Experimental and theoretical formability analysis using strain and stress based forming limit diagram for advanced high strength steels

In this study, experimental and numerical analyses of Forming Limit Diagram (FLD) and Forming Limit Stress Diagram (FLSD) for two Advanced High Strength Steel (AHSS) sheets grade DP780 and TRIP780 were performed. Initially, the forming limit curves were experimentally determined by means of the Nakazima forming test. Subsequently, analytical calculations of both FLD and FLSD were carried out based on the Marciniak–Kuczinsky (M–K) model. Additionally, the FLSDs were calculated using the experimental FLD data for both investigated steels. Different yield criteria, namely, von Mises, Hill’s 48, and Barlat2000 (Yld2000-2d) were applied for describing plastic flow behavior of the AHS steels. Both Swift and modified Voce strain hardening laws were taken into account. Hereby, influences of the constitutive yield models on the numerically determined FLDs and FLSDs were studied regarding to those resulted from the experimental data. The obtained stress based forming limits were significantly affected by the yield criterion and hardening model. It was found that the forming limit curves calculated by the combination of the Yld2000-2d yield criterion and Swift hardening law were in better agreement with the experimental curves. Finally, hole expansion tests were conducted in order to verify the different failure criteria. It was shown that the stress based forming limit curves could more precisely describe the formability behavior of both high strength steel sheets than the strain based forming limit curves.     

Stress and strain based fracture forming limit curves for advanced high strength steel sheet

In this work, strain based fracture forming limit curve (FFLC) of advanced high strength (AHS) steel grade 980 was determined by means of experimental Nakajima stretch-forming test and tensile tests of samples under shear deformation. During the tests, a digital image correlation (DIC) technique was applied to capture the developed strain histories of deformed samples up to failure. The gathered fracture strains from different stress states were used to construct the FFLC. Subsequently, the FFLC in the strain space was transformed to a principal stress space by using plasticity theories. As a result, the fracture forming limit stress curve (FFLSC) of examined steel was obtained. Furthermore, fracture locus (FL) as a relationship between stress triaxialities and critical plastic strains was determined. Hereby, two anisotropic yield functions, namely, the Hill’48 and Yld89 model were taken into account and their effects on the calculated curves were investigated. To verify the applicability of the obtained limit curves, rectangular cup drawing test and forming tests of so-called Diabolo and mini-tunnel samples were performed. Obviously, the FFLSCs and FLs more accurately described the failure occurrences of 980 steel sheets than the FFLCs. In addition, it was found that the drawing depths predicted by the FLs and the Yld89 yield criterion slightly better agreed with the experimental results than those from the FFLSCs and the Hill’48 model, respectively.     

B550CL高强钢轮辐反拉深-翻边复合成形损伤开裂研究

针对高强钢汽车轮辐在实际生产过程中经常出现反拉深-翻边复合工序中中心孔翻边开裂这一问题,采用有限元数值模拟的方法,建立了有限元模型,并通过试制轮辐验证其可靠性.模拟获得了新型高强钢材料B550CL在用于轮辐翻边成形时的应力应变的分布和变化规律,并进一步对轮辐的损伤和壁厚分布情况进行了分析.研究表明:反拉深-翻边复合工序中翻边区在成形过程中应力应变集中明显,变形量较大;同时,材料的损伤和壁厚减薄在翻边区域也比较严重,导致实际成形中翻边区可能出现开裂等缺陷.     

The equivalent plastic strain-dependent Yld2000-2d yield function and the experimental verification

An equivalent plastic strain-dependent anisotropic material model was developed for 5754O aluminum alloy sheet. In the developed model, the anisotropy coefficients for Barlat’s Yld2000-2d anisotropic yield function were established as a function of the equivalent plastic strain. The developed anisotropic material model was implemented into the commercial FEM code ABAQUS as a user material subroutine (UMAT) for simulations. In order to evaluate the accuracy of the developed material model, biaxial tensile tests were carried out using cruciform specimens and a biaxial loading testing machine. The results show that the developed material model predicts the experimental results better than the other three material models (Yld2000-2d, Mises and Hill48 yield functions). It is also found that the developed material model describes the uniaxial tensile test curves better than Yld2000-2d yield function. The deep drawing test for 5754O aluminum alloy sheet was carried out and was simulated with different material models. The comparison between the experimental and simulation results indicates that the developed material model predicts the earing profile better than other material models. It is concluded that the equivalent plastic strain-dependence of the material coefficients should be considered for the accurate prediction of the anisotropic deformation behavior of materials.     

