Springback simulation of advanced high strength steels considering nonlinear elastic unloading–reloading behavior

The stress–strain response of some materials, such as advanced high strength steels, during unloading is nonlinear after the material has been loaded into the plastic deformation region. Upon reloading, the response shows a nonlinear elastic response that is different from that in unloading. Therefore, unloading–reloading of these materials forms a hysteresis loop in the elastic region. The Quasi-plastic–elastic model (Sun and Wagoner, 2011) was modified and combined with both isotropic-nonlinear kinematic hardening and two-surface plasticity models to simultaneously describe the nonlinear unloading response and complex cyclic response of sheet metals in the plastic region. The model was implemented as user-defined material subroutines, i.e. UMAT and VUMAT, for ABAQUS/Standard and ABAQUS/Explicit finite element codes, respectively. Uniaxial loading-unloading tests were performed on three common grades of automotive sheet steel: DP600, DP980 and TRIP780 steel. The model was verified by comparing the predicted material response with the corresponding experimental response. Finally, the model was used to predict the springback of a U-shape channel section formed in a plane-strain channel draw process. The results showed that the model was able to considerably improve springback predictions compared to the usual assumption of linear elastic unloading.     

Investigation of Springback of Metallic Sheets at Small Strains

Abstract:  The bending process of an aluminium alloy and a high‐strength steel is analysed using the cylindrical bending test of 1 (Proceedings of the 5th International conference and workshop on numerical simulation of 3D sheet forming processes, 2002 , Jeju Island, South Korea). Despite its simplicity, it is now well known that this test is difficult to reproduce numerically. Indeed, it involves small plastic strains but large springback and exhibits complex contact boundary conditions providing severe benchmark characteristics. In order to obtain reliable results to be used for the validation of finite element models or simulations, particular attention has been paid to the fine measurement of several experimental parameters using a high‐resolution video camera. Several geometrical and contact parameters, as well as the springback angle, are determined. The springback results are compared with analytical results obtained using a classical bending model. It is shown that the agreement is good if the work‐hardening is identified within a small strain range, corresponding to the one covered during the test, as it mainly involves small deformations, pure bending and a weak anticlastic effect. Moreover, the decrease in the apparent modulus as a function of plastic strain leads to a more accurate measurement of the variation in the springback angle.     

Effect of prestrain on tensile properties and ratcheting behaviour of Ti-stabilised interstitial free steel

Effect of prestrain ranging between 2.5 and 15 percent on tensile properties, and ratcheting behaviour of an interstitial free steel has been studied at two different stress combinations. It is found that while yield strength increases in two distinctly different stages, the increase of tensile strength follows perfect linear relationship with increase in the amount of prestrain. The ratcheting strain accumulation direction during initial stage of asymmetric cyclic loading at constant tensile mean stress depends upon imposed maximum stress and the amount of prestrain. Number of cycles for accumulation of 16.30 pct true ratcheting strain increases with the amount of prestrain following perfect exponential relationships for both the stress combinations; but it increases in a perfectly bilinear manner with tensile yield strength of prestrained specimens. With 16.30 pct accumulated ratcheting strain the amount of back stress is found as 110 MPa irrespective of the amount of prestrain. Marginal variation in post-ratcheting tensile properties as a function of tensile prestrain has been observed.     

Application of the cohesive zone model for analysing the edge crack propagation of steel sheet in the cold rolling process

In the cold rolling process, the expansion and coalescence of micro‐defects can make steel sheet quality descend and create edge crack in the steel sheet. And the edge crack can cause the strip rupture completely. In this research, the cohesive zone model (CZM) was used to analyse the initiation and propagation of edge crack in the cold rolling process with the non‐reversing two‐high mill. A bi‐linear traction–separation law was utilized which is primarily given by the CZM parameters including the cohesive stress, T, and the cohesive energy, Γ. Compared with other popular models such as the Gurson–Tvergaard–Needleman (GTN) model, the CZM presents certain advantages because it requires a smaller number of parameters to be defined. Comparison results of the experiments and simulation illustrated that the CZM can provide accurate prediction for the propagation of edge crack in the cold rolling process. Parametric analysis was carried out and showed that the extent of the crack propagation increases with the increasing of the reduction ratio.     

