An anisotropic damage-based plastic yield criterion and its application to analysis of metal forming process

Based on the continuum damage mechanics (CDM), both the total damage growth force and the damage equivalent stress has been firstly derived. Material and damage are considered to be anisotropic in this theoretical model. A new damage-based plastic yield criterion, where the damage equivalent stress is taken as the yield function, and the corresponding damage evolution equation has been established. A relation between the damage variable and the plastic deformation has been proposed. This relation can be applied to obtain the damage history in terms of Moiré photo-mechanics technology in this work. In addition, from the anisotropic damage-based plastic yield criterion presented, the large-scale finite element codes ABAQUS/Explicit in conjunction with the damage experimental results has been then used to calculate some mechanical variable fields in metal forming process. It has finally been verified that the position corresponding to the maximum value in damage equivalent stress field accurately coincides with the initial place of macro-crack given by Erichsen cupping tests carried out by authors.     

Forming limits of AL 6022 sheets with material damage consideration—theory and experimental validation

Characteristics of localized necking in sheet metals are examined with anisotropic plasticity as well as anisotropic damage developed progressively after load application. The vertex theory is employed to describe basic mechanisms of localized necking. Anisotropic plasticity is accounted for by Hill's quadratic yield criterion. An anisotropic damage model based on Continuum Damage Mechanics is reviewed and expanded. The anisotropic damage model is combined with a modified vertex theory to generate damage-coupled localized necking criteria on both sides of forming limit diagram (FLD). The criteria lead to explicit expressions of critical hardening modulus on both sides of FLD. It is shown that the damage-coupled FLD model can be readily reduced and used to predict the forming limit strains of damage-free materials satisfying power hardening law given by other researchers (Hill, J. Mech. Phys. Solids. (1952)1,19; Zhu et al., ASME J. Eng. Mater. Tech. (2001) 123, 329). Critical damage value at localized necking can be computed from the damage-coupled localized necking criteria as a function of stress/strain states and strain paths. Tests on the formability and material properties of Al 6022, such as hardening and damage law, anisotropic plasticity parameter, have been performed. The measured FLD of the material are compared with the predictions based on the damage-coupled localized necking criteria for validation of the proposed FLD model. Material damage is observed to have a definite effect on the forming limits of Al 6022, thus providing a more accurate prediction than that of the conventional models.     

10CrMo910钢的疲劳损伤演变与寿命估算

对在周期加载条件下运行的锅炉、压力容器、压力管道等设备,不可避免地产生低周疲劳的失效。本文从损伤力学基本理论出发,以10CrMo910钢作为研究对象,采用应力幅法测量材料的低周疲劳损伤,提出了低周疲劳各向同性连续损伤模型,在有效应力和应变等价假设基础上建立了低周疲劳损伤演变和寿命估算式。     

An anisotropic creep damage model for multiaxial cyclic loading histories

The purpose of this study is the development of an anisotropic creep damage theory within the continuum damage mechanics, applicable to creep-dominated cyclic loading histories. A damage distribution is expressd in rate form as a symmetric tensor of rank necessary to match physically measured damage. A theoretical model which expresses general anisotropic creep damage phenomena with power law cavity growth is proposed. The coupling of damage with a bounding surface cyclic viscoplasticity theory is also accomplished. Comparison with experimental results are made for weakly anisotropically damaging materials, type 304 stainless steel at 593°C. Good correlation of rupture time, secondary creep, and tertiary creep has been obtained for proportional and nonproportional, isothermal, constant isochronous nominal stress loading histories. A modification of the isochronous stress (the set of stress state which have a same rupture time) for compressive hydrostatic stress state has been offered.     

Constitutive law for AZ31B Mg alloy sheets and finite element simulation for three-point bending

The constitutive law to describe the anisotropic and asymmetric mechanical behavior of AZ31B magnesium (Mg) alloy sheets at room temperature has been developed here for the plane stress condition, based on the orthotropic yield criterion proposed by Cazacu O, Plunkett B, Barlat F. [Orthotropic yield criterion for hexagonal closed packed metals. International Journal of Plasticity 2006;22:1171–94] and different isotropic hardening laws for tension and compression. Experimental procedures to obtain the material parameters of the yield surface and the hardening laws have been discussed for the AZ31B Mg alloy sheet. For verification purposes, finite element simulation results based on the developed constitutive laws have been compared with experimental results for a three-point bending test.     

