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.     

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.     

A numerical analysis of sheet metal formability for automotive stamping applications

A theoretical failure model is presented for the numerical prediction of the forming limit strains of automotive sheets. The model uses the Swift’s diffuse necking and Hill’s localized necking concepts in describing tearing-type sheet metal failures and a computational scheme is proposed in which the failure conditions are expressed in incremental forms. The Bauschinger effect is included properly in the deformation modeling using an additive backstress form of the nonlinear-kinematic hardening rule. The necking conditions and plasticity model are transformed into a set of algebraic equations that may be applied both for proportional and non-proportional strain-controlled loadings. An iterative approach is employed in the incremental solution of algebraic equations. The formability analyses are conducted using the proposed theoretical model and the forming limit strains of two new generation auto sheets (Trip600 1.4 mm, DP980 1.15 mm) are estimated. The numerically generated FLC are compared with the experimental data and the FLC calculated with the Keeler–Brazier equation. For both steels, the model produced conservative plain–strain intercept values, FLC₀, when compared with the predictions of Keeler–Brazier equation. Also the negative minor strain part of the experimental FLD’s is estimated with sufficient accuracy. For the positive minor strain side, however, the predictions are lower than both the experimental fit and the standard curve.     

Path-independent forming limit models for multi-stage forming processes

Many practitioners of the metal forming community remain faithful to the idea that strain metrics are useful for formability assessment. However, it is only valid when deformation occurs along linear strain paths. Current simulations of multi-stage sheet forming processes for rigid-packaging and automotive components result in higher rejection rates due to the inaccuracy of forming and fracture limit models. In this work, we establish a new approach considering path-independency in forming limits based on the stress-based forming limit and polar EPS (Effective Plastic Strain) diagram which appear to be an effective solution for nonlinear effects. The related theory has been implemented into a user material model in commercial software.     

利用最小厚度准则预测高强热轧钢板的FLC曲线

为更便捷地获得成形极限图(FLD)中的成形极限曲线(FLC),用最小厚度准则,通过少量成形极限试验结合数值模拟来预测FLC.采用Barlat1989屈服准则对QStE340TM、SAPH370、ZStE260P三种高强度热轧钢板进行成形极限模拟,并以最小厚度准则作为极限判据,根据数值模拟结果绘制FLC图.计算结果表明,采用平面应变路径下的成形极限实验数据作为最小厚度准则的已知参数时,数值预测结果与实验结果能较好吻合.故采用平面应变路径下的成形极限实验数据,结合最小厚度准则和数值模拟,即可得到材料完整的FLC曲线.     

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 experimental and numerical study of the effects of length scale and strain state on the necking and fracture behaviours in sheet metals

Within sheet metal forming, crashworthiness analysis in the automotive industry and ship research on collision and grounding, modelling of the material failure/fracture, including the behaviour at large plastic deformations, is critical for accurate failure predictions. In order to validate existing failure models used in finite element (FE) simulations in terms of dependence on length scale and strain state, tests recorded with the optical strain measuring system ARAMIS have been conducted. With this system, the stress–strain behaviour of uniaxial tensile tests was examined locally, and from this information true stress–strain relations were calculated on different length scales across the necking region. Forming limit tests were conducted to study the multiaxial failure behaviour of the material in terms of necking and fracture. The failure criteria that were verified against the tests were chosen among those available in the FE software Abaqus and the Bressan–Williams–Hill (BWH) criterion proposed by Alsos et al, 2008. The experimental and numerical results from the tensile tests confirmed that Barba's relation is valid for handling stress–strain dependence on the length scale used for strain evaluation after necking. Also, the evolution of damage in the FE simulations was related to the processes ultimately leading to initiation and propagation of a macroscopic crack in the final phase of the tensile tests. Furthermore, numerical simulations using the BWH criterion for prediction of instability at the necking point showed good agreement with the forming limit test results. The effect of pre-straining in the forming limit tests and the FE simulations of them is discussed.     

