Analytical solutions of reduced rigidity plastic bifurcation of circular plates

Summary Two analytical solutions of reduced rigidity plastic bifurcations, respectively for simply supported and clamped circular plates, are found in the way of a post-confirmed presumption for the distribution of unloading area at the bifurcation. TheJ ₂ flow theory is used. While the solutions, by their bifurcation modes and distributions of the unloading areas, recall many aspects of Engesser bifurcation, the corresponding bifurcation loads lie above the true Engesser loads.     

Investigation into the wrinkling behaviour of thin sheets in the cylindrical cup deep drawing process using bifurcation theory

The initiation and growth of wrinkles are influenced by many factors such as stress ratios, the mechanical properties of the sheet material, the geometry of the workpiece, and contact condition. It is difficult to analyse wrinkling initiation and growth while considering all the factors because the effects of the factors are very complex and studies of wrinkling behaviour may show a wide scattering of data even for small deviations in factors. In the present study, the bifurcation theory is introduced for the finite element analysis of wrinkling initiation and growth. All the above mentioned factors are conveniently considered by the finite element method. Wrinkling initiation is found by checking the determinant of the stiffness matrix and wrinkling behaviour is analysed by successive iteration with the perturbed guess along the eigenvector. Wrinkling initiation and growth in the cylindrical cup deep drawing process are analysed by using the proposed bifurcation algorithm. The mechanism of wrinkling initiation and growth in the cylindrical cup deep drawing process is investigated in detail. The results of the finite element analysis are compared with the experimental results. Through the finite element analyses of wrinkling growth for various material constants, the effects of plastic anisotropy and the strain hardening coefficient on the wrinkling behaviour of sheet metal are investigated. Copyright © 2003 John Wiley & Sons, Ltd.     

Vibration of circular and annular plates using finite elements

The finite element method is applied to the free transverse vibration of circular and annular plates of varying thickness. An annular element is derived which incorporates the number of diametral modes in the deflection function. This results in an element having only four degrees of freedom, these being the deflection and slopes at the inner and outer radii of the element at an anti-node of the particular vibration mode. Thickness variation in the radial direction is readily introduced, and stiffness and inertia matrices are presented for elements with linear and parabolic variations in thickness. The method is checked with several numerical examples. Calculations of free vibration of circular and annular plates of constant, linear and parabolic thickness variation are compared with available exact solutions.     

Homogenization and bifurcation of perforated plates

This short paper aims to present an application of the homogenization method to a nonlinear situation. We show that the buckling and post buckling analysis of a multiperforated plate can be reduced to that of an homogeneous plate, the constitutive law of which is given. We use a nonclassical formulation of the Von Karman equations which is adapted to multiconnected domains and seems to be useful for others problems.     

Particulate flow simulations using lubrication theory solution enrichment

A technique for the numerical simulation of suspensions of particles in fluid based on the extended finite element method (X‐FEM) is developed. In this method, the particle surfaces need not conform to the finite element boundaries, so that moving particles can be simulated without remeshing. The finite element basis is enriched with the Stokes flow solution for flow past a single particle and the lubrication theory solution for flow between particles. The latter enrichment allows the simulation of particles that come arbitrarily close together without refining the mesh in the gap between them. Example problems illustrating both types of enrichment are shown, along with a study of a 50% solution in channel flow. Copyright © 2003 John Wiley & Sons, Ltd.     

Derivation of a modified lubrication equation for hydromagnetic non-Newtonian cylindrical squeeze films and its application to circular plates

On the basis of the hydromagnetic flow model and the Stokes microcontinuum fluid model, a modified lubrication equation for cylindrical squeeze films is derived. To guide the use of the derived equation, an example with parallel circular plates is illustrated. Comparing with the case of a non-conducting Newtonian lubricant, the hydromagnetic non-Newtonian effects provide better performance for circular squeeze films. The improved characteristics are further emphasized for circular plates operating with a larger magnetic field parameter and non-Newtonian parameter.     

