Anisotropic Yield Function Research Articles

Prediction of DP600 and TRIP780 yield loci using Yoshida anisotropic yield function

The continual demand for vehicle weight reduction, improved fuel efficiency and crashworthiness has driven the automotive industry to increasingly fabricate automotive body parts from advanced high strength steel (AHSS) sheet, such as dual phase (DP) and transformation induced plasticity (TRIP) steels. It is therefore essential to carefully investigate the forming behaviour of these sheet materials under various forming conditions. In this work, the quasi-static tensile flow behaviour of DP600 and TRIP780 sheet specimens was obtained in three orientations (RD, DD, and TD) with respect to the sheet rolling direction. A 3-parameter Voce hardening function was then fitted to each flow curve in order to determine true stress and true strain based on constant amount of plastic work per unit volume to calculate the normalized yield stress as well as the r-value for each material orientation. Yoshida’s 6th-order polynomial anisotropic yield function, expressed as a function of the second and third invariants of the deviatoric stress tensor (J2 and J3, respectively), was used to predict the mechanical response of these two sheet materials. A new optimization method based on the Markov chain Monte Carlo (MCMC) MetropolisHastings (MH) algorithm was employed to calibrate the anisotropic yield function and determine the anisotropic coefficients. The yield loci for both materials were then derived as a function of only, and also as a function of both J2 and . The performance of each function is evaluated and validated by comparing the numerical predictions of r-value and flow stress directionality with the experimental results. And the effects of J2 and J3 in predicting the shape of the yield locus of DP600 and TRIP780 are also discussed.

Warm Forming Response of ZEK100 Sheet obtained under Biaxial Stretching with Full-Field Displacement Measurements

The warm forming response of ZEK100 sheet was studied between 150 °C and 250 °C using hydrostatic bulging coupled with full-field displacement mapping. Due to the strain-rate sensitivity of magnesium alloys at elevated temperature, it was important to ensure that the strain-rate remained reasonably constant during the bulge test. Various gas pressure versus time profiles were used to achieve strain-rates in the range of 0.01 s−1 to 0.1 s−1. The results from equi-biaxial bulge testing will be detailed in addition to bulge testing performed using elliptical dies with various aspect ratios to generate data under different biaxial stretching conditions. Supplementary characterization was provided using shear specimen data and tension test data from the rolling and transverse directions of the sheet. The biaxial, tensile, and shear data was used to calibrate a linear transformation-based anisotropic yield function at 200 °C. The yield function coefficients were optimized by matching the surface displacement data between experiments and finite element simulations of the bulge test. Agreement of the surface displacement data for a given applied pressure ensured that both the stress and strain in the bulged dome were accurately captured. Finite element simulations of the bulge tests were compared to experimental data to confirm accuracy of the calculated yield function.

Measurement of Differential Hardening under Biaxial Stress of Pure Titanium Sheet

Biaxial stress tests of a commercially pure titanium sheet (JIS #1) using the cruciform specimen and the servo-controlled biaxial testing machine have been carried out in order to elucidate its anisotropic plastic deformation behavior. The geometry of the cruciform specimen is identical to that regulated by the ISO 16842. Nine linear stress paths, σx (rolling direction): σy (transverse direction) = 1:0, 4:1, 2:1, 4:3, 1:1, 3:4, 1:2, 1:4, and 0:1 in the first quadrant of the principal stress space are applied to the cruciform specimens. Contours of plastic work in the principal stress space and the directions of plastic strain rates at selected levels of plastic work have been precisely measured. The range of the equivalent plastic strain applied to the specimens is 0.002 ≤ ≤ 0.01. The shapes of the work contours significantly change with increasing the test material exhibits differential hardening (DH). Using the data of the work contours and the directions of plastic strain rates, the applicability of selected anisotropic yield functions to the accurate prediction of the plastic deformation behavior of the test material is examined.