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。     

A different approach for considering the effect of non-proportional loading path on the forming limit diagram of AA5083

In this paper, the effects of strain path change on the forming limit diagram (FLD) of AA5083 sheet were investigated. The aim is to predict the forming limit curve (FLC) with non-proportional loading path by ductile fracture criteria. For this purpose, some square blanks were pre-strained by uniaxial tension in rolling direction (RD) and transverse direction (TD), and some others were pre-strained by biaxial stretching over a hemispherical punch. Then, the FLD test specimens were prepared by trimming the pre-strained blanks with the longitudinal axis in the RD and TD directions. The out-of-plane formability test was used for obtaining the FLD. The commercial finite element software ABAQUSE 6.9 was used for simulation in accordance with the experimental procedure. For trimming in the simulation environment, a program was written in MATLAB 7.6 that could determine the elements and introduce their properties to the new simulation model. Ductile fracture criteria were used for predicting the failure, and the Hill’79 criterion was used for applying the anisotropic coefficients. The results show that pre-straining in biaxial tension generally reduces the FLC and shifts it to the right-hand side of the FLD, whereas pre-straining in uniaxial tension raises the FLC and shifts it to the left-hand side. The numerical results were compared with the experimental findings, and relatively good agreement was achieved.     

Formability prediction of high-strength steel sheet using experimental,analytical and theoretical analysis based on strain and stress forming limit curves and its application to automotive forming parts

In this work, the high-strength steel (HSS) sheet dual-phase 440 (DP440) were conducted to establish the forming limit curve (FLC) and analytical forming limit stress curve (FLSC) obtained from experimental forming limit curve. First, the Nakajima stretch forming examination was carried out to obtain forming limit curve of investigated sheet. Afterwards, the theoretical Marciniak–Kuczinsky (M–K) model was developed and calculated to evaluate localized necking limits both in strain and stress spaces combination with anisotropic yield criteria. Then, the analytical forming limit stress curves were plastically calculated by using experimental forming limit curve data combination with Swift hardening model and anisotropic yield criteria namely, Hill’48 and Yld2000-2d for representing anisotropic plastic deformation behavior on examined steel sheet. Finally, automotive stamping parts were performed in order to verify an applicability of all developed curves. It was observed that the analytical forming limit stress curves could more precisely predict the formability of automotive parts better than the forming limit curve based on strain. Particularly, the one based on Yld2000-2d predict better than the one based on Hill’48. Simultaneously, the experimental forming limit curve and analytical forming limit stress curve were also evaluated comparing with the theoretical calculated forming limit curve and forming limit stress curve using the Marciniak–Kuczinsky model. It should be noted again that the experimental forming limit curve and analytical forming limit stress curve are the best one. Therefore, the Yld2000-2d yield function better represented the anisotropic behavior of the high-strength steel sheet dual-phase 440 than Hill′ 48 yield function, and can suitable be used for the analysis prediction and design of bumper automotive parts under forming processes.     

On the use of effective limit strains to evaluate the forming severity of sheet metal parts after nonlinear loading

The conventional forming limit curve (FLC) is significantly strain path-dependent and therefore is not valid for formability evaluation of sheet metal parts that undergo nonlinear loading paths during the forming process. The stress-based forming limit curve (SFLC) is path-independent for all but very large prestrains and is a promising tool for formability evaluation. The SFLC is an ideal failure criterion for virtual forming simulations but it cannot be easily used on the shop floor as there is no straightforward experimental method to measure stresses in stamped parts. This paper presents a theoretical basis for predicting the effective limit strain curve (ELSC) using the Marciniak and Kuczynski (MK) analysis (Int J Mech Sci 9:609–620, 1967, Int J Mech Sci 15:789–805, 1973). Since the in-plane strain components are sufficient to calculate the effective strain, the ELSC can easily be determined from strains measured in the stamping plant, and therefore it is a better alternative to the SFLC for formability evaluation. This model was validated using experimental data for AISI-1012 steel (Molaei 1999) and AA-2008-T4 aluminum alloys Graf and Hosford (Metall Trans 24A:2503–2512, 1993). Predicted results showed that, similar to SFLC, the ELSC remains practically unchanged for a significant range of prestrain values under various bilinear loading paths, but some strain-path dependence can be observed for significant magnitudes of the effective prestrain (ε _e ?≥?0.37 for AISI-1012 steel and ε _e ?≥?0.25 for AA-2008-T4 aluminum).     

