Stress integration method for a nonlinear kinematic/isotropic hardening model and its characterization based on polycrystal plasticity

Sheet metal forming processes generally involve non-proportional strain paths including springback, leading to the Bauschinger effect, transient hardening, and permanent softening behavior, that can be possibly modeled by kinematic hardening laws. In this work, a stress integration procedure based on the backward-Euler method was newly derived for a nonlinear combined isotropic/kinematic hardening model based on the two-yield’s surfaces approach. The backward-Euler method can be combined with general non-quadratic anisotropic yield functions and thus it can predict accurately the behavior of aluminum alloy sheets for sheet metal forming processes. In order to characterize the material coefficients, including the Bauschinger ratio for the kinematic hardening model, one element tension–compression simulations were newly tried based on a polycrystal plasticity approach, which compensates extensive tension and compression experiments. The developed model was applied for a springback prediction of the NUMISHEET’93 2D draw bend benchmark example.     

Spring-back evaluation of automotive sheets based on isotropic–kinematic hardening laws and non-quadratic anisotropic yield functions,part III: applications

In order to improve the prediction capability of spring-back in the computational analysis of automotive sheet forming processes, the modified Chaboche type combined isotropic–kinematic hardening law was formulated to account for the Bauschinger and transient behavior in Part I. As for the yield stress function, the non-quadratic anisotropic yield potential, Yld2000-2d, was utilized under the plane stress condition. Experimental procedures to obtain the material parameters of the combined hardening law and the yield potential were presented in Part II for three automotive sheets. For verification purposes, comparisons of simulations and experiments were performed here for the unconstrained cylindrical bending, the 2-D draw bending and the modified industrial part (the double-S rail). For all three applications, simulations showed good agreements with experiments. Simplified one-dimensional plane strain analytical and numerical methods were also developed here to better understand the spring-back in forming processes.     

An elasto-plastic damage constitutive theory and its prediction of evolution of subsequent yield surfaces and elastic constants

Based on pair functional potentials, Cauchy-Born rule and slip mechanism, a material model assembling with spring-bundle components, a cubage component and slip components is established to describe the elasto-plastic damage constitutive relation under finite deformation. The expansion/shrink, translation and distortion of yield surfaces can be calculated based on the hardening rule and Bauschinger effect defined on the slip component level. Both kinematic and isotropic hardening are included. Numerical simulations and predictions under tension, torsion, and combined tension-torsion proportional/non-proportional loading are performed to obtain the evolution of subsequent yield surfaces and elastic constants and compare with two sets of experimental data in literature, one for a very low work hardening aluminum alloy Al 6061-T6511, and another for a very high work hardening aluminum alloy annealed 1100 Al. The feature of the yield surface in shape change, which presents a sharp front accompanied by a blunt rear under proportional loading, is described by the latent hardening and Bauschinger effect of slip components. Further, the evolution law of subsequent yield surfaces under different proportional loading paths is investigated in terms of their equivalence. The numerical simulations under non-proportional loading conditions for annealed 1100 Al are performed, and the subsequent yield surfaces exhibit mixed cross effect because the kinematic hardening and isotropic hardening follow different evolution tendency when loading path changes. The results of non-proportional loading demonstrate that the present model has the ability to address the issue of complex loading due to the introduction of state variables on slip components. Moreover, as an elasto-plastic damage constitutive model, the present model can also reflect the variation of elastic constants through damage defined on the spring-bundle components.     

Modeling multi-axial deformation of planar anisotropic elasto-plastic materials,part I: Theory

In sheet metal forming processes local material points can experience multi-axial and multi-path loadings. Under such loading conditions, conventional phenomenological material formulations are not capable to predict the deformation behavior within satisfying accuracy. While micro-mechanical models have significantly improved the understanding of the deformation processes under such conditions, these models require large sets of material data to describe the micromechanical evolution and quite enormous computation expenses for industrial applications. To reduce the drawbacks of phenomenological material models under the multi-path loadings a new anisotropic elasto-plastic material formulation is suggested. The model enables the anisotropic yield surface to grow (isotropic hardening), translate (kinematic hardening) and rotate (rotation of the anisotropy axes) with respect to the deformation, while the shape of the yield surface remains essentially unchanged.Essentially, the model is formulated on the basis of an Armstrong–Frederick type kinematic hardening, the plastic spin theory for the reorientation of the symmetry axes of the anisotropic yield function, and additional terms coupling these expressions. The capability of the model is illustrated with multi-path loading simulations in ‘tension-shear’ and ‘reverse-shear’ to assess its performance with ‘cross’ hardening and ‘Bauschinger’ effects.     

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.     

