Nonlinear buckling and postbuckling of a shear-deformable anisotropic laminated cylindrical panel under axial compression

The nonlinear buckling and postbuckling of a shear-deformable anisotropic laminated cylindrical panel of finite length is investigated based on a boundary-layer theory for buckling. The layers of the panel are assumed to be linearly elastic. The governing equations are based on Reddy’s higher-order shear deformation theory of shells and include the von Karman-type kinematic nonlinearity and extension/twist, extension/flexure, and flexure/twist couplings. The nonlinear prebuckling deformations and the initial geometric imperfections of the panel are both taken into account. The postbuckling behavior of the panel under axial compression is analyzed. A singular perturbation technique is employed to determine its buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of perfect and imperfect moderately thick anisotropic laminated cylindrical panels with different geometric parameters and stacking sequences. The new finding reveals that there arises a compressive stress along with an associate shear stress and twisting when a moderately thick anisotropic laminated cylindrical panel is subjected to axial compression.     

Free-edge effects in a cylindrical sandwich panel with a flexible core and laminated composite face sheets

In the present study, the static response of cylindrical sandwich panels with a flexible core is investigated. The face sheets are considered as composite laminates with a cross-ply lay-up and the core as a flexible elastic medium. The flexibility of the low-strength core leads to high stress concentrations in terms of peeling stresses between the face sheets and the core at edges of the sandwich panel. To take into account the compressibility of the core and to determine the free-edge stresses of sandwich structures accurately, the Reddy layerwise theory (LWT) is used in this paper. The paper outlines the mathematical formulation, along with a numerical study, of a cylindrical sandwich panel with two simply supported and two free edges under a transverse load. The formulation includes the derivation of field equations along with boundary conditions. A Levy-type solution procedure is performed to determine the distributions of stresses and strains. In the numerical study, first a comparison is made with results from the commercial finite-element software ANSYS to verify the LWT results. Finally, a parametric study is conducted, and the effect caused by varying different parameters, such as the radii of curvature and the core to face sheet thickness ratio, on the results are investigated. The results obtained demonstrate a good agreement between LWT and FEM solutions and show increasing interlaminar stresses in the free edge of the sandwich panel     

Nonlinear response of shear deformable FGM curved panels resting on elastic foundations and subjected to mechanical and thermal loading conditions

This paper presents an analytical investigation on the nonlinear response of thick functionally graded doubly curved shallow panels resting on elastic foundations and subjected to some conditions of mechanical, thermal, and thermomechanical loads. Material properties are assumed to be temperature independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. The formulations are based on higher order shear deformation shell theory taking into account geometrical nonlinearity, initial geometrical imperfection and Pasternak type elastic foundation. By applying Galerkin method, explicit relations of load-deflection curves for simply supported curved panels are determined. Effects of material and geometrical properties, in-plane boundary restraint, foundation stiffness and imperfection on the buckling and postbuckling loading capacity of the panels are analyzed and discussed. The novelty of this study results from accounting for higher order transverse shear deformation and panel-foundation interaction in analyzing nonlinear stability of thick functionally graded cylindrical and spherical panels.     

新型空间薄壁梁单元

基于Timoshenko梁理论和Vlasov薄壁杆件约束扭转理论,建立了具有内部结点的新型空间薄壁截面梁单元．通过对弯曲转角和翘曲角采取独立插值的方法,考虑了横向剪切变形,扭转剪切变形及其耦合作用,弯曲变形和扭转变形的耦合以及二次剪应力等因素影响,由Hellinger-Reissner广义变分原理,推得单元刚度矩阵．算例表明所建模型具有良好的精度,可用于空间薄壁杆系结构的有限元分析．     

Longitudinal Flexure as a Method for Determining the Flexural Strength of Composite Materials

A method is presented for determining the flexural strength of composites from tests on pinned specimens in axial compression. This scheme is close to the so-called Euler buckling model, but the load is applied eccentrically, and the bar is bent practically from the beginning of loading. An analytical model is proposed for calculating the flexural stress in longitudinal flexure of a thin bar in the case of large displacements. The problem is solved in elliptic integrals of the first and second kinds. The analytical model shows an excellent agreement with experiments. The results obtained are compared with experimental tension data. A device is also designed for testing composite bars in longitudinal flexure. The method offered has an obvious advantage over the conventional three- or four-point bending — the gage zone in this scheme is far away from the loading zone of the test specimen.     

地下结构与岩体动力相互作用的一种解析解

本文根据大量试验和数值分析结果指出:在侧向爆荷下岩洞在厚度为跨度三分之一围岩中具有厚壁受弯构件的力学特征,而在该围岩区外即基本上接近自由场应力状态,并可用厚板理论方程在自由场压力的外载下进行求解.因此,地下结构与围岩动力相互作用,可用分别代表衬砌及介质的基于薄板与厚板理论的受弯构件动力方程宋描述.围岩和衬砌二者之间的相互作用力用接触压力函数q(x,t)加以联系.通过解一组联立方程,给出了计入与弹性半空间相互作用效应的直墙拱顶衬砌时动力分析的解析解,同时列出了函数q(x,t)的解析表达式. 本解析解将有助于从理论上探讨地下结构与介质动力共同作用的一些本质问题.     

