Cross-sectional deformations of rectangular hollow sections in bending: Part I — experiments

This investigation deals with deformations of individual cross-sectional members as flanges and webs in bending of rectangular hollow sections. Part I describes the experimental work, while analytical models developed to predict pre- and post-buckling deformations are presented in a paper to follow (Part II). The experimental program involved rectangular single- and double-chamber aluminium alloy AA6060 extrusions with three different wall thicknesses. The profiles were given two distinct heat treatments to obtain different hardening characteristics. Multiaxial tests were performed to determine the mechanical properties of the materials. The profiles were then bent into a number of different bend radii. Measurements of strains, curvatures, deflections and bending forces were taken. The results show that cross-sectional distortions take place from the very beginning of bending; at first in the form of a uniform sagging-like deformation along the entire length of both sides of the bend until the inner (compressive) flange buckles into several half-waves, superimposing the pre-deformation modes. The instant at which buckling occurs is found to be mainly associated with the width-to-thickness ratio of the flange and the strain hardening characteristic of the material. The magnitude of pre- and post-deformations, however, appears to be more directly related to the actual width of the flange than to its slenderness. The material stress–strain curve is shown to have an increasingly effect on the distortions of members directly sustained to buckling as bending proceeds beyond the onset of buckling, leading to severely concentrated deformations for sections made of low hardening materials. The material has less impact on sagging of the outer (tensile) flange.     

Evaluation of bending strain measurements in a composite sailboat bowsprit with embedded fibre Bragg gratings

Acrylate-coated Fibre Bragg Gratings (FBGs) were embedded in a sailing boat bowsprit tubular section manufactured using the vacuum bagging process in autoclave with the aim of measuring strains under realistic loading conditions. In order to establish an effective procedure for sensor integration, flat tensile specimens with embedded sensors were firstly produced under different processing conditions, using advanced composite material employed in the marine industry. Information obtained in this way was used to manufacture a sailboat bowsprit with an array of gratings embedded between the last inner plies. In order to assess the sensitivity and accuracy of the embedded sensors, the bowsprit was subjected to bending tests and the results were compared to analytical values and to data obtained from electrical strain gages. Results are very encouraging but reveal that great attention must be paid both to the integration process and to the in situ calibration of the embedded sensors.     

Optimization of bending sequence in roll forming using neural network and genetic algorithm

In the roll forming process, the bending sequence plays a major role in the product quality. The optimal bending sequence results in the smallest number of passes and the flawless process. This paper presents a new optimization procedure of bending sequence in a roll forming process. The multilayer perceptron is used to build the neural network (NN), which models the variation of longitudinal strain in process while the genetic algorithm (GA) is employed to optimize the bending sequence. The data used for training the network is automatically obtained by the integration between CAD and CAE. The values of peak longitudinal strains are maximized while the number of passes is reduced to the smallest and the constraint conditions being set on the maximal longitudinal strain to avoid buckling. The overbending at final pass after spring back is also considered in this paper. Two roll forming processes are optimized in order to prove applicability and efficiency of the optimization procedure. This method maintains the longitudinal strain less than the buckling limit, whereas reducing the number of passes to the smallest. Thus, the advantages of the proposed method show the high applicability in designing and optimizing the bending sequence in the roll forming process.     

Cyclic loading of beams based on the Prager and Frederick–Armstrong kinematic hardening models

The kinematic hardening theory of plasticity based on the Prager and Frederick–Armstrong models are used to evaluate the cyclic loading behavior of a beam under the axial, bending, and thermal loads. The beam material is assumed to follow non-linear strain hardening property. The material's strain hardening curves in tension and compression are assumed to be both identical for the isotropic material and different for the anisotropic material. A numerical iterative method is used to calculate the stresses and plastic strains in the beam due to cyclic loadings. The results of the analysis are checked with the known experimental tests. It is concluded that the Prager kinematic hardening theory under deformation controlled conditions, excluding creep, results into reversed plasticity. The load controlled cyclic loading under the Prager kinematic hardening model with isotropy assumption results into reversed plasticity. Under anisotropy assumption of tension/compression curve, this model predicts ratcheting. On the other hand, the Frederick–Armstrong model predicts ratcheting behavior of the beam under load controlled cyclic loading with non-zero mean load. This model predicts reversed plasticity under the load controlled cyclic loading with zero mean load, and deformation controlled cyclic loading.     