材料参数对板材胀形过程综合影响的数值研究

本文将Hosford 与Hill 各向异性屈服函数应用于刚粘塑性有限元方法,分析了圆形薄板液压胀形过程.研究了材料性能参数:硬化指数n、速率敏感指数m、厚向各向异性参数R、屈服函数非多项式指数M 对液压胀形过程的综合影响.并通过数值分析,找出了临界厚向断裂应变-ε_3~(ov·)与材料参数关系的经验方程式.屈服表面形状对极限厚度应变的影响,可以用Barlat 提出的包含了R 和M 影响的p 值表示出来.     

Investigation of changes in the spherical shell shape under the action of pulsed loading due to contact interaction with a rigid block

An axisymmetric problem of high strains in a spherical lead shell enclosed into an aluminum “spacesuit” under the action of pulsed loading is considered. The shell straining is described with the use of equations of mechanics of elastoviscoplastic media in Lagrangian variables, and the kinematic relations are determined in the current state metrics. Equations of state are taken in the form of equations of the flow theory with isotropic hardening. The problem is solved numerically by using the variational difference method and the “cross” explicit scheme of integration with respect to time. The influence of the yield stress as a function of the strain rate on changes in the shell shape is studied for different values of loading. The calculated final shape and residual strains are demonstrated to be in good agreement with experimental data.     

An endochronic model of yield surface accounting for deformation induced anisotropy

In this article, an endochronic model of yield surface is proposed. Based on this model, the yield surface is simulated such that the forward and rear parts of the yield surface are described by different ellipses which are characterized by corresponding aspect ratio functions, respectively. Verification of the endochronic theory used the experimental results of yield surfaces obtained by Wu and Yeh for 304 stainless steel (Wu, H.C., Yeh, W.C., 1991. On the experimental determination of yield surfaces and some results of annealed 304 stainless steel. Int. J. Plasticity 7, 803–826). The experiments were performed cyclically under uniaxial, torsional, and combined axial–torsional loading conditions. The result has shown that the agreement between the prediction and experiments is quite satisfactory. In addition to the distortion of the yield surface plastically behaving a sharp front accompanied by a blunt rear, the anisotropic kinematic hardening effect has been addressed in this investigation. Although the experimental results of yield surfaces subjected to non-proportional loading conditions can be found in the literature, lack of information about the plastic strain history makes it impossible to verify the theory under such complicated loading conditions. The domain of applicability and validity of the theory, which is defined in terms of plastic strain increments, need be further investigated with the aim to set up related experiments.     

On the performance of kinematic hardening rules in predicting a class of biaxial ratcheting histories

It is well known that strain-symmetric axial cycling of thin-walled metal tubes in the presence of pressure results in a progressive accumulation (ratcheting) of circumferential strain. It was previously demonstrated that the prediction of the rate of ratcheting under constant internal pressure, by nonlinear kinematic hardening models, is very sensitive to the hardening rule adopted. It was shown that the Armstrong-Frederick hardening suitably calibrated and used in a class of models can yield reasonably good predictions of the rate of ratcheting for a range of cycle parameters. In this paper, the subject is revisited in the light of further experimental results involving simultaneous cycling of the internal pressure and the axial strain. Experiments and analyses were performed for a family of five such biaxial loading histories. A similar sensitivity to the kinematic hardening rule used in the models was observed in all the new loading histories. Furthermore the hardening rule calibrated in the constant pressure experiments was found to yield accurate predictions for three of the loading histories considered and poor predictions for the other two. The reasons for this varied performance are analyzed and some recommendations for implementation of such models in structural applications are made.     

A more comprehensive method for yield locus construction for metallic alloys and composites

Conventional methods for constructing yield loci rely on the assumption that nonlinear strains are permanent strains, which is not always the case. A nickel-base alloy, SiC fiber-reinforced titanium, an aluminum alloy, and particlereinforced aluminum have been observed to violate this assumption. We present a method for constructing yield loci using a proof strain criterion for the permanent strain that relies on cyclic, proportional, probes of the yield surface. Two criteria are implemented: one for stress reversal and one for yielding. The method is demonstrated by the construction of initial and subsequent yield loci in the axial-shear stress plane using thin-walled tubular specimens. Results are presented for 6061-T6 aluminum as well as for 6092/SiC/17.5p-T6, which is 6092 aluminum reinforced with 17.5 volume percent silicon carbide particulate. The centers of the initial yield loci for the composite are eccentric to the origin of the stress plane most likely because of the residual stresses induced during processing. Material hardening due to multiaxial stress states can be described by tracking evolution of the subsequent yield surfaces and here hardening of the particulate composite was primarily kinematic     

