Phase boundary motion in an elastic bar of finite length

Using the continuum mechanical model of solid-solid phase transitions of Abeyaratne and Knowles, this paper examines the large time dynamical behavior of a phase boundary. The problem studied concerns a finite elastic bar initially in an equilibrium state that involves two material phases separated by a phase boundary at a given location. Interaction between the moving phase boundary and the elastic waves generated by an impact at the end of the bar and subsequent reflections is studied in detail by using a finite difference scheme. The numerical results show that the phase boundary in a finite bar returns to an equilibrium state after a disturbance of finite duration, whether the two-phase material is trilinear or not.     

Dynamic behavior of the interface between two solid phases in an elastic bar

Using the continuum mechanical model of solid-solid phase transitions of Abeyaratne and Knowles, this paper examines the large time dynamical behavior of a phase boundary. The problem studied concerns a semi-infinite elastic bar initially in an equilibrium state that involves two material phases separated by a phase boundary at a given location. Interaction between the phase boundary and the elastic waves generated by an impact at the end of the bar and subsequent reflections is studied in detail, and an exact solution of the dynamical problem, which is governed by a nonlinear resursive formula, is obtained. It is shown that the phase boundary reaches a new equilibrium state for large time. Numerical calculations based on the recursive formula are carried out to illustrate analytical results.Address after August 15, 1995: Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA24061, USA.     

变截面压杆稳定非线性微分方程边值问题的最优化算法研究

针对任意约束类型的变截面受压杆件的稳定临界载荷计算问题,结合非线性微分方程数值算法和最优化方法,以起点边界的初始条件、待求临界荷载和附加约束力为设计变量;以终点边值条件满足的函数关系与位型条件构建目标函数,提出变截面压杆临界载荷和稳定位型的优化求解算法。应用VB编制通用的优化计算程序,分析了典型算例;通过对比发现,本文以较少设计变量实现了临界载荷的高精度计算,为工程应用提供参考。     

Stress wave propagation in a rectangular bar

The process of propagation of nonstationary waves in a rectangular bar is studied from the viewpoint of three-dimensional elasticity. The motion arises owing to the action of normal impact forces applied at the end face of a half-infinite bar all of whose four lateral surfaces are force-free. Precisely these one-type conditions complicate the solution of this problem. The already known solutions were obtained under the assumption that conditions of mixed type are partially or completely posed on the lateral sides, and precisely this fact permits separating the boundary values of distinct waves on these surfaces. In the absence of this simplifying factor, it is rather problematic to construct a solution satisfying all free lateral conditions.     

Stability of a straight bar of variable rigidity

We consider the longitudinal compression of a straight bar whose rigidity is a periodic integrable function of the longitudinal coordinate. For a hinged bar with one clamped end, we obtain approximate analytic formulas that permit obtaining the critical compressing loads under which an adjacent, curved form of equilibrium is possible. In the case of a bar of stepwise varying rigidity that consists of a single period (the limit case), we compare the results obtained by our formulas with the already known exact solutions of the stability equation. A good agreement between the approximate and exact results is shown.     

Stability of bars with variable rigidity

A variable cross-section bar is considered. The bar is not uniform in length. The bar axis through the mass centers of all cross sections is a straight line. The bar is compressed by a longitudinal force applied to the mass center of the boundary cross section. The stability loss of the straight-line shape of the bar’s equilibrium is discussed when a curved shape is also possible. Approximate analytical formulas are obtained for the critical compressive force when four types of end fixing are used for a periodically nonuniform bar. The numerical results obtained by these formulas are compared with the known exact solutions to the stability equation for a bar whose cross section is stepwise variable and whose nonuniformity consists of only one period (the limiting case).     

