Characterizing the mechanical deformation response of AHSS steels: A comparative study of cyclic plasticity models under monotonic and reversal loading

1. INTRODUCTION

Advanced high-strength steels (AHSS) are extensively utilized in the automotive industry due to their high strength, good ductility, and reduced weight (Keeler et al., 2017Keeler, S., Kimchi, M., Mooney, P.J. (2017). Advanced High-Strength Steels Application Guidelines. Version 6.0, World Auto Steel. https://www.worldautosteel.org/download_files/AHSS%20Guidelines%20V6/00_AHSSGuidelines_V6_20170430.pdf ). The increased usage of AHSS in car body structures allows for weight reduction while maintaining structural integrity (Shan et al., 2018Shan, T.K., Li, S.H., Zhang, W.G., Xu, Z.G. (2018). Prediction of martensitic transformation and deformation behavior in the TRIP steel sheet forming. Mater. Des. 29 (9), 1810-1816. https://doi.org/10.1016/j.matdes.2008.03.023.). In this context, understanding the formability and mechanical behavior of AHSS steels is paramount. During sheet metal forming processes, such as bending and unbending over die/punch radii or drawing beads, the sheet materials experience cyclic loading. Accurately describing the hardening behavior during cyclic loading is crucial for reliable finite element analyses (FEA) of these processes.

The Bauschinger effect, characterized by a decrease in yield strength when the load is reversed, is a significant phenomenon observed during cyclic and reversal loading conditions (Dieter, 1988Dieter, G.E. (1988). Mechanical Metallurgy. SI metric ed. New York, USA, Mc-Graw Hill Book Company.).

Traditional isotropic hardening models fail to adequately capture the Bauschinger effect, as they assume a uniform expansion of the yield surface during hardening. To accurately represent the Bauschinger effect, kinematic hardening models are employed in cyclic loading cases. These models translate the yield surface within the stress space during hardening, allowing for a more precise description of the material’s behavior. The location of the yield surface is determined by the back stress tensor (α) (Banabic, 2010Banabic, D. (2010). Sheet Metal Forming Processes - Constitutive Modelling and Numerical Simulation. Berlin, Germany, Springer.). Several kinematic hardening models have been proposed, including linear models by Prager (1956)Prager, W. (1956). A new method of analyzing stresses and strains in work hardening plastic solids. ASME J. Appl. Mech. 23 (4), 493-496. https://doi.org/10.1115/1.4011389. and Ziegler (1959)Ziegler, H.A. (1959). A modification of Prager’s hardening rule. Q. Appl. Math. 17 (1), 55-65., multilinear models by Besseling (1958)Besseling, J.F. (1958). A theory of plastic and creep deformations of an initially isotropic material Showing Anisotropic Strain-Hardening, Creep Recovery, and Secondary Creep. J. Appl. Mech. 25 (4), 529-536. https://doi.org/10.1115/1.4011867., Mroz (1967)Mróz, Z. (1967). On the description of anisotropic work hardening. J. Mech. Phys. Solids. 15 (3), 163-175. https://doi.org/10.1016/0022-5096(67)90030-0., Dafalias and Popov (1976)Dafalias, Y.F., Popov, E.F. (1976). Plastic internal variables formalism of cyclic plasticity. J. Appl. Mech. 43 (4), 645-651. https://doi.org/10.1115/1.3423948., and the nonlinear models by Armstrong and Frederick (2007)Armstrong, P.J., Frederic, C.O. (2007). A mathematical representation of the multiaxial Bauschinger effect. Mater. High Temp. 24 (1), 1-26. https://doi.org/10.1179/096034007X207589. and Chaboche and Rousselier (1983)Chaboche, J.L., Rousselier, G. (1983). On the plastic and viscoplastic constitutive equations - Part I: Rules developed with internal variable concept. J. Press. Vessel Technol. 105 (2), 153-158. https://doi.org/10.1115/1.3264257..

