Forming severity concept for predicting sheet necking under complex loading histories

A new method of predicting neck formation in sheets under non-proportional loading is proposed, based on the concept of “cumulative forming severity”. This concept is borrowed from a macroscopic model of ductile fracture where the crack initiation is governed by the accumulated equivalent plastic strain modified by the stress triaxiality and the Lode angle parameter. Such an approach necessitates a representation of the forming limit diagram (FLD) in the space of the equivalent strain to neck and the Lode angle parameter.Another new factor is the assumption of the non-linear accumulation of forming severity for non-proportional and complex loading histories. A class of non-linear weighting function is proposed with only one free parameter. A starting point in the derivation is the known FLD corresponding to proportional loading. This can be determined from Hill's and Stören and Rice analytical solutions, from numerical simulation, or else taken directly from experiments. In the case of proportional loading, necking depends on the final state of stress or strain, so it does not matter if necking severity index is accumulated in a linear or non-linear way. For non-proportional loading, the unknown free parameter of the non-linear accumulation rule must be determined from a test.Experimental data on FLDs under complex strain paths for two types of material, aluminum alloy 6111-T4 [Graf A, Hosford W. The influence of strain-path changes on forming limit diagrams of A1 6111 T4. International Journal of Mechanical Sciences 1994;36(10):897–910.] and aluminum-killed sheet steel [Muschenborn W, Sonne HM. Influence of the strain path on the forming limits of sheet metal. Archiv fur das Eisenhuttenwesen 1975;46:597–602], found in the literature are revisited by the proposed model. Calibrated from only one test with non-proportional loading condition, the model is able to predict the remaining tests of complex loading paths with good accuracy.     

Fracture limit prediction using ductile fracture criteria for forming of an automotive aluminum sheet

Forming limit curves at neck and at fracture have been experimentally determined, and surfaces of fractured dome specimens have been observed optically and in the SEM, for an automotive AA6111-T4 sheet material. Various continuum ductile fracture criteria from the literature along with the assumptions of power law hardening, Hill’s quadratic yield criterion, and proportionality of stress and strain paths have been utilized for prediction of forming limit curve at fracture and compared with the experimental curve to assess the applicability of the different fracture criteria. The maximum shear stress criterion by Tresca predicts reasonably well the fracture limits of AA6111-T4 sheet material for a range of strain ratios, and is consistent with the microstructural observations. The criterion can be used to predict fracture limit curves from uniaxial tensile data and plane strain limit at fracture. A methodology for incorporating such a ductile fracture criterion into FE simulations of sheet stampings for prediction of fracture is discussed.     

Sheet metal formability analysis for anisotropic materials under non-proportional loading

Sheet metal formability is conventionally assessed in a two-dimensional plot of principal strains or stresses in comparison to a forming limit curve. This method of assessment implicitly assumes that the forming limit is isotropic in the plane of the sheet. While the assumption of isotropy in the forming limit is perhaps a good engineering approximation, it is intrinsically inconsistent with the use of material models that are anisotropic. Since the trend today is to utilize models with full anisotropy in order to more accurately capture the physics of material behavior, the issue of anisotropy of forming limits must also be addressed. The challenge is that the forming limit is no longer defined by a curve but requires the definition of a surface in strain or stress space, and therefore it is no longer appropriate to view these limits with the convenience of two-dimensional diagrams. Furthermore, recent developments in the characterization of sheet forming limits under non-proportional loading suggest that is advantageous to view forming limit behavior in terms of stresses rather than strains, a view that is adopted in this paper. A solution to the challenge of assessing formability for an anisotropic material is proposed that rescales the stresses by a factor so that the scaled stresses have the same relationship to a single forming limit curve in a 2D plot in stress-space, as the actual stresses have to the true anisotropic forming limit in 3D space. The rescaling enables engineers to accurately view the formability of all the elements at the same time for a given finite element analysis of an application. This paper also discusses other challenges of using stresses in the assessment of formability, focusing on an analysis of the 2-Stage Forming Benchmark highlighted in the Numisheet ’99 Conference. Stresses are found in this application to unload to non-critical values after reaching critical levels earlier in a forming process, which suggests that a full integration of the stress-based forming limit criterion with FE simulation is required to detect critical states that may temporarily occur during the forming process.     