形状记忆合金驱动梁的变形分析及试验研究

将预拉伸的形状记忆合金 (Shapememoryalloy ,简称SMA)薄片作为驱动器 ,粘贴在构件表面。加热SMA ,当其发生相变时 ,会产生很大的恢复力 ,驱动构件发生变形。建立了粘有SMA薄条应变驱动器的简化机翼—梁的力学模型 ,分析了单边粘贴SMA梁的压弯复合变形 ,给出了其应变分布及弯曲变形的解析表达式。同时通过试验对理论结果进行了验证。     

Low-angle subgrain misorientations in deformed NaCl

The development of subgrain boundary misorientations with strain in NaCl polycrystals has been investigated. At low strains, a power law relationship exists between strain and average misorientations. The accuracy of this relationship is assessed in terms of material and electron backscattered diffraction (EBSD) processing parameters and is found to hold for a material of constant grain size deformed in compression, providing EBSD mapping and processing conditions were similar. Average misorientations are strongly influenced by grain orientation, suggesting that the misorientation–strain relationship may also be texture dependent in materials with high plastic anisotropy. A slight grain size dependency of the average misorientations was observed.     

油(气)管用钢板的包辛格效应

研究了输油（气）管用X70、X80钢板在不同的应变类型（轴向加载和弯曲加载）、不同预应变量和不同反向应变量下所导致的应变抗力的减小（称为包辛格效应,简称BE）,并探讨了影响钢板BE的主要因素。结果表明：强度较高的X80钢比强度较低的X70钢有较大的BE,说明材料的形变硬化能力影响到BE的大小;预应变量和反向应变的大小都对BE有显著的影响。弯曲加载方式引起试样横截面上弹、塑性应变分布的不均匀性,在反向应变时产生的宏观残余应力对BE也有不可忽视的影响。     

Experimental studies of the strain distributions in bends from aluminium sheet

The use of microhardness measurements to follow the development of the plastic-strain distributions during bending is described. The microhardness values are calibrated in terms of true, equivalent plastic strain in subsidiary experiments. The technique is shown to be sufficiently sensitive to enable the determination of the strains present in shear bands that form during bending.     

Effects of process variables on V-die bending process of steel sheet

This investigation aims to clarify the process conditions of the V-die bending operation of steel sheet. It provides a model which predicts the correct punch load for bending and the precise final shape of products after unloading, in relation to the tensile properties of the material and the geometry of tools. The process variables are punch radius (R_p), die radius (R_d), punch width (W_p), punch speed (V_p), friction coefficient (μ), strain hardening exponent (n) and normal anisotropy (R).This investigation is carried out by performing some experiments and by finite-element simulation. Experiments determine the punch for bending for various process variables, such as punch radius, punch speed and lubrication, were carried out. As a result it was found that punch load increases as punch radius and punch speed increase or lubrication decreases.An elasto-plastic incremental finite-element computer code based on an updated Lagrangian formulation was developed to simulate the V-die bending process of sheet metal under the plane-strain condition. Isotropic and normal anisotropic material behavior was considered including nonlinear work hardening. A modified Coulomb’s friction law was introduced to treat the alternation of sliding–sticking state of friction at the contact interface. Simulation results, such as the punch load of bending and the bend angle of the bent part after unloading, are compared with experimental data; satisfactory agreement was observed. The simulation clearly demonstrates that the code to simulate V-die bending process was efficient.Simulations were made to evaluate the effects of die radius, punch width, strain hardening exponent and normal anisotropy on punch load of bending. The punch load for bending is smaller for materials with a larger strain hardening exponent. The effect of punch width on punch load is limited. The punch load decreases in the early stage and increases in the final stage of the bending process as the die radius increases. The influences of all of the process variables on the final bend angle of the bent parts of sheet after unloading were also evaluated. The effects of process variables except die radius on the bend angle after unloading are also limited. The angle of spring-back is greater for tools with larger die radius.     

Thickness and grain size dependence of limit strains in sheet metal stretching

A theoretical analysis is presented of thickness and grain size dependence of the limit strains in isotropic sheet metals under biaxial tension. Sheet surfaces are assumed to consist of asperities which increase proportionally with the equivalent strain and the grain diameter of the material. The limit strains are given as the sum of the strain up to Swift instability modified by taking into consideration the effect of surface asperities and the subsequent strain up to a localized neck which is based on the Marciniak analysis. The thickness ratio of the material in the Marciniak model is determined making use of the magnitude of the surface roughness at the modified Swift instability.Numerical results show that the limit strains decrease in all cases as thickness to grain size ratio decreases. The increase in surface roughness with plastic strain varies from material to material and this is taken into account in predicting limit strains.     

