Crack buckling in soft gels under compression

Recent interest in designing soft gels with high fracture toughness has called for simple and robust methods to test fracture behavior. The conventional method of applying tension to a gel sample suffers from a difficulty of sample gripping. In this paper, we study a possible fracture mechanism of soft gels under uni-axial compression. We show that the surfaces of a pre-existing crack, oriented parallel to the loading axis, can buckle at a critical compressive stress. This buckling instability can open the crack surfaces and create highly concentrated stress fields near the crack tip, which can lead to crack growth. We show that the onset of crack buckling can be deduced by a dimensional argument com- bined with an analysis to determine the critical compression needed to induce surface instabilities of an elastic half space. The critical compression for buckling was verified for a neo-Hookean material model using finite element simulations.     

Buckling of an Axially Compressed Incompressible Half Space∗

Symmetric and nonsymmetric buckling solutions are obtained for an axially compressed half space. The half space is assumed to be made of a general in compressible isotropic elastic material. Solutions are made possible via the use of a direct approach that does not employ conventional Fourier-analysis calculations.     

Creep buckling considering material damage

The influence of Kachanov-Rabotnov type damage on creep buckling of a compressed column is investigated. All deformation and damage of the column is assumed to take place at a “creep and damage hinge”. Instantaneous buckling as well as creep buckling are shown to be represented by a certain instability surface in strain-damge-load space. The special case of purely brittle, i.e. non deforming, compressive instability is studied in some detail.     

黏弹性薄板蠕变屈曲的载荷-时间特性研究

通过对黏弹性薄板压屈的稳定性分析,着重讨论了蠕变屈曲载荷－时间的特性,理论分析表明：黏弹性薄板蠕变屈曲与材料的力学性能密相关,屈曲载荷不像弹性薄板为一定值,而是与时间相关的;在一定载荷下,经过一段时间后出现延迟屈曲,相关的实验研究也证实这一重要结论;这种延迟失稳问题在工程中有其重要的意义。     

An analysis of the effect of tension induced damage on creep buckling of columns

The influence of tension induced Kachanov-Rabotnov type damage on the buckling of a Shanley column model is investigated. Both instantaneous and delayed buckling are shown to occur when the state of the column, represented by load-deflection-damage, reaches a certain surface, the instability surface. The special case of purely brittle buckling is studied in some detail.     

高速荷载下多孔饱和地基的动力响应

研究高速荷载作用下梁与多孔饱和半空间的动力响应。由Fourier变换求解多孔饱和固体的动力基本方程，根据梁与半空间的接触条件得出多孔饱和半空间上梁的垂直位移的表达式。文中的数值算例考虑了荷载移动速度对梁的动力位移的影响，并与相应的弹性半空间问题作了对比。从算例中可以发现荷载移动速度对动力位移有很大的影响，当移动速度与半空间的表面波速相近时，地面会当产生很大的振动，同时还发现当速度大于介质的剪切波速时，多孔饱和半空间上梁的动力响应与弹性半空间上梁的动力响应有很大的差别。     

Qualitative criteria in non-linear dynamic buckling and stability of autonomous dissipative systems

A general qualitative approach for dynamic buckling and stability of autonomous dissipative structural systems is comprehensively presented. Attention is focused on systems which under the same statically applied loading exhibit a limit point instability or an unstable branching point instability with a non-linear fundamental path. Using the total energy equation, the theory of point and periodic attractors of the basin of attraction of a stable equilibrium point, of local and global bifurcations, of the inset and outset manifolds of a saddle and of the geometry of the channel of motion, the stability of the fundamental equilibrium path and the mechanism of dynamic buckling are thoroughly discussed. This allows us to establish useful qualitative criteria leading to exact, approximate and upper/lower bound buckling estimates without integrating the highly non-linear initial-value problem. The individual and coupling effect of geometric and material non-linearities of damping and mass distribution on the dynamic buckling load are also examined. A comparison of the results of the above qualitative analysis with those obtained via numerical simulation is performed on several two- and three-degree-of-freedom models of engineering importance.     

On Bifurcation in Finite Elasticity: Buckling of a Rectangular Rod

Although there is an extensive literature on the linearization instability of the nonlinear system of partial differential equations that governs an elastic material, there are very few results that prove that a second branch of solutions actually bifurcates from a known solution branch when the known branch becomes unstable. In this paper the implicit function theorem in a Banach space setting is used to prove that the quasistatic compression of a rectangular elastic rod between rigid frictionless plates leads to the buckling of the rod as is observed in experiment and as first predicted by Euler. This work was supported in part by the National Science Foundation under Grant No. DMS–8810653 and DMS–0405646.     