考虑初始残余应力的板料弯曲回弹理论分析

在实际弯曲加工过程中,板料内部如果带有初始残余应力,将与弯曲应力发生叠加,对板料的回弹产生一定的影响。由于传统的回弹理论都没有考虑初始残余应力的影响,该文基于平面应变假设,采用服从Mises屈服准则和线性强化材料模型,推导了考虑初始残余应力的板料弯曲回弹角近似公式并基于有限元软件ABAQUS进行了残余应力板料弯曲回弹仿真对比分析。理论计算与仿真结果具有较好的一致性,验证了理论模型的正确性。研究结果表明,残余应力和厚度对板料回弹均有较大影响:沿宽度方向,不同初始残余应力处的板料回弹并不均匀;增大初始残余应力峰值和减小板料厚度均使不同初始残余应力处板料的回弹差值增大。     

型材拉弯的力学与回弹分析

回弹是弯曲成形中普遍存在的现象,是由卸载过程中内力重新分布引起的,回弹的存在直接影响弯曲件的成形精度.本文针对转台拉弯成形过程,对等边型材等曲率拉弯进行了应力-应变分析,并按照卸载预拉力与不卸载预拉力两种情况对工件回弹进行了研究与探讨,得到两组半径回弹率理论曲线.通过与试验结果对比,不卸载拉力计算的结果与试验值比较吻合.     

Influence of size effect on the springback of sheet metal foils in micro-bending

To analyze the unloading springback of sheet metal foils after micro-bending process, a constitutive model is proposed based on the surface layer model by which the sheet foil is divided into surface layer and inner portions. For the inner portion, each grain is envisaged as a composite, comprised of grain interior and grain boundary work-hardened layer. The classical composite model is then used to calculate its flow stress. For the surface layer portion, a model without grain boundary strengthening is constructed to represent the flow stress in this zone. The developed method is verified through the comparison of the calculated strain–stress curves with the tensile test results of four kinds of pure copper sheet foils with different thicknesses ranging from 0.1 mm to 0.6 mm. To investigate the effect of thickness and grain size on the springback of pure copper sheet foils, three-point bending tests are carried out. A finite element (FE) model for predicting the springback in micro-bending process is further developed, which takes into account the deformation behavior and orientation of each grain. The influences of grain size and thickness on the springback of sheet foils are investigated. The research results show that the decrease of sheet foil thickness or the increase of grain size results in a big springback. The scatter of springback angle is mainly attributed to the elastic anisotropy of surface grains and increases with the reduction of grains along the thickness direction. A good agreement between the experimental results and the analytical calculations shows that the developed FE model can predict the springback of sheet metal foils well in micro-bending process.     

基于正交试验的S形板料折弯工艺研究

针对典型的S形板料折弯工艺进行了分析.应用DYNAFORM有限元模拟软件,分析了S形板料折弯后的内外层等效应力、等效应变分布及回弹前后的应力应变变化,对其折弯过程中的回弹机理进行了 研究.通过正交试验的直观及方差分析确定了影响板料回弹若干关键因素的主次顺序,节省了现场试验的成本,对S形折弯件的精密化生产有一定的指导意...     

Strain mechanisms in grey cast iron

Grey cast iron exhibits a continuous stress-strain curve on which elastic and plastic strains cannot be identified. Tensile straining is analysed here in the case of a grey cast iron solidified by continuous casting. It is demonstrated that the main damaging processes are directly related to strain components which can be macroscopically identified. The amount of microcracking is evaluated by the decrease in elastic stiffness. On complete unloading, a small increase in the elastic stiffness is recorded; however, the initial value is not recovered. Deformation due to localized plastic strains is evaluated by recording the tangent modulus. For stresses inducing negligible creep strains the tangent modulus decreases linearly for increased applied stress. Under low stress amplitude, however, an elastic range is clearly observed. Both mentioned strain mechanisms are fully responsible for the non linear behaviour of the material. Furthermore, it is demonstrated that as a consequence of these strain mechanisms, the stress-strain curve of a prestrained grey cast iron is directly related to the initial stress-strain curve of the same material.     