Numerous studies have investigated the definition of material behavior under cyclic loading using kinematic hardening models. Koo and Lee (2007)Koo, G.H., Lee, J.H. (2007). Investigation of ratcheting characteristics of modified 9Cr-1Mo steel by using the Chaboche constitutive model. Int. J. Press. Vessel Pip. 84 (5), 284-292. https://doi.org/10.1016/j.ijpvp.2006.12.002. examined the ratcheting characteristics of modified 9Cr-1Mo steel using the Chaboche constitutive model, demonstrating the model’s capability to describe cyclic behavior. Meyer et al. (2018)Meyer, K.A., Ekh, M., Ahlström, J. (2018). Modeling of kinematic hardening at large biaxial deformations in pearlitic steel. Int. J. Solids Struct. 130, 122-132. https://doi.org/10.1016/j.ijsolstr.2017.10.007. investigated kinematic hardening models for large biaxial deformations in pearlitic steel. They found that combining the Ohno and Wang (1993)Ohno, N., Wang, J.-D. (1993). Kinematic hardening rules with critical state of dynamic recovery. Part I: formulation and basic features for ratchetting behavior. Int. J. Plast. 9 (3), 375-390. https://doi.org/10.1016/0749-6419(93)90042-O. model with the Burlet and Cailletaud (1986)Burlet, H., Cailletaud, G. (1986). Numerical techniques for cyclic plasticity at variable temperature. Eng. Comput. 3 (2), 143-153. https://doi.org/10.1108/eb023652. radial evanescence term improved the prediction of material response during cyclic loading. Joo and Huh (2017)Joo, G., Huh, H. (2017). Modeling of rate-dependent hardening behaviors of the TWIP980 steel sheet in tension and compression. Procedia Eng. 207, 143-148. https://doi.org/10.1016/j.proeng.2017.10.752. and Joo and Huh (2018)Joo, G., Huh, H. (2018). Rate-dependent isotropic-kinematic hardening model in tension-compression of TRIP and TWIP steel sheets. Int. J. Mech. Sci. 146-147, 432-444. https://doi.org/10.1016/j.ijmecsci.2017.08.055. developed rate-dependent isotropic-kinematic hardening models based on Zang’s isotropic-kinematic model (Zang et al., 2011Zang, S.L., Guo, C., Thuillier, S., Lee, M.G. (2011). A model of one-surface cyclic plasticity and its application to springback prediction. Int. J. Mech. Sci. 53 (6), 425-435. https://doi.org/10.1016/j.ijmecsci.2011.03.005.) to accurately describe the hardening behavior of TWIP980 and TRIP980 steels at high strain rates. Skibicki et al. (2022)Skibicki, D., Pejkowski, L., Karolczuk, A., Seyda, J. (2022). Verification of the Tanaka non-proportional isotropic cyclic hardening model under asynchronous loading. Int. J. Solids Struct. 254-255, 111896. https://doi.org/10.1016/j.ijsolstr.2022.111896. coupled Tanaka and Chaboche hardening models to extend the usage of the Chaboche model under asynchronous loading for E235 and E335 steels. They concluded that the proposed model can be implemented for fatigue applications. Apart from the cyclic deformation of the materials, kinematic hardening is also essential for sheet metal forming operations. One of the critical issues in forming processes is the springback defect, and the kinematic hardening approach is also able to predict this behavior. Firat (2008)Firat, M. (2008). A numerical analysis of sheet metal formability for automotive stamping applications. Comput. Mater. Sci. 43 (4), 802-811. https://doi.org/10.1016/j.commatsci.2008.01.068. and Firat et al. (2013)Firat, M., Karadeniz, E., Yenice, M., Kaya, M. (2013). Improving the accuracy of stamping analyses including springback deformations. J. Mater. Eng. Perform. 22, 332-337. https://doi.org/10.1007/s11665-012-0257-5. successfully applied plasticity models incorporating material anisotropy and the Bauschinger effect to predict springback and formability in sheet metal forming processes. He et al. (2022)He, W., Meng, B., Song, B., Wan, M. (2022). Grain size effect on cyclic deformation behavior and springback prediction Ni-based superalloy foil. Trans. Nonferrous Metal. Soc. China 32 (4), 1188 - 1204. https://doi.org/10.1016/S1003-6326(22)65866-7. conducted a comparative study of kinematic hardening models of springback prediction in the U-draw bending process. They concluded that the prediction performance was affected adversely with increased grain size. Yu et al. (2023)Yu, K., Choi, H., Ha, J., Yoon, J.W. (2023). Bauschinger effect calibration by the different types of loading/reverse loading tests for springback prediction in sheet metal forming. Int. J. Mater. Form. 16, 16. https://doi.org/10.1007/s12289-023-01738-3. employed a nonlinear kinematic hardening model to predict the springback behavior of AHSS steels in the U-draw bending process under different blank holder forces and friction coefficients. They obtained accurate results for predicting the Bauschinger effect and springback profile of AHSS steels. Besides, kinematic hardening can also be utilized in cup drawing processes to predict earing defects. Aksen and Firat (2023)Akşen, T.A., Firat, M. (2023). Effect of the yield surface evolution on the earing defect prediction. Rev. Metal. 59 (1), e235. https://doi.org/10.3989/revmetalm.235. also implemented an anisotropic yield criterion in conjunction with the isotropic hardening model and combined isotropic-Armstrong-Frederic kinematic hardening model separately to capture the earing defect of AA6016-T4 alloy in a cup drawing process. It was seen that the earing prediction performance was noticeably increased when the combined isotropic-kinematic hardening model was assumed.