Finite element simulations of sheet metal forming under complex strain paths

Fracture is a common defect in sheet metal forming and it is essentially caused by tensile instability. This paper analyzes some experiments and theories for building forming limit diagrams of sheet metal and points out the advantages and disadvantages of current experiments and theories. According to this, a method that integrates the finite element simulation and experiment was used to research the forming limit diagrams of the sheet metal under complex strain paths. Taking the rear hanger that undergoes twice stamping as an example, the strain paths of the dangerous point of the rear hanger is investigated. Finally, the forming method of the rear hanger is confirmed. Results indicate that finite element method (FEM) can achieve the complex strain paths and different strain paths will have great impacts on the result of the sheet metal forming. __________ Translated from Journal of Jiangsu University (Natural Science Edition), 2005, 26(4): 289–293 [译自: 江苏大学学报 (自然科学版)]     

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.     

不同强化模型下的板料成形极限

介绍Hill48屈服准则下基于不同强化模型的屈服方程。推导出能够用来确定随动强化模型和混合强化模型中参数的方程。采用单向拉伸曲线上所取得的数据,对所得方程进行拟合,得到参数值,并使用所得参数值得出三种强化模型下的单向拉伸曲线。结果表明采用上述方法能够准确地确定强化模型中的参数。给出随动强化模型和混合强化模型下成形极限的计算方法。基于三种强化模型,针对分散性失稳准则、Hill集中性失稳准则、凹槽失稳准则和平面应变漂移失稳准则,得到简单加载路径下的成形极限图和成形极限应力图。从这些图中可以看出,强化模型对成形极限图和成形极限应力图影响明显。因此应当确定板料在成形过程中的强化规律,选择合适的强化模型进行成形极限预测。     

Prediction of sheet forming limits with Marciniak and Kuczynski analysis using combined isotropic-nonlinear kinematic hardening

The forming limit curve (FLC), a plot of the limiting principal surface strains that can be sustained by sheet metals prior to the onset of localized necking, is useful for characterizing the formability of sheet metal and assessing the forming severity of a drawing or stamping process. Both experimental and theoretical work reported in the literature has shown that the FLC is significantly strain-path dependent. In this paper, a modified Marciniak and Kuczynski (MK) approach was used to compute the FLC in conjunction with two different work-hardening models: an isotropic hardening model and a mixed isotropic-nonlinear kinematic hardening model, which is capable of describing the Bauschinger effect. Predictions of the FLC using the MK analysis have been shown to be dependent on the shape of the initial yield locus and on its evolution during work hardening; therefore the hardening model has an influence on the predicted FLC. In this investigation, published experimental FLCs of AISI-1012 low carbon steel and 2008-T4 aluminum alloy sheets that were subjected to various nonlinear loading paths were compared to predictions using both hardening models. The predicted FLCs were found to correlate quite well with experimental data and the effects of strain path changes and of the hardening model on predicted FLCs are discussed.     

Characterising material and process variation effects on springback robustness for a semi-cylindrical sheet metal forming process

Variation in the incoming sheet material and fluctuations in the press setup is unavoidable in many stamping plants. The effect of these variations can have a large influence on the quality of the final stamping, in particular, unpredictable springback of the sheet when the tooling is removed. While stochastic simulation techniques have been developed to simulate this problem, there has been little research that connects the influence of the noise sources to springback. This paper characterises the effect of material and process variation on the robustness of springback for a semi-cylindrical channel forming operation, which shares a similar cross-section profile as many automotive structural components. The study was conducted using the specialised sheet metal forming package AutoForm™ Sigma, for which a series of stochastic simulations were performed with each of the noise sources incrementally introduced. The effective stress and effective strain scatter in a critical location of the part was examined and a response window, which indicates the respective process robustness, was defined. The incremental introduction of the noise sources allows the change in size of the stress–strain response window to be tracked. The results showed that changes to process variation parameters, such as BHP and friction coefficient, directly affect the strain component of the stress–strain response window by altering the magnitude of external work applied to forming system. Material variation, on the other hand, directly affected the stress component of the response window. A relationship between the effective stress–strain response window and the variation in springback was also established.     

Neck growth and forming limits in sheet metals

An analytical model that is capable of predicting the development of nonuniform flow in sheet metals under various plane-stress loading conditions is presented. The model is based on the previously formulated idea that the neck grows from the initial geometric or material inhomogeneity. Applying a rate-dependent flow theory of plasticity and a simplified constitutive equation to the nonuniform section where the state of stress is assumed to be uniform, successive stages of neck profile are computed under an imposed state of strain. The computation of the detailed neck growth is then repeated over a wide range of proportional loading conditions to establish the forming limit diagram. The specific magnitude of the initial inhomogeneity that is used in the analysis has been estimated from the measured variations in sheet thickness in a representative sheet of AK steel. The necessary input material parameters have also been determined from the same sheet material. Predicted forming limit diagrams based on these input parameters compare favorably with the general trend observed in the published data on AK steel. Some discrepancies are observed between the predicted and experimental forming limits for 2036 aluminum alloy when the published material parameters are used in the neck growth model. Based on these results, some of the limitations of the analytical model are discussed in light of available experimental observations.     