Limit Strains Analysis of Advanced High Strength Steels Sheets Based on Surface Roughness Measurements

The limit strains of dual-phase steels DP600 and 800 were evaluated in this work with a localization model formulated in plane-stress conditions using elasto-plastic constitutive equations. In this model, a geometrical imperfection parameter is defined from the sheet nominal thickness, initial ferrite grain size and average surface roughness. The proposed identification procedure provided a more physically meaning for this parameter and at best more conservative predictions in the drawing Forming Limit Curve (FLC) range of both investigated dual-phase steels. Nevertheless, the corresponding limit strains in the biaxial stretching region are underestimated with the present theoretical model. Thus, more detailed anisotropic yield function and hardening descriptions must be implemented to improve the accuracy of the FLC prediction of advanced high strength steels.

Improved formability prediction by modeling evolution of anisotropy of steel sheets

Depending on the experimental observations, the Marciniak–Kuckzinki (MK) model was modified incorporating the evolution in the Yld2000-2d anisotropic yield function as a function of plastic work for two ferritic stainless steel sheets. The numerically estimated FLDs were validated with the strain limits evaluated by stretch forming experiments. Moreover, FE simulations were conducted to predict limiting dome heights (LDH) and limiting drawing ratios (LDR) by both models with and without considering the anisotropy evolutions. Additional formability performances such as surface strain distributions over the deformed cup surfaces and earing profile were also compared with experimental results. It was observed that the accuracy of predictions for formability could be significantly improved in the FE simulations when both initial and its subsequent evolution in yield function were included in the modeling. Further, microstructural analysis of parent sheets and stretch formed cups were performed to understand the effect of microstructure change on the anisotropy and formability. It was found that the preferred orientations along rolling and transverse directions changed differently with deformation. Orientation distribution function and Taylor factor maps were analyzed to confirm non-proportional evolution in stress directionality for both the materials.

Effect of Anisotropic Yield Functions on the Accuracy of Material Flow and its Experimental Verification

This paper evaluates the performance of anisotropic yield functions based on mathematical methods which consider the prediction accuracy of yield stresses and R-values. The anisotropic yield functions being evaluated include one quadratic yield function and six non-quadratic ones. These yield functions are applied to describe the anisotropy of steel sheets and aluminum alloy sheets to evaluate their predictability. The root-mean-square errors (RMSEs) are computed to quantitatively assess their performance. The computation results of RMSEs demonstrate that the Yld2004-18p yield function exhibits the highest accuracy but requires extensive tests to calculate anisotropic parameters. The Yld2000-2d and BBC2000 yield functions are the same and thus have the same prediction accuracy. The application to cylindrical cup drawing shows that, associated with the Yld2004-18p and Yld2000-2d yield functions, the finite element (FE) simulations of cup drawing process can predict cups with six or eight ears. Considering both efficiency and accuracy, the Yld2000-2d and BBC2000 yield functions, which have less anisotropic parameters to be calculated, are recommended for metals with intermediate anisotropy.

Numerical modeling for accurate prediction of strain localization in hole expansion of a steel sheet

The hole expansion of a low carbon steel sheet shows an interesting feature that localized thinning and subsequent crack initiation are observed inside the specimen, and not at the hole edge as is typically expected. The present work investigated a numerical modeling approach to predict this localization behavior within the framework of a finite element (FE) analysis. Plastic anisotropy of the sheet was taken into account using the anisotropic yield functions Yld2000-2d and Yld2004-18p for the plane stress and three-dimensional elements, respectively. Careful examination of the FE model revealed that the influence of the out-of-plane stress is very small, suggesting that shell elements can be efficiently used in the analysis. The influence of friction was also found to be negligibly small. However, the constitutive description exhibited a significant influence in that even a slight change in the yield function parameters resulted in a considerable difference in the prediction. For this reason, several sets of parameters were obtained based on the different material properties, and their influences on the hole expansion simulation were analyzed. In particular, the prediction accuracy could be greatly improved when the yield function parameters were optimized such that the flow stresses and plastic strain rate ratios in uniaxial and plane strain states were well captured.