This study investigates the plastic behavior of TWIP980 and TRIP980 steels, representative of AHSS, under monotonic and reversal loading conditions. TRIP steel is a first-generation AHSS, while TWIP steel is a second-generation. In the structure of the TRIP steels, ferrite, martensite, residual austenite, and bainite are present. During the deformation, the residual austenite transforms to the martensite. Thus, the hardening rate is increased, and high ductility and strength are ensured. This type of transformation is also called the TRIP effect in the literature (Jacques et al., 2001Jacques, P.J., Delannay, F., Ladriere, J. (2001). On the influence of interactions between phases on the mechanical stability of retained austenite in transformation-induced plasticity multiphase steels. Metal. Mater. Trans. A 32, 2759-2768. https://doi.org/10.1007/s11661-001-1027-4.; Muransky et al., 2008Muransky, O., Sittner, P., Zrník, J., Oliver, E.C. (2008). In situ neutron diffraction investigation of the collaborative deformation-transformation mechanism in TRIP assisted steels at room and elevated temperatures. Acta Mater. 56 (14), 3367-3379. https://doi.org/10.1016/j.actamat.2008.03.026.; Fu et al., 2014Fu, B., Yang, W.Y., Wang, Y.D., Li, L.F., Sun, Z.Q., Ren, Y. (2014). Micromechanical behavior of TRIP-assisted multiphase steels studied with in situ high-energy X-ray diffraction. Acta Mater. 76, 342-354. https://doi.org/10.1016/j.actamat.2014.05.029.).

On the other hand, TWIP steels contain a high ratio of manganese to maintain the austenitic structure. This structure ensures the twinning mode of the deformation along with the dislocations. Accordingly, the hardening rate is increased stemming from both twinning and dislocation coalescence, which postpones the localization of the deformation (Grassel et al., 2000Grassel, O., Krüger, L., Frommeyer, G., Meyer, L.W. (2000). High strength Fe-Mn-(Al, Si) TRIP/TWIP steels development - properties - applications. Int. J. Plast. 16 (10-11), 1391-1409. https://doi.org/10.1016/S0749-6419(00)00015-2.; Prakash et al., 2008Prakash, A., Hochrainer, H., Reisacher, E., Riedel, H. (2008). Twinnig models in self-consistent texture simulations of TWIP steels. Steel Res. Int. 79 (8), 645-645. https://doi.org/10.1002/srin.200806178.).

A combined hardening model comprises an isotropic von Mises yield criterion and a single-term Chaboche nonlinear kinematic hardening rule. The model is implemented in a commercial finite element code, and FE analyses of tension and tension-compression tests are performed. The predicted stress-strain curves under monotonic and reversal loading conditions are compared with experimental results to evaluate the model’s predictive capability.

The accuracy of the presented cyclic plasticity model is assessed by examining the agreement between the predicted and experimental stress-strain curves. The outcomes of this research contribute to a better understanding of the mechanical behavior of AHSS steels under different loading conditions and provide insights for improving the modeling of their response.

This study contains 5 sections. In section 2, the governing equations of the cyclic plasticity modeling and the construction of the FE model were presented. Section 3 summarizes the prediction results of monotonic and cyclic loading conditions. In section 4, the results, which were compared with the experimental outcomes, were discussed and the conclusions were summarized in section 5.

2. MATERIALS AND METHODS

In this section, relevant information related to plasticity modeling and finite element modeling was given. In this study, TRIP980 and TWIP980 steels were selected to investigate the cyclic response of the material. The TRIP steel contains C (0.21%), Mn (1.5%), Si (0.24%), Al (1.6%), with iron balance, while the TWIP steels contain high Mn content (16%). Apart from the Mn, these steels contain high C content of 0.66%, Si (0.035%), P (0.01%), with iron balance.

