On the modelling of the bending-unbending behaviour for accurate springback predictions

The prediction of springback is probably the area in sheet forming simulation where the least success has been achieved in terms of solution accuracy. The springback is caused by the release of residual stresses in the workpiece after the forming stage. An accurate prediction of residual stresses puts, in turn, high demands on material modeling during the forming simulation. Among the various ingredients that make up the material model, the hardening law is one of the most important ones for an accurate springback prediction. The hardening law should be able to consider some, or all, of the phenomena that occurs during bending and unbending of metal sheets, such as the Bauschinger effect, the transient behaviour, and permanent softening. The complexities of existing hardening laws do of course vary within quite wide ranges. One of the purposes of the present study was to try to identify a model of reasonable complexity that at the same time can fulfill the requirements concerning accuracy. Five different hardening models have been evaluated in the present investigation. The simplest model, the isotropic hardening one, involves only one history variable, while the most advanced model involves ten history variables and four additional material parameters. In the current report, results for four different materials will be accounted for. The kinematic hardening parameters have been determined by inverse modeling of a three-point bending test. A response surface method has been used as an optimization tool, together with a finite-element model of the bending test set-up. The springback of a simple U-bend has been calculated for one of the materials, and from the results of these simulations some conclusions regarding the choice of hardening law are drawn.     

Influence of constitutive model in springback prediction using the split-ring test

The aim of this paper is to compare several plastic yield criteria to show their relevance on the prediction of springback behavior for a AA5754-0 aluminum alloy. An experimental test similar to the Demeri Benchmark Test [Demeri MY, Lou M, Saran MJ. A benchmark test for springback simulation in sheet metal forming. In: Society of Automotive Engineers, Inc., vol. 01-2657, 2000] has been developed. This test consists in cutting a ring specimen from a full drawn cup, the ring being then split longitudinally along a radial plan. The difference between the ring diameters, before and after splitting, gives a direct measure of the springback phenomenon, and indirectly, of the amount of residual stresses in the cup. The whole deep drawing process of a semi-blank and numerical splitting of the ring are performed using the finite element code Abaqus. Several material models are analyzed, all considering isotropic and kinematic hardening combined with one of the following plasticity criteria: von Mises, Hill’48 and Barlat’91. This last yield criterion has been implemented in Abaqus. Main observed data are force-displacement curves during forming, cup thickness according to material orientations and ring gap after splitting. The stress distributions in the cup, at the end of the drawing stage, and in the ring, after springback, are analyzed and some explanations concerning their influence on springback mechanisms are given.     

Elastic-plastic analysis of a rolling disk by finite elements

An elastic-plastic finite element method is employed to determine the stress distribution in a circular disk rolling on a rigid track and subjected to hub loads. This model can be assumed to represent a railroad wheel. Of particular interest are the stresses that exceed the initial yield limit near the point of contact. The material constitutive laws considered herein represent elastic-perfectly plastic and strain hardening materials that obey Mises' yield condition. It is shown that unloading of the previously yielded elements due to rotation can be easily handled by the direct stiffness tangent modulus approach. Total and plastic strains, and the plastic work in the disk model are given for the entire rolling history of the disk. It is also demonstrated that the disk will shake down to a purely elastic steady state for load values beyond the initial yield if the material of the disk exhibits even a small amount of strain hardening.     

基于包辛格效应的回弹仿真分析

金属材料在一个方向上的应变硬化降低了反方向的屈服强度,材料包辛格效应的存在对车身成形仿真精度产生了重要影响,尤其现今高强钢和铝合金的大量应用,使车身成形件的回弹问题日益突出,车身模具制造对有限元回弹预测的准确性提出更高的要求。为了提高回弹的仿真精度有必要对材料的包辛格效应进行研究,利用一套夹具对DC06和DP600两种材料的薄板进行拉伸压缩试验,获得不同预应变下的位移加载曲线,通过拉伸压缩试验结果与仿真结果的对比分析,得到能反映材料包辛格效应的非线性混合硬化模型材料参数。开展U形件成形试验,并建立试验的仿真模型,计算DC06和DP600薄板的U形件成形回弹量,分析等向强化、混合强化和随动强化本构模型对回弹预测精度的影响,针对回弹仿真结果和试验结果的差别,对影响仿真精度的材料模型因素进行分析。结果表明,DC06和DP600的包辛格效应大小存在差别,考虑包辛格效应有助于回弹仿真精度的提高,但小曲率弯曲成形回弹计算对材料本构模型的敏感性,限制了回弹仿真精度的提高。     

