Coupled thermo-mechanical analysis of shape memory alloy circular bars in pure torsion

Pure torsion of shape memory alloy (SMA) bars with circular cross section is studied by considering the effect of temperature gradient in the cross sections as a result of latent heat generation and absorption during forward and reverse phase transformations. The local form of energy balance for SMAs by taking into account the heat flux effect is coupled to a closed-form solution of SMA bars subjected to pure torsion. The resulting coupled thermo-mechanical equations are solved for SMA bars with circular cross sections. Several numerical case studies are presented and the necessity of considering the coupled thermo-mechanical formulation is demonstrated by comparing the results of the proposed model with those obtained by assuming an isothermal process during loading–unloading. Pure torsion of SMA bars in various ambient conditions (free and forced convection of air, and forced convection of water flow) subjected to different loading–unloading rates are studied and it is shown that the isothermal solution is valid only for specific combinations of ambient conditions and loading rates.     

An Exact Solution of Torsion Problem for an Incomplete Torus with Application to Helical Springs

An exact analytical solution for the torsion problem of an incomplete torus with a particular form of non-circular cross-section has been found. The solution extends for the close-coiled helical spring the known solution of the torsion problems for straight cylinders with circular and elliptical cross-sections. The pitch of helix is ignored. The hollow cross-sections of the particular form also demonstrate a closed form of analytical solution. The solution can be used for analysis of helical springs with non-circular wire profile.     

环形截面螺旋管道内二次流动特性的研究

从曲线柱坐标系下的N-S方程出发,以曲率和挠率为小参数,采用摄动法求解了环形截面螺旋管道内的黏性流动,给出了完全二阶摄动解,结果表明：当挠率为零时,二次流表现为上下对称的四个涡;当挠率不为零,涡的对称性遭到破坏,二次涡的强度和个数受De数和环形截面内外径之比δ的影响,轴向速度最大值在De数较小时靠近管道的内侧,随着De数的增加,其最大值向外侧移动。     

Optimization of a Thin-Walled,Anisotropic Curved Bar for Maximum Torsional Stiffness∗

ABSTRACT

ABSTRACT A curved bar in the form of a circular ring sector is under uniform torsion when acted upon by two equal and opposite forces directed alone the axis passing through the center of the ring and perpendicular to its plane, i.e., forces acting along the axis of rotation. The exact torsion theory can be extended to this case when the material of which the bar consists is cylindri-cally anisotropic, with the axis of anisotropy directed along the axis of rotation and having an elastic symmetry about any plane of the transverse cross section. In this paper, a thin-walled curved bar having the loading conditions and material properties described above is optimized so as to maximize its torsional stiffness. The optimization is carried out with respect to the cross-sectional shape of the bar subject to constraints on the transverse area (single-purpose design) and bending stiffness (multipurpose design). In the special case of an orthotropic material, the angle of inclination of the ortho-tropy axes with respect to the middle plane is optimally determined for a cross section with constant thickness. A perturbation method is used to obtain analytical solutions, and numerical results are presented indicating the efficiency of the designs and the optimal cross-sectional shapes.     

Modeling and dynamics analysis of helical spring under compression using a curved beam element with consideration on contact between its coils

Helical springs are indispensable elements in mechanical engineering. This paper investigates helical springs subjected to axial loads under different dynamic conditions. The mechanical system, composed of a helical spring and two blocks, is considered and analyzed. Multibody system dynamics theory is applied to model the system, where the spring is modeled by Euler–Bernoulli curved beam elements based on an absolute nodal coordinate formulation. Compared with previous studies, contact between the coils of spring is considered here. A three-dimensional beam-to-beam contact model is presented to describe the interaction between the spring coils. Numerical analysis provides details such as spring stiffness, static and dynamic stress for helical spring under compression. All these results are available in design of helical springs.     

Equilibrium equations for 3D critical buckling of helical springs

In most cases,the research on the buckling of a helical spring is based on the column,the spring is equivalent to the column,and the torsion around the axial line is ignored.A three-dimensional(3D) helical spring model is considered in this paper.The equilibrium equations are established by introducing two coordinate systems,the Frenet and the principal axis coordinate systems,to describe the spatial deformation of the center line and the torsion of the cross section of the spring,respectively.By using a small deformation assumption,the variables of the deflection can be expanded into Taylor’s series,and the terms of high orders are ignored.Hence,the equations can be simplified to the functions of the twist angle and the arc length,which can be solved by a numerical method.The reaction loads of the spring caused by the axial load subjected to the center point are also discussed,giving the boundary conditions for the solution to the equilibrium equations.The present work is useful to the research on the behavior of the post-buckling of the compressed helical spring.     

