Homogeneous Anisotropic Hardening Research Articles

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.

A Homogeneous Anisotropic Hardening Model in Plane Stress State for Sheet Metal under Nonlinear Loading Paths.

In sheet metal forming, the material is usually subjected to a complex nonlinear loading process, and the anisotropic hardening behavior of the material must be considered in order to accurately predict the deformation of the sheet. In recent years, the homogeneous anisotropic hardening (HAH) model has been applied in the simulation of sheet metal forming. However, the existing HAH model is established in the second-order stress deviator space, which makes the calculation complicated and costly, especially for a plane stress problem such as sheet metal forming. In an attempt to reduce the computational cost, an HAH model in plane stress state is proposed, and called the HAH-2d model in this paper. In the HAH-2d model, both the stress vector and microstructure vector contain only three in-plane components, so the calculation is significantly simplified. The characteristics of the model under typical nonlinear loading paths are analyzed. Additionally, the feasibility of the model is verified by the stress-strain responses of DP780 and EDDQ steel sheets under different two-step uniaxial tension tests. The results show that the HAH-2d model can reasonably reflect the Bauschinger effect and the permanent softening effect in reverse loading, and the latent hardening effect in cross loading, while the predictive accuracy for cross-loading softening remains to be improved. In the future, the HAH-2d model can be further modified to describe more anisotropic hardening behaviors and applied to numerical simulations.

A new constitutive model to describe evolving elastoplastic behaviours of hexagonal close-packed sheet metals

This study develops a new phenomenological constitutive model to capture the unique evolving cyclic elastoplastic behaviours of hexagonal close-packed (HCP) sheet metals. This new constitutive model is developed by adopting the concepts of multiple-yield surface approaches. Four phenomenological deformation modes, including Monotonic Compression (MC), Monotonic Tension (MT), Reverse Compression (RC), and Reverse Tension (RT), are considered to represent the hardening evolution of the materials, including the twining/untwining behaviours. Reference flow stress equations are introduced, and a Cazacu-Barlat 2004 (CB2004) type yield surface is employed to each deformation mode. In addition, the RT hardening parameters are defined as functions of plastic pre-strains to mitigate the interpolation error caused by parameter determination processes of existing models. For validation, the calculated stress–strain curves of AZ31B magnesium alloy are compared with experimental curves available from literature. Moreover, to show the accuracy of the proposed analytical model, the reproduced stress–strain curves are compared with those of an existing model—the modified homogeneous anisotropic hardening (HAH) model. The obtained results show that the new constitutive model can successfully reproduce experimental Tension–Compression-Tension (TCT) and Compression-Tension–Compression (CTC) stress–strain curves of HCP sheet metals with considerably less percentage errors.

Experimental characterization and modeling of complex anisotropic hardening in quenching and partitioning (Q&P) steel subject to biaxial non-proportional loadings

Quenching and partitioning (Q&P) steels, as one of the third-generation advanced high strength steels, exhibit complex anisotropic hardening behavior under biaxial non-proportional loadings. In this study, a wide range of strain path changes (SPCs) defined as an angle (χ) between stress deviators before and after the SPC is experimentally applied to characterize the anisotropic hardening of a Q&P steel with a strength grade of 980 MPa (QP980). The investigated SPCs include uniaxial tension followed by compression (UT-UC; χ=180°), uniaxial compression followed by tension (UC-UT; χ=180°), uniaxial tension followed by plane strain (UT-PS; χ=30°), equi-biaxial tension followed by plane strain (EBT-PS; χ=30°), followed by uniaxial tension (EBT-UT; χ=60°) and followed by simple shear (EBT-SH; χ=90°). These SPCs are achieved by the program-controlled biaxial tensile testing equipment using novel arm-reinforced cruciforms. The experimental findings in the QP980 steel sheet are summarized as follows: (1) Strong strain-dependent Bauschinger effect, transient hardening and permanent softening are observed in both UT-UC and UC-UT loadings; (2) Bauschinger effect increases as strain increases, and it exhibits obvious tension-compression asymmetry; (3) The transient hardening behaviors are path-dependent under SPCs with various χ; (4) The Bauschinger coefficient is approximately a parabola function of cosχ; (5) The most significant strength softening occurs under EBT-SH loading. In order to describe the complex hardening behavior of the QP980 steel, a distortional hardening model is proposed on the basis of non-associated flow rule. A coupled quadratic and stress-invariant-based yield function is used to describe stress anisotropy, strength differential effect, and anisotropic hardening for proportional loadings. Then, the Bauschinger effect related hardening and path-dependent permanent softening are captured by a modified homogeneous anisotropic hardening model. The calibrated model parameters and experimental validations demonstrate that the strain- and path-dependent Bauschinger effect can be well predicted by the proposed constitutive model. Especially, the model enables to accurately describe both the asymmetric anisotropic hardening and yield surface distortions of the QP980 steel at various biaxial SPCs.