管材弯曲中应变中性层位移的分析

在管材弯曲变形系统研究中,为了进一步揭示管材弯曲的变形机理并促进管材精确弯曲和数字化弯曲技术的快速发展,利用板弯曲理论在平面应变条件下推导出弯管应变中性层半径和外侧管壁厚变薄量的近似计算公式,并且在考虑弯曲外侧管壁厚变薄的情况下分析弯管应变中性层沿半径方向的变化.根据应变中性层半径与弯曲半径之比小于1的关系,证明弯曲过程中应变中性层向弯曲中心移动.指出应变中性层的内移量与相对弯曲半径成反比.根据部分弯曲试验和有限元模拟结果的比较,证明管壁厚变薄计算相对准确,但由于忽略了垂直于弯曲平面方向的变形影响,且未计入材料性能参数而导致计算值偏大,有待进一步修正.     

粗糙体变形特性对接触过程的应力与应变的影响

运用W-M函数生成分形粗糙表面,建立一个新的双粗糙体接触模型,采用有限元方法模拟仿真了在粗糙体不同变形特性条件下的接触过程,并分析了接触表面的应力分布及不同接触位置的塑性应变随深度的变化规律．结果表明双粗糙接触表面的应力主要集中在个别的较高微凸体上,其应力最大值出现在微凸体肩部区域的位置;等效塑性应变在不同位置沿深度的变化,呈现出不同的规律,微凸体顶部区域沿深度方向的最大等效塑性应变均发生在次表层,材料表层下的塑性应变将会导致材料表层中的夹杂或微观缺陷周围萌生微孔和裂纹源,对比不同变形特性的模型,得出弹塑性一刚体模型的最大应力及应变值都大于弹塑性一弹塑性模型。     

Evaluation of thermal deformation model for BGA packages using moiré interferometry

A compact model approach of a network of spring elements for elastic loading is presented for the thermal deformation analysis of BGA package assembly. High-sensitivity moiré interferometry is applied to evaluate and calibrated the model quantitatively. Two ball grid array (BGA) package assemblies are employed for moiré experiments. For a package assembly with a small global bending, the spring model can predict the boundary conditions of the critical solder ball excellently well. For a package assembly with a large global bending, however, the relative displacements determined by spring model agree well with that by experiment after accounting for the rigid-body rotation. The shear strain results of the FEM with the input from the calibrated compact spring model agree reasonably well with the experimental data. The results imply that the combined approach of the compact spring model and the local FE analysis is an effective way to predict strains and stresses and to determine solder damage of the critical solder ball.     

内压弯管受对称面外弯曲时的棘轮应变有限元分析

利用有限元法,采用弹性理想塑性模型,对受对称面外弯曲的内压弯管进行循环塑性分析。计算结果表明,在一定载荷作用下,弯管中部两顶线与内缘线之间有较大的塑性应变累积,且塑性应变区域随加载次数增加而增大。分析发现,棘轮应变沿多个方向发生,但以径向与环向为主,棘轮应变可用等效塑性应变描述。通过计算不同载荷下棘轮应变速率,得出结构的棘轮应变边界。由于采用弹性理想塑性模型,棘轮应变速率恒定,所得棘轮应变边界偏于安全,可供工程设计参考。     

基于等刚度条件的薄壁结构的一种材料替代轻量化设计分析方法

通过对板壳结构有限单元的弯曲应变能和膜应变能进行分析,得到一个采用不同材料的薄壁结构的刚度之间的近似解析关系式。将其与结构刚度的有限元计算分析过程相结合,提出一种不同材料但等刚度的薄壁结构的迭代设计方法和材料替代轻量化分析方法。分析得到的薄壁结构关于板壳的弯曲应变能占总的应变能的比例  的等刚度曲线,可用于直观地评价材料替代得到的等刚度结构的轻量化效果,可指导薄壁结构的轻量化设计。分析表明,材料替代对薄壁结构刚度的影响不仅与材料的弹性特性和板壳厚度相关,还与给定载荷下结构的  参数相关。对于铝板替代钢板的薄壁结构设计问题, 越高,则等刚度铝板结构的轻量化效果越明显。通过对一个悬臂薄壁钢板曲梁结构和一个典型车身接头结构的材料替代设计和轻量化分析,验证了提出的设计、分析方法的有效性和结论的正确性。     

Modelling drawbeads in sheet metal forming

In this paper, we describe the numerical modelling of drawbead forces required to draw a sheet metal through a bead with a constant cross section. The model is formulated using an elasto-plastic large strains finite element method combined with an improved numerical technique taking into account the contact and friction conditions, based on linear programming techniques and fixed point conditions. The numerical simulations were carried out with various drawbead geometries and the results are expressed in terms of drawing restraining force versus drawing movement, as a function of the drawbead geometry, gap conditions and friction conditions. With this procedure it is also possible to determine the main deformation paths in the drawbead region where the bending strain predominates over the membrane strain. The results obtained are compared with experimental data and the agreement proves to be good. The calculated results can be used as a basis to obtain better approximations of the drawbead constitutive equations to be used in general 3D FE simulation codes in cases where it is impossible to exactly determine the drawbead geometry.     