大幅面板材渐进折弯成形的本构建模及模拟

为了提高大幅面板材成形的模拟精度,在板材折弯平面应变假设条件下,推导出基于Hill各向异性屈服准则的弹塑性本构方程.借助ABAQUS有限元软件本构模块用户子程序接口,通过编程将上述推导的应力-应变本构关系显示表达式嵌入ABAQUS分析平台.以超长大开口半椭圆形工件成形为例,建立了大幅面钢板渐进折弯的三维弹塑性有限元模型,并数值模拟了多道次渐进折弯成形及回弹全过程.模拟效果和工程应用结果表明,与传统的基于平面应力假设的本构关系模型相比,采用平面应变假设的本构关系模型的模拟结果更接近实验值.     

Dedicated linear – Voce model and its application in investigating temperature and strain rate effects on sheet formability of aluminum alloys

This paper proposes a method to investigate the effects of temperature and strain rate on the forming limit curves (FLCs) by combining a modified Voce constitutive model (Lin-Voce model) with the numerical simulation of Marciniak test. The tensile tests are firstly carried out at different forming temperatures (20, 230 and 290 °C) and strain rates (2.5, 120 and 150 s⁻¹) for AA5086 sheet. A modified Voce constitutive model (named Lin-Voce model) is proposed to describe the deformation behavior of AA5086 and its material parameters are identified by inverse analysis technique. Then, the proposed constitutive model is verified by comparing numerical and experimental results obtained by tensile tests and Marciniak test, respectively. Finally, the numerical simulation of Marciniak test is carried out at different temperatures (100, 200 and 300 °C) and strain rates (2.5, 120 and 150 s⁻¹), and the effects of temperature and strain rate on the FLCs of AA5086 are investigated and discussed.     

Evaluation of yield criteria for forming simulations based on residual stress measurement

Process induced anisotropy in sheet metal is accounted for in analytical modeling by anisotropic yield criteria. The suitability of a yield criterion for predicting sheet metal forming process is generally validated by way of its ability to predict surface strains. However, the sensitivity of surface strains to yield criteria is dependent upon strain modes, with plane strain mode exhibiting higher sensitivity. To eliminate dependency on strain modes, stresses are used to evaluate yield criteria, since forming stresses are less sensitive to strain modes. In the study, the residual stresses remaining in a hemispherical cup formed in plain strain mode is predicted using Hill48 and Barlat89 criteria. The residual stresses are experimentally characterized by using X-Ray diffraction method. Suitable yield criterion for forming simulation is validated based on the correlation of theoretical predictions with experimental residual stress values.     

Analytical and finite element simulation studies on earing profile of Ti-6Al-4V deep drawn cups at elevated temperatures

Rolled sheet metal alloys exhibit plastic anisotropy, which leads to the formation of ears during the deep drawing process. An analytical function proposed by Yoon et al. (Int J Plast 27(8):1165–1184, 2011) predicts earing profile based on yield stress and r value directionalities for circular cup drawing. In this study, this analytical approach is applied for a deep drawing of Ti-6Al-4V at elevated temperatures up to 400 °C. Three yield criteria namely, Hill 1948, Barlat 1989 and Barlat Yld2000-2d are used to obtain the directionality inputs for the analytical formula. The analytical model is validated using experimental results and FE simulations and is found to be closely matched while requiring very less CPU time. FE simulation has been also conducted with various yield functions. Barlat Yld2000-2d is considered to be the most suitable yield criterion for very accurate earing prediction in deep drawing of Ti-6Al-4V as the inputs for both the analytical and FEM models.     

屈服准则用于高强度热轧钢板的适用性分析

为确定适合描述高强度热轧钢板变形行为的屈服准则,采用Hollomon流动应力方程和三种屈服准则对几类高强度热轧钢板在不同应变路径下达到成形极限的成形过程进行了模拟.比较了Barlat(1989)、Hill(1948)、Barlat六参数3种屈服准则,对热轧酸洗板QStE340TM、SAPH370和热轧镀锌板ZStE260P在单向拉伸、平面应变和双向等拉3种应变路径下的变形过程进行了比较.结果表明,Barlat(1989)屈服准则能较好地描述单元的变形行为,且在平面应变路径下的模拟结果最符合实验结果.     

