Anisotropic Yield Function Research Articles

Extension of quasi-plastic–elastic approach to incorporate complex plastic flow behavior – application to springback of advanced high-strength steels

Materials modeling and numerical formulations were conducted to describe the complex material behavior upon strain path change in order to enhance the prediction accuracy of springback in advanced high strength steels (AHSS). An approach without kinematic hardening rule, or the homogeneous anisotropic hardening (HAH) model, was incorporated to the newly conceived quasi-plastic–elastic strain (QPE) formulations. The HAH model is able to capture complex plastic flow behavior of sheet metals such as the Bauschinger effect, transient behavior, work-hardening stagnation and permanent softening. The QPE approach was developed to reproduce the nonlinear elastic behavior during unloading and reloading. The two models were independently validated for predicting springback, with better performance than conventional constitutive models. In this study, the two models are combined and extended to enhance the prediction capability of springback in AHSS. For this purpose, fully implicit numerical algorithms were re-formulated to link the two modeling approaches using general anisotropic yield function and hardening for shell element. The original model was only valid for continuum isotropic element with analytical stress integration procedure. Simulations of 2D draw bending test were performed to validate the developed approach for two AHSS, DP780 and TRIP780, sheets. The springback prediction was significantly improved if most of the complex material behavior relating to elasticity and plasticity were taken into account in the finite element simulations.

A new macroscopically anisotropic pressure dependent yield function for metal matrix composite based on strain gradient plasticity for the microstructure

Metal matrix composites with long aligned elastic fibers are studied using an energetic rate independent strain gradient plasticity theory with an isotropic pressure independent yield function at the microscale. The material response is homogenized to obtain a conventional macroscopic model that exhibits anisotropic yield properties with a pressure dependence. At the microscale free energy includes both elastic strains and plastic strain gradients, and the theory demands higher order boundary conditions in terms of plastic strain or work conjugate higher order tractions. The mechanical response is investigated numerically using a unit cell model with periodic boundary conditions containing a single fiber deformed under generalized plane strain conditions. The homogenized response can be modeled by conventional plasticity with an anisotropic yield surface and a free energy depending on plastic strain in addition to the elastic strain. Hill's classical anisotropic yield criterion is extended to cover the composite such that hydrostatic pressure dependency, Bauschinger stress and size-effects are considered. It is found that depending on the fiber volume fraction, the anisotropic yield surface of the composite is inclined compared to a standard pressure independent yield surfaces. The evolution of the macroscopic yield surface is investigated by quantifying both anisotropic hardening (expansion) and kinematic hardening (translation), where the coefficients of anisotropy and the Bauschinger stress are extracted.

The Effect of Normal Anisotropy on Thin-walled Tube Bending

Thin-walled tube bending has common applications in the automobile and aerospace industries. The rotary-draw-bending method is a complex physical process with multi-factor interactive effects and is one of the advanced tube forming processes with high efficiency, high forming precision, low consumption and good flexibility for bending angle changes. However, it may cause a wrinkling phenomenon, over thinning and cross-section distortion if the process parameters are inappropriate. Wrinkles propagate in thin-walled tube, but in some cases, localize in a finite zone and lead to failure. Wrinkling prediction in thin-walled tube bending processes has been an important and challenging subject in the related industry. In this paper, the plastic deforming behavior and wrinkling mechanism for a thin-walled tube is simulated and the results are compared with the available experimental ones. Next, the effect of anisotropy on ovalization, thickness and wrinkling of tube is investigated using FEM. Numerical results are presented showing the effects of the various kinds of materials and geometric parameters on wrinkling using anisotropic yield function.

Springback Prediction Using the Split-Ring Test Based on a Combined Anisotropic Hardening Model

Material model which could describe the Bauschinger effect is essential to accurately predict springback since strain path reverse is quite common in sheet metal forming. In this paper, a combined isotropic-kinematic hardening law which can capture the Bauschinger effect, transient behavior and permanent softening was used to model the hardening behavior. Also, the non-quadratic anisotropic yield function, Yld2000-2d, was chosen to describe the anisotropy. An inverse identificat- ion method was carried out to calibrate the material parameters by using uni-axial tension and Bauschinger simple shear tests. Experiments of cylindrical cup drawing and following split-ring tests were performed. Similar processes were carried out by numerical simulation, and springback predicted by present model was compared with experiments and that of vonMises model. The result shows that the current model significantly improves the accuracy of springback prediction.

