Limit point buckling of a finite beam on a nonlinear foundation

In this paper, we consider an imperfect finite beam lying on a nonlinear foundation, whose dimensionless stiffness is reduced from 1 to k as the beam deflection increases. Periodic equilibrium solutions are found analytically and are in good agreement with a numerical resolution, suggesting that localized buckling does not appear for a finite beam. The equilibrium paths may exhibit a limit point whose existence is related to the imperfection size and the stiffness parameter k through an explicit condition. The limit point decreases with the imperfection size while it increases with the stiffness parameter. We show that the decay/growth rate is sensitive to the restoring force model. The analytical results on the limit load may be of particular interest for engineers in structural mechanics.     

A novel nonlinear contact stiffness model of concrete–steel joint based on the fractal contact theory

The heavy-duty machine tool is usually assumed in the concrete foundation, in which the machine tool-foundation joints have a critical effect on the working accuracy and life of heavy-duty machine tool. This paper proposed a novel contact stiffness model of concrete–steel joint based on the fractal theory. The topography of contact surface exist in concrete–steel joint has a fractal feature and can be described by fractal parameters. Asperities are considered as elastic, plastic deformation in micro-scale. However, the asperities of concrete surface will be crushed when the stress is larger than their yield limit. Then, the force balance of contact surfaces will be broken. Here, an iteration model is proposed to describe the contact state of concrete–steel joint. Because the contact asperities cover a very small proportion (less than 1%), the load on crushed asperities is assumed carried by other no contact asperities. This process will be repeated again and again until the crushed asperities are not being produced under external load. After that, the real contact area, contact stiffness of the concrete–steel joint can be calculated by integrating the asperities of contact surfaces. Nonlinear relationships between contact stiffness and load, fractal roughness parameter G, fractal dimension D can be revealed based on the presented model. An experimental setup with concrete–steel test-specimens is designed to validate the proposed model. Results indicate that the theoretical vibration mode shapes agree well with the experimental variation mode shapes. The errors between theoretical and experimental natural frequencies are less than 4.18%. The presented model can be used to predict the contact stiffness of concrete–steel joint, which will provide a theoretical basis for optimizing the connection characteristic of machine tool-concrete foundation.     

On the coupling between macroscopic material degradation and interfiber bond fracture in an idealized fiber network

The numerical analysis performed here, using a finite element network model, provides a number of important results regarding the evolution of micro fractures in planar random fiber networks where the only active microscopic fracture mechanism is bond fracture. The fibers are randomly distributed in the network meaning that the network is considered having in-plane isotropic properties on the macroscopic scale. The network is loaded so that, in an average sense, homogenous macroscopic stress and strain fields are present.Several conclusions are drawn. It is found that the development of macroscopic material degradation follows an exponential two-parameter law, consisting of an onset parameter and a fracture rate parameter, justifying a previous theory derived by the authors. The fracture rate parameter is linearly related to the inverse of the bond density above a certain density limit (percolation) and increases with increasing slenderness ratio of the fibers when keeping the bond density at a constant level. The strain energies stored in interfiber bonds are exponentially distributed over the whole network. The numerical analysis reveals that there is a linear relation between the ratio of fractured and initial number of loaded bonds, and the network’s macroscopic material stiffness normalized with its pristine stiffness, confirming earlier findings based on experimental observations. At localization the analyzed theory looses its validity because the fracture process is no longer randomly distributed over the whole network. Localization coincides with location of peak load in force–displacement tensile tests.     

BIFURCATIONS OF AIRFOIL IN INCOMPRESSIBLE FLOW

Bifurcations of an airfoil with nonlinear pitching stiffness in incompressible flow are investigated. The pitching spring is regarded as a spring with cubic stiffness. The motion equations of the airfoil are written as the four dimensional one order differential equations. Taking air speed and the linear part of pitching stiffness as the parameters, the analytic solutions of the critical boundaries of pitchfork bifurcations and Hopf bifurcations are obtained in 2 dimensional parameter plane. The stabilities of the equilibrium points and the limit cycles in different regions of 2 dimensional parameter plane are analyzed. By means of harmonic balance method, the approximate critical boundaries of 2-multiple semi-stable limit cycle bifurcations are obtained, and the bifurcation points of supercritical or subcritical Hopf bifurcation are found. Some numerical simulation results are given.     