Delayed presetting of helical springs

Abstract

This article establishes the calculation of remaining deformation and residual stress for helical springs after long-lasting presetting process. The article extends the model for the immediate presetting process accounting the creep deformation of the spring. The method is based on plasticity theory for the instant flow overexposed by the relaxation over the long-term presetting. In this article, the following method is used. The plastic deformation of the helical spring with the circular cross section occurs instantly. If the shortening of the spring in the tool holder persists, the relaxation of stresses occurs and the force of the spring reduces. As the consequence, after the elastic unloading of the long-time presetting, the residual stresses spring reduce gradually with the squeezing time as well. The final length of the springs considerably shortens with the increasing preset duration. The advantage of the discovered closed form solutions is the calculation without the necessity of complex finite-element simulation of spring length loss and residual stresses after presetting process. The analytical expressions are proposed and the exact calibration applied for evaluation of factors for presetting processes.     

Analysis of the rate-dependent coupled thermo-mechanical response of shape memory alloy bars and wires in tension

In this paper, the coupled thermo-mechanical response of shape memory alloy (SMA) bars and wires in tension is studied. By using the Gibbs free energy as the thermodynamic potential and choosing appropriate internal state variables, a three-dimensional phenomenological macroscopic constitutive model for polycrystalline SMAs is derived. Taking into account the effect of generated (absorbed) latent heat during the forward (inverse) martensitic phase transformation, the local form of the first law of thermodynamics is used to obtain the energy balance relation. The three-dimensional coupled relations for the energy balance in the presence of the internal heat flux and the constitutive equations are reduced to a one-dimensional problem. An explicit finite difference scheme is used to discretize the governing initial-boundary-value problem of bars and wires with circular cross-sections in tension. Considering several case studies for SMA wires and bars with different diameters, the effect of loading–unloading rate and different boundary conditions imposed by free and forced convections at the surface are studied. It is shown that the accuracy of assuming adiabatic or isothermal conditions in the tensile response of SMA bars strongly depends on the size and the ambient condition in addition to the rate dependency that has been known in the literature. The data of three experimental tests are used for validating the numerical results of the present formulation in predicting the stress–strain and temperature distribution for SMA bars and wires subjected to axial loading–unloading.     

Multiscale mass-spring models of carbon nanotube foams

This article is concerned with the mechanical properties of dense, vertically aligned CNT foams subject to one-dimensional compressive loading. We develop a discrete model directly inspired by the micromechanical response reported experimentally for CNT foams, where infinitesimal portions of the tubes are represented by collections of uniform bi-stable springs. Under cyclic loading, the given model predicts an initial elastic deformation, a non-homogeneous buckling regime, and a densification response, accompanied by a hysteretic unloading path. We compute the dynamic dissipation of such a model through an analytic approach. The continuum limit of the microscopic spring chain defines a mesoscopic dissipative element (micro-meso transition) which represents a finite portion of the foam thickness. An upper-scale model formed by a chain of non-uniform mesoscopic springs is employed to describe the entire CNT foam. A numerical approximation illustrates the main features of the proposed multiscale approach. Available experimental results on the compressive response of CNT foams are fitted with excellent agreement.     

高应力岩体爆破卸荷过程中应变率及应变能特征

针对高应力岩体爆破开挖卸载问题,自制了一台轴向加、卸载实验测试平台,通过实验测试获得了爆破卸荷过程中岩杆的动态应变及应变率数据。实测数据表明:开挖面附近岩体的爆破加、卸载以及初始应力卸载应变率均在10?1 s?1量级以上,验证了高地应力区岩体爆破开挖卸荷是一动态过程。建立了初始应力卸载一维力学模型,揭示了卸载波的传播机制;通过分析爆破卸荷过程应变能密度的时空分布特征,建立了应变能密度与各阶段应变率变化规律的联系。结合实测数据,采用隐式-显式顺序求解方法,进一步分析了高应力区岩体爆破卸荷荷载各阶段应变率沿岩杆的变化规律。结果表明:爆破加载阶段的平均应变率沿杆件逐渐衰减,且衰减速度逐渐减小;爆破卸阶段平均应变率沿杆件也呈衰减趋势;而初始应力的应变能稳定释放,其平均应变率无衰减趋势。     