Consistency condition of isotropic–kinematic hardening of anisotropic yield functions with full isotropic hardening under monotonously proportional loading

For the combined type isotropic–kinematic hardening law to account for the Bauschinger, transient and permanent softening behaviors observed in reverse loading, formulations have been initially developed for isotropic yield functions, mainly based on the von Mises criterion, and then later extended for anisotropic yield functions. Among the efforts to introduce anisotropic yield functions to the combined type hardening formulation, however, some inconsistency has been encountered in manipulating the kinematic hardening law, especially for the nonlinear type law, even though their von Mises yield function versions have been consistent. Therefore, theoretical clarification and clearance of such inconsistency were attempted in this work by imposing the following consistency condition: the combined type hardening law is expected to behave the same as the full isotropic hardening for monotonously proportional loading, regardless of anisotropic yield functions which are coupled with the combined type hardening law. An example to account for the anisotropic hardening of an anisotropic yield function utilizing the combined type hardening law, but for which the consistency condition was partially released, was also demonstrated.

The forming limit diagram of ferritic stainless steel sheets: Experiments and modeling

The forming limit diagrams (FLDs) of two ferritic stainless steel sheets of thicknesses 1 and 0.1mm were determined experimentally. For the 0.1mm thick sheet, the modified Marciniak test and the conventional ASTM standard test were used for the FLD determination. However, the latter produced wrinkles and buckles on the sheet specimens and undesired fractures for some strain paths. The results showed that the modified Marciniak test is a more robust method for the FLD determination of thin sheet materials. Nevertheless, in spite of the issues associated with the ASTM standard test for the low thickness sheet, the FLDs determined by the two methods led to similar results. In addition to the experimental approach, the FLD was predicted using a modification of the Parmar–Mellor–Chakrabarty (PMC) model, which incorporates the effect of surface roughness. A non-quadratic anisotropic yield function, Yld2000-2d was implemented in this model to represent the anisotropy of the sheet metals. The FLD predicted with the conventional M–K (Marciniak–Kuczyński) and the modified PMC models were compared to the FLD determined experimentally. The FLD calculated with this modified model was in better agreement with the measured data than that computed with the M–K model for both thin and thick sheets.

The effects of anisotropic yield functions and their material parameters on prediction of forming limit diagrams

In this paper, the capability of five different yield functions to predict the forming limit diagram (FLD) of the continuous cast 5754 Aluminum sheet has been compared with focus on the differences in the predicted limit strains based on the method of determining the yield function parameters. The yield functions proposed by Hill (1948, 1990, 1993), Barlat and Lian (1989), and Plunkett et al. (2008) have been considered in this study. The FLDs have been calculated numerically based on the above mentioned yield functions together with the Marciniak–Kuczynski (M–K) approach. Simulations show that the predicted FLDs strongly depend on the method of determining the material parameters (used in the yield functions) and the corresponding shape of the yield surface. Furthermore, simulations show that for biaxial stretching, the shape of the yield surface is the most dominant factor affecting the FLD.

Plasticity of formable all-metal sandwich sheets: Virtual experiments and constitutive modeling

The mechanical behavior of a metallic sandwich sheet material composed of two flat face sheets and two bi-directionally corrugated core layers is analyzed in detail. The manufacturing of the sandwich material is simulated to obtain a detailed unit cell model which accounts for the non-uniform thickness distribution and residual stresses associated with the stamping of the core layers. Virtual experiments are performed by subjecting the unit cell model to various combinations of bi-axial in-plane loading including the special cases of uniaxial tension, uniaxial compression, equi-biaxial tension and shear. The results demonstrate that the core structure’s contribution to the in-plane load carrying capacity of the sandwich sheet material is similar to that of the face sheets. The numerical results are also used to identify the effective yield surface and hardening response of both the core layer and the face sheets. An anisotropic yield function with linear pressure dependency is proposed to approximate the equal-plastic work surfaces for the core structure and face sheets. Furthermore, a new two-surface model with non-linear interpolation based on plastic work density is presented to describe the observed combined isotropic-distortional hardening of the core structure.

