A mixed hardening rule coupled with Hill48’ yielding function to predict the springback of sheet U-bending

A mixed hardening model has been used to model the Bauschinger effect. This hardening model is based on Lemaitre and Chaboche nonlinear kinematic hardening theory to consider cyclic behavior and the Bauschinger effect. Hill’48 yielding criterion is used because of the general stress state and relative ease of formulation. The backward Euler return mapping algorithm is applied to calculate the stress and strain increment. The mixed hardening model is implemented based on UMAT subroutine of FEA code ABAQUS. The NUMISHEET’93 benchmark shows that the mixed hardening model coupled with anisotropic yield criteria can give a favorable springback angle prediction.     

Springback investigation of anisotropic aluminum alloy sheet with a mixed hardening rule and Barlat yield criteria in sheet metal forming

A mixed hardening model has been implemented based on Lemaitre and Chaboche non-linear kinematic hardening theory to consider cyclic behavior and the Bauschinger effect. The Chaboche isotropic hardening theory is incorporated into the non-linear kinematic hardening model to introduce a surface of nonhardening in the plastic strain space. The bending and reverse bending case study has verified the effectiveness of the mixed hardening model by comparison with the proposed experiment results. Barlat’89 yielding criterion is adopted for it does not has any limitation while Hill’s non-quadratic yield criterion is for the case that the principal axes of anisotropy coincides with principal stress direction. The Backward–Euler return mapping algorithm was applied to calculate the stress and strain increment. The mixed hardening model is implemented using ABAQUS user subroutine (UMAT). The comparisons with linear kinematic hardening model and isotropic hardening model in NUMISHEET’93 benchmark show that the mixed hardening model coupled with Barlat’89 yield criteria can well reflect stress and strain distributions and give a more favorable springback angle prediction.     

晶体塑性模型描述多晶体循环加载中的Bauschinger效应

为了研究循环加载过程中织构对多晶材料Baushinger效应的影响,利用经典晶体塑性模型及含随动硬化的晶体塑性模型模拟AA6104铝合金循环加载力学行为.研究了多晶体中晶粒取向差异对材料宏观塑性行为的影响.详细分析了经典晶体塑性模型可描述多晶体循环加载Bauschinger效应机理,定量分析了多晶有限元模型中晶体取向差异对模拟结果的影响.结果表明多晶体中由于晶粒取向差异而造成的晶粒间相互作用力使得多晶体模型宏观卸载时晶粒内的残余应力是产生Bauschinger效应的主要原因,采用含随动硬化的晶体塑性模型能够较好地模拟具有织构的AA6014铝合金的循环加载过程.     

A study of the relationship between variable level fatigue crack growth and the cyclic constitutive behaviour of steel

Sensitivity of fatigue crack growth to the material behaviour was studied in two previous numerical studies (Pommier S. Plane strain crack closure and cyclic hardening. Eng Fract Mech, in press; Pommier S, Bompard P. Bauschinger effect of alloys and plasticity-induced crack-closure: a finite element analysis. Fatigue Fract Eng Mater Struct 2000;23:129–39). It was shown, in particular, that material hardening induces a rotation of the crack tip plastic zone from the front to the back of the crack, which enhances the effects of crack closure (Pommier S. Plane strain crack closure and cyclic hardening. Eng Fract Mech, in press). The type of hardening is also of key importance: Isotropic hardening is found to lower the effective part of the fatigue cycle, while kinematic hardening (Pommier S, Bompard P. Bauschinger effect of alloys and plasticity-induced crack-closure: a finite element analysis. Fatigue Fract Eng Mater Struct 2000;23:129–39) is found to increase it. This study is devoted to check the validity of those numerical results in a 0.4% C carbon steel, which displays a high Bauschinger effect and a moderate amount of isotropic hardening. The comparison between numerical results and experiments is satisfactory.     