Limit models in the analysis of three different layered elastic strips

The mechanical behavior of three types of laminated strips is investigated. They are made of three layers filled with homogeneous, isotropic and elastic materials; the upper and lower layer are called adherents, the middle layer is called adhesive. The first model studies a strip consisting of three layers made of materials with similar stiffness; the second one concerns with a strip in which the adhesive is soft; in particular, we suppose that the elastic stiffness of the middle layer is two orders of magnitude smaller than that of the upper and lower layers; the third case is a strip in which the core is thinner and stiffer than the two adherents: the elastic modula of the adherents are one order of magnitude bigger that those of the adhesive. After identifying a parameter of smallness ε (which measures the thickness and the stiffness of each layer), the limit of the solution when ε tends to zero has been considered. Afterwards, it has been shown that each solution of the simplified models verifies the so-called limit problems, written using a “weak” and a “strong” formulation. The existence and uniqueness of the solutions of each limit problem have been established. The strong convergence of the exact solutions towards the solution of the limit problem of the first model has been established, too.     

依据各向异性分形理论的固定结合面椭圆弹塑性法向接触刚度建模及仿真分析

依据各向异性分形几何理论,将结合面接触点拓展为椭圆形,结合微凸体接触点面积大小分布函数以及概率论相关理论获得关于椭圆形接触点接触面积以及离心率的二维联合分布密度函数,应用赫兹接触理论,进而建立了依据各向异性分形理论的结合面椭圆弹塑性法向接触刚度模型,并采用MATLAB软件对影响结合面法向刚度的相关因素进行了数值仿真及结...     

移动荷载作用下非局部地基梁动力响应分析的有限元法Dynamic response of beams with nonlocal foundations subjected to moving loads using finite element method

基于非局部地基理论,推导了移动荷载作用下非局部地基梁动力响应问题的有限元解,分别讨论了地基的非局部参数、刚度、阻尼系数以及移动荷载速度对非局部地基梁动力响应的影响,并比较了非局部结果与局部结果的差异。结果表明,地基的非局部参数、刚度和阻尼是地基梁的动力响应的主要影响参数,地基梁最大响应及其发生的时刻与移动荷载速度有关。研究成果可为轨道地基系统设计提供参考。     

环板在局部线性荷载作用下的塑性极限分析的计算方法探讨

本文结合环板在线性荷载作用下的塑性极限分析问题,给出一种简便的计算方法,即奇异函数法。文中用此法计算了具有外边界或内边界支承的环板的极限荷载,并画出几何参数对极限荷载的影响曲线。所得的结果具有良好的规律性而且是比较合理的。     

重力坝坝基沿软弱结构面滑动失稳的非线性机制

对于存在单组软弱结构面的坝基深层抗滑问题,考虑了坝基软弱结构面介质在不同环境因素条件下的复杂弹脆性(应变硬化)和应变软化属性,用突变论建立了坝基沿单软弱结构面滑动的尖点突变模型。通过建立的模型分析,发现坝基滑动失稳主要取决于软弱面上复杂弹脆性介质和应变软化介质在应变软化拐点处的刚度比k、材料的均匀性指标m及几何—力学参数ξ;并给出了发生突变失稳的充分力学判据,指出传统的刚体极限平衡分析法存在一定的缺陷。指出库水水位上升还存在一种降低介质材料刚度比k的新机制,从理论上阐明,坝基滑动突变失稳的发生相应于水位峰值,在时间上可能具有滞后性;而工程处理措施的实施除提高坝基软弱结构面抗滑能力外,还具有一种增大材料刚度比k的新效应,使得滑动失稳更不易发生。     

The Strongest Structures with Macrostiffness Constraints∗

An example of a truss structure with limited global stiffness, measured by some parameter k, is discussed. The optimal structural design problem of maximizing the external load intensity, with axial stresses in the rods of the truss remaining within an allowable range, is well known and leads to a statically determinate solution with uniformly loaded elements. However, if one takes into account the additional constraint on stiffness k, the problem is open and the solution leads, in general, to a redundant truss. The paper presents this problem and proposes its application to optimal design of supporting structures     

Dramatically stiffer elastic composite materials due to a negative stiffness phase?

