On the refined first-order shear deformation plate theory of Kármán type

A new refined first-order shear-deformation plate theory of the Kármán type is presented for engineering applications and a new version of the generalized Kármán large deflection equations with deflection and stress functions as two unknown variables is formulated for nonlinear analysis of shear-deformable plates of composite material and construction, based on the Mindlin/Reissner theory. In this refined plate theory two rotations that are constrained out in the formulation are imposed upon overall displacements of the plates in an implicit role. Linear and nonlinear investigations may be made by the engineering theory to a class of shear-deformation plates such as moderately thick composite plates, orthotropic sandwich plates, densely stiffened plates, and laminated shear-deformable plates. Reduced forms of the generalized Kármán equations are derived consequently, which are found identical to those existe in the literature. Foundation item: the National Natural Science Foundation of China (59675027) Biography: Zhang Jianwu (1954-)     

Asymptotic solutions for the Föppl – von Kármán equations governing deflections of thin axisymmetric annular plates

We are concerned with the deformation of thin, flat annular plates under a force applied orthogonally to the plane of the plate. This mechanical process can be described via a radial formulation of the Föppl – von Kármán equations, a set of nonlinear partial differential equations describing the deflections of thin flat plates. We are able to obtain analytical solutions for the radial Föppl – von Kármán equations with boundary conditions relevant for clamped, loosely clamped, and free inner and outer. This permits us to study the qualitative behavior of the out-of-plane deflections as well as the Airy stress function for a number of cases. Provided that an appropriate non-dimensionalization is taken, we find that the perturbation solutions are surprisingly valid for a wide variety of parameters, and compare favorably with numerical simulations in all cases (rather than just for small parameters). The results demonstrate that the ratio of the inner to outer radius of the annular plate will strongly influence the properties of the solutions, as will the specific boundary data considered. For instance, one may choose to fix the plate in place with a specific set of boundary conditions, in order to minimize the out-of-plane deflections. Other boundary conditions may result in undesirable behaviors.     

Modified von Kármán equations for elastic nanoplates with surface tension and surface elasticity

In this paper, modified von Kármán equations are derived for Kirchhoff nanoplates with surface tension and surface tension-induced residual stresses. The simplified Gurtin-Murdoch model which does not contain non-strain displacement gradients in surface stress-strain relations is adopted, so that the von Kármán strain-compatibility equation can be expressed in terms of the stress function and deflection. The modified von Kármán equations derived here are different than the existing related models especially for elastic plates with in-plane movable edges. Unlike the existing models which predict a surface tension-induced tensile pre-stress for an elastic plate with in-plane movable edges, the present model predicts that this tensile pre-stress is actually cancelled by the surface tension-induced residual compressive stress. Our this result is consistent with recent clarification on similar issue for cantilever beams with surface tension, which implies that the existing models have incorrectly predicted an invalid tensile pre-stress for an elastic plate with in-plane movable edges which leads to significant overestimation of postbuckling load and free vibration frequencies. In addition, our numerical examples indicated that surface stresses can moderately increase or decrease postbuckling load and free vibration frequency of Kirchhoff nanoplate with all in-plane movable edges, depending on the surface elasticity parameters and the geometrical dimensions of nanoplates.     

Dynamical Behavior of Viscoelastic Cylindrical Shells Under Axial Pressures

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｏｒｙｏｆｖｉｓｃｏｅｌａｓｔｉｃｉｔｙａｎｄｉｔｓａｐｐｌｉｃａｔｉｏｎｓｈａｖｅｂｅｅｎｂｅｃｏｍｉｎｇａｖｅｒｙａｔｔｒａｃｔｉｖｅｒｅｓｅａｒｃｈｆｉｅｌｄｓｉｎｃｅｖｉｓｃｏｅｌａｓｔｉｃｓｔｒｕｃｔｕｒｅｓｗｅｒｅｗｉｄｅｌｙａｐｐｌｉｅｄｉｎｔｏｖａｒｉｏｕｓｅｎｇｉｎｅｅｒｉｎｇ .Ｂｕｔｔｏｔｈｅａｕｔｈｏｒｓ’ｋｎｏｗｌｅｄｇｅ,ｔｈｅｒｅａｒｅｒａｒｅｆｏｒｐａｐｅｒｓｃｏｎｃｅｒｎｉｎｇｗｉｔｈｄｙｎａｍｉｃａｌｂｅｈａｖｉｏｒｏｆｖｉｓｃｏｅ…     