Large-strain Bauschinger effect in austenitic stainless steel sheet

Large-strain Bauschinger effect in cold-rolled austenitic stainless steel sheet is investigated after large amounts of prestrain. The material is prestrained in uniaxial tension, and the tensile properties of the prestrained material are measured in different angles with respect to the prestraining direction. By comparing the differences in the yield stresses in different orientations, the effect of prestraining on material anisotropy is studied. The method is applied to AISI 304-type stainless steel sheet. The test results are analyzed using a combined isotropic–kinematic hardening model. The results indicate that this kind of material shows a considerable Bauschinger effect. Transient and permanent softening is observed in the experiments. The experimental Young's modulus also seems to decrease with prestrain.     

On the identification of an effective cross section for a cruciform specimen

Biaxial tensile testing of sheet metals is becoming increasingly popular for sheet metal forming. Determining equivalent stresses in biaxial tensile specimens is more complicated than in conventional uniaxial tensile specimens. In the present study, we compare four different approaches to calculate effective stresses during biaxial tensile loading of a cruciform specimen: (a) partial unloading method, where stresses are determined based on force–strain curves; (b) identification with uniaxial tensile testing; (c) an analysis of equivalent biaxial tests; and (d) numerical simulations. Considering experimental results for an AA1050 aluminium alloy and for a low‐carbon steel DC06, we show that, for the cruciform sample studied here, two methods do not yield physically reasonable results: The uniaxial approach does not properly take into account the effect of transverse loading, and the equivalent biaxial approach exhibits uncertainties in strain measurement data. The most comprehensible approach is the numerical method, because it also yields detailed information about the local stress and strain states. The numerical results are in excellent agreement with the partial unloading method in terms of the initial flow stress and of effective stress–strain curves for strains up to 0.02, with both methods predicting a similar effective cross section of 18.0 mm² for the considered specimen.     

Effects of approximations in analyses of beams of open thin‐walled cross‐section—part II: 3‐D non‐linear behaviour

In a companion paper, the effects of approximations in the flexural‐torsional stability analysis of beams was studied, and it was shown that a second‐order rotation matrix was sufficiently accurate for a flexural‐torsional stability analysis. However, the second‐order rotation matrix is not necessarily accurate in formulating finite element model for a 3‐D non‐linear analysis of thin‐walled beams of open cross‐section. The approximations in the second‐order rotation matrix may introduce ‘self‐straining’ due to superimposed rigid‐body motions, which may lead to physically incorrect predictions of the 3‐D non‐linear behaviour of beams. In a 3‐D non‐linear elastic–plastic analysis, numerical integration over the cross‐section is usually used to check the yield criterion and to calculate the stress increments, the stress resultants, the elastic–plastic stress–strain matrix and the tangent modulus matrix. A scheme of the arrangement of sampling points over the cross‐section that is not consistent with the strain distributions may lead to incorrect predictions of the 3‐D non‐linear elastic–plastic behaviour of beams. This paper investigates the effects of approximations on the 3‐D non‐linear analysis of beams. It is found that a finite element model for 3‐D non‐linear analysis based on the second‐order rotation matrix leads to over‐stiff predictions of the flexural‐torsional buckling and postbuckling response and to an overestimate of the maximum load‐carrying capacities of beams in some cases. To perform a correct 3‐D non‐linear analysis of beams, an accurate model of the rotations must be used. A scheme of the arrangement of sampling points over the cross‐section that is consistent with both the longitudinal normal and shear strain distributions is needed to predict the correct 3‐D non‐linear elastic–plastic behaviour of beams. Copyright © 2001 John Wiley & Sons, Ltd.     

