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

https://doi.org/10.1016/j.ijplas.2019.07.007Get rights and content

Highlights

  • Both anisotropic hardening and non-proportional loading are modeled simultaneously.

  • CQN yield model is to capture the directional hardening, and the HAH model is to follow the Bauschinger effect.

  • The present model can describe the yield surface distortion with Bauschinger and anisotropic hardening response.

Abstract

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.

Introduction

When material goes through a cycling loading, for example, tension followed by compression, the material presents an asymmetric behavior between the tension and compressive responses (Yoshida et al., 2002; Cardoso and Yoon, 2009; Vladimirov et al., 2010; Zhu et al., 2014, 2017; Marcadet and Mohr, 2015). This asymmetry is called the Bauschinger effect, and it is important to capture the Bauschniger effect for predicting springback. In the meanwhile, the anisotropic behavior is the difference of yield stress with respect to the angle to the rolling direction (RD) of the sheet, and it occurs even under monotonic loading conditions. The stress anisotropic behavior affects the springback prediction (Kuwabara et al., 2004; Zang et al., 2011) and the stress-based forming limit (Stoughton, 2000; Stoughton and Yoon, 2005). The forming limit of the Marciniak-Kuczynski (M-K) model is also affected by anisotropic yield functions (Ozturk et al., 2014; Bandyopadhyay et al., 2017). Consequently, the anisotropy in yielding is one of the key factors to predict the formability and springback.

In order to conduct an accurate numerical analysis, both the Bauschinger effect and directional hardening response should be modeled. Furthermore, the Bauschinger effect is sometimes in coupling with hardening models. Since the isotropic hardening model is not able to describe the Bauschinger effect, Prager (1956) and Ziegler (1959) introduced the kinematic hardening principle in order to capture the Bauschinger behavior through the concept of the back stress which makes a translation of yield surface. Since the above two models only have the linear kinematic hardening behavior, Armstrong and Frederick (1966) came with a nonlinear kinematic hardening term, and Chaboche (1986) brought a general nonlinear kinematic hardening model. Later, different approach is proposed with two-yield surface models (Yoshida and Uemori, 2002, 2003; Lee et al., 2007) to improve the accuracy of the simulation. In addition, Sun and Wagoner (2011) proposed the quasi-plastic-elastic (QPE) model having two surfaces in order to capture the nonlinear elastic behavior. Lee et al. (2017a) also tried to capture both the nonlinear elastic and asymmetric plastic behaviors with only one yield surface. The mentioned models are based on the kinematic hardening principle that lead to a translation of the yield surface. Alternatively, Barlat et al. (2011, 2013; 2014) introduced the homogeneous anisotropic hardening (HAH) model in order to capture the Bauschinger effect without kinematic hardening, and the HAH model have been employed in several studies (Fu et al., 2016; Liao et al., 2016). Choi and Yoon (2019) recently showed that the HAH model works with the non-associated flow rule as well.

For describing the anisotropic (or directional) yielding behavior, it needs to employ an appropriate yield function. Hill (1948) proposed a quadratic yield function, and Hosford (1979) came with a non-quadratic yield function with anisotropic coefficients to follow the anisotropy of yielding behavior. Since non-quadratic yield function has the ability to control the curvature of the yield surface while quadratic model only has a fixed curvature, some studies have improved non-quadratic yield function (Barlat and Lian, 1989; Chung and Shah, 1992; Barlat et al., 1991, 1997). One of the most widely used yield functions is also the one proposed by Barlat et al. (2003). This yield function is called the Yld2000-2d model, and has been applied to many sheet metal forming simulations (Dick and Korkolis, 2015; Zhang et al., 2016; Kuwabara et al., 2016). The advantage of the Yld2000-2d model is to simultaneously account for anisotropy of both the yield stresses and r-values along 0°, 45°, 90° to the RD and the equal biaxial (EB) condition. The limitation is that this model is not able to capture the directional hardening response because the ratios of anisotropy are fixed in this model.