X-ray residual stress measurement on mechanical components with high curvature

In this paper we deal with the definition of an appropriate X-ray diffraction procedure for residual stress determination on samples with high curvature radius. We choose as the application high-strength hot worked coil springs for car suspensions (wire diameterd=12 mm). Different methods of X-ray measurement area limitation are compared, taking into account the measurement errors, for the determination of stress in one and in three directions. After the identification of the irradiated area limits for plane samples, further limits are identified due to the sample curvature (torsion bar). We describe loading devices purposely designed, constructed and calibrated. In each case, the sample is loaded so that the stress state is determined at the same time both by strain gages and by X-ray measurement.     

Warping effect in free vibration analysis of unidirectional composite non-cylindrical helical springs

The differential equations of motion for unidirectional composite non-cylindrical helical springs including warping, which consist of 14 first-order partial differential equations with variable coefficients, are first derived based on arbitrary spatially curved anisotropic beam theory. An analytical formula for the warping function of Saint-Venant’s torsion of unidirectional composite beams with rectangular cross-section is also obtained. The natural frequencies are determined using improved Riccati transfer matrix method. The element transfer matrix is calculated by the use of the Scaling and Squaring method and Pad’e approximations. Comparisons are made with the EF-results on the natural frequencies of the springs, made from rectangular wire, with inclusion of the warping effect. Information is given on the effect on the natural frequencies of the ratio of radii of the minimum cylinder to the maximum cylinder, the helix pitch angle and the number of active turns. Numerical results reveal that the warping deformation has a significant influence on the natural frequencies, which should be considered in the free vibration analysis of such springs.     

Fluid flow in a helical vessel in presence of a stenosis

Large arteries are not straight and rather present curvature and torsion. The present study analyzed fluid flow in a helical vessel without and with a stenosis in comparison with an analogous rectilinear vessel. The analysis is performed by three-dimensional numerical simulation of the Navier–Stokes equations under steady conditions considering stenosis as an axially symmetric reduction of vessel lumen. Results show that the double curvature gives rise to persistent secondary motion which combines with the vorticity separated behind the constriction to develop a complex three-dimensional vorticity structure. The curved streamlines and the three-dimensional vortex wake result in a increase of energetic losses in helical vessels. However, the same symmetry break due to the double curvature improves the capacity of self-cleaning and allows a more rapid wash-out of the flowing blood.     

The warping functions of regular prismatic bars under torsion evaluated by caustics

Reflection of a bundle of coherent light on the warped cross section of a prismatic bar submitted to torsion forms a caustic on a receiver plane. From the mathematical expression of this curve and the theory of reflected caustics, it is possible to evaluate accurately the warping function of the cross section. Using this idea, it was possible to study the torsion problem in prismatic bars with sections which were equilateral triangles and squares. It was observed that the shape of the caustic is an hypocycloid curve with three or four cusps respectively. By evaluating the warping function by using elements from the respective caustics it was possible to find out that, for the triangular cross section, the expression for the warping function coincided exactly with the expression given by the exact solution of the problem. For the square cross section, a closed-form solution for its warping function was readily derived, to which the series approximation solution differed only by a few percent at maximum for the shear stresses. Since the method can be readily extended to any canonical polygonic cross section, it constitutes a general solution for the torsion of prismatic bars, which approximates their exact deformations better than the solutions based on the Saint-Vénant assumptions.     

有限弹簧法在钢筋混凝土细观断裂分析中的应用

叙述了依据计算几何学的分割理论划分多边形单元的计算前处理系统，及其有限弹簧法在钢筋混凝土构件细观断裂数值模拟中的应用。该模拟系统的主要特色为：一是利用有限弹簧法的单元形状任意性来考虑混凝土断裂模拟时单元形状的影响；二是利用其变位不连续性模拟构件从微细裂缝到断裂的全过程；三是把钢筋和混凝土分别看成具有不同性质的微小构造单元来考虑其界面性质。数值模拟钢筋混凝土构件在拉伸作用下的断裂全过程，考察钢筋的螺纹抵抗作用及其保护层的约束机理。     