Transverse flexure of a physically nonlinear rectangular grid plate elastically fixed at its contour

The transverse flexure of a rectangular grid plate consisting of two families of physically nonlinear rods is considered. On the basis of the calculation scheme, the hypothesis of a continuum calculation model is adopted. A numerical algorithm based on the elastic-solution method is proposed for the investigation of the initial nonlinear boundary problem. The calculation results for a plate with a rhombic grid formed by H-beam rods with a tension-compression diagram in the form of a cubic binomial are subjected to detailed analysis. The influence of physical nonlinearity of the material, as well as the parameters of the grid and elastic fixing of the contour, on the stress-strain state and the limiting load is shown.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 49–55, 1987.     

Identification of a Model of Transverse Viscoplastic Deformation for a UD Composite from Curvature Changes of Unsymmetric Cross-Ply Specimens

Mechanics of Composite Materials - A novel testing methodology for the characterization of viscoplastic (VP) deformation of unidirectional (UD) composites under transverse tensile loading is...     

Transverse shear stress relaxed zigzag theory for cylindrical bending of laminated composite and piezoelectric panels

Cylindrical bending is studied by developing a new zigzag theory which relaxes the zero transverse shear stress condition on the outer surfaces of the panels subjected to transversely applied electromechanical load. The mechanical portion of the transverse displacement approximation in this new shear deformation theory is considered constant as well as non-constant through the development of three models. Unlike the existing zigzag theories which enforce the condition of vanishing transverse shear stresses on outer surfaces of laminates, these new theories relax it. Though the number of primary mechanical variables get increased by four or five or six, the computational cost does not increase appreciably. Approximating the electric potential in each piezoelectric layer as sublayerswise linear, variational principle is applied in deriving equilibrium equations and boundary conditions. Accuracy of the new base model as well as two augmented models is assessed by comparing with elasticity and piezoelasticity solutions. While it is observed that the new base model is highly accurate than the existing zigzag model, the two augmented models do not aid in its further improvement. This is attributed to the fact that layerwise consideration of the transverse displacement, not global consideration, is needed to correctly establish the effect of transverse normal deformation in the laminated composite and smart panel.     

Mathematical modeling of the stress state of a transverse plastic layer in a round rod

We construct mathematical models of plastic deformations of a continuous round rod containing a transverse (less strong) inhomogeneous layer under an axial load. We thoroughly study the local strengthening of such layers by involving the base material of the rod in the plastic deformation process. We obtain explicit formulas for the critical stress states in the layer and the critical axial load on the rod.     

复合材料层合剪切圆柱曲板在侧压作用下的后屈曲

基于Reddy高阶剪切变形理论的Kármám-Donnell型非线性壳体方程,给出复合材料层合剪切圆柱曲板在侧压作用下的后屈曲分析。将壳体屈曲的边界层理论推广到复合材料层合剪切圆柱曲板受侧压作用的情况。相应的奇异摄动法,用于确定圆柱曲板的屈曲荷载和后屈曲平衡路径。分析中同时考虑非线性前屈曲变形和初始几何缺陷的影响。数值算例给出完善和非完善,中等厚度正交铺设层合圆柱曲板的后屈曲荷载-挠度曲线。讨论了横向剪切变形,曲板几何参数,铺层数,铺展方式和初始几何缺陷等各种参数变化的影响。     

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Solar cell is the basic component of satellite photovoltaic panels with complicated redundant system structure. Its reliability plays an important role in the system, and its performance shows a degradation trend over time. In this paper, study is conducted for the solar cell degradation modeling and reliability analysis based on practical testing results. Specifically, we illustrate an accelerated test for the attenuation ratio character test under different accumulative irradiation levels, focusing on the heteroscedasticity of the collected testing data. A heteroscedastic linear model is proposed, and the life distribution of the photovoltaic panel is obtained by using Fiducial method. A numerical example is shown for the purpose of illustration.     

Analysis of a Compliant Mechanism for Positioning in the cm-Range

The kinematics and kinetics of a compliant mechanism are analyzed quasi-statically. The mechanism consists of a parallelkinematic (Biglide type) with the particular feature, that its revolute joints are implemented by flexure hinges. In the considered example the flexure hinges are uniform beams similar to leaf springs. The preliminary considerations with a rigid body system and by linear theory of elasticity result in an initial geometry. In order to predict the large displacement behavior more precisely the theory of Elastica is used. An optimization method is applied to solve the mechanism's state of deformation. It was found that some aspects can be approximated by a pseudo rigid body system, while others, namely the rotation, cannot be rendered. The decreased stiffness in the highly deformed operating state appears as key problem. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Flexure Hinge Mechanisms Modeled by Nonlinear Euler-Bernoulli-Beams