A new concept for continuum distortional plasticity

This paper presents hexah, a framework based on the homogeneous anisotropic hardening (HAH) model. It aims at recreating the effects of arbitrary strain path changes over the plastic behavior of metals. Its core relies on the concept of microstructure deviator, a tensor representing one set of active slip systems. The present theory makes use of an arbitrary number of microstructure deviators spawned by abrupt changes of loading paths. A formalism is detailed to describe a smooth evolution from one set of activated slip systems to another. This new approach carries significant advantages: evolution equations are simplified, the number of parameters decreased, and the behavior near cross-loading stabilized.

Non-associated anisotropic plasticity of metal sheets based on the distortional concept

The anisotropic plastic behavior of metals is important and commonly necessary in industry for accurate sheet metal forming simulations. An efficient model would greatly improve the final simulation results for forming processes and component designs. In this paper, an alternative plastic method is proposed where a distortional yield surface is adopted under the non-associated flow rule (NAFR) to accommodate the complex anisotropic behavior of the stress and strain evolution. This plastic model bypasses a multistep numerical procedure in calculating the first derivative of homogeneous anisotropic hardening (HAH) [1] function. Thus, it is an efficient model with low computational cost. The capability of prediction and a detailed application are investigated considering material data of dual phase steel sheets. The plastic constraints are derived to obtain stable flow conditions during plastic deformation accumulation. The experimental calibration and nonlinear verification through tension-compression and tension-tension (including cross-loading) tests are performed, where a good prediction and agreement are observed. The integration procedure, as well as its corresponding material subroutine, is developed and implemented into the finite element program for structural simulations.

Calibration of Distortional Plasticity Framework and Application to U-draw Bending Simulations

A new version of a distortional plasticity framework, the so-called homogeneous anisotropic hardening (HAH), was investigated regarding its calibration and its application to the numerical analysis of the U-draw bend test. First, the mechanical properties of a dual-phase steel sheet sample, DP780, were characterized using uniaxial tension, bulge and elastic loading-unloading tests to provide the data for the calibration of a conventional constitutive description assuming isotropic hardening. Then, tension-compression with different load reversal numbers and two-step tension tests were performed to produce strain path changes that lead to anisotropic hardening effects. The coefficients of this new model, called HAH20, were determined with an optimization procedure. Several sets of coefficients were identified depending on the number of reversals considered in tension-compression. U-draw bend test simulations were also carried out to validate the HAH20 model and its implementation in a finite element code, to compare the results with an older version of the HAH model family, and to assess the influence of the input data used in the coefficient calibration. In addition, the influence of the cross-loading effect on the U-draw bending predictions was examined. The main conclusion of this work is that the HAH20 coefficients calibrated for the description of reverse loading depend on the number of reversals during the tension-compression tests. The results of the U-draw bending simulations indicate that this dependence may lead to significant differences in the predicted amounts of springback.

Calibration of a strain path change model for a dual phase steel

Dual phase steels are largely used to form automotive structural parts by deep drawing involving complex loading paths, which influence the material mechanical behavior and formability, as well as subsequent service life. The aim of this paper is to investigate the calibration of a strain path change model from an experimental database involving simple shear tests. The enhanced Homogeneous Anisotropic Hardening (e-HAH) is an advanced constitutive model that can take account of the strain path change influence on the material behavior by using a stress-based indicator. The performance of such a model depends highly on the material parameter identification and the experimental database. The mechanical behavior of a dual phase steel DP600 is first characterized under different linear strain paths, uniaxial and biaxial tension and simple shear, in order to quantify the initial anisotropy. Then, the influence of strain path changes on the material behavior is investigated during sequences involving simple shear, uniaxial tension and compression. Finally, the parameters of e-HAH model are identified and the influence of the experimental database on the optimised parameters is highlighted with the use of different strain paths and different strain ranges.