Characterization and modeling of the mechanical behavior and formability of a 2008-T4 sheet sample

A 2008-T4 sheet sample has been characterized and its mechanical behavior and formability have been modeled. Uniaxial tensile and equal biaxial tensile stress-strain data, compressive yield strengths, crystallographic texture, earing and the forming limit curve were experimentally determined. Bulge test specimen shape and thickness profiles were also measured after various amounts of biaxial strain. A recently developed phenomenological constitutive model of anisotropic mechanical behavior was used to predict the directionality of strength, plastic strain ratio (R) and shear strain ratio (Г) values. In addition, this model was used to predict the forming limit curve for this sample. Predictions made with the recent model generally compare favorably with experimental results and with predictions made using the Taylor/Bishop and Hill theory. According to the data obtained in hydraulic bulge testing, the 2008-T4 exhibited apparent isotropic behavior. However, in cup drawing—another axisymmetric deformation mode—this material exhibited anisotropic behavior, as indicated by the formation of ears and troughs.     

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.     

A theory of anisotropic creep after plastic pre-straining

Plastic pre-strain induces directional material response. It has been observed that uni-directional pre-strain results in anisotropic creep. Creep tests in various directions with respect to the direction of pre-strain have shown different steady creep rates and times to rupture.A theory of anisotropic creep is proposed employing the properties of tensor functions. The creep rate in the steady creep range is expressed as a tensor valued tensor function of the stress and pre-strain. In comparison with the Odqvist theory additional material constants appear in the law proposed. The constants are established and predictions from the theory advanced are compared with the results of experiments concerning two pre-strain magnitudes and seven inclinations of the uniaxial creep tests with respect to the pre-strain direction.     

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 theoretical prediction of the strain path of anisotropic sheet metal deformed under uniaxial and biaxial stress state

A model based on a combination of the micro- and macroscopic theories of plasticity has been built to predict the strain path of a textured sheet metal for a given imposed stress state. By applying the flow rule to a crystallographically based anisotropic continuum yield locus, the deformation strain tensor is determined. For each small increment of deformation, the change in the crystal rotation of each grain is followed and the strain tensor recalculated. The successive changes in the strain state with strain increment give the strain path followed by a material element. Analyses are made for different crystallographic orientations and typical sheet textures of commercially pure aluminium and a Cu-20% Zn alloy deformed in either the uniaxial or equibiaxial stress states. It is found that the simulated strain paths often deviate from those based on isotropic assumptions. The significance of the finding to the study of the formability of sheet metal is discussed.     

Approximate yield criterion for voided anisotropic ductile materials

As most fractures of ductile materials in metal forming processes occurred due to the results of evolution of internal damage — void nucleation, growth and coalescence. In this paper, an approximate yield criterion for voided (porous) anisotropic ductile materials is developed. The proposed approximate yield function is based on Gurson’s yield function in conjunction with the Hosford’s non-quadratic anisotropic yield criterion in order to consider the characteristic of anisotropic properties of matrix material. The associated flow rules are presented and the laws governing void growth with strain are derived. Using the proposed model void growth of an anisotropic sheet under biaxial tensile loading and its effect on sheet metal formability are investigated. The yield surface of voided anisotropic sheet and void growth with strain are predicted and compared with the experimental results.     

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.     

Role of plastic anisotropy and its evolution on springback

Springback angles and anticlastic curvatures reported for a series of draw-bend tests have been analyzed in detail using a new anisotropic hardening model, four common sheet metal yield functions, and finite element procedures developed for this problem. A common lot of 6022-T4 aluminum alloy was used for all testing in order to reduce material variation. The new anisotropic hardening model extends existing mixed kinematic/isotropic and nonlinear kinematic formulations. It replicates three principal characteristics observed in uniaxial tension/compression test reversals: a transient region with low yield stress and high strain hardening, and a permanent offset of the flow stress at large subsequent strains. This hardening model was implemented in ABAQUS in conjunction with four yield functions: von Mises, Hill quadratic, Barlat three-parameter, and Barlat 1996. The simulated springback angle depended intimately on both hardening law after the strain reversal and on the plastic anisotropy. The springback angle at low back forces was controlled by the hardening law, while at higher back forces the anticlastic curvature, which depends principally on yield surface shape, controlled the springback angle. Simulations utilizing Barlat's 1996 yield function showed remarkable agreement with all measurements, in contrast to simulations with the other three yield functions.     

Limit strain predictions for strain-rate sensitive anisotropic sheets

The combined effects of material strain-rate sensitivity and anisotropy on necking or “limit” strain predictions are examined for thin sheets with transversely isotropic properties. Various rate dependent constitutive laws based on flow theory and deformation theory of plasticity are considered.The strong effect of material strain-rate sensitivity in increasing the amount of straining prior to localized necking is first emphasized. We then discuss the joint influence of rate dependence and anisotropy on the theoretical limit strains and forming limit curves. Both strain-rate sensitivity and the local shape of the anisotropic yield surface are shown to significantly affect the predicted limit strains.A necking-band bifurcation analysis is also carried out to reveal in an explicit manner the remarkable sensitivity of overall forming limit diagram shapes to the parameters in the anisotropic yield function.     