子弹冲击下TiNi固支梁的结构动态响应特性研究

采用改装的霍普金森压杆装置结合数值模拟对伪弹性TiNi合金固支梁的结构动态响应特性进行了研究。结果表明,在子弹冲击下,撞击点和固定端附近首先发生相变,并随着载荷增加,进一步产生相变铰,梁演变为二杆铰接机构。由于轴力作用,此处相变铰为拉伸侧的单边铰。与传统塑性铰不同,卸载后相变铰完全消失,梁回复原状没有残余变形。此外,对固支边界条件的实现及其对实验结果的影响进行了专门研究。     

The improvement of the general expression for the stress function Φ of the two-dimensional problem

In this paper, it is pointed that the general expression for the stress function of the plane problem in polar coordinates is incomplete. The problems of the curved bar with an arbitrary distributive load at the boundries can’t he solved by this stress function. For this reason, we suggest two new stress functions and put them into the general expression. Then, the problems of the curved bar applied with an arbitrary distributive load at r=a,b boundaries can be solved. This is a new stress function including geometric boundary constants.     

端部切向载荷作用下的压杆失稳分析及其应用

细直杆件在压应力作用下会产生横向屈曲即失稳．直杆撞击刚性平面或拉断卸载后将形成压缩波，因承载压缩载荷的长度增加可以引起失稳．冲击速度转换的压应力沿着杆件切线方向，该处弯矩和剪力为零；而众多文献设定的失稳段固支边界条件并不准确．基于精确的杆件变形曲率方程得到端部载荷指向杆件中固定点时的受压失稳条件，得到其极限状态即载荷沿杆端切向作用时失稳长度相当于两端简支的1.5 倍．对于钢丝绳拉断形成的冲击失稳，载荷恒定而长度增加，可以产生高阶屈曲即在侧向出现多次曲折，并基于尼龙-橡胶带的模拟试验给出了定性说明．     

On stretching of a bar capable of undergoing phase transitions

The dynamic processes in an elastic bar composed of a material, which is capable of undergoing phase transitions, are under consideration. We use a model of an elastic body with non-convex strain energy potential. The bar with variable cross-sectional area is considered. Propagation of the phase boundary along the bar loaded with time-dependent tensile forces is investigated. It is assumed that the phase boundary moves at a variable speed. The problem is solved analytically by using two approaches, namely, the full dynamic approach and the quasi-static (kinetic) approach. The results obtained by means of these two methods are compared.     

Sudden tensile loading of a rubberlike bar

In uniaxial tension, the stress–strain curve for rubber changes curvature from concave to convex as the strain increases. For sudden tensile loading of a bar, a one-dimensional model that reflects this behavior leads to an under-determined problem reminiscent of that arising in materials capable of undergoing phase transitions. In the latter setting, adding the kinetic relation underlying the phase change to the conventional statement of the problem removes the indeterminacy; the same is true when such a relation is used in a formal way in the problem for rubber. This presents a physical question: What is the evolutionary process at the microscale whose kinetics are needed in the dynamics of rubber?     

Static and dynamic behaviour of nonlocal elastic bar using integral strain-based and peridynamic models

The static and dynamic behaviour of a nonlocal bar of finite length is studied in this paper. The nonlocal integral models considered in this paper are strain-based and relative displacement-based nonlocal models; the latter one is also labelled as a peridynamic model. For infinite media, and for sufficiently smooth displacement fields, both integral nonlocal models can be equivalent, assuming some kernel correspondence rules. For infinite media (or finite media with extended reflection rules), it is also shown that Eringen's differential model can be reformulated into a consistent strain-based integral nonlocal model with exponential kernel, or into a relative displacement-based integral nonlocal model with a modified exponential kernel. A finite bar in uniform tension is considered as a paradigmatic static case. The strain-based nonlocal behaviour of this bar in tension is analyzed for different kernels available in the literature. It is shown that the kernel has to fulfil some normalization and end compatibility conditions in order to preserve the uniform strain field associated with this homogeneous stress state. Such a kernel can be built by combining a local and a nonlocal strain measure with compatible boundary conditions, or by extending the domain outside its finite size while preserving some kinematic compatibility conditions. The same results are shown for the nonlocal peridynamic bar where a homogeneous strain field is also analytically obtained in the elastic bar for consistent compatible kinematic boundary conditions at the vicinity of the end conditions. The results are extended to the vibration of a fixed–fixed finite bar where the natural frequencies are calculated for both the strain-based and the peridynamic models.     