Reverse-loading coefficients identification for updated homogeneous anisotropic hardening model

The homogeneous anisotropic hardening (HAH) model formulation, which is solely based on distortional plasticity, was improved recently. In this study, a robust coefficient identification scheme was developed for this updated model. Tension-compression tests were conducted on a mild steel sheet sample to assess the hardening behavior in the forward-reverse loading mode. An inverse identification procedure was employed to calibrate the coefficients of the model. The Nelder-Mead simplex and genetic optimization methods were investigated and the results regarding the accuracy of the calibration were compared. As a result, both optimization methods led to reasonable coefficients for the new model.

Numerical integration algorithm of updated homogeneous anisotropic hardening model through finite element framework

In this work, an updated version of the homogeneous anisotropic hardening (HAH20) model proposed by Barlat et al. (2020) is implemented into a finite element framework through a stress integration algorithm. To improve the convergence of this model inside the integration algorithm, the algorithmic step sizes of the solution variable increments are controlled by the line-search method. The HAH20 implementation is validated by comparing the simulation results obtained from the finite element analysis with those calculated by stand-alone HAH20 code. In addition, the simulation results of tests, including strain path changes such as tension–compression and cross-loading, are compared with experimental measurements, and the effectiveness of the step size control in the current HAH20 model is investigated. The performance of the implemented algorithm is analyzed through convergence maps for critical strain path change simulations.

Distortional plasticity framework with application to advanced high strength steel

An improved distortional plasticity framework that describes the anisotropic hardening occurring during strain path changes, such as the Bauschinger and cross-loading effects, is developed. This approach is a modified version of the homogeneous anisotropic hardening (HAH) framework proposed almost a decade ago. In the present formulation, the yield condition includes an alternative description of the distortion suggested by crystal plasticity simulation results. In addition, it incorporates the influence of the hydrostatic pressure, which manifests itself by a higher flow stress in uniaxial compression than in tension. The state variable evolutions are modified compared with the previous HAH version to improve the material response when the strain path changes occur near a pure cross-loading condition. The model is calibrated on a DP780 dual-phase steel sheet sample using the data of a tension-compression test with three full cycles, as well as a sequence of two uniaxial tension segments in different directions. After calibration, predicted and experimental stress-strain curves obtained for independent loading sequences are compared and shown to be good agreement. Finally, theoretical predictions of two-step tension tests using the constitutive coefficients of DP780 are discussed to highlight the improvement offered by this enhanced framework.

On the potential of recurrent neural networks for modeling path dependent plasticity

The mathematical description of elastoplasticity is a highly complex problem due to the possible change from elastic to elasto-plastic behavior (and vice-versa) as a function of the loading path. Advanced physics-based plasticity models usually feature numerous internal variables (often of tensorial nature) along with a set of evolution equations and complementary conditions. In the present work, an attempt is made to come up with a machine-learning based model that can replicate the predictions anisotropic Yld2000-2d model with homogeneous anisotropic hardening (HAH). For this, a series of modeling problems of increasing complexity is formulated and sequentially addressed using neural network models. It is demonstrated that basic fully-connected neural network models can capture the characteristic non-linearities in the uniaxial stress-strain response such as the Bauschinger effect, permanent softening or latent hardening. A neural network with gated recurrent units (GRUs) and fully-connected layer is proposed for the modeling of plane stress plasticity for arbitrary loading paths. After training and testing the model through comparison with the Yld2000-2d/HAH model, the recurrent neural network model is also used to model the multi-axial stress-strain response of a two-dimensional foam. Here, the comparison with the results from unit cell simulations provided another validation of the proposed data-driven modeling approach.

