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.     

Numerical modelling of cyclic plasticity and fatigue damage in cold-forging tools

Industrial cold-forging tools with complex geometry are very likely to be exposed to local plastic deformation near stress concentrating details. The accumulation of plastic deformation resulting from the cyclic loading conditions leads to fatigue damage and eventually to generation of a crack in the surface of the die. To study the effect of die prestressing on fatigue damage development, a plane strain finite element model of a cold-forging die is analysed. A complex material model, combining kinematic and isotropic hardening with continuum damage mechanics, is used to simulate the elastic–plastic material behaviour and damage development. Furthermore, a range of uncoupled damage measures are applied in the comparison of conventional and new prestressing concepts.     

Influence of reverse yielding on residual stresses induced by autofrettage

The paper investigates the influence of reverse yielding on residual stresses induced by autofrettage. On the basis of reverse loading tests, a material model is developed and implemented into analytical procedures capable of treating the elasto-plastic deformation behaviour of thick-walled tubes during both loading and unloading phases. The results show that residual hoop stresses are drastically reduced near the tube bore as compared with residual stresses obtained from conventional isotropic hardening analysis. Pure kinematic hardening analysis is also shown to overestimate residual hoop stress induced by autofrettage.     

晶体塑性模型在单晶铜低周疲劳行为中的应用

关注循环载荷作用下晶体滑移变形中的非线性硬化现象,用改进的晶体塑性模型刻画单晶金属在循环载荷作用下的变形机理.通过试验和数值模拟,研究单晶铜在循环载荷作用下的Bauschinger效应、滞回特性和循环硬化等循环塑性现象.所得结果表明,文中提出的模型可以有效地描述单晶金属在循环载荷下的非线性变形过程,可以与有限元方法结合,用数值模拟再现试样在循环加载中的Bauschinger效应、滞回特性和循环硬化等塑性变形行为.     

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.     

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.     

Effect of continuum damage mechanics on springback prediction in metal forming processes

The influence of considering the variations in material properties was investigated through continuum damage mechanics according to the Lemaitre isotropic unified damage law to predict the bending force and springback in V-bending sheet metal forming processes, with emphasis on Finite element (FE) simulation considerations. The material constants of the damage model were calibrated through a uniaxial tensile test with an appropriate and convenient repeating strategy. Holloman’s isotropic and Ziegler’s linear kinematic hardening laws were employed to describe the behavior of a hardening material. To specify the ideal FE conditions for simulating springback, the effect of the various numerical considerations during FE simulation was investigated and compared with the experimental outcome. Results indicate that considering continuum damage mechanics decreased the predicted bending force and improved the accuracy of springback prediction.     

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 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.     

Effect of kinematic hardening on localized necking in biaxially stretched sheets

For elastic-plastic sheets under biaxial stretching localized necking is investigated assuming that the material follows a kinematic hardening rule. The investigation is mainly based on the plane stress approximation, but includes a few results obtained in the context of the three-dimensional theory. It is found that the forming limit curves predicted by kinematic hardening are in far better agreement with experimental results than the similar curves predicted by standard flow theory with isotropic hardening. For a high hardening material quite good agreement is obtained with predictions of deformation theory of plasticity, which may be considered as a simple model of a solid that develops a vertex on the yield surface.     

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%。     

A two surface plasticity model for the simulation of uniaxial ratchetting response

In this study, a two surface plasticity model was developed and used to simulate the uniaxial ratchetting response of CS 1026 steel. Most cyclic plasticity models used in ratchetting simulations are Chaboche-type nonlinear kinematic hardening models, which deal with dynamic recovery terms considering the back stress tensor. This paper describes the ratchetting simulation of steel by the two surface model based on yield theory following both isotropic and kinematic hardening rules in order to obtain enhanced ratchetting response. The parameters used in the simulation were obtained from a parametric study and were determined from the initial range and stabilized range of CS 1026 steel. In addition, the two surface model was validated by comparing the results of a ratchetting simulation with experimentally determined maximum axial strain per cycle. The ratchetting responses obtained from the two surface model are an improved simulation results compared with results from bilinear and kinematic hardening models.     

