Comparison of homogeneous anisotropic hardening models in the case of the direct redrawing of a DP600 steel

The use of DP600, an advanced high strength steel, has gained significant attention in automotive industry, especially for complex structures that require multi-step forming operations, leading to non-linear strain path changes. From a numerical modelling perspective, the use of advanced constitutive equations has enabled a precise representation of a large range of behaviors, encompassing reverse and orthogonal strain path changes. Within this context, this study is dedicated to the numerical simulation of a two-step deep drawing process based on distortional plasticity. Two models developed within the Homogeneous Anisotropic Hardening (HAH) framework are considered. This study presents a comparison of the model predictions, calibrated over the same experimental database, in terms of their ability to predict the strain path changes and mechanical behavior of the material during the forming process. Several outputs like the punch load evolution and the strain field are compared with experimental data.


Introduction
The evolution of dislocation structures during sheet metal forming has been the driving motivation for numerous studies related to complex strain paths since the 1980s.As a consequence, the mechanical behavior of thin sheet metals has been investigated under multiaxial loadings and strain path changes [1].The first studies were dedicated to the microstructure evolution at the grain scale and later, to the modelling of the specific features at the macroscopic level in reverse and cross loadings [2].In a simplified way, though representative of the forming process, sequences composed of combinations of tension, simple shear, equibiaxial tension, plane strain tension have been used.A unified approach has led to a quantification of the strain path change magnitude, originally via the strain-based Schmitt's indicator [3].
Several phenomenological mechanical models have been developed to reproduce the features associated with strain path changes, among which distortional hardening based models referred to as Homogeneous Anisotropic Hardening (HAH) models [4,5,6].HAH-2011 is dedicated to reverse loading features, such as the Bauschinger effect and was further enhanced (HAH-2014) to consider cross loading, as well as permanent softening.Further modifications extend the model (HAH-2020) to pressure-sensitive yield criterion and improve the evolution laws of several internal variables, to provide a better description of the permanent softening.They were IOP Publishing doi:10.1088/1757-899X/1307/1/012029 2 implemented in a user subroutine for Abaqus with a non-iterative stress projection method [7,8].Though HAH-2020 was successfully used for the virtual forming of automotive parts, there is still a need to compare the performance of both models when dealing with multi-step processes.
Indeed, although strain path changes may occur in single stage process such as the cross-die test [9], multiple stage forming is closer to real industrial process and may exhibit a large range of strain path change magnitudes [10].A dedicated forming tool in two stages, inspired from the reverse redrawing test and proposed as one of Numisheet 1999 benchmark [11], was designed and used to produce cylindrical cups of DP600 [12].Convergence issues were encountered in the numerical simulation of the two stage process, using HAH-2014 model.To overcome this issue, a modification of the tool geometry in stage 2 was performed but has an influence on the load evolution.
In this paper, HAH-2014 and HAH-2020 models, both implemented with a non-iterative stress projection method, are used for the virtual forming of the 2 stage drawing process of cylindrical cups.The prediction of the mechanical behavior of a DP600 steel under reverse and cross loadings, using these 2 models, are compared.Then, the numerical predictions of the two stage forming process are compared with experimental data.

Homogeneous Anisotropic Hardening models
HAH-2014 model, previously referred to as e-HAH, is an enhanced version of the original HAH model [4] developed as an alternative to the kinematic hardening model to capture the Bauschinger effect and account for cross loading effects.HAH 20 , called HAH-2020 in this paper, includes several enhancements related to permanent softening after reverse loading and cross loading singularity highlighted in [13]; moreover, HAH-2020 has shown its superiority in the cross-loading predictions [14].A pressure-independent formulation is considered in this study.HAH-2014 and HAH-2020 depend on respectively 13 and 23 material parameters.The equations for both models are briefly described below.
A tensorial internal variable h, also called microstructure deviator, is introduced in HAH models.The deviatoric stress tensor s is projected on the microstructure deviator to quantify the magnitude of the strain path change.Normalized quantities ĥ and ŝ are also defined.The deviatoric stress tensor is additively split into two components, namely s c , which is collinear to ĥ and s o which is orthogonal to it, such that: The scalar H is equal to respectively 8/3 and 1 for HAH-2014 and HAH-2020.For latent hardening and cross loading contraction effects, two additional tensors, defined by linear transformations of the stress deviator, are defined: For HAH-2014, the values ξ L = 0 and ξ C = 4 are fixed.g L and g C are internal variables.