Water waves in a suspended container

Analysis and experiments are carried out on a horizontal rectangular wave tank which swings at the lower end of a pendulum. The walls of the tank generate waves which affect the motion of the pendulum. For small displacements of the tank, linearised shallow water equations are used to model the motion, and there exist time-periodic solutions for the system whose periods are governed by a transcendental relation. Numerical and analytic solutions of this relation show that the fundamental period is greater than both the period of the empty tank (moving like a simple pendulum) and the fundamental period of the standing wave which occurs when the tank is removed from its supports and held fixed. For a rectangular tank the theory compares well with some experimental measurements. Qualitative observations are also made of the effect of breaking waves on the tank motion: for a tank which has a mass small compared with its load the energy dissipated by breaking waves can rapidly reduce the amplitude of swing of the tank. Potential flow theory is used with linearised free-surface boundary conditions to find time periodic motions for a tank with a hyperbolic cross section.     

Constant velocity motion of a strip die on the boundary of a viscoelastic base

We consider a contact problem on the interaction of a rigid strip die with the boundary of a viscoelastic base. We assume that the die moves at a constant velocity on this boundary and is indented into it by a constant normal force. Friction in the die—surface contact region is neglected. The die base is corrugated in the direction perpendicular to the direction of motion. At the first stage, we determine the displacement of the base boundary due to the normal load applied to it. Then, at the second stage, we derive the integral equation of the contact problem for determining the contact pressure. At the third stage, we construct an approximate solution of this integral equation by using the modified Multhopp—Kalandiya method.     

Modeling of macroscopic response of phase transforming materials under quasi-static loading

Based on a continuum model of solid-solid phase transformations, the macroscopic response of a bar of a thermoelastic phase transforming material loaded quasistatically is investigated. A critical loading rate is identified for the evolution of a single phase boundary in the bar during an isothermal process. It is shown that, when the loading rate is larger than this critical loading rate, nucleation occurs either continuously or at multiple sites; when the loading rate is lower than this critical loading rate, the size of the hysteresis loop increases with increasing loading rate, and decreases with an increase in the mobility of the phase boundary. The heat conduction due to the heat generated by the latent heat of the phase transformation is considered for a special case.     

On the encounter of an acoustic shear pulse with a phase boundary in an elastic material: reflection and transmission behavior

The fully dynamical motion of a phase boundary is considered for a specific class of elastic materials whose stress-strain relation in simple shear is nonmonotone. It is shown that a preexisting stationary phase boundary in a prestressed layer composed of such a material can be set in motion by a finite amplitude shear pulse. An infinity of solutions is possible according to the present theory, each of which is characterized by different reflected and transmitted waves at the phase boundary. A global analysis gives exact bounds on the size of the solution family for different shear pulse amplitudes. For certain ranges of shear pulse amplitudes a completely reflecting solution will exist, while for an in general different range of shear pulse amplitudes a completely transmitting solution will exist. The properties of these different solutions are examined. In particular, it is observed that the ringing of a shear pulse between the external boundaries and the internal phase boundary gives rise to periodic phase boundary motion for both the case of a completely reflecting phase boundary and a completely transmitting phase boundary.     