The effect of tension compression asymmetry on modelling the bending response of sheet moulding compound composites

Phenomenological models are required to predict the behaviour of a glass fibre reinforced Sheet Moulding Compound (SMC) composite material for use in the automotive industry. Material testing is conducted in tension, compression, in-plane shear, and three-point bending. The SMC composite exhibits tension-compression asymmetry and in-plane anisotropy. A model, which incorporates an anisotropic and asymmetric yield function, is developed. The model is calibrated to the experimental tension, compression and in-plane shear tests and are validated using the three-point flexure test. The model captures the flexure response within 8.1% of the experimental observations. The importance of including tension compression asymmetry within the model is demonstrated.

Development of the User Subroutine Library “Unified Material Model Driver for Plasticity (UMMDp)” for Various Anisotropic Yield Functions

Many yield functions have been proposed in academia to describe the complicated shapes of yield surfaces of metals. However, many of the commercial FEM codes provide only classical yield functions. Moreover, it takes a long time for engineers to implement newly proposed yield functions to commercial FEM codes. The Japan Association for Nonlinear CAE (JANCAE), a non-profit organization, developed the Unified Material Model Driver for plasticity (UMMDp) subroutine suite with the cooperation of industry users of CAE and the engineers of software venders. This subroutine provides several anisotropic yield functions, and is applicable to most of the commercial FEM codes. The users can implement their own anisotropic yield functions easily using the UMMDp. This paper presents the basic framework of UMMDp and the development activities performed with many volunteers.

Earing prediction of AA 2008-T4 with anisotropic Drucker yield function based on the second and third stress invariants

This paper applies anisotropic Drucker yield function [14] (Lou and Yoon, 2018, Int. J. Plasticity 101, 125-55) to predict earing profile after cup deep drawing of AA 2008-T4. The new yield function is implemented into ABAQUS/Explicit using semi-implicit integration algorithm to calculate the increment of plastic strain. Both associated and non-associated flow rules are incorporated in this study. The predicted anisotropy and cup height profile are compared with the experimental results and those predicted from the Hill48, Yld91 and Yld2004-18p yield functions. It is observed that the earing profile predicted from the new yield function shows high accuracy. The computation time is also compared to investigate the computation cost of different yield functions. The comparison reveals that it takes the shortest time for the Hill48 function, the anisotropic Drucker yield function reduces 30%∼40% of computation cost compared with the Yld91 and Yld2004-18p functions. It is concluded from the simulation of cup deep drawing that the new yield function can provide high accurate numerical analysis of plastic deformation for anisotropic metals with competitive computation cost.

Implementation of anisotropic yield functions into the subroutine library “UMMDp”

To implement new yield functions into the user subroutines of advanced FE codes, professional skills and knowledge are required. The Japan Association for Nonlinear Computer Aided Engineering (JANCAE) has been acting as an organization that provides information and knowledge on finite element analysis (FEA) to the researchers and engineers in various positions involved in CAE. Since 2009, as one of the subcommittee activities for applying FEA to practical metal-forming problems, we have not only studied elastoplastic theory, but have also developed the Unified Material Model Driver for Plasticity (UMMDp), a user subroutine library. Basic yield functions, such as the von Mises and Hill’s quadratic yield functions, which almost all advanced FE code already has, were implemented in the UMMDp as a basic study at its beginning of the development. Up to now, many anisotropic yield functions, including Yoshida’s 6th-order polynomial, Barlat’s Yld2000 and Yld2004, Comsa and Banabic’s BBC2008, Cazacu’s CPB2006, and Vegter’s spline yield functions, have been implemented in the UMMDp, and it is possible to add a new yield function. In this presentation, the techniques for implementing yield functions in the UMMDp are explained. Moreover, the problem of BM1 in NUMISHEET 2016 is discussed as an example applying selected yield functions to the implementation procedure.

An elasto-plastic constitutive model of magnesium alloy sheet at warm forming temperature under strain path changes

In the current work, a continuum-based approach is discussed in order to describe the plastic hardening behavior of magnesium alloy sheets under strain path changes at warm forming temperature. The constitutive model is represented to use an anisotropic distortional yield function combining a stable component and a fluctuating component. The stable component includes the yield function that represents the material’s anisotropy or the strength effect between tension and compression. The evolution of the fluctuating component is reformulated based on recently developed HCP metal’s distortional hardening model at room temperature. The modified models were implemented into finite element (FE) software via user-material subroutine. Predicted stress-strain curves of tension-compression-tension (TCT) and compression-tension-compression (CTC) loadings for two pre-strains were compared with experimental results. Overall, the proposed model could give reliable agreement with experimental data.