树脂复合减振钢板U型弯曲回弹实验与数值模拟研究

通过带法兰边的U型弯曲成形实验研究,考察了树脂复合减振钢板在不同压边力下的回弹特性.实验结果表明:压边力对树脂复合减振钢板回弹特性影响显著.较大的压边力有利于减小回弹缺陷.其次,考虑树脂层的粘弹性特性,采用非线性粘弹性模型来描述树脂层的力学变形行为,并采用Cohesive单元和固体壳单元分别对树脂层和表层钢板进行离散,进行了树脂复合减振钢板在不同压边力下的U型弯曲有限元数值模拟研究.和实验结果比较表明,所建立的有限元模型能够较好的模拟U型弯曲成形过程.最后,基于建立的有限元模型,考查了成形速度,树脂层厚度和表层钢板初始屈服应力对回弹的影响.参数分析结果表明:这三个参数对回弹角的影响显著.该研究对树脂复合减振钢板冲压工艺设计具有一定的指导意义.     

板料弯曲回弹影响因素的有限元模拟研究

通过静态力学、动态力学实验方法,研究了热致性液晶聚合物(LCP)的种类对环氧树脂共混物在不同温度下的拉伸强度和应力-应变曲线的影响,通过TEM观察了共混物的相形态结构.结果表明,反应型液晶聚合物(LCPU)比其它种类的液晶聚合物对环氧树脂的改性效果更好;在不同温度下,其拉伸强度和应力～应变行为均比其它材料优越;固化物的动态力学结果表明:反应型的液晶聚合物键入了固化网络,出现新的松弛,TEM结果表明,反应型的液晶聚合物在基体材料中形成大小在nm数量级的液晶聚集微区,没有反应基团的液晶聚合物PHBHT在10%的加入量下,与环氧的共混物结构也有液晶聚集微区产生,但是聚集区大小在微米量级.     

Post-forming monotonic and cyclic behavior in a HSLA steel sheet after large deformations by in-plane compression

The effects of large prestrains (18–40%), produced by in-plane compression, on the asymmetry and the anisotropy of the stress response and on the fatigue life are investigated under fully reversed axial strain for a 345 MPa yield strength V–N high strength low alloy steel sheet. After prestraining, the hysteresis loops are asymmetric and the stress response is anisotropic, i.e., the response differs in directions parallel and perpendicular to that of the compressive prestrain. To understand the cyclic flow stress asymmetry, monotonic tension and compression tests were conducted in these two directions after prestraining. It is shown that the loop asymmetry is related to the Bauschinger effect after prestraining. Two cyclic stress strain curves, one corresponding to the tension side of the hysteresis loops and the other to the compression side, are defined to accurately describe the post-prestraining behavior. The amount of strengthening gained by prestraining is partially retained after cycling. Prestraining increases the fatigue life at low strain amplitudes but decreases it at high strain amplitudes.     

Numerical simulation of prestrain history effect on ductile crack growth in mismatched welded joints

Pipe reeling may lead to plastic pre‐deformation (prestrain) around existing cracks in components; therefore, investigating whether this process accelerates or counteracts ductile crack growth, especially for strength mismatched welded joints, is warranted. This study focused on the effect of prestrain history on ductile crack growth in mismatched welded joints. A single‐edge‐notched tension specimen was selected for numerical study, and the crack was assumed to have existed before a prestrain history was applied. Crack growth resistance curves for plane strain and mode I crack growth under large‐scale yielding conditions have been computed using the complete Gurson model. Meanwhile, symmetrical and non‐symmetrical prestrain cycle modes with different loading levels were applied to the overmatched specimens. The outcome demonstrated that the mismatch ratio (the ratio between the yield stress of the weld metal and base metal) showed a significant effect on fracture resistance regardless of the stage at which the prestrain cycle loading was located. By contrast, the processing of the crack growth was weakened by the increase of prestrain values, and the symmetrical prestrain cycle resulted in greater plastic damage than the non‐symmetrical prestrain cycle did. However, the initial crack length had a non‐significant effect on the ductile fracture considering the prestrain and mismatch effects.     

