A finite element formulation based on non-associated plasticity for sheet metal forming

In the present paper, a finite element formulation based on non-associated plasticity is developed. In the constitutive formulation, isotropic hardening is assumed and an evolution equation for the hardening parameter consistent with the principle of plastic work equivalence is introduced. The yield function and plastic potential function are considered as two different functions with functional form as the yield function of Hill [Hill, R., 1948. Theory of yielding and plastic flow of anisotropic metals. Proc. Roy. Soc. A 193, 281–297] or Karafillis–Boyce associated model [Karafillis, A.P. Boyce, M., 1993. A general anisotropic yield criterion using bounds and a transformation weighting tensor. J. Mech. Phys. Solids 41, 1859–1886]. Algorithmic formulations of constitutive models that utilize associated or non-associated flow rule coupled with Hill or Karafillis–Boyce stress functions are derived by application of implicit return mapping procedure. Capabilities in predicting planar anisotropy of the Hill and Karafillis–Boyce stress functions are investigated considering material data of Al2008-T4 and Al2090-T3 sheet samples. The accuracy of the derived stress integration procedures is investigated by calculating iso-error maps.     

Incorporation of yield surface distortion in finite deformation constitutive modeling of rigid-plastic hardening materials based on the Hencky logarithmic strain

In this paper a constitutive model for rigid-plastic hardening materials based on the Hencky logarithmic strain tensor and its corotational rates is introduced. The distortional hardening is incorporated in the model using a distortional yield function. The flow rule of this model relates the corotational rate of the logarithmic strain to the difference of the Cauchy stress and the back stress tensors employing deformation-induced anisotropy tensor. Based on the Armstrong–Fredrick evolution equation the kinematic hardening constitutive equation of the proposed model expresses the corotational rate of the back stress tensor in terms of the same corotational rate of the logarithmic strain. Using logarithmic, Green–Naghdi and Jaumann corotational rates in the proposed constitutive model, the Cauchy and back stress tensors as well as subsequent yield surfaces are determined for rigid-plastic kinematic, isotropic and distortional hardening materials in the simple shear deformation. The ability of the model to properly represent the sign and magnitude of the normal stress in the simple shear deformation as well as the flattening of yield surface at the loading point and its orientation towards the loading direction are investigated. It is shown that among the different cases of using corotational rates and plastic deformation parameters in the constitutive equations, the results of the model based on the logarithmic rate and accumulated logarithmic strain are in good agreement with anticipated response of the simple shear deformation.     

Modelling inelastic behaviour of orthotropic metals in a unique alignment of deviatoric plane within the stress space

A finite strain constitutive model to predict the deformation behaviour of orthotropic metals is developed in this paper. The important features of this constitutive model are the multiplicative decomposition of the deformation gradient and a new Mandel stress tensor combined with the new stress tensor decomposition generalized into deviatoric and spherical parts. The elastic free energy function and the yield function are defined within an invariant theory by means of the structural tensors. The Hill’s yield criterion is adopted to characterize plastic orthotropy, and the thermally micromechanical-based model, Mechanical Threshold Model (MTS) is used as a referential curve to control the yield surface expansion using an isotropic plastic hardening assumption. The model complexity is further extended by coupling the formulation with the shock equation of state (EOS). The proposed formulation is integrated in the isoclinic configuration and allows for a unique treatment for elastic and plastic anisotropy. The effects of elastic anisotropy are taken into account through the stress tensor decomposition and plastic anisotropy through yield surface defined in the generalized deviatoric plane perpendicular to the generalized pressure. The proposed formulation of this work is implemented into the Lawrence Livermore National Laboratory-DYNA3D code by the modification of several subroutines in the code. The capability of the new constitutive model to capture strain rate and temperature sensitivity is then validated. The final part of this process is a comparison of the results generated by the proposed constitutive model against the available experimental data from both the Plate Impact test and Taylor Cylinder Impact test. A good agreement between experimental and simulation is obtained in each test.     

室温下镁合金板循环塑性随动强化本构模型研究

本文在具有各向异性屈服强度和拉压不对称的CPB06屈服准则的基础上,建立了基于随动强化的循环塑性本构模型．通过引入滑移、孪晶以及去孪等不同变形模式下的背应力演化方程,对室温下镁合金板材异常循环硬化行为进行了模拟．选取了AZ31B-O和AZ31B两种镁合金板材,通过拉伸-压缩-拉伸(T-C-T)和压缩-拉伸(C-T)等不同加载路径下的部分实验曲线确定模型的参数,采用三次插值多项式建立了背应力参数与上一变形模式中累积的等效塑性应变（即预应变）之间的函数关系．使用本模型对剩下的实验曲线进行了预测,发现预测结果与实验结果有良好的一致性,说明了当前模型的正确性．     

