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.     

金属板料冲压成形的数值模拟

本文采用有限元动力显式算法模拟金属板料冲压成形的加工过程。四结点蜕化壳单元和刚体壳单元分别用来建立权和模具的有限元模型;更新Ｌａｇｒａｎｇｅ法和速率型本构关系被用来处理板料变形中的大应变和大转动;材料模型采用塑性各向异性屈服与等向强化模型;通过主从面模型定义板料和模具的接触,接触算法采用运动约束法,摩擦力用库仓定律计算;并利用动力松弛法对回弹过程进行了计算。模拟结果和实际零件比较,证明模型合理,算法稳定,结果可靠,具有良好的应用价值。     

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。     

Cyclic loading of beams based on the Prager and Frederick–Armstrong kinematic hardening models

The kinematic hardening theory of plasticity based on the Prager and Frederick–Armstrong models are used to evaluate the cyclic loading behavior of a beam under the axial, bending, and thermal loads. The beam material is assumed to follow non-linear strain hardening property. The material's strain hardening curves in tension and compression are assumed to be both identical for the isotropic material and different for the anisotropic material. A numerical iterative method is used to calculate the stresses and plastic strains in the beam due to cyclic loadings. The results of the analysis are checked with the known experimental tests. It is concluded that the Prager kinematic hardening theory under deformation controlled conditions, excluding creep, results into reversed plasticity. The load controlled cyclic loading under the Prager kinematic hardening model with isotropy assumption results into reversed plasticity. Under anisotropy assumption of tension/compression curve, this model predicts ratcheting. On the other hand, the Frederick–Armstrong model predicts ratcheting behavior of the beam under load controlled cyclic loading with non-zero mean load. This model predicts reversed plasticity under the load controlled cyclic loading with zero mean load, and deformation controlled cyclic loading.     

A new model for springback prediction in which the Bauschinger effect is considered

Springback prediction is an important issue for the sheet metal forming industry. Most sheet metal elements undergo a complicated cyclical deformation history during the forming process. For an accurate prediction of springback, the Bauschinger effect must be considered to determine accurately the internal stress distribution within the sheet metal after deformation. Based on the foundations for isotropic hardening and kinematic hardening, Mroz multiple surface model, plane strain assumptions, and experimental observations, a new incremental method and hardening model is proposed in this paper. This new model compares well with the experimental results for aluminum sheet metal undergoing multiple-bending processes. As is well known, aluminum is one of the most difficult sheet metals to simulate. The new hardening model proposed in this paper is not only a generic model for springback prediction but also a hardening model for sheet metal forming process simulation.     

A critical examination of nesting, bounding and memory surface/s plasticity theories under non-proportional loading conditions

This paper assesses the capabilities of Mroz's nesting surfaces plasticity model, the bounding surface model and the memory surface model under conditions of non-proportional loading path that involves a sharp bend. Simple kinematic and isotropic hardening models are also examined. Comparisons with independent test results of a bench mark problem on 304 stainless steel at room temperature show that both the nesting surfaces model and memory surface model are equally capable of adequately predicting experimental data. The next best model is the simple isotropic hardening, equipped with properly chosen yield stress and associated plastic modulus.On the other hand, the bounding surface model which employs the same material constants as the memory surface model, and the simple kinematic hardening model which employs the same constants as the isotropic hardening model, fail to predict the test results. This result indicates the importance of an isotropic hardening component in any successful constitutive plasticity theory. Furthermore, isotropic hardening characteristics should be clearly related to basic material behaviour. These features are provided by the memory surface model, which appears to be supported by experimental observations.It is observed that exact contact between yield and bounding surface cannot be ensured, owing to problems of numerical instability in the near contact position. The origins of this behaviour are briefly discussed.     

Shakedown analysis of beams using nonlinear kinematic hardening materials coupled with continuum damage mechanics

The nonlinear kinematic hardening theory of plasticity based on the Armstrong-Fredrick model and isotropic damage was used to evaluate the cyclic loading behavior of a beam under the axial, bending, and thermal loads. Damage and inelastic deformation were incorporated and they were used for the beam shakedown and ratcheting analysis. The beam material was assumed to follow the nonlinear strain hardening property coupled with isotropic damage. The effect of the damage phenomenon coupled with the elastoplastic nonlinear kinematic hardening was studied for deformation and load control loadings. The Bree's diagram was obtained for two different types of loading, and all numerical results confirmed the reduction of the safe loading domain due to material damage.     

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.     

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.     

The influence of plastic-strain-induced anisotropy modeled as combined isotropic-kinematic hardening in the tension-torsion straining problem (I)

In order to express the plastic-strain-induced anisotropy at finite deformation of ductile metals, a combined isotropic-kinematic hardening model which is a particular form of anisotropic hardening, is an appropriate model because of its simple and convenient mathematical formulation. This paper examines the applicability of the model in the computation of general straining problems by performing a numerical tension-torsion test. The anisotropy generated by plastic flow is expressed by back stress. The evolution equation contains form invariant isotropic functions of plastic strain rate and back stress and also involves the spin associated with induced anisotropy. A numerical fitting procedure allowed us to show that circules modeled as combined isotropic-kinematic hardening around the loading nose are in good agreement with the experimental yield loci taken from the nonproportional straining. The measure of checking the applicability of combined isotropic-kinematic hardening by analyzing the total stress history has also been demonstrated by simulating an extrusion process using the finite-element method. From the computed results. the angle variations between the principal stress direction and the material direction, initially axial, were observed if they are small enough in the active plastic deformation region to ensure that the stress point will move along the part of the yield locus exhibiting nearly uniform curvature. This indicated that stress and deformation can be predicted with combined isotropic-kinematic hardening as long as the loading is not reversed.     