Springback prediction and reduction in deep drawing under influence of unloading modulus degradation

Springback is considered as one of the major problems in deep drawing of high-strength steels (HSS) and advanced high-strength steels (AHSS) which occurs during the unloading of part from the tools. With an ever increasing demand on the automotive manufactures for the production of lightweight automobile structures and increased crash performance, the use of HSS and AHSS is becoming extensive. For the accurate prediction of springback, unloading behavior of dual phase steels DP600, DP1000 and cold rolled steel DC04 for the deep drawing process is investigated and a strategy for the reduction of springback based on variable blankholder force is also presented. Cyclic tension compression tests and LS-Opt software are used for the identification of material parameters for Yoshida-Uemori (YU) model. Degradation of the Young’s modulus is found to be 28 and 26 and 14 % from the initial Young’s modulus for DP600, DP1000 and for the DC04 respectively for the saturated value. A finite element model is generated in LS-DYNA based on the kinematic hardening material model, namely Yoshida-Uemori (YU) model. The validation of numerical simulations is also carried out by the real deep drawing experiments. The springback could be predicted with the maximum deviation of 1.1 mm for these materials. For DP1000, the maximum springback is reduced by 24.5 %, for DP600 33.3 and 48.7 % for DC04 by the application of monotonic blankholder force instead of a constant blankholder force of 80 kN. It is concluded that despite the reduction of Young’s modulus, the springback can be reduced for these materials by increasing the blankholder force only in last 13 % of the punch travel.     

A new anisotropic elasto-plastic model with degradation of elastic modulus for accurate springback simulations

Springback, a phenomenon that is governed by elastic strain recovery after the removal of forming loads, is of great concern in sheet metal forming. There is no doubt that in this regard, physically reliable numerical modelling of the forming process and predictions of springback obtained by respective computer simulations are crucial for controlling this problem. Unfortunately, by currently available approaches, springback still cannot be adequately predicted in general. In this paper, a new constitutive model is proposed which considers simultaneously sheet anisotropy, damage evolution and strain path-dependent stiffness degradation during sheet metal forming. For parameter identification of the built constitutive model, a particular experimental procedure is developed and an optimization procedure is employed to solve the inverse problem that arises. The proposed approach to constitutive modelling is validated in the end by a simulation of the springback in the formed HSS steel sheet. The simulation results, which prove to be in good agreement with the experimental ones, lead to the conclusion that accurate modelling only of anisotropic yielding is not enough to accurately predict the springback phenomenon; the constitutive model should also include the strain path-dependent change of the elastic moduli.     

Effect of continuous strain path changes on forming limit strains of DP600

The strain path may change in actual sheet metal‐forming processes, so the determination of formability of sheet metal should consider the nonlinear strain path. For identifying the forming limit (FL) strains under nonlinear strain path, a conventional two‐step procedure with unloading is classically used to produce the strain path change, which results in no continuous measure of strain. The in‐plane biaxial tensile test with a cruciform specimen is an interesting alternative to overcome the drawbacks of conventional method. The strain path change can be made without unloading during a single test. In this work, the experimental FL strains of DP600 sheets under two types of nonlinear strain path are investigated and then compared with those under linear strain paths. The Oyane ductile fracture criterion is used in the finite element simulation to predict the experimental results.     

An analytical model for bending and springback of bimetallic sheets

Springback is an inevitable phenomenon due to elastic redistribution of internal stresses occurring in sheet metal forming operations. Most of the research reported in this area has been concerned with the components formed from single metal. This article deals with the analytical solution for prediction of springback in bending of bimetallic sheets. A mathematical model is derived based on Woo and Marshall's constitutive equation, considering logarithmic strain (nonlinear) distribution across the thickness and thickness change during bending. Analytical modeling, based on logarithmic strain distribution across the thickness, can be used for accurate springback predictions in the case of smaller bend radius to the thickness ratio. The results of the springback and thickness change are validated using experimental results for the aluminum sheet layered with steel. Further, springback variation in bimetallic sheets is studied, with a change in material properties and thickness of each layer.