Multiaxial ratcheting with advanced kinematic and directional distortional hardening rules

Ratcheting is defined as the accumulation of plastic strains during cyclic plastic loading. Modeling this behavior is extremely difficult because any small error in plastic strain during a single cycle will add to become a large error after many cycles. As is typical with metals, most constitutive models use the associative flow rule which states that the plastic strain increment is in the direction normal to the yield surface. When the associative flow rule is used, it is important to have the shape of the yield surface modeled accurately because small deviations in shape may result in large deviations in the normal to the yield surface and thus the plastic strain increment in multi-axial loading. During cyclic plastic loading these deviations will accumulate and may result in large errors to predicted strains.This paper compares the bi-axial ratcheting simulations of two classes of plasticity models. The first class of models consists of the classical von Mises model with various kinematic hardening (KH) rules. The second class of models introduce directional distortional hardening (DDH) in addition to these various kinematic hardening rules. Directional distortion describes the formation of a region of high curvature on the yield surface approximately in the direction of loading and a region of flattened curvature approximately in the opposite direction. Results indicate that the addition of directional distortional hardening improves ratcheting predictions, particularly under biaxial stress controlled loading, over kinematic hardening alone.     

准脆性材料的弹塑性各向异性损伤耦合本构模型研究

针对准脆性材料的非线性特征：强度软化和刚度退化、单边效应、侧限强化和拉压软化、不可恢复变形、剪胀及非弹性体胀,在热动力学框架内,建立了准脆性材料的弹塑性与各向异性损伤耦合的本构关系。对准脆性材料的变形机理和损伤诱发的各向异性进行了诠释,并给出了损伤构形和有效构形中各物理量之间的关系。在有效应力空间内,建立了塑性屈服准则、拉压不同的塑性随动强化法则和各向同性强化法则。在损伤构形中,采用应变能释放率,建立了拉压损伤准则、拉压不同的损伤随动强化法则和各向同性强化法则。基于塑性屈服准则和损伤准则,构建了塑性势泛函和损伤势泛函,并由正交性法则,给出了塑性和损伤强化效应内变量的演化规律,同时,联立塑性屈服面和损伤加载面,给出了塑性流动和损伤演化内变量的演化法则。将损伤力学和塑性力学结合起来,建立了应变驱动的应力-应变增量本构关系,给出了本构数值积分的要点。以单轴加载-卸载往复试验识别和校准了本构材料常数,并对单轴单调试验、单轴加载-卸载往复试验、二轴受压、二轴拉压试验和三轴受压试验进行了预测,并与试验结果作了比较,结果表明,所建本构模型对准脆性材料的非线性材料性能有良好的预测能力。     

The strain rate intensity factor in the plane strain compression of thin anisotropic metal strip

Using an available analytic solution for instantaneous plane strain compression of a plastically anisotropic strip between two parallel plates the strain rate intensity factor is found assuming Hill’s quadratic yield criterion. The distribution of material properties is uniform. The effect of parameters characterizing plastic anisotropy of the strip on the magnitude of the strain rate intensity factor is demonstrated. A possibility to replace the strain rate intensity factor with the plastic work rate intensity factor is discussed. Singular behavior of the plastic spin in the vicinity of the friction surface is revealed and discussed.     

Forming simulation of aluminum sheets using an anisotropic yield function coupled with crystal plasticity theory

This paper describes the application of a coupled crystal plasticity based microstructural model with an anisotropic yield criterion to compute a 3D yield surface of a textured aluminum sheet (continuous cast AA5754 aluminum sheet). Both the in-plane and out-of-plane deformation characteristics of the sheet material have been generated from the measured initial texture and the uniaxial tensile curve along the rolling direction of the sheet by employing a rate-dependent crystal plasticity model. It is shown that the stress–strain curves and R-value distribution in all orientations of the sheet surface can be modeled accurately by crystal plasticity if a “finite element per grain” unit cell model is used that accounts for non-uniform deformation as well as grain interactions. In particular, the polycrystal calculation using the Bassani and Wu (1991) single crystal hardening law and experimental electron backscatter data as input has been shown to be accurate enough to substitute experimental data by crystal plasticity data for calibration of macroscopic yield functions. The macroscopic anisotropic yield criterion CPB06ex2 (Plunkett et al., 2008) has been calibrated using the results of the polycrystal calculations and the experimental data from mechanical tests. The coupled model is validated by comparing its predictions with the anisotropy in the experimental yield stress ratio and strain ratios at 15% tensile deformation. The biaxial section of the 3D yield surface calculated directly by crystal plasticity model and that predicted by the phenomenological model calibrated with experimental and crystal plasticity data are also compared. The good agreement shows the strength of the approach. Although in this paper, the Plunkett et al. (2008) yield function is used, the proposed methodology is general and can be applied to any yield function. The results presented here represent a robust demonstration of implementing microscale crystal plasticity simulation with measured texture data and hardening laws in macroscale yield criterion simulations in an accurate manner.     

