Axisymmetric bonded punch problem: A complete solution

Summary Explicit expressions are derived for the field of stresses and displacements around an axisymmetric punch bonded to a transversely isotropic elastic half-space. The method is based on the new results in potential theory obtained by the author earlier. A flat centrally loaded circular punch is considered as an example. Specific computations were performed in order to compare the elastic field in the vicinity of a bonded punch with similar parameters for a smooth punch.

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Eine vollständige Lösung des Stempelproblems bei Axialsymmetrie und Haften

Übersieht Für die Spannungs- und Verschiebungsfelder eines transversal-isotropn elastischen Halbraums, der unter Haftbedingung von einem axialsymmetrischen Stempel belastet ist, werden explizite Formeln hergeleitet. Die dazu benutzte Methode beruht auf neueren Ergebnissen des Autors in der Potentialtheorie. Als Beispiel wird ein zentrisch belasteter, flacher Kreisstempel behandelt. Als numerische Ergebnisse werden die Verschiebungsfelder in der Umgebung eines haftenden und glatten Stempels bei gleichen Parameterbedingungen zum Vergleich vorgestellt.

     

A convergent method of solution for clamped rectangular plate

The conventional method of solution for clamped rectangular plate lies in releasing the constraints for rotation at the clamped sides and substituting them by equivalent distributed bending moments. This paper gives a general formulation of this method for arbitrary load with proof of the convergence, and discusses the rate of convergence for displacements and bending moments. The method and its convergence can be extended to the case of arbitrary boundary conditions and to plate on elastic foundation.The author expresses his sincere gratitude to Prof. Hu Hai-chang for guidance in this work.     

Uniform indentation of the elastic half-space by a rigid rectangular punch

The problem considered is that of a rigid flat-ended punch with rectangular contact area pressed into a linear elastic half-space to a uniform depth. Both the lubricated and adhesive cases are treated. The problem reduces to solving an integral equation (or equations) for the contact stresses. These stresses have a singular nature which is dealt with explicitly by a singularity-incorporating finite-element method. Values for the stiffness of the lubricated punch and the adhesive punch are determined: the effect of adhesion on the stiffness is found to be small, producing an increase of the order of 3%.     

Plane contact problem for an elastic rectangular punch and a half-plane with initial stresses

International Applied Mechanics -     

矩形板屈曲问题的一个小波解

利用Wavelet-Galerkin法分析了四边固支与四边简支矩形板的屈曲问题.以小波作为基函数表示板的挠度,推导出屈曲系数及屈曲模态的计算过程.数值计算给出了不同边长比的矩形板的屈曲系数及屈曲半波数.与传统的三角函数作基函数的Galerkin法及有限元法结果比较,结果表明在一定条件下小波可以作为试函数解决结构力学的屈...     

A perturbation solution of non-linear vibration of rectangular orthotropic plates

Classical perturbation theory is applied to the non-linear dynamic response of orthotropic plates. Expressions are derived for the ratio of non-linear to linear frequency, membrane stress, and the ratio of the maximum total stress to the maximum bending stress. Where possible the analysis is compared to other available numerical solutions, and excellent agreement is shown.     

Vibrations and self-heating of a layered elastic-viscoelastic rectangular prism under a vibrating punch

The vibrations and self-heating of a layered metal-polymer rectangular prism kinematically excited by a normally vibrating punch are analyzed numerically. The effect of reinforcement layers on the rate of variation in and the spatial distribution of the temperature field is examined. It is established that thin metallic layers in a polymer matrix change the dissipation mechanism from shear to bulk. The effect of contact stress concentration on the localization of temperature field is studied. It is established that the energy flux through the load application area decreases due to softening of the material. Several reinforcement effects are revealed. It is shown that they are determined by the value of the load parameter. The numerical results are in good agreement with experimental data on self-heating kinetics __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 8, pp. 71–79, August 2007.     

