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.     

混凝土结构I型裂纹裂尖塑性区研究

应用双剪统一强度理论,研究了I型裂纹的塑性变形问题。给出了包含反映材料拉压性能差异的参数拉压比及反映中间主应力效应的参数b的I型裂纹裂尖塑性区形状和大小的统一解。已有的Tresca准则、Mises准则和Mohr-Coulomb准则解均是本文的特例或线性逼近。针对混凝土结构,画出了不同参数b情况下的裂尖塑性区半径变化图。得出了材料拉压比对I型裂纹裂尖塑性区影响很大。b对I型裂纹裂尖塑性区影响随拉压比的不同而不同,拉压比较大时,b对塑性区影响大,拉压比较小时,b对塑性区影响小的结论。该统一解可以适应于各种不同材料,能充分发挥材料潜力,具有普遍性和广泛的适应性,有一定的工程应用价值。结论对于研究各种材料的断裂问题有参考作用。     

Influences of grain size and precracking load on the critical stress intensity factor of mild steel

The effects of grain size and precracking load on the critical stress intensity factor are studied. A plane stress model of elastic-plastic stress distribution which includes the strain hardening effects is used. The effects of residual stresses and strain hardening due to fatigue load are calculated by choosing plastic zone size as fracture criterion. Experimental results are obtained to demonstrate the reliability of theoretical calculations.     

双层组合厚壁圆筒弹脆塑性极限内压统一解

基于三剪统一强度准则和弹脆塑性软化模型,考虑材料的脆性软化和中间主应力效应,推导了双层组合厚壁圆筒弹脆塑性极限内压统一解,探讨了粘聚力、内摩擦角、半径比、强度理论参数和中间主应力系数的影响特性,克服了以往基于Tresca屈服准则、Mises屈服准则或双剪强度理论的理想弹塑性解的不足。研究结果表明:中间主应力、材料模型和脆性软化对厚壁圆筒的极限内压均有显著影响。该文所得统一解具有广泛的适用性和理论意义,不但可退化为现有公式,而且还能得到系列化的新解答,对组合厚壁圆筒的设计及工程应用有重要参考价值。     

GM屈服准则求解Ⅰ型裂尖塑性区

用几何中线(GM)屈服准则求解了Ⅰ型裂尖塑性区的形状与尺寸,对比了基于Mises和Tresca准则的求解结果。表明在平面应变条件下,GM准则求解的塑性区面积在Tresca和Mises结果之间,Tresca塑性区面积最大,Mises面积最小,GM塑性区与Mises塑性区非常接近,三者的塑性区均成哑铃状。在平面应力下,GM和Mises塑性区二者仍最接近并为豆芽状,Tresca的塑性区最大。无论平面应力还是平面应变,GM准则计算结果与Mises结果均有最佳接近度。     

Approximate yield criteria for anisotropic porous ductile sheet metals

An approximate macroscopic yield criterion for anisotropic porous sheet metals is developed under plane stress conditions in this paper. The metal matrices are assumed to be rigid perfectly plastic and incompressible. The Hill quadratic and non-quadratic anisotropic yield criteria are used to describe the matrix normal anisotropy and planar isotropy. The voids in sheet metals are assumed in the form of through-thickness holes. Under axisymmetric loading, a closed-form upper bound macroscopic yield criterion is derived as a function of the anisotropy parameter R, defined as the ratio of the transverse plastic strain rate to the through-thickness plastic strain rate under in-plane uniaxial loading conditions. The plane stress upper bound solutions for different in-plane strain ratios can be fitted well by the closed-form macroscopic yield criterion.     