An application of homogeneous anisotropic hardening to springback prediction in pre-strained U-draw/bending

In this work, a constitutive model based on anisotropic hardening was used in the finite element (FE) simulations of springback and its performance was compared with that of conventional hardening laws. The homogeneous yield function-based anisotropic hardening (HAH) model (Barlat et al., 2011), considered in this work, describes a partial distortion of the yield surface under plastic loading. Although it does not use the concept of kinematic hardening, the HAH model was able to predict the complex material behavior upon load reversals such as the Bauschinger effect, transient hardening and permanent softening. For the application to springback, FE simulations were conducted for U-draw/bending of base (as-received) and pre-strained DP780 steel sheets, which was recently proposed as one of the Numisheet 2011 benchmark problems. The predictions with the HAH model, combined with a non-quadratic anisotropic yield function and a plastic strain-dependent unloading modulus, were in good agreement with experimental results for both as-received and pre-strained DP780 sheets.

Improvement on CAE Model for Accurate Torsional Springback Prediction in High Strength Steel Part Forming

It is difficult to predict springback, torsion in particular, with great accuracy by FE simulation. In this study, the effects of anisotropic yield functions of the blank and elastic deformation of the tools on springback are investigated in order to improve the accuracy of torsional springback prediction in high strength steel part forming. The effects of anisotropic yield functions on the torsional springback in curved hat forming were compared, using parameter sets derived from various experimental methods: uniaxial tensile tests only; hydrostatic bulge tests and uniaxial tension tests; uniaxial and biaxial tensile tests. FE analysis of press forming taking account of tool deformations was carried out and the results were compared with the ones of FE simulation with rigid tools and experimental results. It is demonstrated that FE simulated torsional springback with deformable tool model has better correlation with experimental one than with rigid tool model.

F012004 板材成形シミュレーションの高精度化に資する材料試験方法と材料モデリング

This work is a review of experimental methods for observing and modeling the anisotropic plastic behavior of metal sheets and tubes under a variety of loading paths, such as in-plane stress reversal tests on metal sheets and biaxial tension tests on metal sheets and tubes using closed-loop feed-back control testing machines. Observed material responses are compared with the predictions by phenomenological plasticity models. Special attention is paid to the plastic deformation behavior of materials commonly used in industry, and to verifying the validity of conventional anisotropic yield functions for these materials and associated flow rules at large plastic strains. The effects of using appropriate anisotropic yield functions on the accuracy of forming simulations are discussed to highlight the importance of accurate material testing and modeling.

Finite Element Analysis of Perforated Sheet Using Anisotropic Gurson's Yield Function

During the plastic deformation, the influence of voids on macroscopic mechanical behaviors is an important consideration, because ductile fracture occurs through the growth and the coalescence of voids. However, in order to analyze the model with distributed voids by finite element method directory, a large number of elements and nodes are required. Mesoscopic model which describes the effect of such voids by internal state variables is needed. In this paper, perforated sheets with randomly distributed holes are regarded as a two-dimensional ideal model of real ductile materials containing voids. The increase in stress around holes was approximated by the net stress tensor which incorporates the damage tensor obtained by the voronoi tessellation and the shape of each hole. Substituting the net stress into Gurson's yield function, an anisotropic Gurson's yield function and its constitutive relation are derived. Moreover, in order to evaluate the proposed mesoscopic model, the comparison was made between the mesoscopic and the usual FE analyses of sheets with randomly distributed holes. As a result, it is found that the proposed mesoscopic model well describes the macroscopic deformation of perforated sheets. The influence of the element size is also examined.