Mass transfer of aerosols with axial diffusion in narrow rectangular channels

The problem of mass transfer of aerosols with axial, as well as radial, diffusion in laminar flow in a narrow rectangular channel is studied. Two cases are investigated. The first case is where all particles enter the channel inlet and none form within the channel; and the second, where no particles enter the channel, and “formation in flight” occurs within the channel. For each case, analyses are made for both slug and Poiseuille flows. The first twenty modes of the eigenvalues, the eigenfunctions, and the coefficients of series expansion are obtained for several diffusion Péclet numbers, Pe. The first twelve of them are presented for Pe=1, 5, 10, 100, and ∞. Asymptotic expressions for the eigenvalues and the eigenfunctions are also given. The effects of axial diffusion on the local particle concentration, the bulk concentration, the Sherwood number, and the fraction of aerosols arriving at any cross-section of the channel are studied for various diffusion Péclet numbers. It was found that, for diffusion with or without formation in flight, the effect of axial diffusion may be neglected at an axial distance from the channel inlet greater than one and a half times that of the channel height for 1<Pe<100.     

A novel solution of rectangular composite laminates under oblique low-velocity impacts

An analytical solution for the responses of composite laminates under oblique low-velocity impacts is presented for a cross-ply, orthotropic, and rectangular plate under oblique low-velocity impacts. The plate is under simply-supported edge conditions, and the dynamic displacement field is expressed in a mixed form by in-plane double Fourier series and cubic polynomials through the thickness as 12 variables for each layer. A system of modified Lagrange equations is derived with all interface constraints. The Hertz and Cattaneo-Mindlin theories are used to solve for the normal and tangential contact forces during the impacts. By further discretizing in the time domain, the oblique impact problem is solved iteratively. While the numerical results clearly show the influence of impact velocity, stacking sequence, mechanical parameters, and geometric parameters, the proposed analytical approach could serve as a theoretical basis for the laminate analysis and design when it is under low-velocity impacts.     

Similarity solution of thermal boundary layers for laminar narrow axisymmetric jets

In this work, the similarity equation describing the thermal boundary layers of laminar narrow axisymmetric jets is derived based on boundary layer assumptions. The equation is solved exactly. Some properties of the thermal jet are discussed. By introducing new-defined non-dimensional coordinates, the similarity solution results in a “universal” format. The results can also be applied in the boundary layer problem of species diffusion.     

CC series solution for bending of rectangular plates on elastic foundations

The analytical solution for the bending problem of the rectangular plates on an elastic foundation is investigated by using the Stockes' transformation of a double variables function. The numerical results for the rectangular plates with free edges on the elastic foundations under a concentrated force are given in the example. First Received Dec. 14 1993     

A comprehensive analytical solution for free vibration of rectangular plates with classical edge coditions: Experimental verification

The problem of obtaining free vibration frequencies and mode shapes of rectangular plates resting on combinations of classical (i.e., clamped, simply supported, or free) edge supports is one that has been investigated for more than one hundred years. More recently, the superposition method has been developed for obtaining accurate analytical-type solutions for this family of problems. The object of this paper is to report on the results of numerous experimental tests carefully performed in order to verify the superposition method and associated computer software. Experimental and computed results are compared for a wide range of plate configurations. Very good agreement between theory and experiment has been obtained with regard to both plate natural frequencies and mode shapes. It is concluded that this computational procedure constitutes a powerful new tool for analysis of rectangular plate vibration problems.     

Enhancement of heat transfer due to bubbles passing through a narrow vertical rectangular channel