Localization conditions and diffused necking for damage plastic solids

We start by generalizing the one dimensional Considère condition to a maximum power condition for arbitrary three dimensional loadings. In particular, proportional loadings with arbitrary confining pressures and Lode angles are considered. It is shown that the maximum power localization criterion for J2 material does not agree with experimental observations. The constitutive relationship of damage plastic solids is then introduced to the maximum power condition for the diffused necking. We show that the onset of localization is a spontaneous consequence of the evolution of plastic damage as the weakening rate increases. Emphasis is given to the extensively studied sheet metal forming. We show that the governing factor of the localization condition for damage plastic solids is not the damage itself, but the resulting effect of the rate of the weakening from the plasticity induced damage, which is a function of the stress states on the loading path. Effects of the pressure sensitivity, the Lode angle dependence, the damage accumulation and weakening are explored through parametric studies. Examples are given for several metallic alloys that exhibit different shapes of the forming limit curve (FLC). The predicted strain components at the onset of localization agree well with the experimental results.     

非线性路径下板料拉深成形过程中破裂预测

在优选模型参数和简化孔洞形核规律的基础上,采用Gurson-Tvergaard (GT) 多孔材料本构模型分析圆筒件拉深过程;根据金属成形工艺特点,综合考虑拉伸型和剪切型2种不同韧性断裂机制,提出一个统一的韧性断裂准则形式.对于未经过预变形和经过预变形的圆筒件拉深试验和数值模拟进行了比较,结果表明:相对于成形极限图,新的韧性断裂准则可以更加准确地预测非线性路径下圆筒件的拉深破裂.     

航空用GH625镍基高温合金带材的成形极限研究

目的 通过理论预测及胀形实验建立GH625高温合金的成形极限曲线,并结合仿真手段揭示其成形性能.方法 首先,通过基本力学性能测试获取不同方向下GH625材料的基本力学参数;然后,基于颈缩理论和M-K理论模型预测GH625材料的成形极限曲线;其次,通过胀形实验建立相应的成形极限图,并与理论结果进行对比;最后,结合有限元方法进一步研究GH625材料的成形特性.结果 准确获得了GH625高温合金的塑性应变比r值、应变硬化指数n值等参数;通过理论模型及胀形实验分别获得了相应的成形极限曲线,基于颈缩理论的集中性失稳预测结果与实验结果吻合较好;建立了可靠的有限元模型,进一步分析了摩擦因数及球头直径对GH625材料成形性能的影响规律.结论 建立了准确的GH625材料成形极限曲线的理论预测模型,并通过半球胀形实验验证了理论结果的可靠性,数值仿真结果发现,较小的摩擦因数或者冲头直径有利于改善GH625材料在胀形实验中的失效位置.     

An extension of Hill’s localized necking model

An extension of the classical localized necking model according to Hill [25] is proposed, with special reference to forming limit strain prediction of orthotropic sheet metals. Due to its computational efficiency the proposed model is an appealing alternative to the popular and more advanced Marciniak-Kuczyński model [34] and [35] with variable imperfection orientation. The proposed model is expected of being very beneficial when computationally demanding constitutive models are used, for example deformation texture models. Application examples demonstrate the capabilities of the developed localized necking model.     

管道失效判据简析

在外部载荷超过管道所能承受范围时管道即发生失效。通过研究管道在外力作用下的力学性能,有助于确定管道失效时的应力或应变的临界值,根据是取应力还是取应变作为衡量管道失效时的指标,分别有基于应力的失效判据和基于应变的失效判据。合理选用管道失效判据,可以节约管道投资、延长管道使用时间。简介材料应力—应变曲线的一般特征,分析管道基于应力的失效判据和基于应变的失效判据的方法。     

Experimental evaluation and prediction of forming limit of FSW blanks made of AA 6061 T6 sheets at different weld orientations and weld locations

