A planar rod model with flexible thin-walled cross-sections. Application to the folding of tape springs

This paper is focused on the modeling of rod-like elastic bodies that have an initially curved and thin-walled cross-section and that undergo important localized changes of the cross-section shape. The typical example is the folding of a carpenter’s tape measure for which the folds are caused by the flattening of the cross-section in some localized areas. In this context, we propose a planar rod model that accounts for large displacements and large rotations in dynamics. Starting from a classical shell model, the main additional assumption consists in introducing an elastica kinematics to describe the large changes of the cross-section shape with very few parameters. The expressions of the strain and kinetic energies are derived by performing an analytical integration over the section. The Hamilton principle is directly introduced in a suitable finite element software to solve the problem. The folding, coiling and deployment of a tape spring is studied to demonstrate the ability of the model to account for several phenomena: creation of a single fold and associated snap-through behavior, splitting of a fold into two, motion of a fold along the tape during a dynamic deployment, scenarios of coiling and uncoiling of a bistable tape spring. This 1D model may also be relevant for future applications in biomechanics, biophysics and nanomechanics.     

Snap-through buckling of two simple structures

The snap-through buckling of two simple structures subjected to quasistatic loading is analyzed by use of the elastica theory of prismatic bars. In the first problem, the deformation of perfect and imperfect three-hinged deep trusses is considered and the results of the previous experimental observations are explained analytically. In the second problem, the snap-through behavior of a column restrained by an elastic wire is studied and a comparison of critical loads are made with the approximate solutions obtained recently by Nachbar [1].     

Snap-through path in a bistable dielectric elastomer actuator

The dielectric elastomer (DE) has attracted significant attention due to its desired features, including large deformation, fast response, and high energy density. However, for a DE actuator (DEA) utilizing a snap-through deformation mode, most existing theoretical models fail to predict its deformation path. This paper develops a new finite element method (FEM) based on the three-parameter Gent-Gent model suitable for capturing strain-stiffening behaviors. The simulation results are verified by experiments, indicating that the FEM can accurately characterize the snap-through path of a DE. The method proposed in this paper provides theoretical guidance and inspiration for designing and applying DEs and bistable electroactive actuators.     

Study on energy absorbing composite structure made of concentric NiTi spring and porous NiTi

An energy absorbing composite structure made of a concentric NiTi spring and a porous NiTi rod is investigated in this paper. Both NiTi spring and porous NiTi rod are of superelastic grade. Ductile porous NiTi cylindrical specimens are fabricated by spark plasma sintering. The composite structure exhibits not only high reversible force–displacement relation for small to intermediate loading but also high energy absorbing property when subjected to large compressive load. A model for the compressive force–displacement curve of the composite structure is presented. The predicted curve is compared to the experimental data, resulting in a reasonably good agreement.     

A Review of External Pressure Testing Techniques for Shells including a Novel Volume-Control Method

A review of conventional testing methods for applying external hydrostatic pressure to buckling-critical shells is presented. A new “volume-control” pressure testing method, aimed at preventing catastrophic specimen failures and improving control of specimen deformation near the critical load, is also introduced. The implementation of conventional and volume-control systems in an experimental program involving the destructive pressure testing of ring-stiffened cylinders is described. The volume control method was found to improve control of the specimen deformations, especially near the critical load, and catastrophic failures observed while using a conventional setup were avoided. The quasi-static tracking of post-collapse load-deformation relationships for snap-through buckling behaviour was possible while using a volume-control system, but precise control of dynamic shell deformations during buckling was not achieved for specimens failing with large buckling lobes. Expressions for estimating the available control over specimen deformations for pressure testing systems are presented.     

具有初始构型的MEMS驱动器的跳跃和吸合现象研究

在曲梁变形后以弧长为参数的自然坐标系中,利用曲梁大变形分析理论,建立了具有任意初始构型的微电驱动器大变形电动力学分析的数学模型,并采用微分求积法(DQM)进行空间离散,得到了一组具有强非线性的微分-代数系统方程,运用Petzold-Gear BDF方法进行时间域内的求解。研究了MEMS驱动器在电场力作用下的瞬态动力学特性,包括跳跃(snap-through)和吸合(pull-in)现象,并与已有实验结果进行了比较。     

Elliptic averaging methods using the sum of Jacobian elliptic delta and zeta functions as the generating solution

The present paper describes an improved version of the elliptic averaging method that provides a highly accurate periodic solution of a non-linear system based on the single-degree-of-freedom Duffing oscillator with a snap-through spring. In the proposed method, the sum of the Jacobian elliptic delta and zeta functions is used as the generating solution of the averaging method. The proposed method can be used to obtain the non-odd-order solution, which includes both even- and odd-order harmonic components. The stability analysis for the approximate solution obtained by the present method is also discussed. The stability of the solution is determined from the characteristic multiplier based on Floquet’s theorem. The proposed method is applied to a fundamental oscillator in a non-linear system. The numerical results demonstrate that the proposed method is very effective for analyzing the periodic solution of half-swing mode for systems based on Duffing oscillators with a snap-through spring.     