Numerical analysis of panels’ dent resistance considering the Bauschinger effect

The Bauschinger effect should be considered in the analysis of panels’ dent resistance, because sheet metal experiences a complex loading history from stamping to denting. This paper studies the modeling and simulation of panels’ static dent resistance, taking the Bauschinger effect into consideration. Our work covers two parts: simulation and experiment. Procedures of drawing, springback I, indenting and springback II are simulated in a multiple step analysis. Different hardening models, including the isotropic hardening model, the linear kinematic hardening model and the nonlinear combined hardening model are used, respectively, in simulation. Comparing the simulation results with the experiment results, we find that the Bauschinger effect has a great influence on panels’ dent resistance. When panels are made of high strength steel or stamped with a high Blank Holder Force (BHF), the Bauschinger effect on panels’ dent resistance is more severe. Considering the effect in numerical analysis would improve the simulation accuracy effectively. The work of this paper is beneficial to material selection and processing optimization for automobile exterior panels.     

Discrete element methods for the plastic analysis of structures subjected to cyclic loading

Methods for the analysis of complex, highly redundant structures subjected to intermittent loads causing biaxial membrane stress and stress reversal into the plastic range are presented. The Bauschinger effect in multi-axial stress is taken into account by the use of Ziegler's modification of Pragers kinematic hardening theory. The implementation of this plasticity theory in the discrete element methods involves the application of the loading in small increments. A linear relationship between increments of plastic strain and of stress, arising out of the theory, is used in conjunction with a linear matrix equation that governs the elastic behaviour of the structure. In the latter equation, plastic strains are interpreted as initial strains. A solution to the linear matrix equation, expressed in terms either of stress or of total strain, may be obtained by utilizing one of two alternative procedures. The methods are capable of treating materials which exhibit elastic–plastic behaviour involving ideal plasticity, linear or non-linear strain hardening, or limited strain hardening. Application is made to several representative structures. Comparison of some of the results with existing test data for both monotonic and reversed loading shows good correlation.     

弹塑性随机有限元在低周疲劳分析中的应用

推导了交变载荷下弹塑性随机有限元的迭代格式,计算了局部多轴应力应变的随机响应.迭代格式中,针对复杂的交变载荷,采用运动强化模型反映塑性变形引起的各向异性和包辛格效应,运用Jhansale模型描述材料的瞬态应力应变关系.弹塑性有限元分析,克服了以往近似方法只能计算单轴局部应力应变响应的缺陷,为多轴疲劳分析奠定了基础.考虑零构件的随机因素,将随机有限元方法引入到交变载荷下弹塑性有限元的迭代格式中,得到局部应力应变的随机响应,为低周疲劳可靠性分析提供了更精确的依据.Mont Carlo模拟结果证实了提出的弹塑性随机有限元方法是可靠的.     

Large-strain Bauschinger effect in austenitic stainless steel sheet

Large-strain Bauschinger effect in cold-rolled austenitic stainless steel sheet is investigated after large amounts of prestrain. The material is prestrained in uniaxial tension, and the tensile properties of the prestrained material are measured in different angles with respect to the prestraining direction. By comparing the differences in the yield stresses in different orientations, the effect of prestraining on material anisotropy is studied. The method is applied to AISI 304-type stainless steel sheet. The test results are analyzed using a combined isotropic–kinematic hardening model. The results indicate that this kind of material shows a considerable Bauschinger effect. Transient and permanent softening is observed in the experiments. The experimental Young's modulus also seems to decrease with prestrain.     

Influence of Bauschinger effect on springback and residual stresses in plane strain pure bending

A finite deformation elastic plastic analysis of plane strain pure bending of a wide sheet is presented. The general closed-form solution is proposed for a kinematic hardening law assuming that the material is incompressible. The stage of unloading is included in the analysis to investigate the influence of the Bauschinger effect, elastic properties of the material and process parameters on the distribution of residual stresses and springback. A simple example is provided to illustrate the procedure for finding the solution and some quantitative features of the process.     

Effect of reverse and cyclic shear on the work-hardening of AISI 430 stainless steel

Sheet metal forming commonly involves various processing steps leading to complex strain paths. The work hardening of the metal under these circumstances is different from that observed for monotonic straining. The effect of the strain path on the hardening of materials is usually studied through sequences of standard mechanical tests, and the shear test is especially well adapted to such studies in sheet forming. Shear straining covering Bauschinger and cyclic strain paths were used in the analysis of the hardening of AISI 430 stainless steel sheets. The tests were conducted at 0°RD, 45°RD, and 90°RD (Rolling Direction) and for three effective strain amplitudes. The results indicate that the material presents Bauschinger effects and strain hardening transients that are sensitive to the testing direction. In addition, the cyclic straining leads to an oscillating stress pattern for the forward and reverse shearing cycles, which depends on the deformation amplitude.     