A numerical method for rapid estimation of drawbead restraining force based on non-linear,anisotropic constitutive equations

Numerical procedures to predict drawbead restraining forces (DBRF) were developed based on the semi-analytical (non-finite-element) hybrid membrane/bending method. The section forces were derived by equating the work to pull sheet material through the drawbead to the work required to bend and unbend the sheet along with frictional forces on drawbead radii. As a semi-analytical method, the new approach was especially useful to analyze the effects of various constitutive parameters with less computational cost. The present model could accommodate general non-quadratic anisotropic yield function and non-linear anisotropic hardening under the plane strain condition. Several numerical sensitivity analyses for examining the effects of process parameters and material properties including the Bauschinger effect and the shape of yield surface on DBRF were presented. Finally, the DBRFs of SPCC steel sheet passing a single circular drawbead were predicted and compared with the measurements.     

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.     

Deformation gradient based kinematic hardening model

A kinematic hardening model applicable to finite strains is presented. The kinematic hardening concept is based on the residual stresses that evolve due to different obstacles that are present in a polycrystalline material, such as grain boundaries, cross slips, etc. Since these residual stresses are a manifestation of the distortion of the crystal lattice a corresponding deformation gradient is introduced to represent this distortion. The residual stresses are interpreted in terms of the form of a back-stress tensor, i.e. the kinematic hardening model is based on a deformation gradient which determines the back-stress tensor. A set of evolution equations is used to describe the evolution of the deformation gradient. Non-dissipative quantities are allowed in the model and the implications of these are discussed. Von Mises plasticity for which the uniaxial stress–strain relation can be obtained in closed form serves as a model problem. For uniaxial loading, this model yields a kinematic hardening identical to the hardening produced by isotropic exponential hardening. The numerical implementation of the model is discussed. Finite element simulations showing the capabilities of the model are presented.     

Evaluation of advanced anisotropic models with mixed hardening for general associated and non-associated flow metal plasticity

The main objective of this paper is to develop a generalized finite element formulation of stress integration method for non-quadratic yield functions and potentials with mixed nonlinear hardening under non-associated flow rule. Different approaches to analyze the anisotropic behavior of sheet materials were compared in this paper. The first model was based on a non-associated formulation with both quadratic yield and potential functions in the form of Hill’s (1948). The anisotropy coefficients in the yield and potential functions were determined from the yield stresses and r-values in different orientations, respectively. The second model was an associated non-quadratic model (Yld2000-2d) proposed by Barlat et al. (2003). The anisotropy in this model was introduced by using two linear transformations on the stress tensor. The third model was a non-quadratic non-associated model in which the yield function was defined based on Yld91 proposed by Barlat et al. (1991) and the potential function was defined based on Yld89 proposed by Barlat and Lian (1989). Anisotropy coefficients of Yld91 and Yld89 functions were determined by yield stresses and r-values, respectively. The formulations for the three models were derived for the mixed isotropic-nonlinear kinematic hardening framework that is more suitable for cyclic loadings (though it can easily be derived for pure isotropic hardening). After developing a general non-associated mixed hardening numerical stress integration algorithm based on backward-Euler method, all models were implemented in the commercial finite element code ABAQUS as user-defined material subroutines. Different sheet metal forming simulations were performed with these anisotropic models: cup drawing processes and springback of channel draw processes with different drawbead penetrations. The earing profiles and the springback results obtained from simulations with the three different models were compared with experimental results, while the computational costs were compared. Also, in-plane cyclic tension–compression tests for the extraction of the mixed hardening parameters used in the springback simulations were performed for two sheet materials.     

An endochronic theory accounting for deformation induced anisotropy of metals under biaxial load

Using the experimental results of yield surfaces obtained by Wu and Yeh [1991] (Int. J. Plasticity, 7, 803) for 304 stainless steel, this work provides a verification of the endochronic theory of plasticity accounting for deformation induced anisotropy. The experiments were performed under proportional loading conditions. The main difference between this paper and other papers that attempt to describe the distortion of a yield surface is that, in addition to distortion, motion of yield surface (kinematic hardening) has also been addressed by this paper. The result has shown that the theory predicts the experimental data with substantial accuracy. However, since in this theory the plastic strain increment, although normal to the initial yield surface, is in the radial direction emanating from the center of the subsequent yield surface, validity of the present model must be further studied for the case involving nonproportional loading conditions.     

