Surface layer stability under force and temperature loading

We study stability of an elastic homogeneous ormultilayer plate with a free surface under the action of force and temperature deformations. We show that possible buckling modes can be of two types. For each of these two types, we study the dependence of the critical strain on the wavelength. It was found that, for a homogeneous layer, a decrease in the wavelength leads to a decrease in the critical strain and, for a multilayer material with alternating hard and soft layers, a strain wave of finite length arises in buckling.     

Experimental and theoretical study of the buckling of narrow thin plates on an elastic foundation under compression

The paper describes the processes of elastic deformation of thin films under mechanical loading. The film is modeled longitudinally by a compressed plate arranged on an elastic foundation. A computer model of the buckling of the narrow thin plate with a delamination portion located on an elastic foundation is constructed. This paper also touches upon the supercritical behavior of the plate–substrate system. The experiments on the axial compression of a metal strip adhered to a rubber plate are performed, and 2 to 7 buckling modes are obtained therein. The critical loads and buckling modes obtained in the numerical calculations are compared with the experimental data. It is shown that there is the possibility of progressive delamination of the metal plate from the foundation if the critical load is exceeded. It is found that the use of the proposed approach, which, in contrast to other approaches, accounts for the elastic deformation of the substrate, causes the dependence between the critical bending stress and the stiffness of the foundation.     

薄宽带材局部纵向屈曲的辛弹性力学求解方法

局部纵向屈曲是普遍存在于薄宽带材生产过程的板形缺陷,是屈曲研究的难点,精确的解析求解方法对局部纵向屈曲形成机理的研究和板形质量的提高具有重要意义。本文将任意位置的局部纵向屈曲分为带材边部和内部两类,采用辛弹性力学方法直接推导得到了局部纵向屈曲区域承受不同边界约束条件时的临界屈曲应力和屈曲挠度函数,并将求解结果与有限元和相关文献结果进行了对比。结果表明：辛弹性力学方法与有限元方法相比具有相同计算精度和更高的计算效率,计算精度高于传统能量法;带材边界的约束条件对临界屈曲应力、屈曲区域几何形状和屈曲挠度函数均存在显著影响,验证了传统能量法求解的不足,有利于提高局部屈曲计算精度。     

Mechanical buckling and free vibration of thick functionally graded plates resting on elastic foundation using the higher order B-spline finite strip method

In this study, the mechanical buckling and free vibration of thick rectangular plates made of functionally graded materials (FGMs) resting on elastic foundation subjected to in-plane loading is considered. The third order shear deformation theory (TSDT) is employed to derive the governing equations. It is assumed that the material properties of FGM plates vary smoothly by distribution of power law across the plate thickness. The elastic foundation is modeled by the Winkler and two-parameter Pasternak type of elastic foundation. Based on the spline finite strip method, the fundamental equations for functionally graded plates are obtained by discretizing the plate into some finite strips. The results are achieved by the minimization of the total potential energy and solving the corresponding eigenvalue problem. The governing equations are solved for FGM plates buckling analysis and free vibration, separately. In addition, numerical results for FGM plates with different boundary conditions have been verified by comparing to the analytical solutions in the literature. Furthermore, the effects of different values of the foundation stiffness parameters on the response of the FGM plates are determined and discussed.     

Elastic stability of inhomogeneous thin plates on an elastic foundation

We study the elastic stability of infinite inhomogeneous thin plates on an elastic foundation under in-plane compression. The elastic stiffness constants depend on the coordinate variable in the thickness direction of the plate. The elastic foundation is represented as a Winkler-type model characterized by linear and nonlinear spring constants. First we derive the Föppl–von Kármán equations by taking variations of the elastic strain energy. Next we develop the linear stability analysis of the plate under uniform in-plane compression and explicitly derive the critical loads and wave numbers for particular three cases. The effects of the material inhomogeneity, material orthotropy and loading orthotropy on the critical states are examined independently. Finally, we perform a weakly nonlinear analysis of the plate at the onset of the buckling instability. With the multiple scales method, the amplitude equations for the unstable modes that provide insight into the mode type and its amplitude are derived and then the effect of the material inhomogeneity on buckling modes are evaluated qualitatively.     