Static and dynamic snap-through behaviour of an elastic spherical shell

The deformation and snap-through behaviour of athin-walled elastic spherical shell statically compressed on aflat surface or impacted against a flat surface are studied theoretically and numerically in order to estimate the influenceof the dynamic effects on the response.A table tennis ballis considered as an example of a thin-walled elastic shell.Itis shown that the increase of the impact velocity leads to avariation of the deformed shape thus resulting in larger deformation energy.The increase of the contact force is causedby both the increased contribution of the inertia forces andcontribution of the increased deformation energy.The contact force resulted from deformation/inertia ofthe ball and the shape of the deformed region are calculated by the proposed theoretical models and compared withthe results from both the finite element analysis and somepreviously obtained experimental data.Good agreement isdemonstrated.     

受圆柱面约束螺旋杆伸展为直杆的动力学分析

以大型空间结构的可伸展机械臂从折叠状态被释放的伸展过程为工程背景, 分析受圆柱面单面约束的弹性螺旋杆在惯性力作用下恢复为直杆的动力学过程. 对弹性杆空间大变形的分析不允许利用小变形假设进行简化. Kirchhoff动力学比拟理论是研究细长弹性杆超大变形的有效工具. 但由于圆柱面约束的存在, 不能直接利用无分布力的Kirchhoff 模型, 而必须在方程中增加分布的约束力. 以表述截面姿态的欧拉角为变量, 建立受圆柱面约束弹性杆的动力学方程. 在圆柱面约束条件下, 认为弹性杆在伸展过程中仍维持半径不变的螺旋线形态, 仅螺旋线倾角和杆的扭率随时间变化. 对简化后的非线性微分方程导出解析积分, 以描述伸展运动的动力学过程, 导出螺旋杆伸展速度的变化规律, 以及从初始状态伸展为直杆所需时间的简明的解析形式计算公式.      

跳跃振子平衡及其分岔的能量分析

用能量方法研究了跳跃振子的平衡与分岔.用势能驻值条件确定了平衡位置所满足的方程,通过势能极值判断平衡的稳定性.在不同的弹簧构型下,数值计算了平衡随系统弹性刚度和质量比变化的分岔图.结果表明,弹性刚度和质量比较小时,系统只有一个稳定平衡点和一个不稳定平衡点;刚度和质量比充分大时,系统分岔出一个新的稳定平衡点和一个新的不稳定平衡点.     

Large rotations in first-order shear deformation FE analysis of laminated shells

The paper deals with the geometrically non-linear analysis of laminated composite beams, plates and shells in the framework of the first-order transverse shear deformation (FOSD) theory. A central point of the present paper is the discussion of the relevance of five- and six-parameter variants, respectively, of the FOSD hypothesis for large rotation plate and shell problems. In particular, it is shown that the assumption of constant through-thickness distribution of the transverse normal displacements is acceptable only for small and moderate rotation problems. Implications inherent in this assumption that are incompatible with large rotations are discussed from the point of view of the transverse normal strain-displacement relations as well as in the light of an enhanced, accurate large rotation formulation based on the use of Euler angles. The latter one is implemented as an updating process within a Total Lagrangian formulation of the six-parameter FOSD large rotation plate and shell theory. Numerical solutions are obtained by using isoparametric eight-node Serendipity-type shell finite elements with reduced integration. The Riks-Wempner-Ramm arc-length control method is used to trace primary and secondary equilibrium paths in the pre- and post-buckling range of deformation. A number of sample problems of non-linear, large rotation response of composite laminated plate and shell structures are presented including symmetric and asymmetric snap-through and snap-back problems.     