Yield function
The yield function contains a stable component ξ(s), which is based on the anisotropic yield function Yld2000-2d noted ϕ in this study, and a fluctuating component ϕ h , which distorts the yield surface: where σ is the effective stress, ε the plastic work-based equivalent strain, σ r a reference flow stress and q a scalar.g P is a state variable that defines the amount of permanent softening for HAH-2020; it is fixed constant and equal to unity for HAH-2014.ϕ h is defined as follows: ϕ h s, f − , f + , ĥ = f q − ĥ : s − ĥ : s q + f q + ĥ : s + ĥ : The stable component ξ is given by: where p is a scalar, fixed equal to 2 for HAH-2014.The two functions f ω , with ω ≡ − or + depend on internal variables g ω and have different expressions for the two models:

Strain path change indicator
To quantify the strain path change magnitude, a parameter cos χ is defined in a similar way to the strain-based one proposed in [3] but with a stress-based definition: cos χ = Hŝ : ĥ This parameter is equal to 1 for monotonic loading, −1 for reverse loading and 0 for the most abrupt case of cross loading.
For HAH-2014, the evolution of the microstructure deviator is given by where In HAH-2020, a second microstructure deviator h ′ is defined, as well as cos χ ′ defined by Equation 8.The evolution equations for these two microstructure deviators ĥ and ĥ′ are given by: d ĥ where λ = sgn(cos χ) and λ ′ = sgn(cos χ ′ ).ĥ evolves only when cos χ or cos χ ′ is close to one.The coefficient ξ R controls the evolution.

Internal variable evolution laws for HAH-2014
State variables g − and g + corresponds to reverse loading and g 3 and g 4 are associated with permanent softening.The evolution laws are detailed below: The two state variables g L and g C correspond respectively to latent hardening and cross loading contraction.In this paper, the latent hardening effect is neglected, based on experimental evidence for DP steels.As a consequence, g L remains constant and equal to its initial value of 1.The evolution law of g C is given by the following equation:

Internal variable evolution laws for HAH-2020
State variables related to the Bauschinger effect g − and g + evolve as: The evolution equation of g C is defined by The evolution of permanent softening in HAH-2020 model is based on the assumption that no permanent softening develops as long as cos χ ≥ 0, mostly occurring in pure reverse loading.State variable g P is introduced to account for permanent softening, as well as g 3 ≤ g P .When the sign of cos χ changes, the variables g P , g 3 and ĥ are denoted as g * P ,g * 3 and ĥ * .One more intermediate variable g ′ P is defined by which becomes the new value of g P with the condition g ′ P − g P < 0 which ensures that g P always decreases.The evolution of g 3 and another intermediate variable g ′ 3 is calculated as follows g ′ 3 becomes the new g 3 with the condition g ′ 3 − g 3 ≤ 0. In other cases, g 3 is set equal to g P .The variable g S in introduced to encounter the decrease of permanent softening when material is not subject to pure reverse loading and its evolution is given by

Reference flow stress
In this study, hybrid Swift-Voce law is used to describe the reference flow stress σ r (ε), defined as where R, K, ε 0 , n, Q 1 , Q 2 , Q 3 are material parameters and take the values provided in [15].5

Material and mechanical behavior
A dual phase steel DP600, provided in sheets of thickness 1.2 mm by ArcelorMittal, is used in this study.The ferrite-martensite microstructure of the material was observed by scanning electron microscopy and a martensite volume fraction of 14% ± 4% was measured, which is consistent with results from the literature.The mechanical behavior at room temperature and under quasi-static conditions was thoroughly characterized in uniaxial tension, biaxial tension (with hydraulic bulging), simple shear, both in monotonic loading but also with sequences involving strain path changes, like simple shear loading followed by unloading and then a reverse shear in the opposite direction.This sequence corresponds to cos χ = −1, while pre-strain in tension on large specimen followed by simple shear in the tensile direction, also included in the database for two pre-strain levels, corresponds to an abrupt change of strain path, associated with cos χ = 0.The detailed description of the specimen geometry and boundary conditions is given in [15].The coefficients for the reference flow stress (cf.Eq. 20) were optimized manually to describe both uniaxial and biaxial data, while the Yld2000-2d parameters were calculated using the wizard tool from the ESI Group.An inverse optimisation was then performed to determine coefficients for strain path changes, using forward-reverse shear tests with two values of forward shear strain, i.e., 0.084 and 0.212.Tests composed of a pre-strain in tension followed by a shear test in the same direction, with two values of tensile pre-strain, i.e., 0.049 and 0.193, were also considered.Concerning the calibration of HAH-2020, a manual identification of the parameters related to the strain path changes was performed.
The mechanical properties and plastic anisotropy coefficients are calculated from tensile tests performed for 7 directions between rolling and transverse ones.It was found that Young's modulus E = 199 GPa, the ultimate tensile strength R m = 650 MPa, the normal and planar plastic anisotropy coefficients are respectively r = 0.99 and ∆r = 0.09.

