A simple integration algorithm for a non‐associated anisotropic plasticity model for sheet metal forming

In this paper, an anisotropic material model based on a non‐associated flow rule and nonlinear mixed isotropic‐kinematic hardening is developed. The quadratic Hill48 yield criterion is considered in the non‐associated model for both yield function and plastic potential to account for anisotropic behavior. The developed model is integrated based on fully implicit backward Euler's method. The resulting problem is reduced to only two simple scalar equations. The consistent local tangent modulus is obtained by exact linearization of the algorithm. All numerical development was implemented into user‐defined material subroutine for the commercial finite element code ABAQUS/Standard. The performance of the present algorithm is demonstrated by numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.     

Simulations of stress distributions in crankshaft sections under fillet rolling and bending fatigue tests

The residual stresses due to fillet rolling and the bending stresses near the fillets of crankshaft sections under bending fatigue tests are important driving forces to determine the bending fatigue limits of crankshafts. In this paper, the residual stresses and the bending stresses near the fillet of a crankshaft section under fillet rolling and subsequent bending fatigue tests are investigated by a two-dimensional plane strain finite element analysis based on the anisotropic hardening rule of Choi and Pan [Choi KS, Pan J. A generalized anisotropic hardening rule based on the Mroz multi-yield-surface model for pressure insensitive and sensitive materials (in preparation)]. The evolution equation for the active yield surface during the unloading/reloading process is first presented based on the anisotropic hardening rule of Choi and Pan (in preparation) and the Mises yield function. The tangent modulus procedure of Peirce et al. [Peirce D, Shih CF, Needleman A. A tangent modulus method for rate dependent solids. Comput Struct 1984;18:875–87] for rate-sensitive materials is adopted to derive the constitutive relation. A user material subroutine based on the anisotropic hardening rule and the constitutive relation was written and implemented into ABAQUS. Computations were first conducted for a simple plane strain finite element model under uniaxial monotonic and cyclic loading conditions based on the anisotropic hardening rule, the isotropic and nonlinear kinematic hardening rules of ABAQUS. The results indicate that the plastic response of the material follows the intended input stress–strain data for the anisotropic hardening rule whereas the plastic response depends upon the input strain ranges of the stress–strain data for the nonlinear kinematic hardening rule. Then, a two-dimensional plane-strain finite element analysis of a crankshaft section under fillet rolling and subsequent bending was conducted based on the anisotropic hardening rule of Choi and Pan (in preparation) and the nonlinear kinematic hardening rule of ABAQUS. In general, the trends of the stress distributions based on the two hardening rules are quite similar after the release of roller and under bending. However, the compressive hoop stress based on the anisotropic hardening rule is larger than that based on the nonlinear kinematic hardening rule within the depth of 2 mm from the fillet surface under bending with consideration of the residual stresses of fillet rolling. The critical locations for fatigue crack initiation according to the stress distributions based on the anisotropic hardening rule appear to agree with the experimental observations in bending fatigue tests of crankshaft sections.     

Modelling direction-dependent hardening in magnesium sheet forming simulations

In this work, a model capturing anisotropic hardening during plastic deformation under monotonic loading is proposed. For this purpose, the anisotropic plastic potential coefficients are assumed to be functions of a measure of the accumulated plastic strain. This model is applied to describe the plastic behavior of a magnesium alloy (ZM21) sheet at room temperature. The selected plastic potential accounts for the main features of Mg alloy plasticity, i.e., anisotropy and strength-differential (SD) effects. All the accumulated plastic strain dependent coefficients of the phenomenological model are determined from input data generated with a crystal plasticity approach. They are optimized to best capture the accumulated strain dependent potentials computed with crystal plasticity. The R-value (Lankford coefficient) anisotropy is used as an independent measure for the assessment of the approximation quality. This model is implemented into a finite element (FE) code and successfully validated through the numerical simulations of the cup drawing test. The calculated earing profile obtained with the proposed hardening model is compared to results assuming isotropic hardening for various plausible shapes of the plastic potential. Although the ear and valley numbers and positions are similar in all cases, the height differences between peaks and valleys are strongly dependent on the type of constitutive approach used in the simulation.     

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.     

Extension of strain–life equation for low‐cycle fatigue of sheet metals using anisotropic yield criteria and distortional hardening model

The fatigue strain–life equation is in general applicable to isotropic materials. It was recently attempted to account for material anisotropy because of crystallographic texture in fatigue modelling. The proposed modification was limited to isotropic hardening. The present work is an extension of the previous work, wherein a general framework to model anisotropy using phenomenological yield criterion and anisotropic hardening is provided. Yld2004‐18p yield criterion and the so‐called homogenous anisotropic hardening model are used to demonstrate the anisotropic cyclic behaviour of low carbon steel. The proposed methodology can be utilized in applications including multiaxial fatigue modelling.     