Modeling of complex cyclic inelasticity in heterogeneous polycrystalline microstructure

Based on the crystallographic slip theory, a micromechanical model for polycrystal elasto–inelasticity is proposed and applied to describe the mechanical behavior of a such structure under monotonic and cyclic loading paths. Small strain theory and isotropic elasticity are assumed. Biaxial cyclic loading is of particular interest in proposed work. The kinematic hardening of the polycrystal is naturally obtained from the averaging scheme. In this scheme, we use a self-consistent interaction law. The proposed modeling effort is tested for a FCC metal under different loading situations in order to elucidate its capability. The obtained results show that this model reproduces appropriately the principle cyclic features such as Bauschinger effect, strain memory effect, ratcheting and additional hardening and other effects.     

Bauschinger effect of alloys and plasticity-induced crack closure: a finite element analysis

The effect of overloads, underloads and stress ratio on plasticity-induced crack opening level is examined for different 'model' materials. This study is focused on the consequences of the Bauschinger effect on the crack opening level. Various finite element analyses were conducted using ABAQUS to test these effects, involving the Chaboche constitutive equations that take into account both the Bauschinger effect of the material and its cyclic hardening or softening. The cyclic plastic behaviour of the material is found to strongly affect the crack behaviour after an overload or an underload. The experimental data obtained on a 0.4% carbon mild steel confirm the numerical results.     

Verification of numerical determination of carrying capacity of large rolling bearings with hardened raceway

For the purpose of the numerical analysis of the actual carrying capacity of rolling contacts in large rolling bearings with surface-hardened raceways, an elasto-plastic constitutive model was used which links the mechanics of material damage with the isotropic and kinematic hardening or softening.A damage material model, implemented into a commercial finite element program, allows us to monitor the elastic strain, plastic strain increase, stress changes and material damage growth, which are closely related to the number of load cycles. In this way, the location and the time of occurrence of bearing raceway damage can be determined along with the growth of damage up to the point when a microcrack is formed. In other words, low cycle life of rotational rolling connection can be assessed.The paper presents the material model, numerical analysis of the actual carrying capacity of the rolling contact in single-row ball bearings and the verification of the numerical material model with experimental results of low cycle carrying capacity.     

Constitutive model and finite element formulation for large strain elasto-plastic analysis of shells

This contribution presents a refined constitutive and finite element formulation for arbitrary shell structures undergoing large elasto-plastic deformations. An elasto-plastic material model is developed by using the multiplicative decomposition of the deformation gradient and by considering isotropic as well as kinematic hardening phenomena in general form. A plastic anisotropy induced by kinematic hardening is taken into account by modifying the flow direction. The elastic part of deformations is considered by the neo-Hookean type of a material model able to deal with large strains. For an accurate prediction of complex through-thickness stress distributions a multi-layer shell kinematics is used built on the basis of a six-parametric shell theory capable to deal with large strains as well as finite rotations. To avoid membrane locking in bending dominated cases as well as volume locking caused by material incompressibility in the full plastic range the displacement based finite element formulation is improved by means of the enhanced assumed strain concept. The capability of the algorithms proposed is demonstrated by various numerical examples involving large elasto-plastic strains, finite rotations and complex through-thickness stress distributions.     

Twist springback characteristics of dual-phase steel sheet after non-axisymmetric deep drawing

This paper aims to investigate the twist springback characteristics of advanced high strength steel sheet subjected to deep drawing. A C-rail benchmark, which leads to a particularly pronounced twist springback characteristics, was developed. For an accurate numerical modeling of the process, a non-quadratic anisotropic yield criterion integrated with combined isotropic and kinematic hardening model was used to describe the strain-stress behavior including anisotropy and Bauschinger effects. The corresponding mechanical experiments, namely uniaxial tension and forward-reverse simple shear tests were performed to determine the material parameters. The digital image correlation technique was applied for component tests as well as the deformation and stress-strain analysis. The experimental validation of the elastic-plastic finite element model was assessed by comparing maximum in-plane strain, thickness reduction distribution and twist springback of the drawn rail. To explore the source of twist springback, the deformation associated with in-plane stress and bending moment was analyzed. The results indicate that the bending moment before springback caused by non-symmetric stress states play an important role in twist springback and control. Certain regions of the die radius were varied in a numerical analysis to control the bending moment for the minimization of twist springback as well as the preliminary results of the relationship between the ratio of variable die radius and twist springback.     