Ductile fracture of AA6111 alloy including the effect of bake-hardening

The thermal cycle of paint baking for auto bodies alters the mechanical properties of aluminum alloys including, possibly, their fracture behavior. In this study, the plasticity and fracture of AA6111 sheet after a 30 min heating cycle at 180 °C is investigated. Uniaxial tension, plane-strain tension and disk compression experiments are performed to assess the plasticity of the material. The results are used to calibrate Yld20004-18p anisotropic yield function. An evolution of the yield locus with plastic deformation is observed. This is represented by evolving the exponent of the yield function, rather than the coefficients themselves. The post-necking hardening curve is identified by numerically simulating a notched-tensile test. The fracture locus is probed using notched-tension, center-hole, and shear experiments. In every case, digital image correlation (DIC) is used to acquire the surface strain fields. In parallel, finite element simulations of the fracture specimens are used to approximate the fracture strains, triaxiality, and Lode angle parameter. Yld2004-18p model shows good overall agreement with the experiments; thus, the fracture locus is probed with this model.

Development of plug-ins for bridging variables between advanced finite element codes and ‘UMMDp’

For advanced finite element codes, we can add original material models by programming ourselves. On the other hand, as for implementation methods, we need unique technique in each advanced finite element codes, and there was a problem that it is difficult to implement unless we fully understand the procedure of these codes. Therefore, the Japan Association for Nonlinear CAE (JANCAE) has started to implement anisotropic yield functions independent of advanced finite element codes in its working group activities, and has created a subroutine library “UMMDp”. The “UMMDp” library is connected to each advanced finite element codes with “plug-ins”, and variables are bridged. In this report, we first explain the concept of “plug-ins” development and verification method in implementation. Next, we introduce some examples that we implemented “UMMDp” into the advanced finite element code “LS-DYNA” via the plug-ins. By comparing with the built-in yield function, we will explain that “plug-ins” is working properly and the usefulness of “UMMDp”.

Verification of accuracy of yield functions of sheet steels under shear strains in uniaxial tensile tests in multiple directions

Sheet steels have more or less plastic anisotropy. It is therefore important to consider the anisotropy of a sheet steel in FE analysis of press forming. There are many anisotropic yield functions proposed such as Hill’48, Blart’s Yld functions and Gotoh’s orthotropic yield function. These yield functions have material parameters to express the anisotropy of a sheet steel, and the parameters need to be calibrated based on r-values and stresses in tension tests in various directions, and/or stresses and strains in bulge tests or in biaxial tension tests. The accuracy of a calibrated yield function should be confirmed by using the results of another material test which is not used in the calibration. However, some functions have many parameters and need many calibration tests. A new method to check parameters of yield functions was developed making use of shear strains in a uniaxial tension test. In this study, shear strains were measured by DIC in uniaxial tension tests in various directions. The results, the ratios of a share strain to the tensile strain, were compared with theoretical values from the calibrated yield functions.

Extension of the DF2016 isotropic model into an anisotropic ductile fracture criterion

The paper extends the recently proposed ductile fracture criterion of DF2016 (Lou, Chen, Clausmeyer, Tekkaya and Yoon, 2017. Modeling of ductile fracture from shear to balanced biaxial tension for sheet metals. International Journal of Solids and Structures 112, 169-184) to consider the anisotropic ductile fracture behaviour of sheet metals. The DF2016 criterion is coupled with the Hill48 yield function to model the anisotropy in ductile fracture. For the verification purpose, experiments are conducted for AA6082-T6 sheet with a thickness of 1.0 mm in various loading conditions: in-plane shear, uniaxial tension, plane strain tension, and the balanced biaxial tension. Tests are carried out in the rolling direction, diagonal direction and transverse direction. Fracture strains are measured by digital image correlation along different loading directions. The anisotropic fracture of the sheet is modelled by the DF2016 criterion by incorporating an anisotropic yield function. The predicted anisotropic fracture loci are compared with experimental results in shear, uniaxial tension, plane strain tension and balanced biaxial tension. The comparison demonstrates that the anisotropic DF2016 criterion accurately models the anisotropic fracture behaviour of AA6082-T6 sheet in wide loading conditions from shear to balanced biaxial tension.