Finite element modeling of plastic anisotropy induced by texture and strain-path change

Consideration of plastic anisotropy is essential in accurate simulations of metal forming processes. In this study, finite element (FE) simulations have been performed to predict the plastic anisotropy of sheet metals using a texture- and microstructure-based constitutive model. The effect of crystallographic texture is incorporated through the use of an anisotropic plastic potential in strain-rate space, which gives the shape of the yield locus. The effect of dislocation is captured by use of a hardening model with four internal variables, which characterize the position and the size of the yield locus. Two applications are presented to evaluate the accuracy and the efficiency of the model: a cup drawing test and a two-stage pseudo-orthogonal sequential test (biaxial stretching in hydraulic bulging followed by uniaxial tension), using an interstitial-free steel sheet. The experimental results of earing behavior in the cup drawing test, maximum pressure and strain distribution in bulging, and transient hardening in the sequential test are compared against the FE predictions. It is shown that the current model is capable of predicting the plastic anisotropy induced by both the texture and the strain-path change. The relative significance of texture and strain-path change in the predictions is discussed.     

The constitutive equations for mixed hardening orthotropic material

IntroductionAyieldingcriterionandtheassociatedflowtheorywereproposedbyHill[1]forinitiallyorthotropicmetalsin 1 948,whichareusedbroadly .AquadraticformofstressesisusedastheplasticpotentialthatisindependentofhydrostaticstressinHillplasticitytheory .Butcompressedbyhydrostaticstress,considerabledeformationwillbeproducedinorthotropicmaterials.Inthecaseofcyclicloading ,duetoBauschingereffect,thekinematichardeningcannotbeneglected .Inthispaper,kinematichardeningandproportionalhardeningareconsidered…     

Plastic potentials for anisotropic porous solids

The aim of this paper is to incorporate plastic anisotropy into constitutive equations of porous ductile metals. It is shown that plastic anisotropy of the matrix surrounding the voids in a ductile material could have an influence on both effective stress–strain relation and damage evolution. Two theoretical frameworks are envisageable to study the influence of plastic flow anisotropy: continuum thermodynamics and micromechanics. By going through the Rousselier thermodynamical formulation, one can account for the overall plastic anisotropy, in a very simple manner. However, since this model is based on a weak coupling between plasticity and damage dissipative processes, it does not predict any influence of plastic anisotropy on cavity growth, unless a more suitable choice of the thermodynamical force associated with the damage parameter is made. Micromechanically-based models are then proposed. They consist of extending the famous Gurson model for spherical and cylindrical voids to the case of an orthotropic material. We derive an upper bound of the yield surface of a hollow sphere, or a hollow cylinder, made of a perfectly plastic matrix obeying the Hill criterion. The main findings are related to the so-called ‘scalar effect’ and ‘directional effect’. First, the effect of plastic flow anisotropy on the spherical term of the plastic potential is quantified. This allows a classification of sheet materials with regard to the anisotropy factor h; this is the scalar effect. A second feature of the model is the plasticity-induced damage anisotropy. This results in directionality of fracture properties (‘directional effect’). The latter is mainly due to the principal Hill coefficients whilst the scalar effect is enhanced by ‘shear’ Hill coefficients. Results are compared to some micromechanical calculations using the finite element method.     

Unit cell debonding analyses for arbitrary orientation of plastic anisotropy

Debonding of rigid inclusions embedded in the elastic–plastic aluminum alloy Al 2090-T3 is analyzed numerically using a unit cell model taking full account of finite strains. The cell is subjected to overall biaxial plane strain tension and periodical boundary conditions are applied to represent arbitrary orientations of plastic anisotropy. Plastic anisotropy is considered using two phenomenological anisotropic yield criteria, namely Hill [Proceedings of the Royal Society of London A 193 (1948) 281] and Barlat et al. [International Journal of Plasticity 7 (1991) 693]. For this material plastic anisotropy delays debonding compared to plastic isotropy except for the case of Hill’s yield function when the tensile directions coincided with the principal axes of anisotropy. For some inclinations of the principal axes of anisotropy relative to the tensile directions, the stress strain responses are identical but the deformation modes are mirror images of each other.     

Anisotropic response of high-purity α-titanium: Experimental characterization and constitutive modeling

This paper presents a comprehensive experimental and theoretical investigation of the deformation behavior of high-purity, polycrystalline α-titanium under quasi-static conditions at room temperature. The initial material in this study was a cross-rolled plate with a strong basal texture. To quantify the plastic anisotropy and the tension–compression asymmetry of this material, monotonic tensile and compressive tests were conducted, on samples cut along different directions of the plate. A new anisotropic elastic/plastic model was developed to describe the quasi-static macroscopic response of the aggregate. Key in its formulation is the use of an anisotropic yield criterion that captures strength-differential effects and an anisotropic hardening rule that accounts for texture evolution associated to twinning. A very good agreement between FE simulations using the model developed and uniaxial data was obtained.     

