Anisotropic Hardening Model Research Articles

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.

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.

Interpolation-based anisotropic yield and hardening models

In this paper, an interpolation-based anisotropic yield model and a corresponding anisotropic hardening model are proposed. The proposed yield model is an improvement on Vegter and van den Boogaard (2006)'s yield model. By introducing shape parameters into Bézier basis function to control the curvature of the curve, the number of interpolation curve segments is effectively reduced. The proposed yield model has very good material applicability (suitable for BCC, FCC, HCP materials) and can directly generate the tricomponent yield surface without any numerical fitting. In addition to being applied to capture tension-compression asymmetry, the model can also be extended to describe non-orthotropy, which is usually not realized by existing yield criteria. Based on the proposed yield model, a hardening model based on characteristic yield points tracking is proposed. The so-called characteristic yield points correspond to the uniaxial loading and equi-biaxial loading yield points for constructing the yield surface. By tracking the evolution curve of yield data (stress and r-value) with specific variables (e.g. equivalent plastic strain), the yield surface at different variable levels can be continuously predicted. The proposed hardening model can not only predict the evolution of yield surface with equivalent plastic strain under proportional loading, but also predict the effects of different pre-strains and preloading paths on yield surface, and can also couple other variables (e.g. strain rate) to comprehensively consider the effects of multiple factors.

An Identification Method for Anisotropic Plastic Constitutive Parameters of Sheet Metals

In this work, the virtual fields method (VFM) was extended to simultaneously identify the constitutive parameters of Hill 1948 anisotropic yield criterion and nonlinear kinematic hardening models for rolled sheet metals from a low cycle tension-compression test. First, simulations of different specimens under loading were conducted to determine a specific configuration which exhibits both insignificant buckling effect and heterogeneous stress state. Then, the simulated data were utilized to retrieve the target anisotropic constitutive parameters in order to validate the proposed VFM identification scheme. The results show that the input parameters can be satisfactorily regained from the designed tension-compression configuration and the VFM algorithm, verifying the feasibility of using this method to real experiments.

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.

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.

Effects of reverse loading on forming limit predictions with distortional anisotropic hardening under associated and non-associated flow rules

The recent development of metal sheet plastic theory shows that it is essential to consider the distortional behavior of yield surfaces when accounting for microstructure evolution and Bauschinger effects during plastic deformation under non-proportional loading. This methodology is especially applicable to plastic flows and reveals the distinct flow direction involved during loading reversal. Since localized necking is heavily affected by plastic flows, the evaluation of its impact in forming limits during reverse loading is necessary and important for the further application and understanding of this distortional behavior of metal sheets. For this work, the Marciniak–Kuczynski (MK) approach (Marciniak and Kuczynski [1]) is adopted and further extended with the homogeneous anisotropic hardening (HAH) model (Barlat et al. [2]) under associated and non-associated flow rule frameworks to accommodate complex reverse loading behavior. The evolution of the yield surface, stress transitions and necking angles is examined and compared in detail. Path independent forming limits are also investigated and probed to test their performance. When loading is reversed, the distortional curvature evolution of the yield surface has a critical impact on stress flow trajectories and yield surface normal and further determines final forming limits under associated flow rules, rendering its performance different from that of traditional isotropic and kinematic hardening. The non-associated flow rule framework can be used to measure the impact of twisted yield behavior and untwisted plastic potential, which is suggested to be a useful numerical approach to the theoretical study and engineering prediction of forming limits.

The strain hardening of micro-sized face-centered-cubic single crystal metals: A crystal plasticity study

Recent experimental and theoretical studies indicate the “external geometry” and “internal structure” are the controlling factors to characterize the size-dependent plastic flow for micro-sized single crystal metals. i.e., increasing yield strength with decreasing sample size. This paper aims to investigate the strain hardening behavior for micro-sized face-centered-cubic (FCC) metals by employing crystal plasticity approach. The size-dependent dislocation density evolution law incorporates the dimensional parameters of internal structure and external geometry in order to consider the dislocation surface annihilation and dislocation source controlled plastic mechanisms. The current shear yield strength for micro-sized FCC metals is formulated by introducing the dislocation source length in classical anisotropic hardening model of Franciosi type. Combined with the above two aspects, crystal plasticity finite element method is developed to examine the strain hardening behavior of single crystal micro-sized metals during uniaxial compression. The validity of the new theoretical formulation and modeling approach is checked by comparing the simulation results with the experimental results available in literatures. The simulation results indicate that the micro-sized FCC metals have significant size effect. Further studies are performed to evaluate the influences of the initial dislocation density and the number of dislocation sources on strain hardening behavior of micro-sized FCC metals.