弹性杆盘绕折叠的力学分析

以可伸展空间结构元件的盘绕折叠过程为工程背景,分析受圆柱面单面约束的弹性直杆变形为螺旋杆,最终压缩为叠放的平面圆环的变形过程.对此空间大变形的分析不允许利用小变形假设进行简化.由于约束力的存在,也不能直接利用忽略分布力的Kirchhoff弹性杆方程.本文以表述截面姿态的欧拉角为变量,建立受圆柱面约束弹性杆平衡的非线性方程.利用方程的初积分计算杆截面的内力和力矩.忽略盘绕过程的惯性效应,将参数连续改变的螺旋线状态作为杆盘绕过程中的准平衡状态.导出为实现盘绕过程需要施加的轴向压力和扭矩随螺旋角的变化规律.根据一次近似稳定性理论分析得出:两端铰支弹性杆当相对扭率为零时不能保证螺旋线平衡的稳定性.若杆端支承允许存在相对扭转,则轴向压力和扭矩按文中确定的规律变化时可以保证盘绕过程的稳定性.     

Stokes flow in an elastic tube—a large-displacement fluid-structure interaction problem

Viscous flow in elastic (collapsible) tubes is a large-displacement fluid-structure interaction problem frequently encountered in biomechanics. This paper presents a robust and rapidly converging procedure for the solution of the steady three-dimensional Stokes equations, coupled to the geometrically non-linear shell equations which describe the large deformations of the tube wall. The fluid and solid equations are coupled in a segregated method whose slow convergence is accelerated by an extrapolation procedure based on the scheme's asymptotic convergence behaviour. A displacement control technique is developed to handle the system's snap-through behaviour. Finally, results for the tube's post-buckling deformation and for the flow in the strongly collapsed tube are shown. © 1998 John Wiley & Sons, Ltd.     

A lower bound on snap-through instability of curved beams under thermomechanical loads

A non-linear finite element formulation (three dimensional continuum elements) is implemented and used for modeling dynamic snap-through in beams with initial curvature. We identify a non-trivial (non-flat) configuration of the beam at a critical temperature value below which the beam will no longer experience snap-through under any magnitude of applied quasi-static load for beams with various curvatures. The critical temperature is shown to successfully eliminate snap-through in dynamic simulations at quasistatic loading rates. Thermomechanical coupling is included in order to model a physically minimal amount of damping in the system, and the resulting post-snap vibrations are shown to be thermoelastically damped. We propose a test to determine the critical snap-free temperature for members of general geometry and loading pattern; the analogy between mechanical prestress and thermal strain that holds between the static and dynamic simulations is used to suggest a simple method for reducing the vulnerability of thin-walled structural members to dynamic snap-through in members of large initial curvature via the introduction of initial pretension.     

智能材料MFC驱动双稳态板的跳变特性研究

智能可变形结构能感知周围环境的变化并做出适当的反应,其主要原理是利用智能材料层产生应变以达到驱动主体结构发生变形。含有智能材料压电纤维MFC的双稳态复合材料层合板可以通过智能驱动实时调整主体结构的形状,实现稳态构型之间的跳变。本文对中心点固支的MFC压电智能驱动双稳态复合材料层合板的稳态构型、跳变规律及其影响因素进行了系统地研究。研究内容主要包括：含MFC的双稳态层合板的稳态构型、分叉行为和驱动跳变分析。利用理论模型分析了层合板的长宽比、厚度尺寸、铺层方式和MFC粘贴位置对双稳态板分叉行为的影响,并预测了双稳态层合板的临界尺寸以及驱动跳变的临界电压。双稳态板的跳变是一个大变形的非线性行为,实现和控制跳变行为的研究是其应用的基础。本文的研究结果为新型压电材料驱动双稳态板的结构设计提供了一定的理论参考。     

基于内时大变形弹塑性本构模型的回弹和二次屈服分析

采用了一个由简单机构建立的材料热力学一致性弹塑性大变形本构模型，该模型满足热力学基本的不可逆约束条件，并考虑到了储存在残余微应力场中的能量，用相应的算法，编制了一个用户子程序UMAT，并将其嵌入到商用软件ABAQUS中，对圆柱形工件在弹性工具中镦粗和闭合模镦粗时的回弹和二次屈服进行了分析。比较模拟和实验所测得的载荷一位移关系，可以看出在镦粗的第一阶段发生塑性变形，工件在轴向加载过程中产生的横向变形受到了由于模具的弹性变形引起的径向约束，载荷迅速增加；第二阶段轴向压力减少，模具回弹使得工件的径向变形得以恢复，但随着塑性应变的增加，发生工作强化；第三阶段模具约束解除，工作的径向尺寸增加。用该本构模型对闭合模镦粗过程的计算，表明计算结果与实验结果是一致的。     

An exact solution in a nonlinear theory of rods

A three dimensional nonlinear equilibrium theory of elastic rods, applicable to large displacements and small strains, and accounting for extensibility and shear deformation is developed. Integrals of the governing equations are determined for the case of specified end force and moment. A class of solutions is obtained for an initially straight, untwisted rod and compared to the classical solution. The effects of extensibility and shear deformation are discussed.     