2.2. Finite element model of tension-compression test

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A tension-compression test is conducted using TWIP980 and TRIP980 steels to validate the proposed cyclic plasticity model. The test data is obtained from previous work (Joo and Huh, 2018Joo, G., Huh, H. (2018). Rate-dependent isotropic-kinematic hardening model in tension-compression of TRIP and TWIP steel sheets. Int. J. Mech. Sci. 146-147, 432-444. https://doi.org/10.1016/j.ijmecsci.2017.08.055.) and the thickness was considered as 1 mm. The dimensions of the specimen are shown in Fig. 1.

Figure 1.  Dimensions of test specimen (Joo and Huh, 2018Joo, G., Huh, H. (2018). Rate-dependent isotropic-kinematic hardening model in tension-compression of TRIP and TWIP steel sheets. Int. J. Mech. Sci. 146-147, 432-444. https://doi.org/10.1016/j.ijmecsci.2017.08.055.)

The finite element model of the specimen is created using an implicit finite element code (MSC Marc). The reduced region of the sample is discretized with fine meshes to capture stress-strain data accurately. Eight-node fully integrated hexahedral elements are utilized (Marc (2018a)Marc (2018a). Volume A: Theory and User Information. and Marc (2018b)Marc (2018b). Volume B: Element library) with the dimensions of the solid element set as 0.86x1x1 mm (Fig. 2).

Figure 2.  FE model of the specimen

A spring-loaded clamping device is incorporated to prevent buckling during the compression stage. The specimen is fixed at one end, and displacement is applied to the other. The loading history for both monotonic and reversal loading conditions is illustrated in Fig. 3.

Figure 3.  Loading history for a) Monotonic, b) Reversal loading

The isotropic hardening parameters for TWIP980 and TRIP980 steels are determined using the Swift hardening law (Eq. (10)).

${\sigma }_{true}=K{({\epsilon }_0+{\epsilon }_p)}^p$  (10)

The Swift parameters are calibrated to fit the experimental flow curve. To this end, the MATLAB Curve Fitting Tool (CFTool) was utilized, and the Levenberg-Marquardt algorithm (Sapna et al., 2012Sapna, S., Tamilarasi, A., Kumar, M.P. (2012). Backpropagation learning algorithm based on Levenberg Marquardt algorithm. Comput. Sci. Inf. Technol., pp. 393-398. https://doi.org/10.5121/csit.2012.2438.) embedded into the software was adopted. Here, p represents the hardening exponent related to the slope of the flow curve, while K and ε_p represent the strength coefficient and the plastic strain values, respectively. Apart from them, the yield stress was dictated by the parameter ε₀. Both materials’ experimental and Swift hardening curves are compared to validate the chosen parameters. The flow curves were given in Fig. 4 and the hardening parameters provided in Table 1.

Figure 4.  The hardening curves of a) TRIP980, b) TWIP980

Table 1.  Hardening curve parameters of the steels

TWIP980	Parameter	TRIP980
1685	K (MPa)	1700
0.26	n	0.19
0.0065	ε0	0.0024

The kinematic hardening part was characterized by the single-term Chaboche model that is given in Eq. (11) and Table 2 presents the kinematic hardening parameters.

$\alpha =\frac{C}{\gamma }(1-e^{-\gamma {\epsilon }_p})$  (11)

Table 2.  Kinematic hardening parameters of the steels

Parameter	TWIP980	TRIP980
C (MPa)	10000	16668
γ	55.56	55.56

Eq. (11) is the integrated form of Eq. (6). The parameters C and γ are the material parameters controlling the translation of the yield locus. C/γ is the threshold at which the back stress saturates. The back stress curves were constructed based on the calibrated parameters and these curves are presented in Fig. 5.

Figure 5.  Back stress curves of the steels

The proposed cyclic plasticity model is implemented in the commercial finite element code with the determined isotropic and kinematic hardening parameters. The stress-strain curves under monotonic and reversal loading conditions are predicted and compared with the experimental results.

3. RESULTS

After determining the hardening parameters, finite element analyses evaluate the stress-strain response of TWIP980 and TRIP980 steels under monotonic and reversal loading conditions. The predicted stress-strain curves are compared with experimental data (Fig. 6).