Experimental and theoretical formability analysis using strain and stress based forming limit diagram for advanced high strength steels

In this study, experimental and numerical analyses of Forming Limit Diagram (FLD) and Forming Limit Stress Diagram (FLSD) for two Advanced High Strength Steel (AHSS) sheets grade DP780 and TRIP780 were performed. Initially, the forming limit curves were experimentally determined by means of the Nakazima forming test. Subsequently, analytical calculations of both FLD and FLSD were carried out based on the Marciniak–Kuczinsky (M–K) model. Additionally, the FLSDs were calculated using the experimental FLD data for both investigated steels. Different yield criteria, namely, von Mises, Hill’s 48, and Barlat2000 (Yld2000-2d) were applied for describing plastic flow behavior of the AHS steels. Both Swift and modified Voce strain hardening laws were taken into account. Hereby, influences of the constitutive yield models on the numerically determined FLDs and FLSDs were studied regarding to those resulted from the experimental data. The obtained stress based forming limits were significantly affected by the yield criterion and hardening model. It was found that the forming limit curves calculated by the combination of the Yld2000-2d yield criterion and Swift hardening law were in better agreement with the experimental curves. Finally, hole expansion tests were conducted in order to verify the different failure criteria. It was shown that the stress based forming limit curves could more precisely describe the formability behavior of both high strength steel sheets than the strain based forming limit curves.     

Mechanical and fracture behaviour of porous materials

The elastic moduli of porous materials represented as a combination of spherical, cylindrical or disk shaped holes or solid elements was calculated using a self consistent method. A yield criterion could be found by stating that the elastic distortion energy evaluated with these moduli was equal to a critical value. Another yield criterion could be approximated from the results of finite elements computations using the homogenization technique. These criteria possess the same homogeneity as the yield criterion of the dense material. In the presence of a secondary population of smaller cavities the yield criterion was found to be different from the one obtained by simply adding the two volume fractions of holes. The hole growth rate deduced from these calculations was proportional to the equivalent strain rate and to the stress triaxiality ratio, inkeeping with the evolution of damage proposed by Lemaitre. This hole growth rate was enhanced by a population of small secondary cavities. In general the strain hardening and the damage are coupled. However for sintered porous nickel the introduction of an initial damage parameter proportional to the porosity sufficed to describe the strain hardening behaviour. The condition for hole coalescence could not be connected to either a critical value of the porosity, or of the damage parameter, or a necking condition. The yield criterion modified by replacing the yield stress by the fracture stress could describe experimental results. Yet a critical local strain criterion also gave a good fit. The fracture toughness of syntactic foams could be explained by the local stress distribution in the glass spheres. In the case of ductile porous nickel the COD at initiation decreased as the fracture strain. This material exhibited a large tearing modulus whose decrease when the porosity was increased could be taken into account by the damage parameter. Fatigue crack propagation rates could also be rationalized with the use of the damage parameter and by reducing the surface of the material to be fractured.     

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.     

Plastic yielding at the tip of a blunt notch during static and fatigue loading

Rice's analytical Mode III solution for the relationship between anti-plane stress and anti-plane strain was used to determine the small scale plastic yielding at the tip of a two-dimensional blunt notch. The results were applied to fatigue loading. The plastic zone size and crack opening displacement derived in the present analysis were determined as functions of applied stress, geometric factors (notch radius and length) and material properties (yield stress and the work hardening rate). The minimum stress intensity required for plastic yielding at a blunt notch tip was postulated to be the experimentally observed threshold stress intensity for fatigue crack initiation. The threshold stress intensity so determined depends not only on the notch geometry but also on material properties. There is good agreement with calculated and measured values of the threshold stress intensity for fatigue crack initiation.     