An exact analytical solution for freely vibrating piezoelectric coupled circular/annular thick plates using Reddy plate theory

Based on third-order shear deformation plate theory of Reddy, the authors aim to provide an exact analytical solution for free vibration analysis of thick circular/annular plates, both upper and lower surfaces of which are in contact with a piezoelectric layer. Natural frequencies are determined by the solution of the coupled electromechanical governing equations for a combination of free, soft simply supported, hard simply supported and clamped boundary conditions at the inner and outer edges of the plate. The electrodes on each piezoelectric layer are assumed to be short-circuited. The Maxwell electrostatics equation is satisfied by adopting a half-sine distribution of the electric potential in the transverse direction of the piezoelectric layers. A comparison of the present exact natural frequencies for piezoelectric coupled circular/annular plates with different boundary conditions is made with previously published results obtained by the Mindlin plate theory and 3-D modified finite element method. The effects of plate parameters such as host thickness to radius ratios, inner to outer radius ratios and piezoelectric to host thickness ratios on the natural frequencies of laminated circular/annular plates are investigated for different combinations of boundary conditions. Results obtained by the present exact closed-form solutions can be served as benchmark data for investigators to validate their numerical and analytical methods in the future.     

The effect of anisotropy on post-buckling behavior of laminated plates using semi-energy finite strip method

The effects of anisotropy (i.e. the ratio of longitudinal modulus of elasticity to transverse modulus of elasticity of a rectangular plate) on buckling and post-buckling performance of laminated plates under uniform end-shortening have been studied in this paper by implementing a Rayleigh–Ritz approach and a finite strip approach based on the concept of a rigorous post-buckling solution for composite plates and plate structures, namely the semi-energy approach. To validate the results, they are compared with those obtained from finite element method of analysis. The study of results has revealed that the buckling and post-buckling response of the laminates is significantly influenced by the changing of the anisotropy ratio.     

复合材料层合板的非线性组合共振特性及分岔

考虑几何非线性项和阻尼的影响, 给出了四边简支的正交各向异性矩形层合板在两项横向简谐激励作用下的非线性振动微分方程, 利用伽辽金法导出了相应的达芬型非线性强迫振动方程。应用多尺度法对组合共振问题进行求解, 得到了系统在稳态运动下的幅频响应方程。基于李雅普诺夫稳定性理论, 得到了解的稳定性判定条件。通过数值算例, 分析了不同参数对系统组合共振及其分岔特性的影响。结果表明, 随着调谐参数、板厚度、阻尼系数以及激励力等参数的改变, 系统存在多幅值现象、滞后现象和跳跃现象, 出现不稳定解, 且在某些参数点处具有运动性态发生变化的分岔特性, 表现出较为复杂的动力学特性。      

Pulsed deformation of circular plates and membranes

A method is presented for solution of the problem of a single pulsed deformation of structural elements based on approximation of the pulse functions by harmonic series. Expressions were obtained from which an approximate calculation was made of the dynamic deflection of an elastic circular plate in loading with a local circular pulsed load and of a circular membrane under the action of a uniformly distributed pressure pulse.Translated from Problemy Prochnosti, No. 8, pp. 6–9, August, 1992.     

On a boundary layer phenomenon in Mindlin-Reissner plate theory for laminated circular sector plates

Summary In this article, the edge-zone equation of Mindlin-Reissner plate theory, for composite plates laminated of transversely isotropic layers is studied. Analytical solutions are obtained for both circular sector and completely circular plates with various boundary conditions. The boundary-layer function and its effect on the stresses are numerically studied. Effects of plate thickness and boundary conditions are investigated. The results for circular and completely circular plates are exactly the same as those of rectangular plates in our previous work. Therefore, this boundary layer phenomenon seems to be geometry independent.     

Phenomena in the bifurcation of unsymmetric composite plates

The phenomena observed in the large scale bifurcation of unsymmetric composite plates are characterized. Such plates can exhibit more than one stable geometry and can be ‘snapped’ from one geometry to another. It is convenient to think of this process as providing sufficient strain energy to overcome an energy maximum and is analogous to an activation energy. To describe the snap-through process, 2-ply, [0, 90] unsymmetric plates were forced from one state to the other in rigid test rigs such that great control could be achieved over the process and loads and deformations were monitored throughout the bifurcation. A hitherto unrecognised phenomenon was noted in that the snap-through process was seen to take place not as a single bifurcation but as two closely coupled bifurcations, as first one side snaps through and then the other. The experimental approach taken here, coupled with the simple energy analysis, should be of value in understanding practical structures of multiple stable states, such as might be used in morphing or adaptive aircraft.     