Buckling and post-buckling of annular plates under non-axisymmetric plane edge forces

On the basis of both the general theory [1,2] and the finite element method [4] of perforated thin plates with large deflection,the buckling and post-buckling of annular plates under non-axisymmetric plane edge forces are studied.The Project supported by the State Education Commission of the People's Republic of China     

BUCKLING AND POST-BUCKLING OF ANNULAR PLATES UNDER xNON-AXISYMMETRIC PLANE EDGE FORCES

On the basis of both the general theory[1,2]and the finite element method[4]ofperforated thin plates with large deflection,the buckling and post-buckling of annular platesunder non-axisymmetric plane edge forces are studied.     

开孔粘弹性薄板的非线性数学模型

应力函数的多值性和位移单值性要求，是开孔薄板大挠度问题中必须注意的两个方面。本文利用线粘性力学中的Ｂｏｌｔｚｍａｎｎ蠕变律，把开孔弹性薄板大挠度问题的一般数学理论推广到开孔弹性薄板。在考察应力函数的多值性和位移单值性条件的基础上，提出了开孔粘弹性薄板的控制方程和初、边值条件，系统地建立了开也粘弹性薄板非线性分析的三类初一边值问题。     

ON THE REFINED FIRST-ORDER SHEAR DEFORMATION PLATE THEORY OF KARMAN TYPE

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｕｓｅｏｆｌａｍｉｎａｔｅｄｃｏｍｐｏｓｉｔｅｓｉｎｔｈｉｎ＿ｗａｌｌｅｄｓｔｒｕｃｔｕｒｅｓｉｎｃｒｅａｓｅｓｓｏｔｈａｔｅｆｆｅｃｔｓｏｆｔｒａｎｓｖｅｒｓｅｓｈｅａｒｄｅｆｏｒｍａｔｉｏｎｓｃａｎｎｏｔｂｅｎｅｇｌｅｃｔｅｄａｎｄｉｎｖｏｋｅｑｕｉｔｅｃｏｍｐｌｅｘｅｓｉｎｎｏｎｌｉｎｅａｒａｎａｌｙｓｉｓ.Ｉｔｉｓｗｅｌｌ＿ｋｎｏｗｎｔｈａｔｔｈｅｎｏｎｌｉｎｅａｒａｎａｌｙｓｉｓｏｆｌａｍｉｎａｔｅｄｐｌａｔｅｓａｎｄｓｈｅｌｌｓｃｏｕｎｔｉｎｇｆｏｒｔｒ…     

孔结构金属复合夹芯板抗爆炸冲击响应及吸能特性研究

采用有限元方法研究爆炸载荷下四边固支孔结构金属复合夹芯板的动力响应及吸能特性，给出了孔结构金属复合夹芯板的动力响应过程，得到夹芯板的变形模式，比较了孔结构金属复合夹芯板与非孔结构金属复合夹芯板的抗爆炸冲击性能，同时讨论了孔大小、间距、排布方式和面板质量分布等因素对孔结构金属复合夹芯板抗爆炸冲击性能的影响。研究结果表明，迎爆面外面板的孔设计使爆炸冲击波穿过孔洞直接作用在芯材上，增强了芯材的压缩，从而提高了夹芯板的能量吸收能力。同等面密度情况下，内外面板厚度比大于1的孔结构金属复合夹芯板变形挠度小于内外面板厚度比小于1的孔结构金属复合夹芯板。进一步研究发现，通过合理设计内外面板的质量分布，可以使孔结构金属复合夹芯板的抗爆炸冲击性能最优。     

Asymptotic analysis of perforated plates and membranes. Part 2: Static and dynamic problems for large holes

Static and dynamic problems for the elastic plates and membranes periodically perforated by holes of different shapes are solved using the combination of the singular perturbation technique and the multi-scale asymptotic homogenization method. The problems of bending and vibration of perforated plates are considered. Using the asymptotic homogenization method the original boundary-value problems are reduced to the combination of two types of problems. First one is a recurrent system of unit cell problems with the conditions of periodic continuation. And the second problem is a homogenized boundary-value problem for the entire domain, characterized by the constant effective coefficients obtained from the solution of the unit cell problems. In the present paper the perforated plates with large holes are considered, and the singular perturbation method is used to solve the pertinent unit cell problems. Matching of limiting solutions for small and large holes using the two-point Padé approximants is also accomplished, and the analytical expressions for the effective stiffnesses of perforated plates with holes of arbitrary sizes are obtained.     