Experimental and theoretical plasticity analyses of steel materials deformed under a nonlinear strain path

In this study, the macroscopic mechanical behavior of extra-deep drawing quality (EDDQ) and transformation-induced plasticity (TRIP1180) steels subjected to nonlinear-strain-path modes of deformation was investigated. For the nonlinear-strain-path experiments conducted on steels, an optimized experimental approach was proposed for deforming the material using two linear loading segments with a single abrupt change in the strain path. A specific medium-scale uniaxial tension specimen was designed for the pre-strain stage. The advantage of this approach over existing methods was that it enabled testing of various strain-path changes for steels with strengths greater than 1 GPa within a reasonable load range. In addition, this approach produced a more uniform strain field, which is important during the pre-strain stage. After the uniaxial pre-strain, various specimens were extracted from the uniformly deformed area for use in experiments involving changes in the strain path. To predict the material behavior under nonproportional loading, the homogeneous anisotropic hardening (HAH) distortional plasticity model was selected. The predictions of the HAH model were compared with the corresponding experimental results to assess the suitability of the constitutive model to accurately reproduce the observed effects of strain-path changes.

Validation of homogeneous anisotropic hardening model using non-linear strain path experiments

We conducted non-linear strain path experiments on a dual-phase steel-sheet sample. The first strain path in each of these experiments, called pre-strain, was always uniaxial tension in the direction transverse to the sheet (90° from the rolling direction). For this purpose, we clamped large tensile specimens in a suitable gripping system and deformed them to produce a sufficiently large area with a uniform strain distribution. We machined new specimens from this area, and we subjected them to different modes of deformation, namely uniaxial tension in different directions from the transverse direction, simple shear, and biaxial tension with different load ratios. In addition, we performed uniaxial tension-compression tests consisting of either one or three full cycles. We obtained the coefficients of a distortional plasticity model—the homogeneous anisotropic hardening (HAH) model—using an inverse analytical identification tool. First, we calibrated two sets of reverse loading coefficients using the tension-compression stress–strain curves obtained with either one or three full cycles. Then we used the 90° tension–45° tension and the 90° tension–0° simple-shear sequences to determine two sets of cross-loading coefficients. We compared the other independent experimental flow curves with simulations using the HAH model with combinations of the different sets of coefficients. These comparisons showed the influence of the selected input data on the calculations of the coefficients calibrated and demonstrated the advantages and limitations of the present HAH model.

A comparative study between elasto-plastic self-consistent crystal plasticity and anisotropic yield function with distortional hardening formulations for sheet metal forming

A comparative study between micro- and macro-mechanical constitutive models is carried out while predicting deformation behavior of an aluminum alloy (AA) 6022-T4 during several loading scenarios of increasing complexity including monotonic tension, large strain cyclic deformation, and drawing of a cylindrical cup. The micro-model is a recently developed implicit formulation of the elasto-plastic self-consistent (EPSC) crystal plasticity, which is coupled with the implicit finite element method (FEM) through a user material subroutine in Abaqus. In the coupled formulation, every finite element integration point embeds the implicit EPSC constitutive law that accounts for the directionality of deformation mechanisms and microstructural evolution. The crystallography based EPSC model integrates a dislocation-based hardening law and accounts for inter-granular and slip system level back-stresses, which make it capable of capturing non-linear unloading and the Bauschinger effect. The macro-model is a recently developed anisotropic yield function incorporating distortional hardening using the homogeneous anisotropic hardening (HAH) approach. The model is also implemented as a user material subroutine in Abaqus. Parameters pertaining to the micro and macro models are identified using experimental data from a set of monotonic and cyclic tests performed for AA6022-T4. Additional experimental data for the alloy in terms of flow stress curves, R-value, and anisotropic yield surface evolution are used to verify the models. Finally, the cup drawing simulations are carried out in the FEM using the two constitutive formulations and geometrical changes including the earing profile and sheet thinning/thickening are compared against each other and with experiments to further verify the predictive characteristics of the models. The two formulations and results are discussed in terms of accuracy and computational efficiency.

Mechanical properties, springback, and formability of W-temper and peak aged 7075 aluminum alloy sheets: Experiments and modeling

High strength aluminum alloys have drawn much attention as the requirement for lightweight vehicle design increases in the automotive industry. However, inferior formability due to higher strength poses a technical hurdle against successful application of high strength aluminum sheets in automotive components. In this study, the mechanical properties of 7075 aluminum alloy sheet after W-temper heat treatment, which consists of solution heat treatment and subsequent quenching, are investigated. Detailed experimental and numerical studies are conducted on the anisotropy, non-proportional deformation behavior, and corresponding constitutive modeling in comparison with the corresponding characteristics of peak aged T6 heat-treated as-received alloy sheets. Based on the experimentally characterized mechanical properties and associated constitutive laws, the formability and springback of the W-tempered sheet are analyzed using the Marciniak–Kuczyinski forming limit diagram and U-draw bending springback simulations, respectively, by employing the Yld2000-2d anisotropic yield function and distortional hardening-based homogeneous anisotropic hardening model. Additionally, the improved constitutive modeling of the W-tempered 7075 aluminum sheets due to the effect of serrated flow characteristics on the hardening and frictional behaviors is discussed.