Investigation on the strain-path dependency of stress-based forming limit curves

Path-dependent forming limits have been computed for sheet metals undergoing various combinations of plane stress loading conditions. This paper presents a theoretical model for prediction of stress-based forming limit curves (SFLC) based on the Marciniak and Kuczynski (MK) model. Acceptable agreement was observed between calculated forming limit curves (FLC) and experimental data for AISI-1012 steel (Molaei 1999) and AA-2008-T4 alloys (Graf and Hosford Metallurgical Trans 24A:2503–2512, 1993). In this paper, the path dependency of SFLCs predicted for different non-proportional loading histories has been investigated. For a range of prestrain values in different bilinear loading paths, the SFLC remains practically unchanged. However, some strain path dependency is observed for large values of prestrain ( $ \bar{\varepsilon } \geqslant 0.35 $ for AISI-1012 steel) and for abrupt changes in strain path. Nevertheless, the SFLC remains a good failure criterion for virtual forming simulations because the path dependency of SFLCs is much less significant than that of strain-based FLCs.     

Springback prediction in sheet metal forming of high strength steels

Both increased weight reduction and improved passive safety have been simultaneously required for components of new vehicle generation. Thus, advanced high strength Dual Phase (DP) steels have been progressively used when making automotive parts. During each sheet metal forming process the high strength steels exhibit distinct springback effect, which is governed by strain recovery of material after load removal. The springback is variably sensitive to materials and process parameters. Considering springback occurred in a formed part is significant for designing tools and dies. In this work, both experiments and Finite Element Analyses (FEA) of a U-shape forming test were performed and compared for investigating the springback effect. Two DP steels (JSC590R and JSC780Y) with different strengths and a mild steel (JSC270C) were taken into account. The planar anisotropic material model according to Hill’s 1948, Barlat’s yield 2000, and Yoshida–Uemori kinematic hardening model were applied in the simulations. Various mechanical testing as hydraulic bulge test, disk compression test, and in particular cyclic test under tension and compression load were carried out in order to determine required materials parameters of the models. Obviously, steel with higher yield and tensile strength definitely showed an increasing in magnitude of both springback and curling. All presented material models restricted ability to predict springback effect of the examined steels, although the Yoshida–Uemori criterion provided more accurate results than other ones. The model is therefore preferred for describing the strain recovery mechanism of high strength steels, while parameter determination plays a decisive role. The cyclic test was verified to successfully describe the kinematic behaviour of material.     

Measurement and analysis of differential work hardening behavior of pure titanium sheet using spline function

Biaxial stress tests and in-plane tension/compression tests of pure titanium sheet (JIS #1) have been carried out in order to elucidate its anisotropic plastic deformation behavior under linear stress paths. Contours of plastic work and the directions of plastic strain rates at different levels of plastic work have been precisely measured in the stress space. The measured work contours bulged significantly toward the equibiaxial direction and showed strong asymmetry, and moreover, changed its shapes significantly with increasing plastic work (differential work hardening). Using the data of the measured work contours, the applicability of selected anisotropic yield functions, i.e., Hill’s quadratic, the Yld2000-2d and Cazacu’s yield functions, to the accurate prediction of the plastic deformation behavior of the pure titanium has been discussed. It was found that these yield functions were not able to reproduce the measured data. A new method for analyzing the differential work hardening behavior of the pure titanium sheet has been developed. This method uses the spline function of Bezier curves which approximates a work contour, inspired by the methodology proposed by Vegter and Boogaard (Int. J. Plasticity 22 (2006) 557-580). The procedure for determining the spline function is described in detail. The calculated results have been in good agreement with the differential work hardening behavior of the pure titanium sheet.     

2219铝合金薄壁曲面件拉形过程变形均匀性EI北大核心CSCD

为改善2219铝合金薄壁拉形曲面件的变形均匀性,建立基于Hill 1990各向异性屈服准则的有限元模型,利用ABAQUS软件对曲面件的应变分布规律进行数值模拟,分析加载路径和板坯形状对拉形变形均匀性的影响规律。结果表明:加载路径和板坯形状对曲面件的变形均匀性有较大影响。采用折线路径,开始加载时使板材发生压缩失稳从而形成一定拱形,不仅可缓解左侧钳口附近的破裂倾向,还可增加曲面件右侧变形量,从而提高其变形均匀性。此外,减小变形量不足位置对应的板坯宽度,如采用中间窄板坯或左侧宽板坯,使其在拉形时所受应力增加,从而提高其变形量,也可实现变形均匀性的改善。最终,利用矩形板坯,经两次转折的实验路径进行拉形,获得了表面质量良好的高性能2219铝合金薄壁曲面件。     