Elasto-plasticity Behavior of IF Steel Sheet with Planar Anisotropy and Its Macro-meso Modeling

Elasto-plasticity behavior of an IF steel sheet was investigated by performing uniaxial tension tests in three directions (0°, 45° and 90° to the rolling direction of the sheet), in-plane cyclic tension compression test and bi-axial tension test. The sheet has strong planar anisotropy (r0 = 2.15, r45 = 2.12 and r90 = 2.89) but very weak flow stress directionality. Equi-biaxial flow stress is as large as 1.23 times of the uniaxial flow stress. These elasto-plasticity deformation characteristics, as well as the Bauschinger effect and cyclic hardening behavior, are well described by a macro-plasticity model (Yoshida-Uemori model incorporating with the 4th-order anisotropic yield function). Further, the simulation of elasto-plasticity stress strain responses of the sheet were conducted by two types of crystal plasticity models, i.e., Taylor hypothesis based model and CPFEM, using the crystallographic orientation distribution data measured by neutron diffraction method. The models capture most of the above-mentioned deformation characteristics qualitatively, but the predicted anisotropy and the Bauschinger effect are weaker than those of the real material. The CPFEM gives more realistic results than the Taylor model.

Numerical Evaluation of Hydro-Formed DP-Steel Tubes on Crash-Performance with Welding Heat Effects

In order to numerically evaluate hydroformed DP-steel tubes on crash performance while considering weld-heat effects, finite element simulations of the crash were performed for hydroformed tubes with and without considering heat treatment effects. Also, finite element simulations were carried out for sequential procedures of bending and hydro-forming of tubes in order to evaluate formability and spring-back for process parameters: boost condition and axial feeding. As for the constitutive law, the isotropic hardening law was used along with non-quadratic anisotropic yield functions Yld2000-2d. Forming limit diagrams were calculated based on Hill’s bifurcation theory and the M–K theory. In order to characterize the material properties of the base material (BM) sheet, tensile tests were performed with various strain rates. Tensile tests were conducted also for tensile specimens that were cut out from tubes for the measurement of material properties of the base material (BM) tube and electric resistance weld (ERW) zone. The mechanical properties of the metal inert gas (MAG) arc-weld zone as well as those of the heat affected zone (HAZ) were also obtained by using the continuous indentation method. Based on the determined material characteristics, crash performance evaluations were then numerically carried out for the tube, formed with optimized process parameters, with and without welding heat effects. With this work, numerical methods to properly implement the welding heat effects on the tube crash performance were established, which will be useful to evaluate the welding heat effects on the tube crash performance under realistic crash conditions.

Effects of Elastic Modulus Degradation on Simulation of Stretch-Bend Springback of Dual Phase Steel

The stretch-bend springback of dual phase steel DP590 was studied experimentally and numerically in this paper. The aim of this work is to investigate the effects of elastic modulus degradation on springback prediction. Two anisotropic yield function yld89 and yld2000 were utilized along with isotropic hardening law. The appropriate material parameters characterizing the elastic modulus decrease were identified by response surface modeling. The current work showed that the accuracy of springback simulation improved when considering elastic modulus decreased with plastic strain and yld2000 can predict springback more accurately considering blank orientation in stretch bending.

On the plastic anisotropy of an aluminium alloy and its influence on constrained multiaxial flow

The influence of plastic anisotropy on the mechanical behaviour of a rolled aluminium plate under quasi-static loading conditions is studied experimentally and numerically. Material tests in different directions with respect to the rolling direction of the plate were carried out on various specimen shapes providing a wide range of stress states. The Yld2004-18p anisotropic yield function was identified through uniaxial tensile tests, shear tests and upsetting tests. This yield function was found to provide an adequate description of the significant anisotropic behaviour of a high-strength AA7075-T651 plate. Numerical simulations of all the material tests were then performed with an elasto-plastic material model using both the anisotropic and an isotropic version of the yield function. The numerical predictions of the mechanical response for notched tensile tests obtained with the isotropic version of the material model clearly over-estimated the experimental results. Similar results have been reported in the literature on other materials using isotropic constitutive relations. This over-estimation was significantly reduced when using the anisotropic version of the material model, and the reason why plastic anisotropy is so important for an accurate prediction of the notch-strengthening effect is explained. It was also established that the exponent of the yield function Yld2004-18p has a strong influence on the results for the notched tensile tests. When the exponent of the yield surface was assigned a sufficiently high value, an almost perfect fit between the numerical predictions and the experimental results was obtained.