夹层FGM圆柱壳在扭转载荷作用下的弹性稳定性

采用半解析方法研究了两端简支的功能梯度夹层圆柱壳在端部扭转载荷作用下的弹性稳定性.考虑圆柱壳的里外表层为均匀材料,中间层为材料性质沿厚度方向连续变化的功能梯度材料,并且在界面处的材料性质保持连续. 基于Flügge薄壳理论,建立了位移形式的结构静态屈曲控制方程.根据边界条件将位移表示为三角级数形式,获得包含柱壳端部扭转载荷参数的近似线性代数特征值问题,并通过数值方法求得了表征结构失稳特征的临界载荷. 数值结果表明,临界载荷随着半径与厚度比的增加而减小,随着功能梯度中间层的弹性模量的平均值的增加而增加.      

Secondary buckling of a flat plate under uniaxial compression: Part 2. Analysis of clamped plate by F.E.M. and comparison with experiments

In Part 1, a theoretical analysis was used to study the secondary buckling of a simply supported plate. In Part 2, a clamped plate is analysed by the finite element method. The stability criterion for a non-linear post-buckling equilibrium state is evaluated by the sign of the determinant of the stiffness matrix. It should be noted that the secondary buckling loads of clamped plates are unexpectedly smaller than those of simply supported plates and are only one and a half times the primary buckling loads. In previous analyses, only a quarter segment of the plate was considered by assuming a stable equilibrium state with symmetrical mode. However, instability can also be predicted by considering the unsymmetrical mode over the whole plate. Results of experimental analysis of secondary instability of clamped square plates under uniaxial compression agreed with the numerical results.     

A problem in second order elasticity

When an incompressible half space of elastic material is twisted by a rigid circular punch bonded to its surface, to first order the azimuthal displacement is zero. In this paper, second order theory is used to investigate the deviation of the surface of the half space from the undeformed configuration.Visiting Simon Fraser University, Burnaby, B.C., V5A 1S6, Canada for the year 1982/83.     

微尺度金属薄膜屈曲稳定性的实验研究

微尺度金属薄膜的脱粘和屈曲严重影响着膜基结构的性能和使用寿命。本文对微尺度的金属铜薄膜在残余应力和外部压力共同作用下的脱粘屈曲和后屈曲模式进行了研究,用自行设计的单轴对称加载装置进行压力加载,用一台光学显微镜观察薄膜表面的屈曲形貌。在外力作用下薄膜会出现垂直于加载方向的直线型屈曲,但在外力卸载过程中该屈曲并不稳定,会演化成电话线型屈曲,完全卸载后形成泡状屈曲。再次加载后,恢复到直线型屈曲。研究表明:直线型屈曲的不稳定现象主要与薄膜的残余应力、基底的泊松比以及薄膜沿纵向与横向的应力比有关。     

Design of buckling constrained multiphase material structures using continuum topology optimization

This study contributes to a possibility of evaluating composite structures configuration such as steel and concrete using buckling and volume constraints based on multi-material topology optimization. A Jacobi active-phase algorithm is used to generate multiphase topology optimization. It provides a rational solution appropriated to the topology optimizer, Method of Moving Asymptotes due to the conflict in updating the design variables. A modified material interpolation scheme solving spurious buckling modes problem which occurs in the multi-material topology optimization process is given and discussed. An investigation of buckling constraint parameter is described. It allows a single-objective minimum compliance topology optimization to obtain two objectives of maximizing both structure stiffness and first buckling load factor. The optimal changing topologies of single material structure and multi-material structure corresponding to different buckling constraints are presented. Numerical examples of compression-only structures and compression-tension structures considering structural instability are performed using both single material and multiple materials to verify the efficiency and superiority of the present method.

     

On dynamic buckling phenomena in axially loaded elastic-plastic cylindrical shells

Some characteristic features of the dynamic inelastic buckling behaviour of cylindrical shells subjected to axial impact loads are discussed. It is shown that the material properties and their approximations in the plastic range influence the initial instability pattern and the final buckling shape of a shell having a given geometry. The phenomena of dynamic plastic buckling (when the entire length of a cylindrical shell wrinkles before the development of large radial displacements) and dynamic progressive buckling (when the folds in a cylindrical shell form sequentially) are analysed from the viewpoint of stress wave propagation resulting from an axial impact. It is shown that a high velocity impact causes an instantaneously applied load, with a maximum value at t=0 and whether or not this load causes an inelastic collapse depends on the magnitude of the initial kinetic energy.     

Buckling of a pre-compressed or pre-stretched membrane in shear flow

Membranes enclosing capsules and biological cells undergo periodic compression and stretching due to an imparted hydrodynamic traction as they rotate in a shear flow. Compression may cause transient or permanent buckling manifested by the onset of wrinkled shapes. To study the effect of pre-compression and pre-stretching on the critical conditions for buckling, the response of an elastic circular plate flush mounted on a plane wall and deforming under the action of a uniform tangential load due to an over-passing simple shear flow is considered. Working under the auspices of the theory of elastic instability of plates governed by the linear von Kármán equation, an eigenvalue problem is formulated resulting in a fourth-order partial differential equation with position-dependent coefficients parametrized by the Poisson ratio. Solutions are computed by applying Fourier series expansions to derive an infinite system of coupled ordinary differential equations, and then implementing orthogonal collocation. The solution space is illustrated, critical values for buckling are identified, the associated eigenfunctions representing possible modes of deformation are displayed, and the effect of the Poisson ratio is discussed.     