A non-associated constitutive model with mixed iso-kinematic hardening for finite element simulation of sheet metal forming

In this paper an anisotropic material model based on non-associated flow rule and mixed isotropic–kinematic hardening was developed and implemented into a user-defined material (UMAT) subroutine for the commercial finite element code ABAQUS. Both yield function and plastic potential were defined in the form of Hill’s [Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc. Lond. A 193, 281–297] quadratic anisotropic function, where the coefficients for the yield function were determined from the yield stresses in different material orientations, and those of the plastic potential were determined from the r-values in different directions. Isotropic hardening follows a nonlinear behavior, generally in the power law form for most grades of steel and the exponential law form for aluminum alloys. Also, a kinematic hardening law was implemented to account for cyclic loading effects. The evolution of the backstress tensor was modeled based on the nonlinear kinematic hardening theory (Armstrong–Frederick formulation). Computational plasticity equations were then formulated by using a return-mapping algorithm to integrate the stress over each time increment. Either explicit or implicit time integration schemes can be used for this model. Finally, the implemented material model was utilized to simulate two sheet metal forming processes: the cup drawing of AA2090-T3, and the springback of the channel drawing of two sheet materials (DP600 and AA6022-T43). Experimental cyclic shear tests were carried out in order to determine the cyclic stress–strain behavior and the Bauschinger ratio. The in-plane anisotropy (r-value and yield stress directionalities) of these sheet materials was also compared with the results of numerical simulations using the non-associated model. These results showed that this non-associated, mixed hardening model significantly improves the prediction of earing in the cup drawing process and the prediction of springback in the sidewall of drawn channel sections, even when a simple quadratic constitutive model is used.     

On non-associative anisotropic finite plasticity fully coupled with isotropic ductile damage for metal forming

In this work, non-associative finite strain anisotropic elastoplasticity fully coupled with ductile damage is considered using a thermodynamically consistent framework. First, the kinematics of large strain based on multiplicative decomposition of the total transformation gradient using the rotating frame formulation, is recalled and different objective derivatives defined. By using different anisotropic equivalent stresses (quadratic and non-quadratic) in yield function and in plastic potential, the evolution equations for all the dissipative phenomena are deduced from the generalized normality rule applied to the plastic potential while the consistency condition is still applied to the yield function. The effect of the objective derivatives and the equivalent stresses (quadratic or non-quadratic) on the plastic flow anisotropy and the hardening evolution with damage is considered. Numerical aspects mainly related to the time integration of the fully coupled constitutive equations are discussed. Applications are made to the AISI 304 sheet metal by considering different loading paths as tensile, shear, plane tensile and bulge tests. For each loading path the effect of the rotating frame, the equivalent stress (quadratic or non-quadratic) and the normality rule (with respect to yield function or to the plastic potential) are discussed on the light of some available experimental results.     

Spring-back evaluation of automotive sheets based on isotropic-kinematic hardening laws and non-quadratic anisotropic yield functions: Part II: characterization of material properties

In order to improve the prediction capability of spring-back in the computational analysis of automotive sheet forming processes, the modified Chaboche type combined isotropic-kinematic hardening law was formulated to account for the Bauschinger and transient behavior in Part I. As for the yield stress function, the non-quadratic anisotropic yield potential, Yld2000-2d, was utilized under the plane stress condition. Experimental procedures to obtain the material parameters of the combined hardening law and the yield potential are presented here in Part II for three automotive sheets: AA5754-O, AA6111-T4 and DP-Steel. The modified Chaboche model was confirmed to well represent the measured hardening behavior including the Bauschinger and transient behavior. While the theoretical and numerical formulations of the constitutive law are discussed in Part I, experimental verifications for spring-back of formed parts are further discussed in Part III.     

Plane stress yield function for aluminum alloy sheets—part 1: theory

A new plane stress yield function that well describes the anisotropic behavior of sheet metals, in particular, aluminum alloy sheets, was proposed. The anisotropy of the function was introduced in the formulation using two linear transformations on the Cauchy stress tensor. It was shown that the accuracy of this new function was similar to that of other recently proposed non-quadratic yield functions. Moreover, it was proved that the function is convex in stress space. A new experiment was proposed to obtain one of the anisotropy coefficients. This new formulation is expected to be particularly suitable for finite element (FE) modeling simulations of sheet forming processes for aluminum alloy sheets.     

Strain-space plasticity analysis of fibrous composites

A strain-space formulation of plasticity theory for metal matrix fibrous composites is discussed. Specific results are obtained for a composite system in which the fibers are elastic until failure, while the matrix is of the Mises type, with kinematic hardening. The material model of the composite is based on the assumption that the fiber diameter is vanishingly small, and the fiber volume fraction is finite. Explicit expressions are obtained for instantaneous strain concentration factors in the phases, for the instantaneous overall stiffness, and for the overall loading surface and hardening rule during mechanical straining along a prescribed path.