Finite element simulation of springback for a channel draw process with drawbead using different hardening models

The objective of this work is to predict the springback of Numisheet’05 Benchmark#3 with different material models using the commercial finite element code ABAQUS. This Benchmark consisted of drawing straight channel sections using different sheet materials and four different drawbead penetrations. Numerical simulations were performed using Hill's 1948 anisotropic yield function and two types of hardening models: isotropic hardening (IH) and combined isotropic-nonlinear kinematic hardening (NKH). A user-defined material subroutine was developed based on Hill's quadratic yield function and mixed isotropic-nonlinear kinematic hardening models for both ABAQUS-Explicit (VUMAT) and ABAQUS-Standard (UMAT). The work hardening behavior of the AA6022-T43 aluminum alloy was described with the Voce model and that of the DP600, HSLA and AKDQ steels with Hollomon's power law. Kinematic hardening was modeled using the Armstrong-Frederick nonlinear kinematic hardening model with the purpose of accounting for cyclic deformation phenomena such as the Bauschinger effect and yield stress saturation which are important for springback prediction. The effect of drawbead penetration or restraining force on the springback has also been studied. Experimental cyclic shear tests were carried out in order to determine the cyclic stress-strain behavior. Comparisons between simulation results and experimental data showed that the IH model generally overestimated the predicted amount of springback due to higher stresses derived by this model. On the other hand, the NKH model was able to predict the springback significantly more accurately than the IH model.     

An experimental and theoretical analysis on the application of stress-based forming limit criterion

In this paper, a detailed study on the stress-based forming limit criterion (FLSD) during linear and complex strain paths is developed. The calculation of stress-based forming limits based on experimental strain data is performed by using the method proposed by Stoughton [A general forming limit criterion for sheet metal forming. International Journal of Mechanical Sciences 2000;42:1–27]. By applying several combinations of different constitutive equations on the required plastic calculation, an analysis on the experimental forming stress limits is performed. The necking phenomenon is simulated by Marciniack–Kuczinsky (M–K) model using a more general code for predicting the forming limits. The selected materials are a bake-hardened steel (BH steel) and an AA6016-T4 aluminium alloy. Several yield criteria such as Von Mises isotropic yield function, quadratic and non-quadratic criterion of Hill (A theory of the yielding and plastic flow of anisotropic metals. Proceedings of the Royal Society of London 1948;A193:281–97; Theoretical plasticity of textured aggregates. Mathematical Proceedings of the Cambridge Philosophical Society 1979;85:179–91) and the advanced Barlat Yld96 yield function are used to show the influence of the constitutive law incorporated in the analysis on the stress-based forming limits. The effect of the hardening model on the FLSD is analysed by using two hardening laws, namely Swift law and Voce law. The influence of work hardening coefficient, strain rate sensitivity and the balanced biaxial yield stress on the theoretical FLSD is also presented. The effect of strain path changes on the stress-based forming limits is analysed. Some relevant remarks about stress-based forming limit criterion concept are presented.     

On the rotating elastic–plastic solid disks of variable thickness having concave profiles

In this paper, an analytical solution for the elastic–plastic stress distribution in rotating variable thickness solid disks is presented. The analysis is based on Tresca's yield criterion, its associated flow rule and linear strain hardening material behavior. It is shown that depending on the shape of the disk profile, the radial stress in the central region may exceed the circumferential stress. The plastic zone which develops away from the axis of the disk consists of three annular regions governed by different mathematical forms of the yield criterion. The propagation of these plastic regions with increasing angular velocity is obtained together with the distributions of stresses and deformations in nondimensional forms.     