Composite materials of extremely high stiffness can be produced by employing one phase of negative stiffness. Negative stiffness entails a reversal of the usual codirectional relationship between force and displacement in deformed objects. Negative stiffness structures and materials are possible, but unstable by themselves. We argue here that composites made with a small volume fraction of negative stiffness inclusions can be stable and can have overall stiffness far higher than that of either constituent. This high composite stiffness is demonstrated via several exact solutions within linearized and also fully nonlinear elasticity, and via the overall modulus tensor estimate of a variational principle valid in this case. We provide an initial discussion of stability, and adduce experimental results which show extreme composite behavior in selected viscoelastic systems under sub-resonant sinusoidal load. Viscoelasticity is known to expand the space of stability in some cases. We have not yet proved that purely elastic composite materials of the types proposed and analyzed in this paper will be stable under static load. The concept of negative stiffness inclusions is buttressed by recent experimental studies illustrating related phenomena within the elasticity and viscoelasticity contexts.     

Kink Surfaces in a Directionally Reinforced neo-Hookean Material under Plane Deformation: I. Mechanical Equilibrium

Stationary kinks (elastostatic shocks) are examined in the context of a base neo-Hookean response augmented with unidirectional reinforcing that is characterized by a single additional constitutive parameter for the additional fiber reinforcing stiffness. Previous work has shown that such a transversely isotropic material can lose ellipticity in plane deformation if the reinforcing is sufficiently large and the fiber direction is sufficiently compressed. Here we show that the same reinforcing levels can give rise to piecewise smooth plane deformations separated by a plane stationary kink. Attention is restricted to deformations in which, on one side of the kink, the load axis is aligned with the fiber axis. Then the fiber stretch on this side of the kink is a natural load parameter. It is found that such a deformation can support a planar kink for a certain range of this load parameter. This range is dependent on the reinforcing parameter, and can even involve fiber extension if the reinforcing is sufficiently large. The set of all deformation states on the other side of the kink is precisely characterized in terms of a one-parameter family of (kink orientation, kink strength)-pairs. The results are interpreted in terms of the associated fiber alignment discontinuity and fiber stretch discontinuity.     

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.     

Frictional energy dissipation in contact of nominally flat rough surfaces under harmonically varying loads

If the nominal contact tractions at an interface are everywhere below the Coulomb friction limit throughout a cycle of oscillatory loading, the introduction of surface roughness will generally cause local microslip between the contacting asperities and hence some frictional dissipation. This dissipation is important both as a source of structural damping and as an indicator of potential fretting damage. Here we use a strategy based on the Ciavarella-Jäger superposition and a recent solution of the general problem of the contact of two half spaces under oscillatory loading to derive expressions for the dissipation per cycle which depend only on the normal incremental stiffness of the contact, the external forces and the local coefficient of friction. The results show that the dissipation depends significantly on the relative phase between the oscillations in normal and tangential load—a factor which has been largely ignored in previous investigations. In particular, for given load amplitudes, the dissipation is significantly larger when the loads are out of phase. We also establish that for small amplitudes the dissipation varies with the cube of the load amplitude and is linearly proportional to the second derivative of the elastic compliance function for all contact geometries, including those involving surface roughness. It follows that experimental observations of less than cubic dependence on load amplitude cannot be explained by reference to roughness alone, or by any other geometric effect in the contact of half spaces.     

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基于计算结构力学理论与递推公式相结合的方法，推导出随机刚架结构在退化成机构 前各阶段的内力与结构处于初始状态时的内力关系. 因此，在基于随机有限元的可靠性分 析过程中，不必反复修改刚阵和进行反向节点力的增加，在极限状态函数对设计变量求导 过程中，导数形式只用到了原结构总刚度矩阵和原结构外载向量，这对编写电算程序是有 益的. 应用所提的方法与其它基于随机有限元的可靠性分析方法所求得的结果是一致 的，且大大提高了计算效率.     

