Identification of material hardening parameters by three-point bending of metal sheets

Calibration of hardening and partly viscoplastic parameters of the previously published material model was the primary concern of this paper. The method used for identification of the material hardening parameters for metals, the three-point cyclic bending of sheets, constitutes a basis for this work. Plastic hardening parameters were determined by comparing load–displacement curves from FE simulations with those from the tests. Since viscoplasticity is assumed, stress–strain curves from uniaxial tension tests at selected strain rates for strain-rate sensitive materials were employed to calibrate corresponding viscoplastic parameters. The optimization problems are solved by means of a commercial optimization code, LS-OPT, using a response surface methodology. The objective is to minimize (by the least-squares method) the sum of the differences between measured and simulated loads. The material parameters were identified for two high-strength steel alloys (ZSte340 and DP600, strain-rate sensitive materials), and one aluminium alloy (AA5182).     

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 model of one-surface cyclic plasticity and its application to springback prediction

This paper presents an elasto-plastic constitutive model based on one-surface plasticity, which can capture the Bauschinger effect, transient behavior, permanent softening, and yield anisotropy. The combined isotropic-kinematic hardening law was used to model the hardening behavior, and the non-quadratic anisotropic yield function, Yld2000-2d, was chosen to describe the anisotropy. This model is closely related to the anisotropic non-linear kinematic hardening model of Chun et al. [2002. Modeling the Bauschinger effect for sheet metals, part I: theory. International Journal of Plasticity 18, 571-95.]. Different with the model, the current model captures in particular permanent softening with a constant stress offset as well as the Bauschinger effect and transient behavior under strain path reversal. Inverse identification was carried out to fit the material parameters of hardening model by using uni-axial tension/compression data. Springback predicted by the resulting material model was compared with experiments and with material models that do not account for permanent softening. The results show that the resulting material model has a good capability to predict springback.     

A new cyclic anisotropic model for plane strain sheet metal forming

A finite difference model was developed for sheet metal subjected to plane strain cyclic bending under tension. The model was used to describe the effects on the mechanics of the deformation of the sheet due to mixed isotropic-kinematic cyclic hardening curves. The analytical results were compared with experimental results obtained under testing conditions closely representative of the analytical model. Two tests were used: a pure bending moment device and a bending/unbending under tension device consisting of three cylindrical pins. The model was used to determine a constitutive curve that best characterize the cyclic behavior of the material tested, as compared with the experimental results. The significance of these results were discussed in relation to the prediction of the restraining forces in the sheet as it is drawn through the blank holder drawbeads.     

Sensitivity of model parameters in stretch bending of aluminium extrusions

Stretch bending is a process of considerable importance for plastic forming purposes. Effective operations in the industry demand sufficient knowledge of how different parameters influence the process as well as the final shape of the product. Numerical simulations are an effective way of investigating these issues. However, the numerical model must be validated in order to obtain reliable results. This paper presents an analysis model in the code LS-DYNA which is shown to represent the behaviour in laboratory tests well. The model is subsequently used to check the influence of certain parameters on the response in a stretch bending process. Simple, analytical methods are also presented. It is shown that the local deformations of the cross-section during bending are primarily controlled by the geometry and tensile force level. The main parameters influencing the springback during unloading are the strain hardening properties of the material, and the tensile force.     

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.     

The effect of ‘material ratchetting’ on the incremental growth of beams subjected to steady axial loads and cyclic bending

An elastic-plastic material behaviour model capable of predicting material ratchetting, cyclic relaxation and cyclic hardening has been incorporated into finite element programs. The resulting programs have been used to predict the behaviour of beams subjected to the steady axial loads and cyclic bending. The results are compared with experimental data obtained from three tests performed on beams made from a model material. Overall, the predictions were found to be good; discrepancies in the results for the most highly loaded beam were attributed to creep effects in the experiment.The results indicate that an elastic-perfectly plastic material behaviour model, with an equivalent yield stress, does not allow acceptable prediction of the beam behaviour to be obtained.     

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.     

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.     

A comprehensive study on a propagating buckle in externally pressurized pipelines