Dynamic response of various von-Kármán non-linear plate models and their 3-D counterparts

Dynamic von-Kármán plate models consist of three coupled non-linear, time-dependent partial differential equations. These equations have been recently solved numerically [Kirby, R., Yosibash, Z., 2004. Solution of von-Kármán dynamic non-linear plate equations using a pseudo-spectral method. Comp. Meth. Appl. Mech. Eng. 193 (6–8) 575–599 and Yosibash, Z., Kirby, R., Gottlieb, D., 2004. Pseudo-spectral methods for the solution of the von-Kármán dynamic non-linear plate system. J. Comp. Phys. 200, 432–461] by the Legendre-collocation method in space and the implicit Newmark-β scheme in time, where highly accurate approximations were realized.Due to their complexity, these equations are often reduced by discarding some of the terms associated with time derivatives which are multiplied by the plate thickness squared (being a small parameter). Because of the non-linearities in the system of equations we herein quantitatively investigate the influence of these a-priori assumption on the solution for different plate thicknesses. As shown, the dynamic solutions of the so called “simplified von-Kármán” system do not differ much from the complete von-Kármán system for thin plates, but may have differences of few percent for plates with thicknesses to length ratio of about 1/20. Nevertheless, when investigating the modeling errors, i.e. the difference between the various von-Kármán models and the fully three-dimensional non-linear elastic plate solution, one realizes that for relatively thin plates (thickness is 1/20 of other typical dimensions), this difference is much larger. This implies that the simplified von-Kármán plate model used frequently in the literature is as good as an approximation as the complete (and more complicated) model. As a side note, it is shown that the dynamic response of any of the von-Kármán plate models, is completely different compared to the linearized plate model of Kirchhoff–Love for deflections of an order of magnitude as the plate thickness.     

Effect of multiperforated plates on the acoustic modes in combustors

The analytical model derived by Howe assessing the acoustic effect of perforated plates has been implemented in a 3D Helmholtz solver. This solver allows one to compute the acoustic modes of industrial chambers taking into account the multiperforated plates present for the cooling of the walls. An academic test case consisting of two coaxial cylinders, with the inner one being perforated is used to validate the implementation in the general purpose AVSP code. This case is also used to show the effects of the presence of the plates. In particular, the sensitivity of the acoustic damping to the bias flow speed will be studied. A maximum absorption speed is shown, and the behaviour towards an infinite speed will be illustrated by the academic case. Computations are also conducted in the case of an industrial helicopter chamber. The value of the maximum absorption speed is discussed to explain why the modes are in fact not much absorbed by the perforated plates, and that the frequencies are the same as for walls. To cite this article: E. Gullaud et al., C. R. Mecanique 337 (2009).     

Variational principles of perforated thin plates and finite element method for buckling and post-buckling analysis

On the basis of the general theory of perforated thin plates under large deflections^{[1, 2]}, variational principles with deflectionw and stress functionF as variables are stated in detail. Based on these principles, finite element method is established for analysing the buckling and post-buckling of perforated thin plates. It is found that the property of element is very complicated, owing to the multiple connexity of the region. Project supported by National Natural Science Foundation of China.     

Boundary-layer flow characteristics on perforated plates and a method of approximately computing their temperature under combined cooling

Results of an experimental investigation of the heat elimination from perforated plates to cooling air during its motion along walls and partial drainage through the perforations are elucidated. Boundary-layer characteristics on perforated plates, their thermal state, and their efflux coefficients are presented for a set of orifices. An approximate method to compute the temperature of the permeable wall under combined cooling is given and its domain of application is estimated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 182–187, March–April, 1977.     