Dynamic plastic response of a circular plate based on unified strength theory

The study and design of structures under dynamic loads require a knowledge of the plastic response and deformation behavior under impact loading. The calculations of dynamic plastic response of structures are useful for the design and investigation of colliding vehicle, engine and various impacting structures. The unified solutions of dynamic plastic load-carrying capacities, moment fields and velocity fields of a simply supported circular plate are introduced. The strength is different in tension and compression and the effect on the yield criteria is taken into account by using the unified strength theory. Upper bound and lower bound plastic responses of the plate, under moderate partial uniformly distributed impulsive loading, are obtained. The static and kinematical admissibility of the dynamic plastic solutions are discussed. The unified solutions of the static plastic load-carrying capacities, moment fields and velocity fields of a simply supported circular plate are also obtained according to the dynamic solutions in this paper. The solutions are suitable for many materials with or without different strengths in tension and compression and the effect of intermediate principal stress.The solutions based on the Tresca, the von Mises, the Mohr–Coulomb theory, and the twin-shear strength theory, as well as the unified yield criterion, are all the special cases of the unified solutions. The influences of the coefficient of failure criteria, b, and tension-compression strength ratio, α, on the dynamic and static solutions, are investigated. It is shown that the effects of different strengths in tension and compression and yield criteria on the dynamic load-carrying capacity are greater than in the static plastic limit state.     

Elastic–plastic void expansion in near-self-similar shapes

For void growth in an elastic–plastic strain hardening material the preferred shape of the void is calculated, dependent on the macroscopic stress state. Axisymmetric cell model analyses are carried out with a very small initial void size relative to the cell dimensions. Large deformations of the material around the void are modeled until the void volume is four orders of magnitude larger than the initial volume. An iterative procedure is used until the final void shape and the initial void shape are identical. Even when this convergence has been obtained, the void shape does not stay constant during the growth. Thus, the shapes found give only approximately self-similar growth. The results are compared with self-similar shapes determined previously for nonlinear viscous solids, subject to power law creep. For the time independent elastic–plastic material considered here the effect of the strain hardening level and of the initial yield strain are studied.     

Combined experimental and theoretical method of determining the state of stress and strain near holes

The article suggests a new combined experimental and theoretical method of determining the state of stress and strain next to a cylindrical hole in a plate subjected to uniaxial plane loading in the elastic strain range. The essence of the method consists in the following. By the interferogram of the local component of the normal displacement component of the surface of the plate in the zone of the hole the radius of the cylindrical surface with zero change of the mentioned displacement component, coaxial with the hole, is measured. This radius is used in the theoretical calculation as criterion for establishing the final dimensions of the deformed annular region around the hole, and consequently all the real parameters of the state of stress and strain of the plate in the zone of the hole. The effect of the loading asymmetry in the testing machine on the kind of interferogram under consideration is also taken into account. It is shown that there is good correspondence between the theoretical patterns of levlel bands of equal displacements of the surface of the plate along the normal and the experimental interferograms. The stress concentration factors for a real specimen are determined.Translated from Problemy Prochnosti, No. 8, pp. 46–50, August, 1990.     

Interpreting strain bursts and size effects in micropillars using gradient plasticity

Size effects and strain bursts that are observed in compression experiments of single crystalline micropillars are interpreted using a gradient plasticity model that can capture the process of sequential slip and heterogeneous yielding of thin material layers. According to in situ experiments during compression sub-grains and significant strain gradients develop, while deformation occurs through slip layers in the gauge region. In the multilayer strain gradient model, the higher order stress is discontinuous across the interface between a plastic layer and an elastic layer, but it becomes continuous across the interface between two plastic layers. Strain bursts occur when two neighboring layers yield. Based on this hypothesis the experimental stress-strain curves with strain bursts observed in micropillars can be fitted by properly selecting the number of layers that yield and the ratio of the internal length over the specimen size; the modulus and the yield stress are obtained from the experimental curves while the hardening modulus evolves during deformation based on the dislocation mechanisms.     