Prediction of history-dependent forming limits by applying different hardening models

The loading history-dependent forming limits have been computed for sheet metals undergoing various combinations of plane-stress loading conditions. The analysis method is essentially an extension of Marciniak and Kucźynski's inhomogeneous model, except that the roles of isotropic and Prager-Ziegler kinematic hardening have been examined in detail while the flow theory of plasticity is applied. A suitable modification of the constitutive equations for the kinematic hardening model converts the rate form of the constitutive equations into the finite-increment form which satisfies the yield criterion precisely. Representative combinations of strain history consisted of an initial proportional straining to either a fixed strain state or different levels of strain state followed by continued loading under different conditions of strain ratios. Comparison of computed forming limits with available experimental data shows that the ultimate choice of either an isotropic or a kinematic hardening model is dependent on a specific combination of strain history and the material properties.     

Prediction of forming limits for anisotropic sheets containing prolate ellipsoidal voids

In sheet metal forming operations, the formability of sheet metals is limited by the occurrence of internal damage evolution that eventually yields a localized neck. Thus, designing and optimizing a sheet metal forming process, requires the precise prediction of the forming limits of the sheet materials. Accordingly, the current work attempts to theoretically predict the forming limit diagrams (FLDs) of voided anisotropic sheets using a new version of the Marciniak and Kuczynski (M–K) model. The analysis employs Gologanu–Leblond–Devaux's yield function for materials containing axisymmetric prolate ellipsoidal cavities with random orientations in conjunction with Barlat and Lian's 1989 anisotropic yield criterion. The effect of a void shape parameter on a ductile material under biaxial tensile loading is introduced and examined within the framework of the M–K model, along with the effect of including a first-order strain gradient term in the flow stress. To confirm the validity of the proposed M–K model, the predicted FLDs were compared with experimental results for steel sheets. The predicted forming limits for the voided sheets were found to agree well with the experimental data.     

Improvement of formability for the incremental sheet metal forming process

In order to obtain competitiveness in the field of industrial manufacture, a reduction in the development period for the small batch manufacture of products is required. In order to meet these requirements, an incremental sheet metal forming process has been developed. In this process, a small local region of a sheet blank deforms incrementally by moving a hemispherical head tool over an arbitrary surface. In this work, an incremental sheet metal forming process controlled three dimensionally by a computer has been accomplished. It has been shown by the experiments that a sheet blank is mainly subject to shear-dominant deformation. Therefore, the final thickness strain can be predicted. The uniformity of thickness throughout the deformed region is one of the key factors to improve the formability in the sheet metal forming processes. Using the predicted thickness strain distribution, the intermediate geometry is decided in the manner that a shear deformation is restrained in the highly shear-deformed region and vice versa. This double-pass forming method is found to be very effective so that the thickness strain distribution of a final shape can be made more uniform.     

Variability of forming limit curves

The forming limit curve indicates the maximum uniform strain which can be achieved in an element of a sheet which is strained in a proportional biaxial straining process; it refers to the strain in a region adjacent to, but not within, any area of localized straining associated with rupture. In this work forming limit curves were determined for four different grades of mild steel; gridded test pieces were stretch formed to failure in elliptical and circular hydrostatic bulge dies. Each test was repeated twenty times for each material and each die, in order to provide data for statistical investigation of scatter in measured forming limits. It is shown that the variation in forming limits is much greater than that due to the experimental error and it is proposed that this scatter reflects an intrinsic property of the material which is important in determining material formability. A three-dimensional forming limit diagram is presented which can be used to determine the probability of failure of an element which is stretch formed to a particular strain level. The experimental data are also employed to determine the number of tests required to determine the mean forming limit curve within a specified accuracy.     

比例/非比例加载路径下5754O汽车用铝合金板各向异性变形行为的精确解析

随着越来越高的汽车轻量化需求,铝合金板在现代汽车工业中的应用越来越广。在不同加载路径下,包括比例和非比例加载,5754O铝合金板在塑性成形过程中具有复杂的各向异性规律。试验表明5754O铝合金板的各向异性规律随变形量的增加会发生改变,因此在常参数屈服准则理论框架下,基于传统的单一曲线假设难以对5754O铝合金板在整个塑性变形过程中的各向异性行为进行精确描述。鉴于上述问题,并同时考虑到大变形过程中材料变形的稳定性,对Yld2000-2d屈服准则进行改进。基于改进的Yld2000-2d屈服准则和单一曲线假设推导不同方向的单向拉伸应力应变曲线,并与试验结果进行了对比。结果表明,与原始的Yld2000-2d屈服准则不同,基于改进的Yld2000-2d屈服准则,传统的单一曲线假设仍然适用于5754O铝合金板各向异性问题。给出不同强化方式在比例加载路径下的统一性和非比例加载路径下的分散性证明。基于改进的Yld2000-2d屈服准则和等向强化和混合两种强化方式,推导非比例加载路径下板料的应力应变曲线。基于试验结果,验证了推导的理论曲线的精度。实现了5754O铝合金板在比例和非比例加载路径下变形行为的精确描述,为其工业应用提供了重要的理论支撑。     