A study of the Yld2004 yield function and one extension in polynomial form: A new implementation algorithm, modeling range, and earing predictions for aluminum alloy sheets

Abstract As shown recently in ( Soare and Barlat, 2010 . Convex polynomial yield functions. J. Mech., Phys. Solids, 58, 1804–1818), the principal values based yield function Yld2004, proposed in ( Barlat et al., 2005 . Linear transformation based anisotropic yield function. Int. J. Plast., 21, 1009–1039), is polynomial for integer exponents. Based on this observation, a new algorithm is proposed for implementing symmetric yield functions formulated in terms of principal values. The algorithm is tested here by simulating with a commercial FE code the cylindrical deep drawing of two aluminum sheets. It is found that the classical description of the in-plane directional properties of the sheet (uniaxial r-values and yield stresses), even if modeled correctly by the yield function, is not sufficient for a unique characterization of the predicted earing profile. For certain combinations of the directional properties the r-value in biaxial stressing has to be considered for a correct calibration of the material model. This in turn requires a finer detail in yield surface modeling and, to achieve it, an ad-hoc extension of Yld2004 is constructed. In combination with the proposed implementation algorithm, the extension is shown to be a useful research tool, having some interesting modeling capabilities and satisfactory FE runtime.

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.

Evaluation of constitutive models for textured aluminium alloys using plane-strain tension and shear tests

The constitutive modelling of the strongly textured aluminium alloys AA6063-T6 and AA7003-T6 is studied. The materials were delivered in the form of flat extruded profiles. Plane-strain tension and shear tests in the plane of the flat profiles are performed. The tests are then used to evaluate a constitutive model including an anisotropic yield function, the associated flow rule and a nonlinear isotropic work-hardening rule. The parameters of the yield criterion and the work-hardening rule were identified primarily from uniaxial tension tests in different in-plane directions. It is suggested how analytical considerations and the results from the plane-strain tension and shear tests may be used to obtain a more accurate calibration of the anisotropic yield criterion. To further assess the constitutive model, finite element simulations of the plane-strain tension and shear tests are carried out and the results compared with the experimental force-elongation curves. Significant deviations in the experimental and predicted results are disclosed, and attributed partly to the parameter identification, primarily based on uniaxial tension tests, and partly to the assumption of isotropic work-hardening. Polycrystal plasticity calculations are carried out for simple shear of the AA7003-T6 material, indicating that texture evolution plays an important role in determining the response in this test already at moderate strains.

A modified damage model for advanced high strength steel sheets

More often than not, better formability in the simple tension test implies better formability performance in other stretching modes, especially in hole expansion performance since deformation in the hole expansion test is perceived to be in the same simple tension deformation mode. However, when the hole expansion formability is evaluated particularly for the twinning induced plasticity (TWIP) steel, its performance is so poor compared to other automotive steels, even though the TWIP steel has significantly superior formability in the simple tension test. Therefore, hole expansion formability was experimentally and numerically studied for advanced high-strength grade steel sheets, TWIP940 and a transformation induced plasticity (TRIP) 590 steel sheet, as well as a high-strength grade 340R steel sheet, particularly in conjunction with formability in the simple tension test and its surface condition sensitivity. In order to characterize mechanical properties, simple tension tests were performed to determine anisotropic properties and strain rate sensitivities. To account for macro-crack formation, an inverse calibration method based on a damage model utilizing a triaxiality-dependent fracture criterion and hardening behavior with stiffness deterioration was developed. In this approach, the damage model was inversely calibrated by performing numerical simulations and experiments for the simple tension test (with specimens prepared by milling and punching). Then, the damage model was applied to formability study in the hole expansion test. The damage model along with the anisotropic yield function Hill (1948) incorporated into the ABAQUS/Explicit FEM code performed well to predict hole expansion ratios (HER) and their surface condition sensitivity, elucidating the cause of the lukewarm hole expansion performance and strong surface condition sensitivity of the TWIP steel compared to the others.