考虑翘曲效应的圆柱螺旋弹簧的振动分析

以空间曲梁理论为基础对簧丝截面为矩形的圆柱螺旋弹簧的自由振动特性进行了研究. 在弹簧的运动微分方程中, 所有的位移函数和广义翘曲坐标均定义在横截面的形心主轴上, 同时考虑了翘曲变形对弹簧固有频率的影响. 通过精确地应用符号运算软件MATHEMATICA可以得到振动模态的显式表达式, 固有频率则可用搜索的方法来确定. 在较宽的范围内, 给出了各种参数变化, 如簧丝截面的宽高比(a/ b = 0.6 ~ 1.7)、螺旋角(\bar {\alpha } = 5^\circ ~ 12.5^ \circ)、弹簧工作圈数(n = 6 ~ 12)和圆柱螺旋线半径(R = 4~ 10mm)对固有频率的影响. 为了证明解析法的有效性, 对两端固支和一端固支、一端自由矩形截面圆柱螺旋弹簧的固有频率和振动模态进行了求解, 并同ANSYS三维实体单元(Solid45)的结果和文献的结果进行了比较. 计算表明: 用解析方法得到的解和用数值方法得到的结果吻合得很好.      

Numerical performance of two formulations of truss topology optimization

The present paper studies topology optimization of truss structures in multiple loading cases and with stress constraints. It is pointed out in the paper that the special difficulty of adding bars and/or deleting bars from structure in the numerical algorithm of truss topology optimization is caused by the discontinuity of stress functions at the zero cross sectional area in the conventional formulation. In a new formulation, we replace the stress constraints by new constraints. The new constraints retain the same feasibility of the stress constraints, but are continuous in the closed interval up to zero cross sectional area. The new formulation enables us to solve topology optimization problem in the frame of the existing FEM software and mathematical programming techniques. Powell constrained variable metric method is applied to a number of examples of truss topology optimization. Numerical performances of the two formulations are compared. It is shown that in the conventional formulation the iteration of numerical algorithm may be blocked by discontinuity of the stress constraint and often stops at a nonoptimum solution. And in the new formulation the bar adding and bar deleting is done rationally and a local optimum, even the global optimum can be obtained by iteration. The project supported by the National Natural Science Foundation of China     

Propagation of a long wave in a curved duct (I) basic analysis of long wave propagation in a curved duct with variable cross section

The propagation of a long wave in a three-dimensional curved duct with variable cross section is studied in this paper. It is shown that a three-dimensional Helmholtz equation can be decomposed into a two-dimensional Laplace (or Poisson) equation and a one-dimensional Webster equation by the curvilinear orthogonal coordinate system, non-dimensionization of reduced wave equation and regular perturbation with small parameterka, wherek is the wave number anda is the characteristic radius of the duct. The influences of the duct's geometric parameters (the area variation of the cross section, the curvature and torsion of the central line) on the asymptotic expansion of the solution are analysed. It is concluded that the effects of the variation of the cross sectional area first appear in the first term of the asymptotic expansion, and when the cross section shape has certain symmetric properties, the effects of the curvature and torsion of the central line first appear in the third and the fourth terms, respectively. An example of long wave propagation in a curved circular duct is also given at the end of this paper.     

弹性曲杆的稳定性问题

本文给出空间任一曲杆在弯扭联合作用下的稳定性问题的一般讨论,并且给出了曲杆某一平衡状态的扰动量所满足的方程组(28)—(36),在适当的边界条件下,这些扰动量的非零解对应于临界状态。文末用这组方程具体讨论了五个实际例子,这些例子有些结果是新的,有些是用新的方法去处理老问题。     

A study of the postbuckling path of cylindrically curved panels of laminated composite materials during loading and unloading

In this paper, Dynamic Relaxation Method is applied to study the postbuckling path of cylindrically curved panels of laminated composite materials during loading and unloading. The phenomenon that loading paths do not coincide with unloading paths has been found. Numerical results are given for cylindrically curved cross-ply panels subjected to uniform uniaxial compression under two types of boundary conditions. The influence of the number of layers, the panels curvature and the initial imperfection on the postbuckling paths is discussed.Project supported by the National Natural Science Foundation of China. The main contents of this paper were presented at the Proceedings of the Seventh International Conference on Composite Materials (November 1989, Guangzhou, China).The authors thank Prof. Zhou Li for his guidance.