Bauschinger effect modelled by yield surface distortions

A model for distorting the yield surface by flattening the part in the reverse of the loading direction, is suggested. As the basis for the distortion, the model applies a pair of second-order back-stress tensors of similar type as in kinematic hardening models. The yield-surface formulation provides a flattening and shrinkage of a given first-order homogeneous yield surface in the reverse directions of the back-stress tensors. The mathematical formulation is based on similar ideas as the HAH (homogeneous yield function-based anisotropic hardening) model, for which the calibration of the equivalent stress-strain curve is independent of the Bauschinger part of the model. Severe mathematical and numerical challenges of the HAH model are pointed out, but are avoided in the new model. Furthermore, the yield surface doesn't have to contain the origin, and the r-value is conserved in stress reversals.

Combined anisotropic and distortion hardening to describe directional response with Bauschinger effect

Directional anisotropic hardening under non-proportional loading is modeled. A recently proposed coupled quadratic-nonquadratic (CQN) yield function (Lee et al., 2017b) successfully captured the directional hardening behavior under the proportional loadings. This paper shows a constitutive equation to capture both directional hardening response and Bauschinger effect through an extended model of the CQN yield function with the homogeneous anisotropic hardening (HAH) model (Barlat et al., 2011). In the present study, the role of the CQN yield model is to capture the directional hardening behavior, and the HAH model is to follow non-proportional loading including the Bauschinger effect. The validation, with some material data, shows that the present model can follow the directional hardening under non-proportional loading in the stress-strain data. It is also shown that the present model can describe the yield surface distortion with Bauschinger and anisotropic hardening response. Finally, it is compared with other pre-existing models in order to analyze the utility of the proposed model.

Pre-strain effect on twist springback of a 3D P-channel in deep drawing

Abstract With the widely use of high-strength steel sheets in the automotive industry, the twist springback phenomenon of the steel sheets under multi-step forming conditions has received extensive attention. In this work, the Dual Phase steel DP500 is taken as the research object to investigate the complex non-linear elastoplastic behaviors and twist springback under two-step loading paths. The large specimen with a pre-strain of 4% true strain in rolling direction is carried out on a large tensile testing machine, and several specific blanks are extracted from it at different directions for a subsequent P-channel forming. The influence of twist springback associated with the pre-strain is analyzed. The finite element model based on the non-linear elastic model and the homogeneous anisotropic hardening model (HAH) is also established for the springback prediction and stress analysis. The results indicate that the pre-strain has a considerable impact on the twist springback. The non-linear strain path changes resulted from pre-straining not only influence the residual stress but also affect the elastic modulus distribution.

Determining the coefficients of a homogeneous anisotropic hardening model for ultrathin steel sheets

In the present work, the constitutive description of a 0.1-mm-thick ferritic stainless steel sheet was optimized for application to springback simulations after bending. Springback simulations require Bauschinger effects to be characterized carefully, which is a challenging task with an ultrathin sheet. The coefficients of the homogeneous anisotropic hardening (HAH) distortional plasticity model were calibrated with Zang's novel approach based on three-point bending conducted on pre-strained sheets (Zang et al., 2014). The Hockett–Sherby isotropic hardening law and the Yld2000-2d non-quadratic yield function were considered in this work to complete the HAH approach. Moreover, the degradation of elastic modulus was also accounted for in the finite-element simulations. The coefficients of the HAH model were calibrated using an inverse method by minimizing the difference between experimental and predicted specimen profiles after three-point bending and springback of pre-strained sheets. To validate the coefficients determined with this three-point bending test, U-draw bending tests were conducted and finite-element simulations were carried out. The springback predictions were found to agree well with the experimental results for all three pre-strains investigated, namely, 0% (as-received material) and 7.5% and 12.5% pre-strained ferritic stainless steel sheets.