高强钢板DP590温热环境下屈服准则适用性分析

板料成形过程中温度的变化会影响其塑性流动行为。以先进高强钢板DP590为研究对象,通过不同温度下单拉试验和不同温度不同加载比例双拉试验得到其在温热环境下的试验屈服轨迹,结果显示温度不仅会影响DP590钢板屈服轨迹的大小,而且影响屈服轨迹的形状。将试验屈服轨迹和几种理论屈服轨迹进行对比分析,表明Yld2000-2D屈服准则与试验结果吻合较好,同时等效应力应变点分析也显示出Yld2000-2D屈服准则满足塑性力学的单一曲线假设规律。针对Yld2000-2D屈服准则,采用不同温度下的试验数据计算出不同温度下的各向异性参数,通过拟合获得各向异性参数与温度的四次多项式函数关系,建立温度相关Yld2000-2D屈服准则,并与常规Yld2000-2D屈服准则对比分析,结果显示温度相关Yld2000-2D屈服准则更加适合描述DP590钢板温热环境下的屈服行为。     

A constitutive model for transversely isotropic foams, and its application to the indentation of balsa wood

An elastic-plastic constitutive model for transversely isotropic compressible solids (foams) has been developed. A quadratic yield surface with four parameters and one hardening function is proposed. Associated plastic flow is assumed and the yield surface evolves in a self-similar manner calibrated by the uniaxial compressive (or tensile) response of the cellular solid in the axial direction. All material constants in the model (elastic and plastic) can be determined from a combination of a total of four uniaxial and shear tests. The model is used to predict the indentation response of balsa wood to a conical indenter. For the three cone angles considered in this study, very good agreement is found between the experimental measurements and the finite element (FE) predictions of the transversely isotropic cellular solid model. On the other hand, an isotropic foam model is shown to be inadequate to capture the indentation response.     

Sectional finite element analysis of forming processes for aluminum-alloy sheet metals

The sectional finite element analysis of the forming processes for the aluminum-alloy sheet metal known to be planar anisotropic was performed. The two-dimensional rigid-viscoplastic FEM formulation based on the bending augmented membrane theory as well as the anisotropic yield criteria was introduced. For modeling the anomalous behavior of aluminum-alloy sheet metals, Barlat's strain rate potential and Hill's (Journal of the Mechanics and Physics of Solids 1990;38:405–17) non-quadratic yield theory with an isotropic hardening rule were employed. Furthermore, a new method to determine anisotropic coefficients of Barlat's strain rate potential was proposed. For evaluating bending effects in the forming process of aluminum-alloy sheet metals, the bending equivalent forces were calculated in terms of the changes in the interior angle at a node between two linear finite elements and were augmented to the membrane stretch forces. In order to verify the validity of sectional finite element formulation based on the bending augmented membrane theory, the plane strain stretch/draw forming processes of a square cup test were simulated and simulation results are compared with experimental measurements. Friction coefficient was obtained from drawbead friction test. The properties of selected material were obtained from uniaxial tensile tests. Simulation shows good agreement with measurements. For the application of the sectional finite element formulation introduced in this research, the drawing process of a rear seat back upper bracket of passenger cars is simulated assuming plane strain condition. The thinning distribution of the simulation agreed well with that of the measurement, so that the sectional analysis is acceptable in the design and analysis of aluminum-alloy sheet stamping dies.     

Computational simulation of deformation behavior of 2D-lattice continuum

Finite element methods for an elastic Cosserat continuum obeying micropolar theory and couple stress theory are developed and applied to the simulation of stress concentration around a circular hole in a lattice continuum plate, in which the lattice continuum is the continuum model for materials with lattice-like microstructure. In the formulation of the finite element method based on couple stress theory, a new method consisting of a kind of selective reduced integration is proposed to remedy the over-constraint problem which arises in the penalty method of constraining the micro- and macrorotation vectors. The proposed finite element methods are validated by comparing the numerical solutions of stress concentration around a circular hole in a uniform tension field to the exact solutions for the isotropic materials obeying both micropolar theory and couple stress theory. Subsequently, the proposed method is applied to the lattice continuum, which is a continuous model of discrete lattice structure obeying couple stress theory such as cancellous bone with trabecular architecture, to analyze the dependence of the stress concentration factor on the microstructural parameters. In the range where the dimensions of the structural parameters are comparable to the hole radius, the stress concentration factor rises when the principal direction of the lattice structure is aligned along the tensile direction, whereas it falls when these directions form an oblique angle. The proposed finite element methods are applicable in investigation of the deformation behavior of materials with microstructures.     

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.