Optimal thermal design of anisotropic plates with arbitrary cutouts using genetic algorithm

Anisotropic plates in different applications may have geometric defects such as openings and cracks. The presence of the opening disturbs the heat flow, which creates significant thermal stress around the opening. When the heat flux is high enough, these extreme stresses can lead to structural failure. This article aims to obtain the optimal parameters for achieving the minimum value of the normalized stress near the cutout's boundary in perforated anisotropic plates utilizing the genetic algorithm. Optimization parameters include the curvature of opening's corners, orientation angle of opening, fibers angle, heat flux angle, and opening's elongation. The plate is under heat flux, and the opening's border is thermally insulated. The stress distribution around the opening is calculated using Lekhnitskii's complex variable method and complex potential functions. The genetic algorithm is then implemented to find the optimal values for design parameters. The results show that by selecting the optimal parameters related to the anisotropic material and the opening's geometry, the stress intensity factor of the perforated anisotropic plates is remarkably reduced. Furthermore, this optimization algorithm can be extended to find the optimized parameters and achieve the optimal designs in anisotropic and isotropic perforated plates under thermal loadings.     

Asymptotic analysis of perforated plates and membranes. Part 1: Static problems for small holes

Static problems for the elastic plates and rods periodically perforated by small holes of different shapes are solved using the asymptotic approach based on the combination of the asymptotic technique and the multi-scale homogenization method. Using the asymptotic homogenization method the original boundary-value problem is reduced to the combination of two types of problems. First one is a recurrent system of unit cell problems with the conditions of periodic continuation. And the second problem is a homogenized boundary-value problem for the entire domain, characterized by the constant effective coefficients obtained from the solution of the unit cell problems. The combination of the perturbation method and the technique of successive approximations is applied for the solution of the unit cell problems. Taking into the account small size of holes the method of perturbation of the shape of the boundary and the Schwarz alternating method are used. The problems of torsion of a rod with perforated cross-section; deflection of the perforated membrane loaded by a normal load; and bending of perforated plates with circular and square holes are considered consecutively. The error of the applied asymptotic techniques is estimated and the high accuracy of the obtained solutions is demonstrated.     

开孔薄板的变分原理及屈曲和过屈曲分析的有限元方法

本文以开孔薄板大挠度问题的一般理论为基础,建立了以挠度 w 和应力函数F为未知量的变分原理,并由此建立了用于分析开孔薄板屈曲和过屈曲问题的有限单元法可以看到由于区域的多连通性,单元性质是较为复杂的。     

On the effect of Reynolds number on von Kármán’s constant

Fully developed turbulence measurements in pipe flow were made in the Reynolds number ranging from 10×10³ to 350×10³ with a hot-wire anemometer and a Pitot tube. Comparisons were made with the experimental results of previous work. The mean velocity profile and the turbulent intensity in the experiments indicate that for the mean velocity profile, in the fully developed turbulent pipe flow, von Kármán's constant κ is a function of Reynolds number, i.e. κ increases slowly with the Reynolds number. The empirical relationships could not be considered to be accurate enough to describe the fully developed turbulence over the whole Reynolds number range in pipe flow. The project supported by the Deutscher Akademische Austauschdienst (DAAD)     

Perturbation analyses for the postbuckling of simply supported rectangular plates under uniaxial compression

In this paper, applving perturbation method to von Kármán nonlinear large deflection equations of plates by taking deflection as perturbation parameter, the posibuckling behavior of simply supported rectangular plates under uniaxial compresion is investigated. Two types of in-plane boundary conditions are now considered and the effects of initial imperfections are also studied. It is found that the theoretical results are in good agreement with experiments. The method suggested in this paper which has not been found in previous papers is rather simple and easy for the postbuckling analysis of rectangular plates.     

Hybrid determination of equivalent characteristics of perforated plates

In this paper, we have adopted the hybrid method, which combines experimental and numerical methods, to investigate the dynamic behavior of perforated plates. It was found that the modal assurance criterion is a useful criterion to judge the degree of similarity between experimental and numerical models. The difference between the experimental and numerical models can be reduced via sensitivity testing and by checking the correlation coefficient. The equivalent material properties of the perforated plates were also obtained by the hybrid method. In addition, the curve-fitting technique was utilized to find the relationship of the mass remnant ratio with the parameter ratio. The functions obtained from the curve fitting can be used to predict accurately the equivalent material properties and resonant frequencies of perforated plates of the diagonal array.