In order to resolve this issue, many models have been introduced with the principle of interpolation and optimization at several levels of plastic strain to change the ratio of the anisotropy (Hill and Hutchinson, 1992; Kuroda and Tvergaard, 2000; Wang et al., 2009; Yoshida et al., 2015; Khadyko et al., 2016; Habib et al., 2017; Raemy et al., 2017). Stoughton and Yoon (2009) proposed an alternative model, which does not need either interpolation or optimization. This model was able to capture the anisotropic hardening by explicitly fitting four hardening data in the different directions, 0°, 45°, 90° to the RD and the EB condition. But the limitation is that this model is not able to control its curvature since it is based on the Hill (1948)'s quadratic function. Min et al. (2016) also adopted the Stoughton and Yoon's (2009) scheme. Lee et al. (2017b) recently came with an improved model, called coupled quadratic-nonquadratic (CQN) yield model in this paper, by coupling the Stoughton and Yoon's model (2009) with the Hosford (1972)'s nonquadratic function. The CQN model is able to control the curvature of yield surface as well as capturing the directional hardening behavior (Lee et al., 2017b). However, this model itself had no ability to follow the Bauschinger effect since the CQN model was based on the isotropic hardening assumption. In order to capture the Bauschinger effect as well as the directional hardening, Lee et al. (2018) proposed a condition function by modifying the Stoughton and Yoon's (2009) function, and this condition function was combined with the Chaboche (1986) hardening model in order to provide different kinematic hardening parameters according to the stress state that is affected by the angle to RD; this model is called directional kinematic hardening model in this work. The results showed that the condition function is able to follow both directional hardening and Bauschinger effect with Yld2000-2d yield function. However, since this condition function has many parameters to account for the back stress terms according to loading direction, it requires the calibration of back stress terms four times according to the direction (0°, 45°, 90° and EB condition). In addition, the computation efficiency of this model is low.

This paper proposes an alternative approach of the CQN yield function (Lee et al., 2017b) by combining with the HAH model (Barlat et al., 2011) in order to capture both the Bauschinger and directional hardening response. The role of the CQN yield function is to capture the directional hardening behavior, and the HAH model is to account for the Bauschinger effect. In this work, the present model was implemented into the User-defined Material model (UMAT) of the ABAQUS software and validated with monotonic tension and cycling loading data of two materials by one shell element simulations. Two materials were 780R AHSS data, from a reference paper (Yoshida et al., 2015), and measured MP980 data. The MP980 material has tension data from 0° to 90°at the interval of 15° and an additional EB tension data. The MP980 sheet also has a compression-tension-compression (C-T-C) data at RD of the MP980 sheet. The 780R AHSS data includes 0°, 45°, 90° angles of monotonic tensions and a cycling of tension-compression-tension (T-C-T) data. In the validation, the original CQN yield model (Lee et al., 2017b), the directional kinematic hardening model (Lee et al., 2018), and the original HAH model (Barlat et al., 2011) were additionally incorporated. The results show that the present model is able to follow both the Bauschinger and directional hardening behaviors. Based on these results, a bulge test simulation with AA5182 sheet was conducted to verify whether the present model can lead to better stress-strain prediction than the original HAH model in forming simulation. Finally, the comparison of the models presents the discussion of an appropriate situation for each model.

Section snippets

Constitutive equation

The CQN yield function is expressed in the plane stress condition as below (Lee et al., 2017b):fCQNσ,εP=fQuadσ,εP·fNonquadσ1n+2,where fQuadσ,εP=σ11σ0n+2εP-σ22σ90n+2εPσ11-σ22+σ11σ22-σ12σ12σEBn+2εP+4σ12σ12σ45n+2εP,fNonquadσ=12|σ||n+12|σ|n+12|σ|-σ|n

When fCQNσ,εP<1, the material is in the range of elastic deformation. But when fCQNσ,εP=1, the material is going through elastic-plastic deformation. The quadratic part employs a function proposed by Stoughton and Yoon (2009) based on the

Model calibration

In order to describe the directional hardening behavior, the flow stress model of Eq. (2) should be fitted to the four hardening data along the 0°, 45°, 90° and the EB condition. The present model is calibrated with 780R AHSS and MP980 sheets. The MP980 data has tensile tests results at 8 directions (from 0° to 90° at 15° interval, and the EB condition). Among them, MP980 sheet has the 0°, 45°, 90° and the EB condition data. The 780R AHSS sheet has only the 0°, 45°, 90° tensile test data from a

Results and discussions

Fig. 2(a–d) compares the reference data with the results of the four models - the present model, original HAH model, original CQN model, and directional kinematic hardening model - in the simple tension conditions of 780R AHSS sheet; all of the model parameters of the directional kinematic hardening can be found in the earlier work (Lee et al., 2018). Fig. 3(a–d) shows the comparison of the cycling loading data of 780R AHSS with the four models. Since the monotonic tension conditions do not

Conclusion

This paper showed a constitutive equation to capture the Bauschinger effect and directional hardening response through a combined model of the CQN yield function with the HAH model. The monotonic and cycling loading conditions of 780R AHSS and MP980 sheets validated the present model's ability to account for the Bauschinger effect and directional hardening. The bulge test simulation showed that the present model can be employed in the forming simulation. The comparison of the present model to

Acknowledgments

This work was supported by Ministry of Trade, Industry and Energy in Korea (Grant number: 20004365). The authors are grateful for the supports.

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