板料成形中韧性断裂准则应用研究进展

对板料成形中的成形极限应力图、最大变薄率、成形极限图以及韧性断裂准则等预测成形极限的方法,进行了综述和分析。指出利用韧性断裂准则,不但能够较好地预测塑性差的板料成形极限,而且还能考虑应变路径的变化。利用有限元方法模拟时,韧性断裂准则既可以应用到完全耦合的弹塑性损伤模型的增量方法中,也可以应用到一步有限元逆算法中。为了准确地预测成形极限,除了提高有限元模拟精度外,应找到一种本质地反映材料性能的韧性断裂准则。     

Analysis of the extended stress-based forming limit curve considering the effects of strain path and through-thickness normal stress

In this study, an approach based on the modified Marciniak–Kuczynski (M–K) method for computation of an extended stress-based forming limit curve (FLC) is presented. The extended stress-based FLC is built based on equivalent plastic stress versus mean stress. This curve has some advantages in comparison with the conventional FLC. This new criterion is much more strain path independent than the conventional FLC. The effect of strain path on the predicted extended stress-based FLC is reexamined. For this purpose, two types of pre-straining on the sheet metal have been loaded. Moreover, the plane stress state assumption is not adopted in the current study. The influence of a through-thickness compressive normal stress is also investigated theoretically. The verifications of the theoretical FLCs are performed by using some available published experimental data.     

Numerical study of the effect of martensite plasticity on the forming limits of a dual-phase steel sheet

The formability prediction of dual-phase steel sheets is highly important in the present automotive industry. In this study, the forming-limit curve (FLC) of a DP-780 steel sheet is predicted based on the well-known Marciniak and Kuczynski (MK) theory using a Visco-Plastic Self-Consistent (VPSC) crystal-plasticity scheme. To calibrate the polycrystal model, the stress–strain curves of the ferritic and martensitic phases are inferred by accounting for three martensitic plastic behaviors. Thus, the effect of martensitic plasticity on the FLC simulation can be analyzed. In addition, two different hardening laws – namely saturation and Voce models – are considered in order to study the effects of the extrapolated hardening behavior on the shape of the predicted FLCs. The best agreement with experimentation is found when the FLCs are calculated using the saturation hardening law and when the martensite deformation is either not allowed or retarded to occur after the point of necking. An analysis of the ferritic/martensitic slip system activity inside and outside the MK instability band suggests that, within the MK-VPSC framework, localization occurs much faster in the ferritic than in the martensitic phase. In addition, it is found that, unlike uniaxial tension, after plane-strain deformation and equi-biaxial stretching there is a strong correlation between the orientation of the ferritic grain and the strain that it accommodates. The predictive capability of the model is also confirmed by comparing the measured and simulated crystallographic textures close to necking.     

Plane strain transversely anisotropic analysis in sheet metal forming simulation using 6-component Barlat yield function

In most FEM codes, the isotropic-elastic and transversely anisotropic-elastoplastic model using Hill??s yield function has been widely adopted in 3D shell elements (modified to meet the plane stress condition) and 3D solid elements. However, when the 4-node quadrilateral plane strain or axisymmetric element is used for 2D sheet metal forming simulation, the above transversely anisotropic Hill model is not available in some FEM code like Ls-Dyna. A novel approach for explicit analysis of transversely anisotropic 2D sheet metal forming using 6-component Barlat yield function is elaborated in detail in this paper, the related formula between the material anisotropic coefficients in Barlat yield function and the Lankford parameters are derived directly. Numerical 2D results obtained from the novel approach fit well with the 3D solution.     

各向异性板料屈服轨迹的研究

针对建立的十字形双向拉伸试验系统,利用有限元模拟优化得到的十字形试件,采用载荷控制方式对SPEN钢板和2024-O铝合金板进行了不同加载路径下的双向拉伸试验,得到了不同硬化阶段下的实验屈服轨迹,并与现有屈服准则Hill48、Hill79、Hill90、Hill93、Gotoh、Hosford、Barlat-Lian以及Mises的理论屈服轨迹进行了对比.结果表明:对于SPEN钢板,Hosford各向异性屈服准则得到的理论屈服轨迹与实验屈服轨迹符合得最好,其次是Mises屈服准则,Hill48屈服准则最差;对2024-O铝合金板,Barlat89、Hosford屈服轨迹与实验屈服轨迹符合得最好,Mises屈服准则最差.