Stress integration-based on finite difference method and its application for anisotropic plasticity and distortional hardening under associated and non-associated flow rules

Stress integration algorithm based on finite difference method (FDM) was proposed to effectively deal with both first and second derivatives of yield and potential functions which are the lengthiest component in stress integration procedure. With the proposed numerical algorithm, both first and second derivatives of yield function are approximated by central difference method, so that finite element modeling using advanced constitutive model could be easily performed no matter how complicated its derivatives are. For the verification purpose, the algorithm was applied for advanced constitutive models: Plastic anisotropy model under associated (AFR) and non-associated flow rule (non-AFR), the homogeneous anisotropic hardening (HAH) model under associated (AFR) and non-associated flow rule (non-AFR). The proposed algorithm was verified with single element loading–unloading and cup-drawing simulations. The Euler backward method based on both the proposed numerical algorithm and analytical derivatives were employed for verification purpose. The accuracy and time efficiency of the proposed numerical algorithm were evaluated by comparing the simulation results from analytical derivatives. In addition, the applicability of the proposed numerical algorithm for the HAH models was estimated with single element loading–unloading, loading–reloading, and deep-drawing/springback simulations. Non-associated flow plasticity for the HAH model is newly proposed to improve numerical efficiency with finite difference method by keeping the same level of accuracy as associated flow rule plasticity. All the simulation results proved that the proposed numerical algorithm can be widely used for the implementation of advanced constitutive models.

Forming Limits Under Stretch-Bending Through Distortionless and Distortional Anisotropic Hardening

The necking behavior of sheet metals under stretch-bending process is a challenge for the forming limit prediction. State-of-the-art forming limit curves (FLCs) allow the prediction under the in-plane stretching but fall short in the case under out-of-plane loading condition. To account for the bending and straightening deformation when sheet metal enters a die cavity or slide along a radius, anisotropic hardening model is essential to reflect the nonproportional loading effect on stress evolution. This paper aims to revisit the M-K analysis under the stretch-bending condition and extend it to accommodate both distortionless and distortional anisotropic hardening behavior. Furthermore, hardening models are calibrated based on the same material response. Then the detailed comparison is proposed for providing better insight into the numerical prediction and necking behavior. Finally, the evolution of the yield surface and stress transition states is examined. It is found that the forming limit prediction under stretch-bending condition through the M-K analysis strongly depends on the employed anisotropic hardening model.

Analysis of yield locus description on springback behaviour of CR700Y980T-DP steel

To precisely predict springback, the stress and strain state has to be described correctly during the forming process and after load removal. In this work, different yield locus descriptions are considered for springback prediction of a CR700Y980T-DP steel with a thickness of 1.5 mm. Experiments and Finite Element Analyses (FEA) with LS-DYNA of a U-bending validation tool were performed and springback predictions compared. In the FEA Hill48, Barlat89 and Barlat’s YLD2000 for isotropic hardening, Yoshida-Uemori as a kinematic hardening model and the homogeneous anisotropic hardening (HAH) model are chosen. Monotonic and two-step tensile tests with a directional change of stress are performed and cyclic tension-compression tests are used to determine the material parameters of the investigated hardening models. It is seen that both Yoshida-Uemori and HAH models provide a significant improvement in the springback prediction. The results, however greatly depend on the identification procedure as well as on the modelling of cross-loading contraction effects.

Stress update algorithm based on finite difference method and its application to homogenous anisotropic hardening (HAH) model

This paper deals with stress update algorithm based on finite difference method. The proposed algorithm is based on Euler backward method with multi-step return mapping approach. Central difference method was utilized for the approximation of the first and second derivatives of yield function. With the proposed algorithm, it is possible to perform elastic-plastic finite element simulation without any analytical derivatives of yield function. General yield functions including Hill’s (1948) and Yld2000-2d, and also the HAH distortion hardening model were implemented to an implicit finite element code (ABAQUS/STANDARD). For the verification purpose, various finite element simulations were performed. With the anisotropic functions, single element loading-unloading and cup-drawing simulation were carried out. The results obtained from the proposed algorithm were compared with the results from analytical derivatives and reference data. The availability of the proposed algorithm for distortional plasticity such as HAH model was evaluated by single element loading-reloading simulations: tension-compression and tension-orthogonal tension. The effectiveness of the proposed algorithm compared to the classical Euler backward method was identified from the simulation results.

FE implementation of HAH model using FDM-based stress update algorithm for springback prediction of AHSS sheets

The homogeneous anisotropic hardening (HAH) model was implemented into a finite element (FE) code in order to predict springback for an advanced high strength steel (AHSS) sheet sample after double-stage U-draw bending. The finite difference method (FDM) was utilized as an alternative way to calculate the derivatives of this advanced distortional plasticity model allowing the update of the equivalent plastic strain and stress tensor at each time step in the user-material subroutines (UMAT and VUMAT). The FDM makes it easier to derive the stress gradient of complex yield surfaces. The proposed FDM-based stress update algorithm was verified by comparing the springback profiles after the single- and double-stage U-draw bending tests for a DP980 sheet sample predicted with analytical and numerical approaches. In addition, the springback measurement parameters and computational efficiencies depending on both approaches were also compared. The results indicate that the computational efficiency and accuracy of the FE simulations with the FDM-based stress update algorithm were similar to those of the analytical method.