Modeling of non-ideal hard permanent magnets with an affine-linear model,illustrated for a bar and a horseshoe magnet

This study is dedicated to continuum-scale material modeling of isotropic permanent magnets. An affine-linear extension to the commonly used ideal hard model for permanent magnets is proposed, motivated, and detailed. In order to demonstrate the differences between these models, bar and horseshoe magnets are considered. The structure of the boundary value problem for the magnetic field and related solution techniques are discussed. For the ideal model, closed-form analytical solutions were obtained for both geometries. Magnetic fields of the boundary value problems for both models and differently shaped magnets were computed numerically by using the boundary element method. The results show that the character of the magnetic field is strongly influenced by the model that is used. Furthermore, it can be observed that the shape of an affine-linear magnet influences the near-field significantly. Qualitative comparisons with experiments suggest that both the ideal and the affine-linear models are relevant in practice, depending on the magnetic material employed. Mathematically speaking, the ideal magnetic model is a special case of the affine-linear one. Therefore, in applications where knowledge of the near-field is important, the affine-linear model can yield more accurate results—depending on the magnetic material.     

基于电子万能试验机的新型压杆稳定实验装置

本文介绍了一种基于电子万能试验机开发的新型压杆稳定实验装置。该装置利用电子万能试验机自身传感设备,通过测量压杆两端受力与绘制端部轴向位移曲线来确定压杆失稳荷载。装置包括上、中、下三个约束部分与压杆试样部分,可实现两端固支、两端铰支、一端固支一端铰支和一端固支一端自由等4种不同端部约束型式,并且能够施加中部固支约束与压杆初始偏心。本文同时讨论了两种铰支型式,即刀刃铰支与轴承铰支对测试精度的影响。实验表明,轴承铰支测量精度比刀刃铰支高。该套装置的整体测量精度高,与理论值的相对误差最高为2%。     

Shear deformable bars of doubly symmetrical cross section under nonlinear nonuniform torsional vibrations—application to torsional postbuckling configurations and primary resonance excitations

In this paper a boundary element method is developed for the nonuniform torsional vibration problem of bars of arbitrary doubly symmetric constant cross section, taking into account the effects of geometrical nonlinearity (finite displacement—small strain theory) and secondary twisting moment deformation. The bar is subjected to arbitrarily distributed or concentrated conservative dynamic twisting and warping moments along its length, while its edges are subjected to the most general axial and torsional (twisting and warping) boundary conditions. The resulting coupling effect between twisting and axial displacement components is also considered and a constant along the bar compressive axial load is induced so as to investigate the dynamic response at the (torsional) postbuckled state. The bar is assumed to be adequately laterally supported so that it does not exhibit any flexural or flexural–torsional behavior. A coupled nonlinear initial boundary value problem with respect to the variable along the bar angle of twist and to an independent warping parameter is formulated. The resulting equations are further combined to yield a single partial differential equation with respect to the angle of twist. The problem is numerically solved employing the Analog Equation Method (AEM), a BEM based method, leading to a system of nonlinear Differential–Algebraic Equations (DAE). The main purpose of the present contribution is twofold: (i) comparison of both the governing differential equations and the numerical results of linear or nonlinear free or forced vibrations of bars ignoring or taking into account the secondary twisting moment deformation effect (STMDE) and (ii) numerical investigation of linear or nonlinear free vibrations of bars at torsional postbuckling configurations. Numerical results are worked out to illustrate the method, demonstrate its efficiency and wherever possible its accuracy.

     

Optimal strength of a compound column

A column of fixed length and variable cross section consists of two homogeneous and isotropic components. The components are joined along their side surfaces and have different Young's moduli, but the same Poisson's ratio. One of the components encloses the other that has the smaller Young's modulus. For different values of the ratio of the moduli, the shape of the column, which has the largest critical buckling load under axial thrust, is determined, assuming that the volumes of the components are prescribed. The problem is solved for the case of pinned ends.It appears that the solution of the most general problem, in which each of the areas of the component cross sections may be varied, is a combination of the solutions of some more elementary problems. Therefore, two types of problems are discussed: the compound bar with an inner component of fixed cross section and the general compound bar.The method of solution may be extended to other boundary conditions.