均布载荷作用下简支圆板塑性极限载荷的解析解

该文建立了受均布载荷作用简支圆板运动许可应变场,并首次以EA(等面积)屈服准则进行了塑性极限分析,获得了极限载荷的解析解。该解为圆板半径a、圆板厚度h以及屈服强度的函数。与Tresca、TSS以及Mises解比较表明,Tresca屈服准则预测极限载荷的下限,TSS屈服准则预测极限载荷的上限,EA和Mises屈服准则预测的极限载荷恰居二者中间,且EA解几乎与Mises解重合。此外,该文还讨论了挠度与相对位置r/a之间的变化关系。     

The study of grain boundary density effect on multi-grain thin film under tension

The grain-size effect on the yield strength and strain hardening of thin film at sub-micron and nanometer scale closely relates to the interactions between grain boundary and dislocation. Based on higher-order gradient plasticity theory, we have systematically investigated the size effect of multi-grain thin film arising from the grain boundary density under tensile stress. The developed formulations employing dislocation density and slip resistance have been implemented into the finite element program, in which grain boundary is treated as impenetrable interface for dislocations. The numerical simulation results reasonably show that plastic hardening rate and yield strength are linear to the grain boundary density of multi-grain thin film. The aspect ratio of grain size and orientation of slip system have distinct influence on the grain plastic properties. The research of slip system including homogeneous and nonhomogeneous distribution patterns reveals that the hardening effect of low-angle slip system is greater than that of high-angle slip system. The results agree well with the experimentally measured data and the solutions by discrete dislocation dynamics simulation.     

矩形板在双向压力与剪切力复合加载下的塑性屈曲

依据Prandtl-Reuss增量理论和Mises屈服准则,推导出完整的复合加载平板的塑性分支屈曲微分方程。应用特征值分解技术,计算了双向压力与剪力复合加载铰支矩形板在塑性流动区和应变强化区的稳定参数,绘制了相应的塑性分支应力曲线。计算结果和理论分析有较好的相关性。     

The anisotropic R-criterion for crack initiation

The anisotropic nature of mixed modes I-II crack tip plastic core region and crack initiation is investigated in this study using an angled crack plate problem under various loading conditions. Hill’s anisotropic yield criterion along with singular elastic stress field at the crack tip is employed to obtain the non-dimensional variable-radius crack tip plastic core region. In addition, the R-criterion for crack initiation proposed by the authors for isotropic materials is also extended to include anisotropy. The effect of Hill’s anisotropic constants on the shape and size of the crack tip plastic core region and crack initiation angle is presented for both plane stress and plane strain conditions at the crack tip. The study shows a significant effect of anisotropy on the crack tip core region and crack initiation angle and calls for further development of anisotropic crack initiation theory.     

The elastic-plastic shrink fit with supercritical interference

Summary An interference, for which Tresca's yield criterion predicts vanishing of the circumferential stress at the bore of the hub of an elastic-plastic shrink fit, is called critical. It is shown that Tresca's criterion does not admit a solution of the supercritical shrink fit problem for perfectly plastic materials. For hardening materials, stress, displacement and plastic strain in the supercritical shrink fit are derived. The occurrence of a supercritical interference as well as the range of validity of the results presented are discussed.With 8 FiguresDedicated to Professor Dr.-Ing. Theodor Lehmann on the occasion of the 65th anniversary of his birthday.     

Elastic-plastic stress analysis in a long functionally graded solid cylinder with fixed ends subjected to uniform heat generation

Elastic-plastic deformation of a solid cylinder with fixed ends, made of functionally graded material (FGM) with uniform internal heat generation is investigated, based on Tresca’s yield criterion and its associated flow rule, considering four of the material properties to vary radially according to a parabolic form. These four material properties are yield strength, modulus of elasticity, coefficients of thermal conduction and thermal expansion, assumed to be independent of temperature as Poisson’s ratio which is taken as constant. The materials which compose the functionally graded cylinder are supposed to be elastic-perfectly plastic materials. Expressions for the distributions of stress, strain and radial displacement are found analytically in terms of unknown interface radii. After determining these radii numerically by means of Mathematica 5.2, the distributions are plotted versus dimensionless radius, increasing heat generation, to compare the FGM cylinder with the homogeneous one. The numerical values used in this work for material parameters are arbitrarily chosen to point out the effect of the non-homogeneity on the stress distribution. The results obtained show that the stress distribution, as well as the development of plastic region radii, is influenced substantially by the material non-homogeneity.