A numerical scheme of convex yield function with continuous anisotropic hardening based on non-associated flow rule in FE analysis of sheet metal

A non-associated flow rule (NAFR) model is developed by adopting the convex function YLD-2004 as the yield stress function and the plastic potential function. The yield stress function coefficients are continuously updated which are associated with the change of directional uniaxial yield stress and biaxial yield stress to simulate the anisotropic hardening behaviour. That was achieved by implementing the numerical identification procedure of coefficients into stress integration procedure. The coefficients of plastic potential function are constant and identified by the uniaxial and biaxial r-value. This constitutive model is capable of describing anisotropic hardening and yield behaviour of strongly textured aluminium alloy sheet metal. In this paper, the model was implemented into the FE code via ABAQUS subroutine to predict the deep drawn cup earing and directional flow stresses of the AA5042-H2 aluminium alloy. The new anisotropic hardening model shows better agreement with experiments compared with the isotropic hardening model.

Constitutive modeling of principal stress rotation by considering inherent and induced anisotropy of soils

In order to simulate the soil response during principal stress rotation, anisotropic unified hardening (UH) model is developed within the framework of elastoplastic theory. Without introducing any additional mechanism to display the role of stress rotation specifically, this model achieves the simulation by considering the material anisotropy. The effect of inherent anisotropy is reflected using the anisotropic transformed stress method, but a new formula for the stress mapping is adopted to keep the mean stress unchanged. Analysis indicates that from the view of the transformed stress tensor, the anisotropic soil is subjected to loading during pure rotation of principal stress axes, so that plastic strains can be calculated. To represent the induced anisotropy, a fabric evolution law is proposed based on laboratory and numerical test results. At the critical state, the fabric tensor reaches a stable value determined by the stress state, while the critical state line is unique in the plane of void ratio versus mean stress. The anisotropic UH model has concise formulation and explicit elastoplastic flexibility matrix and can provide reasonable predictions for the deformation of anisotropic soils when principal stresses rotate.

Influence of Yield Stress Determination in Anisotropic Hardening Model on Springback Prediction in Dual-Phase Steel

In this study, a numerical sensitivity analysis of the springback prediction was performed using advanced strain hardening models. In particular, the springback in U-draw bending for dual-phase 780 steel sheets was investigated while focusing on the effect of the initial yield stress determined from the cyclic loading tests. The anisotropic hardening models could reproduce the flow stress behavior under the non-proportional loading condition for the considered parametric cases. However, various identification schemes for determining the yield stress of the anisotropic hardening models significantly influenced the springback prediction. The deviations from the measured springback varied from 4% to 13.5% depending on the identification method.

A constitutive model for transversely isotropic material with anisotropic hardening

Many materials such as wood, as well as rigid and flexible foams, exhibit transverse isotropic or even anisotropic mechanical properties in terms of their elastic and inelastic responses. However, there appear to be few constitutive models that are able to appropriately and accurately describe them. In this study, constitutive model that describes transversely isotropic material displaying anisotropic hardening, based on a weak form of definition of anisotropic hardening, is proposed. This form of hardening is incorporated into a modified yield function by introducing a matrix comprising dimensionless hardening functions. A modified stress is also defined, such that the yield surface remains stationary in the associated stress space, thus enabling the proposed anisotropic hardening function to be evaluated. The model is then used to predict the stress-strain curves and deformation of a transversely isotropic polyethylene foam exhibiting transversely isotropic hardening, subjected to compression at different angles to the axis perpendicular to the plane of isotropy. Good agreement between predictions based on the proposed model and Jebur’s experimental data (Jebur, 2013) is observed; for comparison, predictions employing an earlier transversely isotropic model but with isotropic hardening (Tagarielli et al., 2005) shows noticeable deviation from the experimental data. The results substantiates the validity of the proposed anisotropic hardening model in describing the mechanical and deformation response of a transversely isotropic material exhibiting transversely isotropic hardening.

Effect of anisotropy evolution on circular and oval hole expansion behavior of high-strength steel sheets

The circular and oval hole expansion behaviors of JSH440W and JSH590R high-strength steel sheets were investigated based on experiments and finite element (FE) simulations using three types of anisotropic yield functions: Hill48-r, Yld2000-2d, and 6-poly models. The effects of anisotropy evolution were specifically examined by performing FE simulation using the 6-poly anisotropic hardening (AH) model. Sheet-direction dependent workhardening characteristics were determined by using an in-plane stretch-bending technique. From this work, it was found that hole expansion behavior was strongly affected by anisotropy and its evolution. Thus, it is concluded that the use of AH model is essential for the accurate simulation of hole expansion processes.