Induction heating of circular ferromagnetic plates

The induction heating of circular ferromagnetic plates is studied to predict power levels under various excitation conditions. A detailed and accurate analysis based on a unique "noninteraction" approximation is presented. A lumped parameter transformer model is developed with the nonlinear equivalent disk resistance expressed as a function of the terminal variables in the system. The transformer model is applied to predict operating characteristics. Experimental results are presented to justify the theoretical model.     

Asymmetric flexural vibration and thermoelastic stability of FGM circular plates using finite element method

Here, asymmetric free vibration characteristics and thermoelastic stability of functionally graded circular plates are investigated using finite element procedure. A three-noded shear flexible plate element based on the field-consistency principle is used. Temperature field is assumed to be a uniform distribution over the plate surface and varied in thickness direction only. Material properties are graded in the thickness direction according to simple power law distribution. For the numerical illustrations, aluminum/alumina is considered as functionally graded material. The variation in critical buckling load is highlighted considering gradient index, temperature, radius-to-thickness ratios, circumferential wave number and boundary condition of the plate.     

Bending of plates with circular holes

Summary The method of point matching is used to solve three problems for the bending of a plate having circular holes. The first two problems consist of a uniformly loaded square plate either simply supported or clamped along the outer boundary. Free edge boundary conditions are satisfied exactly along the internal hole and the conditions along the outside contour are satisfied by point matching in the least squares sense. Deflection and bending moment curves along the hole and along the outer edges are presented for various ratios of hole diameter to size of plate. Results for the case when the outer edges are simply supported can be compared with the less extensive results ofDedic.The last problem solved is that of an infinite plate having equally spaced circular holes. The plate is loaded by its own weight and supported at points equidistant from the hole centers. Three different approaches to the problem are used, all satisfying the boundary conditions by point matching. Results for deflections and bending moments for various hole diameters are presented.

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Zusammenfassung Mittles der Methode der Randkollokation werden drei Biegungsprobleme für eine Platte mit kreisförmigen Löchern gelöst. In den beiden ersten Problemen wird eine quadratische Platte unter Gleichlast behandelt, die am Außenrand entweder frei aufliegt oder eingespannt ist. Die Bedingungen am freien Lochrand werden exakt erfüllt, jene am Außenrand durch Kollokation im Sinne der kleinsten Quadrate. Durchbiegung und Biegemoment am Lochrand und am Außenrand werden für verschiedene Verhältnisse von Lochdurchmesser zur Seitenlänge der Platte angegeben. Im Falle des frei aufliegenden Außenrandes können die Ergebnisse mit jenen—allerdings weniger umfassenden—vonDedic verglichen werden.Als letztes wird das Problem einer unbegrenzten Platte mit gleichmäßig verteilten kreisförmigen Löchern gelöst. Die Platte wird durch ihr Eigengewicht beansprucht und wird an Punkten unterstützt, die äquidistant zu den Mittelpunkten der Löcher liegen. Es werden drei verschiedene Näherungsmethoden verwendet, die alle die Randbedingungen mittels Kollokation befriedigen. Durchbiegungen und Biegemomente werden für verschiedene Lochdurchmesser angegeben.

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With 14 Figures

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The results contained in this paper are a partial product of research supported by the U.S. Air Force under Contract No. AF 33 (615) 2504, monitored byG. E. Maddux, FDTR-13, Wright Patterson Air Force Base.     

Dynamic analysis of isotropic composite plates using a layerwise theory

The paper is concerned with small amplitude vibrations of moderately thick composite plates. Polygonal and simply supported plates of three asymmetrically arranged layers in perfect bond are considered. The homogeneous layers are assumed to consist of isotropic linear elastic materials. The Mindlin–Reissner kinematic assumptions are implemented separately to each layer, and as such model both the global and local elastic response of composite plates. The continuity of the transverse shear stress across the interfaces is prescribed according to Hooke's law. For plates composed of layers with a uniform ratio of mass density to shear modulus and a common Poisson's ratio the lateral deflection can be calculated independently from the in-plane displacements. It is shown that the lateral deflection is governed by a boundary value problem of the fourth order. Alternatively, by neglecting the longitudinal as well as rotary inertia the complex problem reduces to the simpler case of a homogenized shear-deformable plate with effective stiffness and corresponding set of boundary conditions. The theory is applied to statically and dynamically loaded rectangular simply supported composite plates with various dimensions and material properties.