A finite element analysis of plane strain dynamic crack growth in materials displaying the Bauschinger effect

In this paper dynamic crack growth in an elastic-plastic material is analyzed under mode I plane strain small-scale yielding conditions using a finite element procedure. The main objective of this paper is to investigate the influence of anisotropic strain hardening on the material resistance to rapid crack growth. To this end, materials that obey an incremental plasticity theory with linear isotropic or kinematic hardening are considered. A detailed study of the near-tip stress and deformation fields is conducted for various crack speeds. The results demonstrate that kinematic hardening does not oppose the role of inertia in decreasing the plastic strains and stresses near the crack tip with increase in crack speed to the same extent as isotropic strain hardening. A ductile crack growth criterion based on the attainment of a critical crack opening displacement at a small micro-structural distance behind the tip is used to obtain the dependence of the theoretical dynamic fracture toughness with crack speed. It is found that for any given level of strain hardening, the dynamic fracture toughness displays a much more steep increase with crack speed over the quasi-static toughness for the kinematic hardening material as compared to the isotropic hardening case.     

Investigation of anisotropy problems in sheet metal forming using finite element method

This paper discusses the results of the numerical study of rectangular cup drawing of steel sheets using finite element methods. To be able to verify the results of the numerical solutions, an experimental study was done where the material behavior under deformation was analyzed. A 3D parametric finite element (FE) model was built using the commercial FE-package ABAQUS/Standard. ABAQUS allows analyzing physical models of real processes putting special emphasis on geometrical non-linearities caused by large deformations, material non-linearities and complex friction conditions. Friction properties of the deep drawing quality steel sheet were determined by using the pin-on-disc tribometer. The results show that the friction coefficient depends on the measured angle from the rolling direction and corresponds to the surface topography. A quadratic Hill anisotropic yield criterion was compared with von Mises yield criterion having isotropic hardening. The sensitivity of constitutive laws to the initial data characterizing material behavior is also presented. It is found out that plastic anisotropy of the matrix in ductile sheet metal has influence on deformation behavior of the material. When the material and friction anisotropy are taken into account in the finite element analysis, this approach gives better approximate numerical results for real processes.     

A generally anisotropic hardening model for big offset-strain yield stresses

Summary In this paper, generally anisotropic quadric models for big offset-strain plastic yield and hardening of rolled sheet-steels are proposed. They are formulated in intrinsic forms such that the effects of isotropic and anisotropic yield coefficients to the hardening factors are naturally decomposed. Such decompositions are supported by the experimental observations and allow further simplifications. The simplified models are capable of describing changeable general anisotropy of hardening, and give predictions in excellent agreement with the experimental data. Two restricted models as special cases of the general model are also examined.     

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...     

基于不同强化模型的GTN损伤模型及其在板料回弹有限元分析中的应用

为了进一步提高板料成形中的回弹预测精度,分别建立了基于Ziegler 线性随动强化模型、Lemaitre-Chaboche 非线性随动强化以及非线性混合强化模型的Gurson-Tvergaard-Needleman(GTN)细观损伤本构模型,并给出有限元数值积分方法。通过用户自定义材料子程序VUMAT 将损伤模型嵌入到有限元软件ABAQUS 中,以NUMISHEET’93 板料U 型弯曲考题为例,应用显隐相结合的方法模拟分析了不同材料强化模型和损伤对板料回弹量的影响。结果表明:在相同GTN 损伤模型情况下,线性随动和非线性随动强化模型预测得到的板料回弹量较小,等向强化预测的板料回弹量偏大,非线性混合强化预测的板料回弹量介于它们之间。材料模型在考虑损伤因素后,预测的回弹严重程度比无损伤情况时略小,与实验值更相近。     

Numerical determination of yield surfaces of Al/SiC laminated composite plates

In this study the elastic-plastic properties of an aluminum alloy reinforced by unidirectional SiC-fibers are computed by using nonlinear finite element simulations. In order to carry out these computations, a representative volume element assuming a regular quadratic arrangement of the fibers is used. The plastic properties of the composite are described by a quadratic flow criterion which is able to consider both direction and tension-compression anisotropy. Additionally, the flow rule takes into account plastic volume changes. The hardening of the material is defined by a mixed isotropic-kinematic hardening model. By using this modeling yield surfaces of multi-layer laminated composite plates under an in-plane loading are determined.     

Numerical investigation of crack tip strain localization under cyclic loading in FCC single crystals

In this work, the crack tip strain localization in a face centered cubic single crystal subject to both monotonic and cyclic loading was investigated. The effect of constraint was implemented using T-stress and strain accumulation was studied for both isotropic and anisotropic elastic cases with the appropriate application of remote displacement fields in plane strain. Modified boundary layer simulations were performed using the crystal plasticity finite element framework. The consideration of elastic anisotropy amplified the effect of constraint level on stress and plastic strain fields near the crack tip indicating the importance of its use in fracture simulations. In addition, to understand the cyclic stress and strain behavior in the vicinity of the crack tip, combined isotropic and kinematic hardening laws were incorporated, and their effect on the evolution of yield curves and plastic strain accumulation were investigated. With zero-tension cyclic load, the evolution of plastic strain and Kirchhoff stress components showed differences in magnitudes between isotropic and anisotropic elastic cases. Furthermore, under cyclic loading, ratcheting was observed along the localized slip bands, which was shown to be affected by T-stress as well as elastic anisotropy. Negative T-stress increased the accumulation of plastic strain with number of cycles, which was further amplified in the case of elastic anisotropy. Finally, in all the cyclic loading simulations, the plastic strain accumulation was higher near the \(55^0 \) slip band.     