Axisymmetric static and dynamics snap-through phenomena in a thermally postbuckled temperature-dependent FGM circular plate

Thermal postbuckling analysis and the axisymmetric static and dynamic snap-through phenomena due to static/sudden uniform lateral pressure in a thermally postbuckled functionally graded material circular plate are performed in this research. Plate is formulated using the first order shear deformation plate theory. Thermo-mechanical properties of the plate are assumed to be temperature dependent where dependency is described according to the higher order Touloukian representation. Two types of temperature loading are considered. Uniform temperature rise and heat conduction across the thickness direction. The one dimensional heat conduction equation in the thickness direction is obtained and discreted via the central finite difference method. The obtained system of equations is nonlinear since the thermal conductivity itself is a function of the unknown nodal temperatures. Using the von-Kármán assumptions, the governing equations of the plate are obtained in a matrix representation with the aid of the conventional Ritz method whose shape functions are developed using the Gram-Schmidt process. At first thermal postbuckling analysis is performed which is a nonlinear problem with respect to both temperature and displacements. Afterwards, response of the bulged thermally postbuckled plate is obtained under the static and dynamic uniform pressure. Snap-through phenomenon may be observed in both static and dynamic loading cases, due to the immovability of the edge of the plate and the initial deflection caused by postbuckling deflection. To capture the snapping phenomenon and trace the path beyond the limit loads, cylindrical arch-length technique is used. In dynamic snap-through analysis, the effect of structural damping is also included. Numerical results of this study reveal that the structure is sensitive to the initial deflection caused by thermal postbuckling load. Increasing the temperature prior to mechanical loads enhances the snap-through intensity and also increases both the upper and lower limit loads. As shown, dynamic snap-through loads are lower than the static ones, however dynamic snap-through intensity is more than the static snap-though intensity. Furthermore, structural damping enhances the dynamic buckling loads of the plate and decreases the dynamic postbuckling deflection of the plate.     

Response of a flexible tube to transverse impact I. Theory

The large non-linear response of a flexible viscoelastic tube under central transverse impact was studied by means of theoretical models, computer simulations and experiments. The present paper is concerned with the analytical solution of the system represented either as an elastic or viscoelastic (standard linear solid) string or Timoshenko beam with the local pinching deformation at the impact point expressed by either a linear elastic spring or a viscoelastic standard linear solid. These four models were solved numerically by the method of characteristics. A few comparisons of the results for the various models are presented here; correlations with corresponding finite element solutions and experimental data are described in a companion paper.     

Considerations on the spring analogy

This paper presents an investigation on the spring analogy. The spring analogy serves for deformation in a moving boundary problem. First, two different kinds of springs are discussed: the vertex springs and the segment springs. The vertex spring analogy is originally used for smoothing a mesh after mesh generation or refinement. The segment spring analogy is used for deformation of the mesh in a moving boundary problem. The difference between the two methods lies in the equilibrium length of the springs. By means of an analogy to molecular theory, the two theories are generalized into a single theory that covers both. The usual choice of the stiffness of the spring is clarified by the mathematical analysis of a representative one‐dimensional configuration. The analysis shows that node collision is prevented when the stiffness is chosen as the inverse of the segment length. The observed similarity between elliptic grid generation and the spring analogy is also investigated. This investigation shows that both methods update the grid point position by a weighted average of the surrounding points in an iterative manner. The weighting functions enforce regularity of the mesh. Based on these considerations, several improvements on the spring analogy are developed. The principle of Saint Venant is circumvented by a boundary correction. The prevention of inversion of triangular elements is improved by semi‐torsional springs. The numerical results show the superiority of the segment spring analogy over the vertex one for a small rotation of an NACA0012 mesh. The boundary correction allows for large deformation of the mesh, where the standard spring analogy fails. The final test is performed on a Navier–Stokes mesh. This mesh contains high aspect ratio mesh cells near the boundary. Large deformation of this mesh shows that the semi‐torsional spring improvement is imperative to retain the validity of this mesh. Copyright © 2000 John Wiley & Sons, Ltd.