Degradation of critical current in Bi2212 composite wire under compression load

Since Bi_2Sr_2Ca_1Cu_2O_(8+x)(Bi2212) wires are subject to mechanical loadings,degradation of critical current will occur. The effect of compressive loadings on the critical current of Bi2212 wire is studied by considering micro-buckling of filament. A Bi2212 wire is regarded as a unidirectional filament-reinforced composite in the theoretical analysis.By considering the influence of inclusion, the micro-buckling wavelength can be derived by using a two-dimensional model. Based on the experimental results, the critical current is fitted as a function of buckling wavelength. It is found that the decrease of the critical current is directly proportional to the reciprocal of square of the buckling wavelength.Change of micro-buckling wavelength with material parameters is discussed. A critical strain in the wire with a filament bridge is analyzed using the finite element method.     

Stability analysis of functionally graded rectangular plates under nonlinearly varying in-plane loading resting on elastic foundation

In this research work, an exact analytical solution for buckling of functionally graded rectangular plates subjected to non-uniformly distributed in-plane loading acting on two opposite simply supported edges is developed. It is assumed that the plate rests on two-parameter elastic foundation and its material properties vary through the thickness of the plate as a power function. The neutral surface position for such plate is determined, and the classical plate theory based on exact neutral surface position is employed to derive the governing stability equations. Considering Levy-type solution, the buckling equation reduces to an ordinary differential equation with variable coefficients. An exact analytical solution is obtained for this equation in the form of power series using the method of Frobenius. By considering sufficient terms in power series, the critical buckling load of functionally graded plate with different boundary conditions is determined. The accuracy of presented results is verified by appropriate convergence study, and the results are checked with those available in related literature. Furthermore, the effects of power of functionally graded material, aspect ratio, foundation stiffness coefficients and in-plane loading configuration together with different combinations of boundary conditions on the critical buckling load of functionally graded rectangular thin plate are studied.     

薄壁截面梁的歪曲屈曲

本文在梁的整体屈曲分析中,废除了传统的刚性截面假定(刚周边假定),允许梁截面自由地歪曲,进而研究了薄壁截面梁的歪曲屈曲性能。分析中采用了样条有限条法,考虑了各种不同的荷载形式、支承条件和边界约束。这一方法与有限元法相比较,计算工作量大大地减少。数值计算结果表明,在梁长细比的较大范围内,歪曲屈曲模型对梁的设计起控制作用。     

Finite strip buckling analysis of curved plate assemblies under biaxial loading

A finite strip method is presented for calculating the linear buckling stresses of structural assemblies of long, thin plate components which, in general, are curved and which are rigidly joined together at their longitudinal edges. It is assumed that on buckling under the action of a biaxial direct stress field the perturbation forces and displacements all vary sinusoidally in the longitudinal direction. A stiffness matrix relating the amplitudes of the perturbation forces and displacements is developed for the curved strip on the further assumption of relatively high-order polynomial variations of the displacement components around the plate width. Numerical results are presented of the application of the curved strip in calculating the buckling stresses of plates, cylinders, panels and formed sections.     

梯度弹性基础上正交异性薄板的屈曲分析

基于双参数弹性基础模型,研究了梯度弹性基础上正交异性薄板的屈曲问题. 首先,基于能量法与变分原理,给出了梯度弹性基础上正交异性薄板的屈曲控制方程,并得到了梯度弹性基础刚度系数K₁ 与K₂的计算式;进而,通过将位移函数采用三角函数展开的方法,给出了单向压缩载荷作用下、四边简支正交异性弹性基础板屈曲载荷的计算式;在算例中,通过将该文的解退化到单纯的正交异性板,并与经典弹性解比较,证明了理论的正确性;最后,求解了弹性模量在厚度方向上呈幂律分布的梯度基础上的薄板屈曲问题,分析了基础上下表层材料弹性模量比与体积分数指数对屈曲载荷的影响.     

Axial buckling of perforated plates reinforced with strips and middle tubes

Global buckling of perforated plates reinforced with circumferential strip or short tube is investigated. Effects of the hole radius, width of the strip, thickness and radius of the tube and boundary conditions are studied numerically and experimentally. Axial buckling loads of the holed plates decrease versus the hole radius. By using the strip or tube, the buckling strength increases significantly. In some cases, the stiffened plate has buckling load greater than the perfect plate. Numerical studies showed that the increasing restraints at the boundaries increase the buckling strength in any case and geometry of the plate.     