Effect of Material Nonlinearity on Spatial Buckling of Nanorods and Nanotubes

We show the importance of incorporating material nonlinearity for accurate determination of spatial buckling of nanorods and nanotubes. Both the nanorods and nanotubes are modeled as a special Cosserat rod whose nonlinear material laws are obtained using the recently proposed helical Cauchy-Born rule. We first present Euler buckling of solid diamond nanorods whose normalized buckling load, obtained from fully atomistic calculations, exhibits an interesting trend. The buckling load starts from unity at large aspect ratio of the nanorod, then as the aspect ratio is decreased, the buckling load increases slowly and finally decreases rapidly. We attribute this trend to material nonlinearity of the nanorod’s core at large compressive strain. We also discuss how surface stress affects buckling in nanorods. We then present the effect of compression and twist on buckling of single-walled carbon nanotubes. Interestingly, for highly twisted nanotubes, fully atomistic calculations show the first buckled mode to be different from a typical Euler buckling mode. Both the observations about nanorods and nanotubes are accurately replicated in the finite element special Cosserat rod simulation when the material nonlinearity is also incorporated. However, the simulation results exhibit completely different trend when only linear material laws are incorporated.     

Mechanism by which a frictionally confined rod loses stability under initial velocity and position perturbations

We present a theory to reveal for the first time the distinct mechanisms by which a compressed rod confined in a channel buckles in the presence of dry friction. Contrary to the case of a frictionless contact, with friction the system can bear substantially enhanced compressive load without buckling after its stiffness turns negative, and the onset of instability is strongly affected by the amount of perturbation set by the environment. Our theory, confirmed by simulations, shows that friction enhances stability by opening a wide stable zone in the perturbation space. Buckling is initiated when the applied compressive force is such that the boundary of the stable zone touches a point set by the environment, at a much higher critical load. Furthermore, our analysis shows that friction has a strong effect on the buckling mode; an increase in friction is found to lead to higher buckling modes.     

Small scale effects on buckling and postbuckling behaviors of axially loaded FGM nanoshells based on nonlocal strain gradient elasticity theory

By means of a comprehensive theory of elasticity, namely, a nonlocal strain gradient continuum theory, size-dependent nonlinear axial instability characteristics of cylindrical nanoshells made of functionally graded material (FGM) are examined. To take small scale effects into consideration in a more accurate way, a nonlocal stress field parameter and an internal length scale parameter are incorporated simultaneously into an exponential shear deformation shell theory. The variation of material properties associated with FGM nanoshells is supposed along the shell thickness, and it is modeled based on the Mori-Tanaka homogenization scheme. With a boundary layer theory of shell buckling and a perturbation-based solving process, the nonlocal strain gradient load-deflection and load-shortening stability paths are derived explicitly. It is observed that the strain gradient size effect causes to the increases of both the critical axial buckling load and the width of snap-through phenomenon related to the postbuckling regime, while the nonlocal size dependency leads to the decreases of them. Moreover, the influence of the nonlocal type of small scale effect on the axial instability characteristics of FGM nanoshells is more than that of the strain gradient one.     

Buckling of cracked composite columns

A crack in a structural element introduces a significant local flexibility which enhances the instability. Buckling of an edge-notched beam is studied for isotropic and anisotropic composites. The local compliance due to the presence of cracks in an anisotropic medium is formulated as a function of the crack-tip stress intensity factors and the elastic constants of the material. The general integration of the non-linear differential equations expressing the buckling model of an eccentrically loaded composite beam is derived for two different types of hinged supports ; namely freely approaching and fixed span. The effect of reducing rigidity on the load-carrying capacity and the post-buckling behavior of the beam is discussed and exemplary numerical solutions are provided. The study indicates that the instability increases with the beam slenderness and the crack length. In addition, the material anisotropy conspicuously reduces the load-carrying capacity of an externally cracked member.     

Non-conservative stability of intermediate spring supported columns with an elastically restrained end support

The non-conservative stability of an intermediate spring supported uniform column clastically restrained at one end and subjected to a follower force at the other unsupported end is studied. It is found that when the intermediate spring support is far from the unsupported end, the instability mechanism is flutter. As the intermediate spring support approaches the unsupported end, the instability mechanism is changed from flutter to divergence with the increase of intermediate spring stiffness. For the hinged-intermediate and guided-intermediatc spring supported columns, the critical buckling load of flutter instability will first decrease, then increase as the intermediate spring stiffness is increased. Nevertheless, when the instability mechanism is divergence, the critical buckling load depends on the location of the intermediate spring support only, whereas for the clamped-intermediate spring supported column the critical buckling load of divergence instability decreases monotonically to a fixed value as the intermediate spring stiffness is increased. Finally, the influence of elastic end restraints on the stability of the column is also investigated.