The main objective of the present work is to predict the forming limit of friction stir welded (FSW) sheets made of AA 6061T6, having different weld orientations, weld locations, and made at two different welding speeds. The predicted forming limit curves (FLCs) are validated with experimental FLCs. The thickness gradient based necking criterion (TGNC) and major strain‐rate ratio based necking criterion (MSRC) are used to predict the forming limit. The significance of single zone model and double zone model in FLC prediction is discussed. A decrease in hardness is witnessed in the weld zone as compared to base material. With increase in shoulder diameter and decrease in rotational speed, hardness has improved in the weld zone. The forming limit predictions of un‐welded sheets and FSW sheets coincide well with experimental results. The predicted FLCs of FSW sheets from TGNC and MSRC are equally accurate as compared to experimental FLCs in all the weld locations. Both TGNC and MSRC predict almost the same forming limit in 90° weld orientation, while TGNC showed better prediction in 45° weld orientation. FSW sheets with double zone models show better prediction accuracy than single zone models in most of the cases, except in the case of weld at centre location and at longitudinal orientation. There is only slight deviation between single zone and double zone model predictions. The failure location and failure pattern predictions are also agreeing well with the experimental FLCs.     

Microstructure and texture evolution during incremental sheet forming of AA1050 alloy

In incremental sheet forming higher limiting strain can be achieved compared to the conventional sheet metal forming process, which results in increased formability. The higher level of strain may be accompanied by non-uniform thinning. Thus, the different sections in a component may undergo different levels of deformation. In the present work a truncated cone of AA1050 H14 alloy was formed using the incremental sheetmetal forming (ISF) technique. The deformation mechanism during ISF was studied by investigating the microstructural and texture evolution in the truncated cone along the thickness of the cone wall. High resolution electron backscatter diffraction was performed at different sections of the formed truncated cone. The results show the formation of subgrains in different sections of the cone. At higher strains, grains become thin and elongated which results in grain fragmentation and formation of small grains. These small grains undergo complete recovery process and new grain boundaries (low and high angle) are formed within the thin elongated grains. Further, the evolution of shear texture shows the evidence of shear mode of deformation during incremental sheet forming. Thus, the presence of through thickness shear could be used for understanding the higher forming limit in the ISF process.

     

Quantitative analysis on the onset of necking in rate-dependent tension

Quantitative analysis on the onset of necking of Zn–5%Al alloy at 340 °C in rate-dependent tension is developed based on Hart’s criterion, through accurately measuring the values of the strain hardening index (n) and the strain rate sensitivity index (m). A critical relation of strain rate to uniform strain is derived and then is verified to be an essential mechanical parameter for characterizing the capability of uniform deformation of rate-dependent materials. The prediction of the onset of necking at specific strain paths with strain rates, tensile velocities and loads agrees well with the critical relation, which suggests the validity of instability analysis in this study. The critical relation indicates that the onset of necking is dependent only on the final values of strain and of strain rate in tension, and is independent of strain path, deformation condition or strain history, which is of great significance for the increase of formability and the reduction of forming time in superplastic forming process. Additionally, it is proved that the tension at constant load is a very effective approach for precisely establishing the critical relation, due to not involving test data at others strain paths.     

Local bifurcation and instability theory applied to formability analysis

The design and optimization of both sheet metal formed parts and processes are nowadays carried out virtually making use of numerical tools by finite element analysis. Such virtual try-out approach contributes with significant savings in terms of money, time and effort in the design, production and process set-up of deep drawn parts. The analysis of either forming success operation or surface defects, in each of the development phases, is generally performed by means of the material’s forming limit diagram (FLD), since it allows to define a safe region that reduces the probability of: (i) necking; (ii) wrinkling and (iii) large deformation occurrence. However, the FLD represented in the strain space is known to present some disadvantages. To overcome this problem, Ito and Goya proposed a local bifurcation criterion that defines the critical state for a local bifurcation to set in as a function of the stress level to work-hardening rate ratio, leading to a FLD represented in the stress space. This suggests that the FLD obtained is completely objective in the sense that it is completely independent of the strain or stress history paths (Ito et al. 2000). In this work the Ito and Goya model is used to evaluate formability, as well as fracture mode and direction on the deep drawing of a square cup. Since the analysis is performed based on the stress state, it is also possible to determine an instability factor that “measures” the degree of acceleration by current stress for the local bifurcation mode towards fracture. The selected example highlights the potential use of the criterion which, once combined with the finite element analysis, can undeniably improve the mechanical design of forming processes.