弹性地基上多孔功能梯度材料矩形板的自由振动与临界屈曲载荷分析

多孔功能梯度材料(FGM)构件的特性与孔隙率和孔隙分布形式有密切关系。本文基于经典板理论,考虑不同孔隙分布形式时修正的混合率模型,研究Winkler弹性地基上四边受压多孔FGM矩形板的自由振动与临界屈曲载荷特性。首先利用Hamilton原理和物理中面的定义推导Winkler弹性地基上四边受压多孔FGM矩形板自由振动的控制微分方程并进行无量纲化,然后应用微分变换法(DTM)对无量纲控制微分方程和边界条件进行变换,得到计算无量纲固有频率和临界屈曲载荷的代数特征方程。将问题退化为孔隙率为零时的FGM矩形板并与已有文献进行对比以验证其有效性。最后计算并分析了梯度指数、孔隙率、地基刚度系数、长宽比、四边受压载荷及边界条件对多孔FGM矩形板无量纲固有频率的影响以及各参数对无量纲临界屈曲载荷的影响。     

冲击载荷作用下矩形薄板的弹性动力屈曲

为了研究冲击载荷作用下考虑应力波效应弹性矩形薄板的动力屈曲,根据动力屈曲发生瞬间的能量转换和守恒准则,导出板的屈曲控制方程和波阵面上的补充约束条件,真实的屈曲位移应同时满足控制方程和波阵面上的附加约束条件。满足上述条件,建立了该问题的完整数值解法,对屈曲过程中冲击载荷、屈曲模态和临界屈曲长度之间的关系进行研究,定量计算了横向惯性效应对提高薄板动力屈曲临界应力的贡献。研究表明:板的厚宽比一定时,临界屈曲长度随冲击载荷的增大而减小;由于屈曲时的横向惯性效应,应力波作用下薄板一阶临界力参数是相应边界板的静力失稳临界力参数的1.5倍;随着边界约束逐渐减弱,板临界力参数逐渐减小,动力特征参数逐渐增大。     

Buckling strength analysis of adhesively bonded aluminum hat sections

This paper provides an analytical solution for the critical buckling stress of adhesively bonded aluminum hat sections under static axial compression. The governing rectangular plate member of the structure is treated based on the differential equation for out-of-plane deflections of thin plates. Finite element eigenvalue buckling analysis is performed to verify the assumed simply supported boundary conditions for common edges between adjacent plate elements. Elastic restraint is applied to the two loaded edges of the rectangular plate, and the relative critical buckling stress is computed according to the transcendental equations. It is found from experiments that there is no adhesive bonding failure in the elastic buckling stage. The analytical solution yields buckling stress predictions which are in reasonable agreement with measured values.     

Nonlocal vibration and buckling of two-dimensional layered quasicrystal nanoplates embedded in an elastic medium

A mathematical model for nonlocal vibration and buckling of embedded two-dimensional(2 D) decagonal quasicrystal(QC) layered nanoplates is proposed. The Pasternak-type foundation is used to simulate the interaction between the nanoplates and the elastic medium. The exact solutions of the nonlocal vibration frequency and buckling critical load of the 2 D decagonal QC layered nanoplates are obtained by solving the eigensystem and using the propagator matrix method. The present three-dimensional(3 D) exact solution can predict correctly the nature frequencies and critical loads of the nanoplates as compared with previous thin-plate and medium-thick-plate theories.Numerical examples are provided to display the effects of the quasiperiodic direction,length-to-width ratio, thickness of the nanoplates, nonlocal parameter, stacking sequence,and medium elasticity on the vibration frequency and critical buckling load of the 2 D decagonal QC nanoplates. The results show that the effects of the quasiperiodic direction on the vibration frequency and critical buckling load depend on the length-to-width ratio of the nanoplates. The thickness of the nanoplate and the elasticity of the surrounding medium can be adjusted for optimal frequency and critical buckling load of the nanoplate.This feature is useful since the frequency and critical buckling load of the 2 D decagonal QCs as coating materials of plate structures can now be tuned as one desire.     

Axonal Buckling Following Stretch Injury

Diffuse brain injury is caused by rapid rotation of the head, and causes strain injury to tissue throughout the brain. Following strain injury, axons exhibit delayed recovery, showing regional buckling behavior immediately after stretch and returning to their original appearance over an extended period of time. This axonal buckling is hypothesized to occur as a result of localized stretching within the axon: Rapid strain causes mechanical damage to microtubules, increasing the effective length of axons. This damage is repaired gradually returning the axon to its initial length.Here, we test the hypothesis that localized stretching is a possible explanation for the regional buckling behavior. An elongated region of axon is modeled as an Euler beam on an elastic foundation, where the foundation represents the surrounding brain tissue, which consists of glial cells and extracellular matrix. After stretch the elastic foundation returns immediately to its pre-stretch length, while the axon is initially elongated and returns to its original length over a longer period of time. The model exhibits solutions similar to those observed experimentally in post-stretch axons, with undulations that have a similar wavelength and amplitude.     