低周循环应力-应变关系和损伤

引入硬化状态变量表征微结构变化对材料循环变形行为的影响，提出应变控制下材料的循环应力一应变关系，由此导出的疲劳损伤演变方程与材料的损伤和硬化状态有关，考虑加载历史的影响。其结果可以为分析多级疲劳加载时，损伤演变过程和剩余寿命估算提供新的思路。     

Numerical simulation of incremental sheet forming with considering yield surface distortion

The incremental sheet forming processes (ISF) are attracting lots of attentions due to their advantages on rapid prototyping, without special dies and short lead time. The numerical simulation can be a valid method to investigate the forming process and predict the defects. In this study, an extended fully coupled ductile damage model with mixed nonlinear hardening was used to simulate the ISF process. At the same time, the yield surface distortion was also considered in this model, which can enhance the capability of modeling metallic material behavior under complex loading paths. Afterwards, some simulations were conducted with the proposed model. Additionally, one tension-shear orthogonal loading test was assigned on the one representative element in order to investigate the loading path effect during ISF process. By comparing the equivalent plastic strain and ductile damage evolution of the blank, the influence of the yield surface distortion on the ISF process was proved.     

304不锈钢两阶段载荷模式下的疲劳寿命

对304不锈钢进行一系列单轴、扭转、比例和非比例循环载荷变化的两阶段低周疲劳实验,比较各阶段载荷下的循环特性.结果表明在第一阶段载荷下,材料在开始几个循环表现出软化特性,在路径转化时有交错强化现象,而在圆路径下产生明显的附加强化.试验结果表明,这种强化对疲劳寿命有明显影响,使两段载荷的总疲劳损伤小于1.文中比较了线性损伤律和几个非线性损伤律(Manson的双线性损伤律、损伤曲线方法、Morrow模型)的寿命预测结果,表明对寿命的预测各模型都给出了不安全的结果.     

Influence of hardening rule on the elasto-plastic behaviour of a simple structure under cyclic loading

The modes of cyclic elasto-plastic deformation of a two-bar structure with unequal areas and lengths under the simultaneous action of sustained mechanical load and cyclic thermal history are investigated analytically using three types of elasto-plastic material models: perfectly plastic, linear kinematic hardening and linear isotropic hardening. This simple structure is shown to exhibit much of the behaviour of interest in design of structural components subjected to repeated thermal loads: viz, elastic shakedown, reversed plasticity and ratcheting. The cyclic plastic behaviour of the structure is developed in closed form and the effects of strain hardening, hardening rule and geometrical parameters of the two-bar assembly on the deformation modes are critically examined.     

An Engineering Model for Three-Dimensional Elastic-Plastic Rolling Contact Analyses

Fatigue life of heavily loaded rolling bearings is strongly dependent on elastic-plastic material properties. For bearing steels these elastic-plastic properties can be accurately obtained by performing monotonic or half-compressive tests. A three-dimensional strain deformation analysis based on the incremental theory of plasticity and the use of Prandtl-Reuss relations in conjunction with the von Mises yield criterion was developed in order to evaluate the permanent deformation in dry contacts loaded above the elastic limit in case of normal loading. The Ramberg-Osgood stress-strain relation for two martensitically hardened variants of SAE 52100 bearing steel considered the nonlinear kinematic and/or isotropic material behavior. Parameters describing the influence of retained austenite are modeled by using a nonlinear isotropic law. Pressure distribution and contact surface displacements during incremental loading are evaluated by using a conjugate gradient method and the internal stress field is derived by using the superposition principle. Further, a fast analysis of smooth surfaces in elastic-plastic static and rolling contact is developed based on analytical relations for the internal stress field. Cyclic evaluation of plastic strains and residual stresses is carried out until shakedown. In order to verify the theoretical model, rolling contact tests under high normal load were performed. Residual stresses and residual profiles measurements show excellent agreement between numerical and measured cyclic values.     

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.     

Numerical improvement of thin tubes hydroforming with respect to ductile damage

Fully coupled constitutive equations, formulated in the framework of the thermodynamics of irreversible processes with state variables, accounting for isotropic hardening as well as the isotropic ductile damage are used to simulate numerically, by the finite element analysis, 3D metal hydroforming processes with damage occurence. An implicit integration scheme for local time integration of the constitutive equations and a dynamic explicit resolution scheme to solve the associated dynamic equilibrium problem are used. The effects of friction coefficient, material ductility and hydro bulging condition, on the hydroformability of various thin tubes are discussed.     

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。