Prediction of forming limit diagrams based on shear failure criterion

Maximum shear stress theory, also called the ‘Third Strength Theory’, is a classical theory used to predict the failure of common metal, but it cannot be used directly to predict sheet metal failure due to anisotropy and the loading path. Therefore, this paper proposes a maximum shear stress calculating method, which has been named “shear failure criterion” for the purpose of this paper. In order to validate the shear failure criterion, a general program was developed, and two typical materials, steel, and aluminum alloy, were used to study the new shear failure criterion in this study. The two materials were modeled by advanced constitutive models, including Hill1948 and Yld2000-2d yield functions and several types of isotropic hardening models. Experimental validation has indicated the accuracy of predicted FLD using shear failure criterion, which is able to provide a new alternative method to numerically predict FLD.     

HILL屈服准则与晶体塑性模型对FCC单晶材料塑性各向异性描述能力的比较

采用宏观HILL模型和晶体塑性模型对面心立方单晶(FCC)材料的非均匀交形进行了数值模拟，意在比较两种不同尺度的模型对塑性各向异性的描述能力的差异。为了使两种模型具有可比性，对于FCC单晶材科，本文提出一种用晶体塑性模型来确定HILL模型中各向异性参数的标定方法。数值分析表明，两类模型对单晶体塑性各向异性的描述能力存在着差异。对FCC单晶材料，HILL模型对各向异性的预测能力没有晶体塑性模型细致，晶体塑性模型更能追踪塑性各向异性的变化。但两种模型对应力应变响应预测的趋势是一致的。对两种模型描述的差异，做了详细的分析。     

Some basic issues in traditional Eulerian formulations of finite elastoplasticity

In traditional Eulerian formulations of finite elastoplasticity, there are some basic issues that still need clarification and further investigation. Among them are the characterization of initial anisotropy for scalar- and tensor-valued constitutive functions, plastic consistency conditions involving time differentiation of anisotropic yield functions, a unified loading criterion for hardening, softening, and perfectly plastic behaviour, etc. Sometimes, it is thought that a satisfactory and complete treatment for these issues could not be achieved within a framework of traditional Eulerian formulation. In this work, efforts are made towards explaining and clarifying these basic issues. Introducing the notion of an Eulerian type rotation-conjugate group of the initial material symmetry group, we show that consistent Eulerian formulations of scalar- and tensorvalued constitutive functions may be achieved for any given type of initial anisotropy. With such consistent Eulerian formulations, we derive plastic consistency conditions in a corotating frame associated with the foregoing rotation-conjugate group. As to the loading conditions for hardening, softening, and perfectly plastic behaviour, we recall and study the unified criterion proposed by Hill [J Mech Phys Solids 6 (1958) 236]. It is pointed out that the tangential elastic stiffness tensor in this unified criterion has to fulfill rather complicated integrability conditions for the elastic rate equation with an objective stress rate. It appears to be far from being a simple or even trivial matter to give an explicit form of the tangential elastic stiffness tensor meeting these conditions. It is indicated that this issue may be resolved by using the logarithmic stress rate. With this rate, an explicit form of Hill's unified loading criterion, together with an explicit form of tangential elastic stiffness tensor, is presented in terms of a general complementary hyperelastic potential.     

A practical two-surface plasticity model and its application to spring-back prediction

A practical two-surface plasticity model based on classical Dafalias/Popov and Krieg concepts was derived and implemented to incorporate yield anisotropy and three hardening effects for non-monotonous deformation paths: the Bauschinger effect, transient hardening and permanent softening. A simple-but-effective stress-update scheme avoiding overshooting was proposed and implemented. Constitutive parameters were fit to 5754-O aluminum alloy using uniaxial tension/compression data. Spring-back predictions using the resulting material model were compared with experiments and with single-surface material models which do not account for permanent softening. The two-surface model improved such predictions significantly as compared with single-surface models, while the differences between two-surface simulations and experiments were insignificant.     

Influence of plastic anisotropy on predictions of some engineering approaches in fracture mechanics

We study the influence of plastic anisotropy on the predictions of some engineering models of fracture mechanics. We consider a welded specimen with a crack in the weld joint under tensile loads. The weld jointmaterial and the mainmaterial are assumed to obey the orthotropic yield condition. A comparison with the isotropic cases shows that the limit load significantly depends on the plastic anisotropy parameters of thematerials. Since this load is one of the basic input parameters of engineering models of fracture mechanics, the plastic anisotropy parameters must strongly affect the predictions of such models, and these should be taken into account in the structural analysis.