On modes of buckling for a plate on an elastic foundation

We study flexural stability of a homogeneous plate compressed in its plane and lying on an elastic foundation. There is an elastic stratum between the plate and the foundation. In the case of homogeneous uniform compression, the linear approximation does not answer the question of the shape of buckling. A geometrically nonlinear approach leads to the conclusion that the buckling shape is of staggered character.     

On out-of-plane buckling of anisotropic elastic plate subjected to a simple shear

Out-of-plane buckling of anisotropic elastic plate subjected to a simple shear is investigated. From exact 3-D equilibrium conditions of anisotropic elastic body with a plane of elastic symmetry at critical configuration, the eqution for buckling direction (buckling wave direction) parameter is derived and the shape functions of possible buckling modes are obtained. The traction free boundary conditions which must hold on the upper and lower surfaces of plate lead to a linear eigenvalue problem whose nontrivial solutions are just the possible buckling modes for the plate. The buckling conditions for both flexural and barreling modes are presented. As a particular example of buckling of anisotropic elastic plate, the buckling of an orthotropic elastic plate, which is subjected to simple shear along a direction making an arbitrary angle of θ with respect to an elastic principal axis of materials, is analyzed. The buckling direction varies with θ and the critical amount of shear. The numerical results show that only the flexural mode can indeed exist. Project supported by the National Natural Science Foundation of China (No. 19772032).     

Stretch-induced stress patterns and wrinkles in hyperelastic thin sheets

Wrinkles are commonly observed in stretched thin sheets and membranes. This paper presents a numerical study on stretch-induced wrinkling of hyperelastic thin sheets based on nonlinear finite element analyses. The model problem is set up for uniaxial stretching of a rectangular sheet with two clamped ends and two free edges. A two-dimensional stress analysis is performed first under the plane-stress condition to determine stretch-induced stress distribution patterns in the elastic sheets, assuming no wrinkles. As a prerequisite for wrinkling, development of compressive stresses in the transverse direction is found to depend on both the length-to-width aspect ratio of the sheet and the applied tensile strain in the longitudinal direction. A phase diagram is constructed with four different distribution patterns of the stretch-induced compressive stresses, spanning a wide range of aspect ratio and tensile strain. Next, an eigenvalue analysis is performed to find the potential buckling modes of the elastic sheet under the prescribed boundary conditions. Finally, a nonlinear post-buckling analysis is performed to show evolution of stretch-induced wrinkles. In addition to the aspect ratio and tensile strain, it is found that the critical condition for wrinkling and the post-buckling behavior both depend sensitively on the sheet thickness. In general, wrinkles form only when both the magnitude and the distribution area of the compressive stresses are sufficiently large. The wrinkle wavelength decreases with increasing strain, in good agreement with the prediction by a scaling analysis. However, as the tensile strain increases, the wrinkle amplitude first increases and then decreases, eventually flattened beyond a moderately large critical strain, in contrast to the scaling analysis.     

ON THE ORIENTATION OF BUCKLING DIRECTION OF ANISOTROPIC ELASTIC PLATE UNDER UNIAXIAL COMPRESSION

The theory of small deformation superimposed on a large deformation of an elastic solid is used to investigate the buckling of anisotropic elastic plate under uniaxial compression. The buckling direction (the direction of buckling wave) is generally not aligned with the compression direction. The equation for determining the buckling direction is obtained. It is found that the out-of-plane buckling of anisotropic elastic plate is possible and both buckling conditions for flexural and extensional modes are presented. As a specific case of buckling of anisotropic elastic plate, the buckling of an orthotropic elastic plate subjected to a compression in a direction that forms an arbitrary angle with an elastic principal axis of the materials is analyzed. It is found that the buckling direction depends on the angle between the compression direction and the principal axis of the materials, the critical compressive force and plate-thickness parameters. In the case that the compression direction is aligned with the principal axis of the materials, the buckling direction will be aligned with the compression one irrespective of critical compressive force and plate-thickness. Project supported by the National Natural Science Foundation of China (No. 19772032).