Buckle propagation is a unique phenomenon occurring in deep-sea pipelines. In previous works, this phenomenon was investigated using a ring technique in which the pipeline was assumed to be in plane strain condition and the energies absorbed in membrane stretching and longitudinal bending were ignored. This paper presents a three-dimensional analysis of the buckle propagation phenomenon with an emphasis to address more complete factors that were not accounted for in the ring analysis. The analyses are based on the available solutions of the transition zone obtained in our previous works. A comprehensive mechanism for buckle propagation phenomenon is described from the point view of plastic stability theory for shells which enables the incorporation of the effects of transverse and longitudinal bending, membrane stretching and material strain hardening. The nondimensionalized buckle propagation pressure is represented in terms of yield coefficient, strain hardening coefficient and membrane stretching factor. It is found that a buckle once initiated in a pipeline may or may not propagate along the pipeline depending on its radius-to-thickness ratio. By comparing with various experimental results the theoretical predictions from this analysis are shown to provide very accurate estimations of the buckle propagation pressure for different materials with diverse geometric parameters and material properties. This paper points to the need for more complete information regarding the effects of transverse bending, membrane stretching and material strain-hardening on the buckle propagation pressure. Upon the requirement of application variations of the yield coefficient, strain hardening coefficient and membrane stretching factor with respect to the radius-to-thickness ratio are sketched out. This eliminates the need for recourse the curves and allows a fast and convenient resolution of buckle propagation pressure for certain pipeline. Most importantly, the present analysis offers the potential for future design of pipelines being at once more rationally and parametrically complete, and yet compact and simple to apply.     

基于虚场法的铝合金各向异性屈服及硬化属性参数同步表征

变形铝合金板材因轻质、高比强和比模量等优点广泛应用于航空航天工业中,其轧制生产过程引起的塑性各向异性可显著影响板材的变形行为,加大零部件成形精度控制和服役行为数值模拟预测的难度。针对目前常规测试方法表征材料各向异性屈服及各向异性塑性硬化属性所需试验数量多、种类复杂、限制条件多的现状,结合全场变形测量和虚场法,通过一种桥型试件的循环拉伸-压缩试验,首次实现2024铝合金板材各向异性屈服与塑性硬化本构参数的同步表征,大幅减少试验数量,简化试验过程。研究表明,采用当前的加载构型,在参数优化目标函数中结合材料0°和90°两个拉伸加载方向的试验数据,并配合多虚场约束,可以在不同参数表征初始猜测值下产生稳定的Hill1948各向异性屈服参数表征结果,保证解的准确性;对于非线性运动硬化模型,采用单材料方向加载和单虚场的目标函数即可获得对应材料方向稳定可靠的非线性运动硬化参数表征结果。研究成果可为铝合金板材成形工艺分析提供理论依据、数据参考和便捷的测试技术支持。     

高强钢变模量随动强化本构模型匹配与解耦标定策略研究

高强钢通过微观组织调控获得高强度,但不同牌号高强钢微观的不均匀变形和微观诱导塑性机理不同,使得高强钢卸载及反向加载行为更加复杂,并且牌号间差异增大,为此给出模型自适应匹配及参数解耦匹配的系统化策略,实现了高强钢回弹精确预测。首先建立幂函数和指数函数混合硬化模型,基于混合模型给出自由弯曲加载的弯矩平衡方程和曲率约束方程,基于变模量模型构建截面弹性弯矩的积分方程,基于加载和卸载解析模型建立逆向识别卸载参数的子优化模型。确定变模量线性随动强化、变模量非线性随动强化模型和含边界面的变模量随动强化模型的匹配策略。基于自由弯曲、单向拉伸和拉压试验数据,确定相应本构的子优化模型参数的优化次序,最终形成本构匹配及其参数解耦标定的系统化策略,并基于Fortran语言开发标定程序库。建立U形弯曲件和弧形弯曲件预测模型,分别对DP980和DH980两种高强钢不同应变水平下的识别结果及回弹预测结果进行对比分析,验证了解耦标定策略不仅提高了不同牌号数据的相关度,而且大幅度提升了同一牌号下的模型精度和稳定性,为基于数据的材料性能统一自辨识方法研究奠定了基础。     

Elastic-plastic characterization of a high strength bainitic roller bearing steel—experiments and modelling

Monotonic and cyclic deformations were studied for a high strength bainitic roller bearing steel. The temperature of 75 °C corresponded to normal roller bearing conditions. The materials showed hydrostatic influence on yielding, but no or marginal influence of plastic deformation on density change. Therefore, a linear elastic constitutive model with pressure dependent yielding, non-associated flow rule, combined non-linear kinematic and isotropic hardening was necessary to characterize the cyclic behaviour. A stepwise process is detailed for determining the material parameters of the pressure dependent model, where particular attention was placed on the hardening parameters. One set of parameters was sufficient to describe all tested load ranges including compressive ratchetting. Some comparative tests were performed at room temperature, 150 °C and on martensitic specimens at 75 °C. The temperature influence was limited to the isotropic hardening parameters.     