Autofrettage of thick-walled pipe bends

The bending of a thick-walled cylinder to a given radius involves an elastic–plastic deformation that results in a residual, axial stress distribution. The latter alternates from maximum tension to maximum compression between top and bottom halves of the cross-section. The residual stress levels depend upon the depth of plastic penetration and may be determined as a closed solution when they arise from a bending moment applied to either a non-hardening or linearly-hardening material. When the bent pipe receives an autofrettage treatment without an intermediate heat treatment, this produces a further residual, triaxial stress state. The interaction between the residual states from bending and autofrettage has an important effect upon the net axial stress and the equivalent stress. It is shown that large plastic penetrations arising from bending and autofrettage can residually stress the section beyond its yield point: in tension and in compression across both its halves. With the unloading from each process, a Bauschinger effect reduces the yield point to assist with the onset of reversed plasticity. The latter is far less beneficial than when unloading is elastic. It is shown how a nonlinear kinematic hardening model can be employed to avoid unloading plasticity at the inner and outer diameters. The consequence of interacting residual stresses is that axial stress can play as important a role as hoop stress when designing for safe service loadings. In general, an enhanced residual stress state is beneficial when compressive but detrimental when tensile. Pre-compression is often employed in practice to reduce tensile stress arising from internal pressure, axial force and self-weight. Here, the compressive residuals arising from an autofrettage treatment have long been exploited to enhance the fatigue life of process piping and weaponry.     

板料屈服行为及强化规律的研究进展

研究板料塑性成形的理论基础是屈服准则、强化规律以及本构模型。随着新材料、新工艺的不断出现,温度和应变速率对塑性成形过程中的影响也不容忽视,原有的塑性理论已无法满足研究和工程应用的需求。从板料屈服准则研究、包辛格效应与强化模型研究、屈服强化规律试验方法研究以及涉及应变速率和温度的板料屈服强化研究4个方面阐述板料屈服行为及强化规律的研究进展,指出常用屈服准则的特点和不足,说明各种强化模型中组合强化模型仍然是研究重点。试验方法主要从研究屈服轨迹的双向拉伸试验及确定强化模型参数试验的2个方面进行介绍。此外,指出针对板料在复杂应力状态下应力张量与应变张量之间的涉及应变率和温度的屈服准则和相应的流动准则的本构关系还有待研究。提出随着新材料、新工艺的不断出现,涉及应变速率和温度的屈服准则和强化规律、试验方法以及在有限元模拟中的应用等研究将是未来的研究热点。     

A model of one-surface cyclic plasticity and its application to springback prediction

This paper presents an elasto-plastic constitutive model based on one-surface plasticity, which can capture the Bauschinger effect, transient behavior, permanent softening, and yield anisotropy. The combined isotropic-kinematic hardening law was used to model the hardening behavior, and the non-quadratic anisotropic yield function, Yld2000-2d, was chosen to describe the anisotropy. This model is closely related to the anisotropic non-linear kinematic hardening model of Chun et al. [2002. Modeling the Bauschinger effect for sheet metals, part I: theory. International Journal of Plasticity 18, 571-95.]. Different with the model, the current model captures in particular permanent softening with a constant stress offset as well as the Bauschinger effect and transient behavior under strain path reversal. Inverse identification was carried out to fit the material parameters of hardening model by using uni-axial tension/compression data. Springback predicted by the resulting material model was compared with experiments and with material models that do not account for permanent softening. The results show that the resulting material model has a good capability to predict springback.     

On analysis and measurement of residual stresses in the bending of sheet metals

A new method for evaluating distribution of residual stresses in bent sheet metals is presented. Due to the non-uniform distribution of strain and stress across thickness, springback on unloading induces residual stress in a bent specimen. The authors have formulated the problem and shown that the springback and residual stresses can be expressed as a function of geometric parameters and material properties of sheet metals, e.g. bending curvature, thickness, Young's modulus, work-hardening index, etc. The layer-removing method was used to determine the residual stresses in the study. By simulating the layer-removing process, analytical measurement of the residual stress was made. The analytical results were compared with the experimental data measured, and a good agreement was found.     