A flexure hinge is an innovative engineering solution for providing relative motion between two adjacent stiff members by the elastic deformation of an arbitrary shaped flexible connector. In the literature, modeling of compliant mechanisms incorporating flexure hinges is mainly focused on linear methods. However, geometrically nonlinear effects cannot be ignored generally. This study presents a nonlinear modeling technique for flexure hinges based on the Euler-Bernoulli beam theory, in contrast to the predominant linear modeling approaches. Higher order beam elements of variable cross-section are employed to model the flexure hinge region. A Newton-Raphson scheme is applied to solve the resulting nonlinear system equations. The proposed approach reduces the overall degrees of freedom and is computationally efficient compared to commonly applied 3D finite element methods. A compliant displacement amplification mechanism is studied by means of the proposed method, where an excellent agreement with results of a reference solution is achieved. The modeling approach is suitable for the structural optimization of compliant mechanisms towards a less intuitive design process. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homoclinic Tangles, Phase Locking, and Chaos in a Two-Frequency Perturbation of Duffing's Equation

Summary. We study a two-frequency perturbation of Duffing's equation. When the perturbation is small, this system has a normally hyperbolic invariant torus which may be subjected to phase locking. Applying a version of Melnikov's method for multifrequency systems, we detect the occurrence of transverse intersection between the stable and unstable manifolds of the invariant torus. We show that if the invariant torus is not subjected to phase locking, then such a transverse intersection yields chaotic dynamics. When the invariant torus is subjected to phase locking, the situation is different. In this case, there exist two periodic orbits which are created in a saddle-node bifurcation. Using another version of Melnikov's method for slowly varying oscillators, we also give conditions under which the stable and unstable manifolds of the periodic orbits intersect transversely and hence chaotic dynamics may occur. Our results reveal that when the invariant torus is subjected to phase locking, chaotic dynamics resulting from transverse intersection between its stable and unstable manifolds may be interrupted. Received November 18, 1993; final revision received September 9, 1997; accepted October 27,1997     

The effect of testing equipment shift on optimal decisions in a repetitive testing process

Repetitive testing process is commonly used in the final testing stage of semiconductor manufacturing to ensure high outgoing product quality and to reduce testing errors. The decision on testing lot size and the number of testing repetitions ultimately determines the effectiveness of the testing process. Setting the retest rule is often difficult in practice due to uncertainties in the incoming product quality and testing equipment condition. In this paper, we study a repetitive testing process where the testing equipment may shift randomly to an inferior state. We develop a cost model that helps us to make optimal decisions on retesting rule. Through numerical analysis, we provide practical insights about the effects of testing equipment shift rate, testing errors, and different costs such as cost of testing and cost of rejecting conforming products on the optimal decision and the system performance. We find that significant penalty may result if the potential testing equipment shift is ignored.     

Flexure of transversely isotropic plates with a low resistance to transverse deformation

This study is concerned with the stress field around holes in plates of a transversely isotropic material, such as a reinforced plastic, with a low resistance to transverse, normal, and tangential deformation. The problem is solved on the basis of the general theory according to I. N. Vekua. It is shown that in the case of a strongly anisotropic material disregarding the transverse deformation will lead to a large error in the determination of the stress field.Institute of Metallurgy, Dnepropetrovsk. Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 1, pp. 89–92, January–February, 1974.     

On equations of the state of stress in a plate of variable thickness

Method of asymptotic integration of three-dimensional equations of the theory of elasticity is used to construct the internal state of stress in a plate of variable thickness /1/. It is shown that it can generally be described by a system of differential equations of eighth order in the components of the displacement vector of the points of a plane projected inside the plate, with the equations of flexure and of plane state of stress not separated. In a particular case when the face surfaces are symmetrical with respect to this plane, the flexure and the plane state of stress are described by separate equations. The accuracy of the equations obtained is of the order of square of relative thickness of the plate away from the edge and other distortion lines of the stress state.

The boundary layer is not considered and conditions at the plate edge are not formulated.     

Modifying a branched surface to carry a foliation

We consider C¹ nonsingular flows on a closed 3-manifold under which there is no transverse disk that flows continuously back into its own interior. We provide an algorithm for modifying any branched surface transverse to such a flow ? that terminates in a branched surface carrying a foliation F precisely when F is transverse to ?. As a corollary, we find branched surfaces that do not carry foliations but that lift to ones that do.     

Eigenwaves in a relativistic heat-conducting viscous fluid according to extended irreversible thermodynamics

We propose an approach to extended irreversible thermodynamics leading to a set of transport equations for the dissipative variables characterizing a relativistic heat-conducting viscous fluid. After this, we study the propagation eigenmodes showing the possible existence of three non-trivial modes: two longitudinal and one transverse. The fast longitudinal wave is a pressure wave associated to a slow longitudinal thermal dissipation wave and a transverse viscous dissipation wave. The general study of the longitudinal case is not possible. Natural special cases are analyzed in detail.