Influence of yield surface curvature on flow localization in dilatant plasticity

In this paper a family of dilatant plasticity theories is introduced by considering yield surfaces which change according to a combination of isotropic expansion and kinematic translation. One limiting member of the family is Gurson's (1977) isotropic hardening model, and the other limiting member is a pure kinematic hardening version. The family of constitutive laws is constructed such that all versions coincide for proportional stressing histories. The differences between any two versions show up only under nonproportional stressing histories, such as those encountered in many plastic instability phenomena. Under nonproportional stressing, the kinematic version is significantly “less stiff” than Gurson's isotropic hardening model due to the relatively higher curvature of the kinematic yield surface. This effect is explored in some basic shear localization calculations and is found to have substantial influence on the localization predictions.     

Prediction of ductile fracture using isotropic and kinematic hardening rules including void nucleation

A formulation combining kinematic translation and isotropic expansion for a yield surface based on Gurson–Tvergaard function is used to describe void growth. By adopting a criterion of internal necking of the ligaments between voids, fracture strains for tensile bars made of conventional alloys and powder metal compacts—with micromechanical parameters mostly identified from experiments—are predicted according to kinematic, isotropic and mixed hardening models. Fracture strains predicted by the kinematic‐hardening model are in closer agreement with experiments whereas those estimated according to isotropic‐hardening model are overestimated. The consideration of either step‐like or continuous void nucleation models indicates its great influence on fracture strains and emphasizes a further need to quantify the statistical parameters involved in these models.     

A robust kinematic hardening rule for cyclic plasticity with ratchetting effects Part II. Application to nonproportional loading cases

Summary Performance of the proposed kinematic hardening rule is examined using several examples of cyclic plasticity phenomena observed in experiments. Results obtained and compared with experimental observations on various loading histories are presented. With the memory effects added to the model, impressive results are obtained without using an anisotropic yield model. Drifting of the yield surface occurs during the numerical computation of the plastic response due to nonproportional loading paths. The drift due to the finite increments of stress or strain is corrected using a simple and efficient method proposed in this paper. The new kinematic hardening rule proposed for the limit surface as being related directly to the yield surface kinematic hardening rule ensures nesting using the blended rule discussed in the part presenting the theoretical formulation [14].     

解析Poly6-I屈服准则对3104-H19铝合金拉深制耳预测的研究

目的 研究解析Poly6-I屈服准则预测具有高各向异性的3104-H19铝合金本构关系的能力,并将其应用于有限元仿真分析中,以实现对3104-H19铝合金拉深制耳的精确预测。方法 分析解析Poly6-I屈服准则的表达形式,减少计算参数所需的试验个数,并与经典的Yld2004-18p屈服准则进行对比,验证它对高各向异性力学性能预测的能力,将其嵌入到有限元软件中进行杯型件拉深制耳模拟,验证模型的精确性和有效性。结果 对于高各向异性材料,解析Poly6-I屈服准则所使用的试验个数可以减少到11,它预测的3104-H19铝合金屈服轨迹的各向异性系数曲线和单向拉伸曲线与Yld2004-18p屈服准则预测的结果基本相同,杯型件拉深有限元模拟结果与试验结果基本一致。结论 与Yld2004-18p屈服准则相比,考虑高各向异性特性的解析Poly6-I屈服准则所使用的试验数据更少,且无须使用优化软件求取参数,更为方便。解析Poly6-I屈服准则能精确地预测3104-H19铝合金材料在杯型件拉深试验中的制耳个数及杯型件杯壁的成形高度。     

A Lagrangian model for hardening behaviour of materials at finite deformation based on the right plastic stretch tensor

In this paper a modified multiplicative decomposition of the right stretch tensor is proposed and used for finite deformation elastoplastic analysis of hardening materials. The total symmetric right stretch tensor is decomposed into a symmetric elastic stretch tensor and a non-symmetric plastic deformation tensor. The plastic deformation tensor is further decomposed into an orthogonal transformation and a symmetric plastic stretch tensor. This plastic stretch tensor and its corresponding Hencky’s plastic strain measure are then used for the evolution of the plastic internal variables. Furthermore, a new evolution equation for the back stress tensor is introduced based on the Hencky plastic strain. The proposed constitutive model is integrated on the Lagrangian axis of the plastic stretch tensor and does not make reference to any objective rate of stress. The classic problem of simple shear is solved using the proposed model. Results obtained for the problem of simple shear are identical to those of the self-consistent Eulerian rate model based on the logarithmic rate of stress. Furthermore, extension of the proposed model to the mixed nonlinear isotropic/kinematic hardening behaviour is presented. The model is used to predict the nonlinear hardening behaviour of SUS 304 stainless steel under fixed end finite torsional loading. Results obtained are in good agreement with the available experimental results reported for this material under fixed end finite torsional loading.     

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.