高强钢变模量随动强化本构模型匹配与解耦标定策略研究

高强钢通过微观组织调控获得高强度,但不同牌号高强钢微观的不均匀变形和微观诱导塑性机理不同,使得高强钢卸载及反向加载行为更加复杂,并且牌号间差异增大,为此给出模型自适应匹配及参数解耦匹配的系统化策略,实现了高强钢回弹精确预测。首先建立幂函数和指数函数混合硬化模型,基于混合模型给出自由弯曲加载的弯矩平衡方程和曲率约束方程,基于变模量模型构建截面弹性弯矩的积分方程,基于加载和卸载解析模型建立逆向识别卸载参数的子优化模型。确定变模量线性随动强化、变模量非线性随动强化模型和含边界面的变模量随动强化模型的匹配策略。基于自由弯曲、单向拉伸和拉压试验数据,确定相应本构的子优化模型参数的优化次序,最终形成本构匹配及其参数解耦标定的系统化策略,并基于Fortran语言开发标定程序库。建立U形弯曲件和弧形弯曲件预测模型,分别对DP980和DH980两种高强钢不同应变水平下的识别结果及回弹预测结果进行对比分析,验证了解耦标定策略不仅提高了不同牌号数据的相关度,而且大幅度提升了同一牌号下的模型精度和稳定性,为基于数据的材料性能统一自辨识方法研究奠定了基础。     

Simulation of springback

Springback, the elastically-driven change of shape of a part after forming, has been simulated with 2-D and 3-D finite element modeling. Simulations using solid and shell elements have been compared with draw-bend measurements presented in a companion paper. Plane-stress and plane-strain simulations revealed the dramatic role of numerical tolerances and procedures on the results. For example, up to 51 integration points through the sheet thickness were required for accuracy within 1%, compared with 5–9 typically acceptable for forming simulations. Improvements were also needed in the number of elements in contact with the tools, and in the numerical tolerance for satisfying equilibrium at each step. Significant plastic straining took place in some cases upon unloading; however the choice of elastic–plastic unloading scheme had little effect on the results. While 2-D simulations showed good agreement with experiments under some test conditions, springback discrepancies of hundreds of percent were noted for one alloy with sheet tension near the yield stress. 3-D simulations provided much better agreement, the major source of error being identified as the presence of persistent anticlastic curvature. Most of the remaining deviation in results can be attributed to inaccuracies of the material model. In particular, the presence of a Bauschinger effect changes the results markedly, and taking it into account provided good agreement. Shell elements were adequate to predict springback accurately for R/t greater than 5 or 6, while solid elements were required for higher curvatures. As R/t approaches 2, springback simulated with solid elements tends to disappear, in agreement with measurements presented in the companion paper and in the literature.     

Role of plastic anisotropy and its evolution on springback

Springback angles and anticlastic curvatures reported for a series of draw-bend tests have been analyzed in detail using a new anisotropic hardening model, four common sheet metal yield functions, and finite element procedures developed for this problem. A common lot of 6022-T4 aluminum alloy was used for all testing in order to reduce material variation. The new anisotropic hardening model extends existing mixed kinematic/isotropic and nonlinear kinematic formulations. It replicates three principal characteristics observed in uniaxial tension/compression test reversals: a transient region with low yield stress and high strain hardening, and a permanent offset of the flow stress at large subsequent strains. This hardening model was implemented in ABAQUS in conjunction with four yield functions: von Mises, Hill quadratic, Barlat three-parameter, and Barlat 1996. The simulated springback angle depended intimately on both hardening law after the strain reversal and on the plastic anisotropy. The springback angle at low back forces was controlled by the hardening law, while at higher back forces the anticlastic curvature, which depends principally on yield surface shape, controlled the springback angle. Simulations utilizing Barlat's 1996 yield function showed remarkable agreement with all measurements, in contrast to simulations with the other three yield functions.     