Robust multi objective optimization of anisotropic yield function coefficients

The coefficients of yield functions have been conventionally determined based on limited number of experiments such as yield stresses and/or R-values along different material orientations. In the present study, a multi objective genetic algorithm (MultiGA) based approach was implemented to obtain coefficients of anisotropic yield functions by simultaneous error minimization for yield stresses and R-values. Three frequently employed yield functions, Hill 1948, Barlat 1989 and Barlat Yld2000-2d, in the sheet metal forming simulations were considered. The performance of the determined coefficients for each yield function was judged by comparing the predicted yield stress and R-values with experimental values. Fundamental questions regarding the effect of experimental data on determining the coefficients were analyzed using the proposed approach. It is generally perceived that increase in experimental data enhances the accuracy of the determination of coefficients. Some counter intuitive results obtained on this regard are discussed. Finally, finite element (FE) simulations were performed to predict earing profile in deep drawn cups and springback profile for split-ring test, which were validated with experimental results. From the comparative study with conventional method and experiment, the yield function coefficients optimized by the MultiGA performed well for predicting the deformation behaviors of various anisotropic sheet metals.

Developing anisotropic yield models of polycrystalline tantalum using crystal plasticity finite element simulations

In this work, plastic anisotropy of polycrystalline tantalum is characterized by meso-scale crystal plasticity-finite element (CP-FE) simulations. Initial texture and grain morphology determined from electron backscatter diffraction (EBSD) measurements were incorporated into a three dimensional polycrystalline microstructure generated by grain growth simulations using kinetic Monte Carlo (kMC) Potts model. CP-FE simulations using such a polycrystalline representative volume element accurately capture anisotropic mechanical behaviors of polycrystalline tantalum. Furthermore, the CP-FE simulations are used to parameterize the classic Hill's anisotropic yield function, which is then employed in dynamic finite element simulations of Taylor impact tests. The predicted impacted cylinder shape agrees well with experimental observation, particularly in regard to anisotropic behavior. This work demonstrates a direct link between grain scale microstructural features and anisotropic mechanical behaviors of polycrystalline metals.

A new temperature-dependent anisotropic constitutive model for predicting deformation and spring-back in warm deep drawing of automotive AA5086-H111 aluminium alloy sheet

A new temperature-dependent anisotropic hardening and yield function based on non-associated flow rule (NAFR) which accurately describes the anisotropic properties of aluminium alloy sheet metal is presented in this paper. This new NAFR coupling yield function has been successfully implemented in commercial FEM software Abaqus via user material subroutine UMAT to model warm forming of AA5086-H111 aluminium alloy and good agreement was found in the predicted earing profiles and spring-back compared with experiments. The new NAFR coupling yield function has been shown be superior to the existing Hill48 (associated flow rule) yield function making it suitable for FEA of warm deep drawing and design of tooling for various automotive panels and components.

Numerical investigation into plastic deformation and failure in aluminum alloy sheet rubber-diaphragm forming

Rubber-diaphragm forming (or flexible forming) is a sheet metal forming technique using a flexible punch or die set with an elastomer. In the current study, a finite element simulation of the rubber-diaphragm forming process for an aluminum alloy sheet aircraft component was carried out. Moreover, the effects of the process and material parameters such as friction and the ductile fracture modeling were studied. Elastoplastic constitutive models of an aluminum alloy sheet with an anisotropic yield function and ductile fracture criteria and a hyperelasticity model of rubber materials were constructed. Comparisons between simulation and experiment were conducted showing good agreement. Furthermore, the numerical simulations were also found to be sensitive to the material and process parameters tested.