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。     

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。     

Ratcheting behavior of cylindrical pipes based on the Chaboche kinematic hardening rule

In this study, cyclic loading behavior of thick cylindrical pipes are described. Effects of internal pressure level and axial strain amplitude on the ratcheting rate under different types of loading histories are investigated. The kinematic hardening theory based on the Chaboche model is used to predict the plastic behavior of the structures. An iterative method is developed to analyze the structural behavior under cyclic loading conditions based on the Chaboche kinematic hardening model.     

GS-20Mn5钢非对称循环应力-应变本构模型及其应用

在实际工程中,机械结构件承受反复载荷时,内部往往是非对称的应力应变状态。在非对称循环加载条件下,材料不仅会表现出循环软/硬化特性,还会表现出平均应力松弛行为。这会影响其在循环稳定状态下的力学性能,进而影响结构在相应工况下承载服役的强度安全性。针对大型压机本体结构常用GS-20Mn5钢进行了单向拉伸及应变比R为0.5,应变幅0.20%、0.25%、0.27%、0.30%和0.40%的非对称应变循环加载试验研究,分别构建了基于单向拉伸试验结果的A-F随动硬化模型,以及基于非对称循环加载的Landgraf模型来描述其平均应力松弛特性,将其应用到Ramberg-Osgood公式中,结合A-F非线性随动硬化模型,建立了非对称循环加载条件下对应于循环应力-应变曲线的本构模型,并确定了相应模型参数。针对承受非对称循环载荷的某大型锻造液压机上横梁,应用所建立的本构模型分别进行了安定性数值分析,评估了其在循环载荷下的弹塑性强度安全性。结果显示,与采用单向拉伸条件下的A-F模型时的计算结果相比,采用非对称循环应力-应变本构模型时上横梁的安定极限载荷提高了约7%。     

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.     

一种超细晶材料的混合硬化模型及其数值模拟

针对超细晶材料强度高、塑性能力不佳以及饱和应力跟晶粒尺寸和应变率等因素有关的特点,在Johnson-Cook模型的基础上引入Hall-Petch关系式,再与Armstrong-Frederick非线性随动硬化规律进行叠加,提出一种同时包含各向同性硬化和非线性随动硬化的混合硬化模型。该数学模型不仅考虑了超细晶材料的尺寸效应,还计及了加工硬化和包辛格效应的组合效应。在推导出该混合硬化模型的积分算法的基础上进行有限元数值分析和试验数据的对比分析。对比结果表明,不同晶粒大小与不同应变率下的超细晶材料的数值仿真结果与试验数据均吻合较好,进而证明该数学模型的合理性。因此,该混合硬化模型不仅丰富了塑性力学的内容,也可为超细晶材料的结构件设计提供一定的理论依据。     

Identification of material parameters in constitutive model for sheet metals from cyclic bending tests

This paper deals with the identification of material parameters in a constitutive model for sheet metals using the bending moment versus curvature diagrams obtained by cyclic bending tests. The model can describe the cyclic strain hardening by the isotropic hardening and the Bauschinger effect by the kinematic hardening. An optimization technique based on the iterative multipoint approximation concept was used for the identification of the material parameters. This paper describes the experimentation, the fundamentals and the technique of the identification problem, and the verification of this approach.     

Axial effects investigation in fixed-end circular bars under torsion with a finite deformation model based on logarithmic strain

In this paper the torsion problem of a circular bar with fixed ends is solved using a finite deformation constitutive model based on the corotational rates of the logarithmic strain. The logarithmic, Green–Naghdi and Eulerian corotational rates of the logarithmic strain are used in the model. The solution is based on a von Mises type yield function that incorporates isotropic and kinematic hardenings. For the kinematic hardening, a modified Armstrong–Fredrick hardening model with the corotational rate of the logarithmic strain is used. Assuming incompressible behavior, the fixed-end torsion problem is simplified to the simple shear problem. Solving the problem, the stress components are integrated to calculate the torque and axial force. It is qualitatively shown that the results based on the logarithmic corotational rate are in good agreement with the experimental results.     

Determination of combined hardening material parameters under strain controlled cyclic loading by using the genetic algorithm method

In this paper, experimental and numerical investigations on mechanical behaviors of SS304 stainless steel under fully reversed strain-controlled, relaxation, ratcheting and multiple step strain-controlled cyclic loading have been performed. The kinematic and isotropic hardening theories based on the Chaboche model are used to predict the plastic behavior. An iterative method is utilized to analyze the mechanical behavior under cyclic loading conditions based on the Chaboche hardening model. A set of kinematic and isotropic parameters was obtained by using the genetic algorithm optimization approach. In order to analyze the effectiveness of this optimization procedure, numerical and experimental results for an SS304 stainless steel are compared. Finally, the results of this research show that by using the material parameters optimized based on the strain-controlled and relaxation data, good agreement with the experimental data for ratcheting is achieved.