动在水中对吸声的影响较大,空气中仅对轻质穿孔板的低频吸声效果有一定影响;同时微穿孔率对周期加筋薄板吸声系数的影响明显,通过改变穿孔率和加筋周期等可有效地提高水中微穿孔薄板结构的吸声性能.