An experimental study has been made of enhancement of heat transfer for water at a subcooling of 70, 40, 20, 10 K and for ethanol at a subcooling of 60, 30 K in a narrow vertical rectangular channel (0.02 m wide, 0.002 m space and 0.2 m long) at atmospheric pressure, in which a vertical heated surface (0.02 m wide and 0.01 m long) is located on one side at a position of 0.15 m from its entrance and air bubbles are forcibly passed through it at a designated period from 0.1 to 10 s. The length of the bubble can be controlled to be equal to 0.03, 0.02, and 0.01 m in length. In addition, a theoretical analysis on the basis of a transient thermal conduction model has been made to elucidate the mechanism of the enhancement. The experiment shows that the heat transfer coefficients at the subcooling of 40 and 70 K are determined only by the period of the passing bubble within the experimental range. The experimental result is a similar tendency to a theoretical one. The difference between both results arises from the effect of the flow which is not taken into account in the analysis.Es wurde eine experimentelle Untersuchung der Verbesserung der Wärmeübertragung von Wasser bei einer Unterkühlung von 70, 40, 20, 10 K und von Ethanol bei einer Unterkühlung von 60 und 30 K in einem schmalen, senkrechten, rechtwinkeligen Kanal (0,02 m breit, 0,002 m Zwischenraum und 0.2 m lang) bei Atmosphären-Druck durchgeführt. In dem Kanal ist eine senkrecht beheizte Oberfläche (0,02 m breit und 0,01 m lang), 0,15 m vom Kanaleingang entfernt, an einer Seite plaziert. Luftblasen strömen in einer Periode von 0,1 bis 10 s an der Oberfläche vorbei. Die Größe der Luftblasen liegt im Bereich von 0,03, 0,02 und 0,01 m. Um die Wirkungsweise der Verbesserung in der Wärmeübertragung erklären zu können, wurde eine theoretische Untersuchung auf der Basis eines transienten thermischen Leitungsmodells durchgeführt. Das Experiment zeigt, daß die Wärmeübertragungskoeffizienten bei einer Unterkühlung von 40 und 70 K nur von der Periode der vorbeiströmenden Blasen abhängen. Das experimentelle Ergebnis zeigt ähnliche Tendenzen wie das Theoretische. Der Unterschied zwischen beiden Ergebnissen entsteht durch den Effekt, den die Strömung auf die Untersuchung ausübt und der bei der Analyse nicht in Betracht gezogen wurde.     

Exact solution of a flat smooth punch on a piezoelectric half plane

This paper derives an exact solution for a flat smooth punch applied on a piezoelectric half plane. The piezoelectric solid occupies the lower half plane, and a flat rigid punch is applied on it. As the permittivity of the air (environment) is far less than that of the piezoelectric material, the electric induction of air may be neglected. The permittivity of the punch is also far less than that of the piezoelectric material and consequently the normal component of the electric displacement vanishes at the contact boundary. The exact solution is obtained by eigen-function representation and analytic continuation. The distribution of pressure under the punch has been found. The electric field along the surface of the lower half plane is extracted in a closed form.     

An exact solution for transversely isotropic,simply supported and layered rectangular plates

Levinson's solution for the problem of a simply supported rectangular plate of arbitrary thickness by normal surface loads is extended to the transversely isotropic and layered case. The exact closed form solution is obtained by using the propagator matrix method in a system of vector functions. As a special case of the layered medium, the normal displacement or deflection of a homogeneous plate of arbitrary thickness by normal surface loads is also given. It is shown that it approaches the classical solution for the transversely isotropic thin plate as the thickness approaches zero on the one hand, and on the other hand reduces to the thick plate expression as given by Levinson when the medium is isotropic.     

The double packer permeameter with narrow packers. Analytical solution for non steady flow

The hydraulic diffusivity and permeability of a rock mass can be measured with a double packer permeameter, placed in a long bore hole. If the flow is steady, the relation between supply pressure and flux is a measure of the permeability whereas if the flow is transient, the relation is a measure of both the hydraulic diffusivity of the rock and the elasticity of the equipment. If the equipment is effectively rigid and the distance between the packers is much bigger than the size of the packers, an approximate closed solution is characterized bytwo parameters: the distance between the packers divided by the radius of the hole and the width of the packers divided by the radius of the hole. The flow is fully 3-D if the hydraulic diffusion distance far exceeds the distance between the packers. If the elasticity of the equipment is not negligible, the transient solution is characterized byone parameter including the elasticity coefficient of the equipment the conductivity of the rock and the radius of the hole. The relations between pressure and flux is given as a closed formula, suitable for engineering practice.