Finite element simulation of springback for a channel draw process with drawbead using different hardening models

The objective of this work is to predict the springback of Numisheet’05 Benchmark#3 with different material models using the commercial finite element code ABAQUS. This Benchmark consisted of drawing straight channel sections using different sheet materials and four different drawbead penetrations. Numerical simulations were performed using Hill's 1948 anisotropic yield function and two types of hardening models: isotropic hardening (IH) and combined isotropic-nonlinear kinematic hardening (NKH). A user-defined material subroutine was developed based on Hill's quadratic yield function and mixed isotropic-nonlinear kinematic hardening models for both ABAQUS-Explicit (VUMAT) and ABAQUS-Standard (UMAT). The work hardening behavior of the AA6022-T43 aluminum alloy was described with the Voce model and that of the DP600, HSLA and AKDQ steels with Hollomon's power law. Kinematic hardening was modeled using the Armstrong-Frederick nonlinear kinematic hardening model with the purpose of accounting for cyclic deformation phenomena such as the Bauschinger effect and yield stress saturation which are important for springback prediction. The effect of drawbead penetration or restraining force on the springback has also been studied. Experimental cyclic shear tests were carried out in order to determine the cyclic stress-strain behavior. Comparisons between simulation results and experimental data showed that the IH model generally overestimated the predicted amount of springback due to higher stresses derived by this model. On the other hand, the NKH model was able to predict the springback significantly more accurately than the IH 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%。     

Deep drawing of a thin-walled hemisphere

A model of deep drawing of a thin-walled hemisphere with a flat bottom from a plane blank by a rigid punch, considering the work hardening and wall thickness variation is developed. Elastic bending and von Mises membrane rigid-plastic strain with different friction coefficients at the punch and die contact boundaries are considered. The computational model determines the distribution of the wall thickness and the material work hardening along the shell generatrix, the drawing force versus punch displacement relation, and the critical parameters of the process in which some defects are probable.     

Kinematic hardening analysis of ratchet strain in the “pulley test”

The classical Bree problem—which represents an uniaxial model of a thin tube subjected to combined internal pressure and cyclic thermal stress across its wall—can be simulated by means of the pulley test in which a wire or strip specimen is subjected to combined steady tensile stress and cyclic bending stress. In this paper, accumulation of ratchet strain in the pulley test is investigated using a linear kinematic hardening material model from which perfect plasticity can be generated as a special case. The results of the investigation show that asymptotic ratchet strains are linearly related to the excess in mean stress σ_D above its value σ*_D at the ratchetting limit regardless of the thermal stress amplitude. Comparisons with test results on copper wire specimens—which exhibit non-linear hardening rate—confirm the qualitative validity of this simple relation. Deviations between theory and experiment are attributed to metallic cyclic creep. Further, perfect plasticity results are shown to be well predicted by a linearized lower bound estimate.     

Material characterization of blanked parts in the vicinity of the cut edge using nanoindentation technique and inverse analysis

Sheet metal blanking is widely used in various industrial applications such as automotive and electrical rotating machines. When this process is used, the designer can be faced with several problems introduced by the change of the material state in the vicinity of the cut edge. In general, blanking operations severely affect mechanical and physical properties of blanked parts. To take into account these modifications during the part design, it is important to assess the influence of the process parameters on the resulting material properties. Previous experimental and numerical investigations of blanking process have been carried out, leading to the development and the validation of a finite element model that predicts the shape of the cut edge and state of the material. The study presented in this paper makes use of nanoindentation technique to improve the validation of the previously cited model. To this end, nanoindentation tests were combined with inverse identification technique to approach some of the characteristics of material state like work hardening near its cut edges. Indentation tests were carried out in the vicinity of several parts of cut edges. Based on the corresponding load versus penetration curves, the evolution of the yielding stress resulting from the material work hardening was estimated and compared to the predictions obtained from the numerical simulation of blanking process. These comparisons show good agreement between the measurements and the predictions from finite element model.     

往复弯曲统一曲率定理及其试验验证

辊式矫直工艺和辊式矫圆工艺均通过往复弯曲方式达到统一曲率的目的,辊式矫形过程中的往复弯曲变形规律是确定矫形工艺参数的理论依据。针对往复弯曲变形过程,建立一套符号系统,将弯曲曲率和弯矩矢量化。基于小曲率平面弯曲弹复方程和应变叠加原理,引入初始当量应变和当量应变的概念,采用图解法对往复弯曲弹复过程进行分析,在考虑稳定金属材料往复弯曲过程中的形变硬化、Baushinger效应和循环软化,分三种情形证明往复弯曲可以湮灭初始曲率的差异,最终使曲率统一到同一方向、同一数值,提出往复弯曲统一曲率定理。进而,设计模压往复弯曲试验装置,选用不同初始形状、不同材料的板坯进行往复弯曲试验,讨论残余曲率半径和拟合圆弧偏差随弯曲次数的变化规律,验证往复弯曲统一曲率定理的正确性,为辊式矫直和辊式矫圆工艺方案和控制策略的制定奠定了理论基础。