Finite element model of direct re-drawing
A two step deep drawing process of a cylindrical cup in the same direction is considered in this study.The tool dimensions are shown in Figures 1 and 2. Starting from a circular blank of diameter 80 mm, a cup of inner diameter 48 mm is first formed and then redrawn to a final cup of inner diameter 36 mm.The ABAQUS explicit solver is used for numerical simulations and both HAH models have been implemented in a user subroutines (VUMAT).In order to reduce the simulation time and taking into account of the low initial anisotropy of DP600, only a 5 • slice of the blank is utilized, as well as von Mises yield criterion.This slice was meshed with shell element (S4R) with 9 thickness integration points.The element size for the radial end of the blank was set to 1 mm while a smaller size of 0.4 mm was used in the middle of the blank.
The element size at the centre of the blank was increased up to 1 mm.The friction coefficient was set to 0.18 for all the contact areas.Symmetry boundary conditions in the global coordinate system were applied to one side of the blank, as well as to the rotated edge, in a local coordinate system.

Model calibration
Calibrated values of material parameters for both models are shown in Table 1.Values highlighted in bold font were optimized for DP600 whereas the other ones correspond to suggested values from [5,6].A comparison of both HAH predictions with experimental results for the reverse simple shear loading case are shown in Figures 3 and 4. The experimental repeatability was investigated only in the forward direction, as the exact value of shear strain when inverting the shear direction is difficult to control.It can be seen that both models give a rather accurate description of the stress level, with some slight differences during reverse loading.
In particular, HAH-2020 predicts a closer curvature in the re-yielding point than HAH-2014.A similar quality of the predictions is obtained for the other tests in the database.
Table 1.HAH-2014 and HAH-2020 parameters for DP600.Although some parameters have the same name for both models, they are related to different equations.

Numerical simulation of direct re-drawing
Numerical simulations of the forming process were performed with the help of the non-iterative stress projection method [8], which provides a reliable convergence of the calculations.Previously [12], a modification was made in the tool geometry to overcome the convergence issues when using HAH-2014.However, the current study provides more consistent results with the original tool geometries.The influence of the material mechanical model on the punch force evolution with respect to the displacement throughout the forming process is illustrated in Figures 5 and  6 respectively for stage 1 and 2. Concerning the experiments, a good repeatability was observed with small differences in the load evolution, as highlighted in Figures 5 and 6.The gap is around 2 kN on the maximum level of 80 kN for the first stage and around 1 kN for the average plateau level of 60 kN for the second stage.In addition to both HAH models predictions, and out of comparison's sake, an isotropic hardening model based on a hybrid Swift-Voce law, referred to as HSV, is also used.It can be observed that there is a fair agreement of the load levels between the experimental and numerical results in the case of stage 1 cup.An over-prediction of the load level can be seen in the case of stage 2 for the 3 models, though HSV model prediction is slightly higher than the two other curves.The influence of the friction coefficient can be a reason behind it as it was set to a constant value of 0.18 for all the contact areas.Similarly to [12], a sensitivity analysis of the friction coefficient would be very interesting to carry out.Concerning the strain field, both HAH models provide very similar distribution of the plastic strain levels.Maximum value of the equivalent plastic strain was noted very close to respectively 0.52 for stage 1 and 1.12 for stage 2.

Strain path change evolution
The stress-based expression for cos χ used in HAH models has proven to be a good indicator of the series of changes in the strain path [15].For stage 1, it seems that HAH-2014 over predicts the occurrence of cross loading, with the value of cos χ approaching 0 in 2 areas in the cup wall, while the HAH-2020 model predicts mainly monotonic or reverse strain paths, which seems reasonable for a single stage cup drawing process.Figures 7 and 8 show the distribution of cos χ at the end of stage 2. A similar over-prediction of cross loading scenario can be observed for HAH-2014 model.Moreover, the reverse strain path change observed with HAH-2020 is at a higher distance from the edge of the cup as compared with the HAH-2014 case.This comes from the principles of surface rotation by the loading path changes that are differently defined in the two models [16].A comprehensive analysis is required to better understand these differences.

Conclusions
Numerical simulations of the direct redrawing process of a DP600 steel were performed using HAH-2014 and HAH-2020 models implemented with a non-iterative stress projection method and numerical results are compared with experiments.Both HAH models estimate the behaviour of the material very accurately, however HAH-2020 seems to give a better prediction of the strain path change with the help of the improved formulation as compared to HAH-2014 model.The non-iterative stress projection method is proven to be stable and efficient, despite the high number of contact areas in this forming process.