有限变形下的后继屈服面演化规律研究

有限变形下的后继屈服面会出现膨胀或收缩,移动和畸变等变形特征。基于塑性变形的滑移机制,建立了适用于有限变形条件下的多晶塑性本构模型。提出了一种混合硬化假设,可以一致描述后继屈服面演化中的等向、运动和畸变硬化以及正或负交叉效应、包氏效应等。预测了2种加工硬化铝合金（A16061-T6511和annexed1100A1）分...     

A note on the combined isotropic–kinematic workhardening rule

The Ziegler modification of the Prager hardening rule has been used in conjunction with a combined isotropic–kinematic workhardening rule in order to predict the Bauschinger effect during cyclic loadings. There has been some controversy as to whether the above workhardening rule predicts a stress increment which is consistent with the yield condition. The purpose of this communication is to establish the combined hardening rule as a consistent statement in order to clarify its correctness.     

Key issues in cyclic plastic deformation: Experimentation

Cyclic plastic deformation phenomena include the Bauschinger effect, cyclic hardening/softening, strain range effect, loading history memory, ratcheting, mean stress dependent hardening, mean stress relaxation and non-proportional hardening. In this work, different cyclic plastic deformation responses of piping materials (SA333 C-Mn steel and 304LN stainless steel) are experimentally explored. Cyclic hardening/softening is depends upon loading types (i.e. stress/strain controlled), previous loading history and strain/stress range. Pre-straining followed by LCF and mean stress relaxation shows similar kind of material response. Substantial amount of non proportional hardening is observed in SA333 C-Mn steel during 90° out of phase tension-torsion loading. During ratcheting, large amount of permanent strain is accumulated with progression of cycles. Permanent strain accumulation in a particular direction causes cross-sectional area reduction and which results uncontrollable alteration of true stress in engineering stress controlled ratcheting test. In this work, true stress control ratcheting on piping materials has been carried out in laboratory environment. Effects of stress amplitude and mean stress on the ratcheting behaviors are analyzed. A comparison has also been drawn in between the true and engineering stress controlled tests, and massive difference in ratcheting life and strain accumulation is found.     

The influence of coldforming on the low cycle fatigue behaviour of the finegrained structural steel Fe E 47 and the age-hardened aluminium alloy AlCuMg2

Fatigue tests were carried out under strain and load control on the cyclically softening structural steel Fe E 47 and on the cyclically hardening aluminium alloy AlCuMg2. The crack initiation period of both materials decreased with increasing degree of coldforming. This reduction in fatigue life is explained by the Bauschinger effect. To describe analytically the life reduction, the Bauschinger effect was introduced into the fatigue life calculation by considering the monotonic stress/strain curves in tension and compression for different degrees of coldforming and by introducing a damage parameter.     

Constitutive equations of single crystals and polycrystalline aggregates under cyclic loading

It is found that the existing dislocation hardening mechanisms can be generally classified into two categories: isotropic and kinematic. This classification leads to the concept of “degree of isotropy” in work hardening. Based on this concept, the cyclic stress-strain relations of single crystals and metals are constructed throughout the entire loading history. The connections between the mechanical behaviour of dislocations and a single crystal, and between those of crystal grains and a polycrystal are explored under monotonic and cyclic loading conditions. These explorations show that the Bauschinger effect of a single crystal is caused by the kinematic hardening mechanisms, and that of a polycrystal is caused by both the kinematic mechanisms and the residual stress in the most favorably-oriented slip system. The cyclic hardening and softening behavior of metals are shown to be intimately related to those of its constituent single crystals. The theory developed is applied to describe the cyclic stress-strain relations of a copper crystal, and to predict the cyclic response of a 2024-T351 aluminum alloy; in both cases the agreement with experiments is considered good.