Homoclinic bifurcation and chaos attractor in elastic two-bar truss

In this paper, we present a dynamic bifurcation analysis of the non-linear Duffing's equation on a simple elastic structure. The structure is a two-bar elastic truss with a damper, and possesses geometrical non-linear stiffness. We consider the dynamic instability of its structure based on Duffing's oscillation, which shows bifurcation behavior of the homoclinic orbit. We could numerically forecast the trajectory near the invariant saddle point of homoclinic bifurcation on this model, and we found that it is possible to solve dynamic bifurcation and strange attractors (chaos) on this non-linear structure. On this truss, we could investigate the dynamic stability of the strange attractor using Lyapunov exponents under the frequency and/or the amplitude parameter of periodic load.     

The size-dependent elastohydrodynamic lubrication contact of a coated half-plane with non-Newtonian fluid

Based on the couple-stress theory, the elastohydrodynamic lubrication(EHL)contact is analyzed with a consideration of the size effect. The lubricant between the contact surface of a homogeneous coated half-plane and a rigid punch is supposed to be the non-Newtonian fluid. The density and viscosity of the lubricant are dependent on fluid pressure. Distributions of film thickness, in-plane stress, and fluid pressure are calculated by solving the nonlinear fluid-solid coupled equations with an iterative method. The effects of the punch radius, size parameter, coating thickness, slide/roll ratio, entraining velocity, resultant normal load, and stiffness ratio on lubricant film thickness, in-plane stress, and fluid pressure are investigated. The results demonstrate that fluid pressure and film thickness are obviously dependent on the size parameter, stiffness ratio, and coating thickness.     

A friction contact stiffness model of fractal geometry in forced response analysis of a shrouded blade

To investigate the nonlinear vibration behavior of a shrouded blade with friction dynamic contact interface, a friction contact stiffness model is proposed to describe the friction force at different rough interfaces and different normal loads. In the proposed model, the friction contact interface is discretized to a series of friction contact pairs and each of them can experience stick, slip, or separate states. Fractal geometry is used to simulate the topography of contact surfaces. The contact stiffness is calculated using the Hertz contact theory and fractal geometry, which is related to contact interfaces parameters including normal load, roughness, Young??s modulus, and Poisson??s ratio. The trajectory tracking method is used to predict the friction force and it is not necessary to judge the transition condition among stick, slip, and separate states. It is suitable for complicated periodic motion of the contact interfaces. The forced response of a real shrouded blade is predicted using the proposed model and the multi-harmonic balance method. The effect of surface roughness, initial normal load, and contact area on the forced response of a shrouded blade is studied. It is shown that contact stiffness increases with normal load and fractal dimension. The resonant amplitude is sensitive to the initial normal load and contact surface roughness. The response can be influenced by the contact area, which is an important parameter for blade designers.     

固定接触界面法向静弹性刚度

基于Hertz接触理论推导了两个微凸体之间互相作用的法向接触静弹性刚度.根据修正后的一个微接触点的平截面积尺寸分布,给出了界面的总法向接触静弹性条件刚度、总条件载荷的解析解.将法向静弹性刚度的解析解嵌入到有限元软件中,获得整机的理论模态.通过实验对解析解进行了定量验证.以机床结合部为研究对象,在理论振型与实验振型一致的...     

Nonlinear bending of corrugated diaphragm with large boundary corrugation under compound load

IntroductionCorrugateddiaphragmisatypeofelasticthinshells .Itsdesignisverycomplicatedbecauseoftoomanyparametersthatinfluenceeachother.Inanumberofinstrumentsmeasuringdisplacements,corrugateddiaphragmissubjectedtoelasticdisplacementthatisatleastthesameorderasitsthickness,sothatitisnecessarytousegeometricalnonlineartheoryofthinshellstoanalyze.Sofarasweknow ,inmostcases,investigatorsdiscussedonlytheproblemofcorrugateddiaphragmwithuniformanddensecorrugationsundertheactionofaunique(uniformlyorconcen…