Influence of reverse yielding on residual stresses induced by autofrettage

The paper investigates the influence of reverse yielding on residual stresses induced by autofrettage. On the basis of reverse loading tests, a material model is developed and implemented into analytical procedures capable of treating the elasto-plastic deformation behaviour of thick-walled tubes during both loading and unloading phases. The results show that residual hoop stresses are drastically reduced near the tube bore as compared with residual stresses obtained from conventional isotropic hardening analysis. Pure kinematic hardening analysis is also shown to overestimate residual hoop stress induced by autofrettage.     

基于三剪统一强度准则的厚壁圆筒自增强分析

基于三剪统一强度准则,考虑材料应变强化效应、包辛格效应、拉压异性及中间主应力的影响,采用双线性强化材料模型对厚壁圆筒进行自增强分析,得到了厚壁圆筒加载应力、残余应力和工作应力的解析解,提出了最佳自增强压力的计算方法,探讨了拉压比、强度准则变化参数的影响,比较了自增强处理和非自增强处理及双线性强化模型和理想弹塑性模型厚壁圆筒的应力分布差异。研究结果表明：厚壁圆筒的最佳自增强压力随半径比和强度准则参数的增大而增大;工作时的最大等效应力随半径比和强度理论参数的增大而减小,随拉压比的增大而增大;自增强等效应力的最大值在弹塑性分界面处,且应力沿壁厚的分布较均匀;与理想弹塑性模型相比,双线性强化模型所对应的弹塑性分界面半径和残余应力较小,且随着自增强压力的增大,两种模型的差值越来越大;等效应力随半径比的变化规律可为厚壁圆筒选择合理的壁厚提供一定的参考;自增强技术可改善厚壁圆筒工作时的实际应力分布,提高其极限承载能力。     

TA2Mδ2.0钛合金外缘翻边成形过程中的形状畸变现象

由于外缘翻边成形过程中应力应变在变形区分布不均匀,以及钛合金板材具有明显的各向异性,导致外缘翻边成形及卸载回弹过程产生诸多形状畸变现象。本文针对料厚为2.0mm的TA2M钛合金板材,采用PAM-STAMP软件对外缘翻边成形及回弹过程进行了有限元数值分析,预测出卸载回弹后的制件将产生翻边壁厚分布不均、翻边高度分布不一致、外翻线端头明显向外隆起、竖边与腹板平面不垂直等畸变现象。室温条件下的一系列外缘翻边成形工艺试验与有限元模拟预测的结果吻合效果良好。     

Wrinkling criterion for an anisotropic shell with compound curvatures in sheet forming

Wrinkling criteria are proposed for an elastic isotropic and plastic anisotropic shell with compound curvatures in the noncontact region of a sheet subjected to internal forming stresses. A quasi-shallow shell is modeled by Donnell-Mushtari-Vlasov (DMV) shell theory. A bifurcation functional from Hill's general theory of uniqueness and bifurcation in elastic-plastic solids is used to model the local wrinkling phenomenon. Both strain hardening and transverse anisotropy are taken into consideration. This wrinkling criterion is especially useful as a failure criterion in three dimensional finite element modeling (FEM) of sheet forming. Given the principal stresses or strains and the geometry provided at each incremental deformation step, the criterion can be used to predict wrinkles in the elements in the unsupported region.     

Finite element modeling of residual stresses in machining induced by cutting using a tool with finite edge radius

A thermal elastic-viscoplastic finite element model is used to evaluate the residual stresses remaining in a machined component. An improvement in the accuracy of the predicted residual stresses is obtained by: (a) using a modified Johnson–Cook material model that is augmented by a linearly elastic component to describe the material behavior as non-Newtonian fluid; (b) using a remeshing scheme to simulate the material flow in the vicinity of the rounded cutting tool edge without the use of a separation criterion; (c) properly accounting for the unloading path, and (d) considering the thermomechanical coupling effect on deformation. Case studies are performed to study the influence of sequential cuts, cutting conditions, etc., on the residual stresses induced by orthogonal machining.