A method of solution for the cyclic bending problem

A new approach to modeling processes of elastoplastic cyclic bending on roller levellers for beam straightening, levelling of long-dimensional profiles with arbitrary cross-sections, and some other applications of a similar kind, is proposed.Standard techniques for modeling the above-mentioned processes do not account for longitudinal motion, and, therefore, are not completely adequate. The stress-strain state (SSS) at every point of the material is determined not only by its position but also by its strain history.In modeling the processes of cyclic bending, use is made of the Euler-Lagrange approach to formulating the governing equations, together with traditional assumptions such as the hypothesis that plane cross-sections remain plane, stationarity of the processes, and the theory of anisotropic hardening. These allow one to take into account the strain history of moving material particles.The method developed makes possible the determination of not only the longitudinal components of the strain and stress tensors, but also the transverse components, as well as the beam profile's curvature, level and distribution of residual stresses.     

Hardening laws, surface roughness and biaxial tensile limit strains of sheet aluminium alloys

The roles of hardening laws and surface roughness have been assessed in the prediction of biaxial tensile limit strains of two A1 alloy sheet materials with different strain hardening characteristics namely AA6111-T4 and AA5754-O, utilizing the surface roughness model proposed by Parmar, Mellor and Chakrabarty [6]. In the work of Parmar et al., the predictions of limit strains were based on the Marciniak—Kuczynski inhomogeneity analysis and utilized the commonly used power hardening law (Hollomon equation) to describe the stress—strain behavior of the material. In the present work, (i) the suitability of a Voce hardening law, (ii) the effect of surface roughness parameters and (iii) the effect of grain size parameters on the prediction of biaxial limit strains has been studied. The biaxial limit strains based on Voce equation were obtained by modifying the set of equations of Parmar et al. and utilizing the experimentally measured surface roughness in 3-D, grain size parameters and stress—strain curves from uniaxial tensile and hydraulic bulge tests for the two A1 sheet materials. The predictions from Voce and Hollomon equations have been compared with the experimental forming limits determined by hemispherical punch stretching of gridded blanks. The discrepancy between predictions from Holloman equation and experiments is small for the low strain hardening AA6111-T4 material but is quite significant for the high strain hardening AA5754-O material. Further, the predictions are also strongly dependent upon the measure of surface roughness and the grain size utilized in the calculations. The results indicate good predictions of limit strains for the two alloys when (i) stress—strain data from tensile or hydraulic bulge tests are fitted to a Voce equation and (ii) half of the maximum peaks-to-valley height and grain thickness are utilized as a measure of surface roughness and grain size respectively. The results are discussed in the context of the characteristics of the hardening laws, assumptions of surface roughness model and surface and grain characteristics of the alloys studied.     

A four variable refined plate theory for nonlinear cylindrical bending analysis of functionally graded plates under thermomechanical loadings

In this paper we present a new application for a four variable refined plate theory to analyse the nonlinear cylindrical bending behavior of functionally graded plates subjected to thermomechanical loadings. This recent theory is based on the assumption that the transverse displacements consist of bending and shear components in which the bending components do not contribute toward shear forces and, likewise, the shear components do not contribute toward bending moments. The theory accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. The material properties are assumed to vary continuously through the thickness of the plate according to a power-law distribution of the volume fraction of the constituents. The non-linear strain-displacement relations in the von Karman sense are used to study the effect of geometric non-linearity. The solutions are achieved by minimizing the total potential energy and the results are compared to the classical and the first-order theories reported in the literature. It can be concluded that the proposed theory is accurate and simple in solving the nonlinear cylindrical bending behavior of functionally graded plates.     

往复弯曲统一曲率定理及其试验验证

辊式矫直工艺和辊式矫圆工艺均通过往复弯曲方式达到统一曲率的目的,辊式矫形过程中的往复弯曲变形规律是确定矫形工艺参数的理论依据。针对往复弯曲变形过程,建立一套符号系统,将弯曲曲率和弯矩矢量化。基于小曲率平面弯曲弹复方程和应变叠加原理,引入初始当量应变和当量应变的概念,采用图解法对往复弯曲弹复过程进行分析,在考虑稳定金属材料往复弯曲过程中的形变硬化、Baushinger效应和循环软化,分三种情形证明往复弯曲可以湮灭初始曲率的差异,最终使曲率统一到同一方向、同一数值,提出往复弯曲统一曲率定理。进而,设计模压往复弯曲试验装置,选用不同初始形状、不同材料的板坯进行往复弯曲试验,讨论残余曲率半径和拟合圆弧偏差随弯曲次数的变化规律,验证往复弯曲统一曲率定理的正确性,为辊式矫直和辊式矫圆工艺方案和控制策略的制定奠定了理论基础。