A simulation of cold-roll forming for elastoplastic materials

A kinematical approach was proposed in a previous paper for predicting the optimal shape and the deformed length of a rigid-plastic metal sheet during cold-roll forming. Because the elastic effects are important in this kind of process, the method has been extended here to elastoplastic materials. In the new formulation, the sheet is still considered as a thin shell and its middle surface is described as a Coons patch depending on one geometrical parameter. Moreover, the material now satisfies a constitutive rate equation in which the corotational rate of stress is used. The Prandtl-Reuss model, including the von Mises yield criterion and the normality flow rule, is used. In order to integrate the elastoplastic constitutive equations, a radial return scheme is adapted so that the plastic power rate is calculated, using a Gauss method. Its minimization gives the optimal shape for a strain hardening elastoplastic material, as well as the optimal velocity field. This approach has been implemented on a workstation and, as for rigid-plastic materials, it gives a very fast simulation of the cold-roll forming process.     

考虑可靠度的成形极限曲线建立方法

实验获得板材成形极限的方法是通过对不同加载路径下获得的极限应变点进行拟合,从而得到成形极限曲线。将百分回归分析理论应用到成形极限实验数据点的曲线拟合方法中,利用该方法可以对成形极限曲线的百分位值进行分析和预测,可以在给定的实验可靠度和置信度下,根据实际需要调节曲线的位置,同时为给定极限应变点分布带的上下限的确定提供理论依据。     

汽车零部件热成形极限裕度云图的构建与应用

在热冲压工艺中,热冲压钢板处于高温状态,不同位置温度分布有明显差异,传统方法难以评价冲压件的成形性,考虑温度的三维成形极限图能有效评价热冲压件的成形性能,但无法判断热冲压件的安全裕度。针对上述问题,在考虑温度的三维成形极限图的基础上,首先提出了成形裕度的计算方法;然后结合数值模拟和理论分析手段,构建了适用于评价高强钢板热冲压的安全裕度云图;最后对某汽车B柱热冲压不同工艺过程进行了仿真分析,并与试验进行了对比。结果表明,在不同工艺参数和板料尺寸下,构建的裕度云图均能有效预测高强钢板热冲压件的开裂部位和程度。     

优化板料成形状态的新技术

为了解决一些形状复杂的薄板零件在实际生产中存在的问题，提出了优化板料成形状态的思想，介绍了一种既能有效地减小薄板成形时法兰面上流动阻力极大区域内的流动阻力，又不会增加起皱趋势，进而优化板料成形状态的新工艺措施。对这一措施进行了较为精确的有限元数值模拟和机理分析。     

复合变形路径下的板材冲压成形极限应变(Ⅰ)——冲压成形极限应变的立论与解析

从4个方面就变形路径对成形极限图影响的研究现状进行综述,并进一步论述了复杂变形路径和单一变形路径的概念,以及基于Hill'48屈服准则的塑性应变几何关系.为了在任意复杂变形路径下计算冲压板材的失稳极限应变,提出"任意复杂变形路径均可简化为线性复合变形路径"、"板材的冲压成形能力亦即板材允许的极限厚度应变",以及"板材在线性复合变形路径下的冲压成形能力取决于其最终变形路径的应变比值"等3个工程简化假设,并对它们进行立论和诠释.同时基于这些假设,在板材承受线性复合变形路径和前后变形路径的应变主轴发生转动的条件下,解析和推导出计算冲压成形极限应变的理论公式.利用这些简化假设和理论公式,可以在任意复杂变形路径下计算板材的冲压成形极限应变并绘制其冲压成形极限图.     

三角冲压件的压边力成形仿真分析

建立了三角冲压件的有限元模型,利用DYNAFORM软件,采用数值模拟的方法研究了三角形冲压件拉深时,压边力随时间及位置变化对成形性能的影响.基于不同的压边力加载模式,对成形极限进行了仿真分析.分析结果表明,对于不同的压边力模式,拉深成形性能完全不同,合理的压边力能改善板材的成形性能.