Figure 6 compares the predicted and experimental hardening curves for monotonic loading conditions. It is observed that the predicted stress-strain curves accurately align with the experimental results for both TRIP980 and TWIP980 steels. Figure 7 compares the predicted and experimental hardening curves for reversal loading conditions.

The model successfully captures the Bauschinger effect, with a decrease in yield strength when the load is reversed, for both materials. The predicted yield strengths in compression for both steels were close to each other and approximately 600 MPa. These values were found compatible with the experimental behavior. 1100 MPa and 920 MPa are the approximate thresholds reached at the end of the loading sequence for TRIP980 and TWIP980 steels, respectively. These values were also the yield stresses if these steels were loaded in the same direction (tension). However, when these steels were subsequently loaded in compression, the change in TRIP980 steel was found to be more apparent. This situation shows that the Bauschinger effect was more dominant for TRIP980 steel when compared to the TWIP980 steel. The stress-strain behavior of TRIP980 steel is well-predicted by the model, while a minor deviation is observed for TWIP980 steel. This discrepancy can be attributed to the permanent softening behavior exhibited by TWIP980 steel, resulting from the dissolution of the cell walls of dislocations (Zang et al., 2011Zang, S.L., Guo, C., Thuillier, S., Lee, M.G. (2011). A model of one-surface cyclic plasticity and its application to springback prediction. Int. J. Mech. Sci. 53 (6), 425-435. https://doi.org/10.1016/j.ijmecsci.2011.03.005.). Based on Armstrong-Frederick-type unified models, the current model cannot fully represent this behavior.

4. DISCUSSION

This work employs a user-defined combined hardening model to characterize the stress-strain response of TWIP980 and TRIP980 steels under monotonic and reversal loading conditions. The model incorporates an isotropic von Mises yield criterion and a single-term Chaboche nonlinear kinematic hardening rule. The predicted stress-strain curves demonstrate good agreement with experimental results for monotonic loading conditions, indicating the accuracy of the proposed model.

Furthermore, the model effectively captures the Bauschinger effect and transient behavior during reversal loading for both materials. Zang et al. (2011)Zang, S.L., Guo, C., Thuillier, S., Lee, M.G. (2011). A model of one-surface cyclic plasticity and its application to springback prediction. Int. J. Mech. Sci. 53 (6), 425-435. https://doi.org/10.1016/j.ijmecsci.2011.03.005. studied the springback response in U-draw bending for an aluminum alloy. Since the kinematic hardening rule is significant for springback prediction, they utilized and calibrated a unified hardening model. They observed that the Armstrong-Frederic type hardening models can capture the Bauschinger effect and transient behavior. Çaylak et al. (2022)Çaylak, M., Akşen, T.A., Fırat, M. (2022). Evaluating the effectiveness of combined hardening models to determine the behavior of a plate with a hole under combined loadings. Eur. Mech. Sci. 6 (2), 97-104. https://doi.org/10.26701/ems.1051057. also utilized the Armstrong-Frederic type hardening model to characterize the deformation behavior of oxygen-free high thermal conductivity copper material under combined tensile and cyclic bending loads. They also observed similar outcomes. In this work, a minor deviation is observed in the stress-strain response of TWIP980 steel, attributed to its permanent softening behavior. Similar results were also recorded in the study of Fu et al. (2016)Fu, J., Barlat, F., Kim, J.H., Pierron, F. (2016). Identification of nonlinear kinematic hardening constitutive model parameters using the virtual fields method for advanced high strength steels. Int. J. Solids Struct. 102-103, 30 - 43. https://doi.org/10.1016/j.ijsolstr.2016.10.020.. They adopted nonlinear kinematic hardening models and observed for the TWIP980 steel that the numerical results deviated from the experimental reciprocals. The inability of the Armstrong-Frederick type unified models to represent this behavior highlights the need for further improvements in modeling the permanent softening observed in TWIP980 steel.

The demonstrated combined hardening model provides a practical and user-friendly approach to characterize the stress-strain response of AHSS steels under different loading conditions. The model successfully predicts the material behavior of TWIP980 and TRIP980 steels. However, modifications or alternative modeling approaches may be necessary to enhance its capability and fully capture the permanent softening behavior (Zang et al. (2011)Zang, S.L., Guo, C., Thuillier, S., Lee, M.G. (2011). A model of one-surface cyclic plasticity and its application to springback prediction. Int. J. Mech. Sci. 53 (6), 425-435. https://doi.org/10.1016/j.ijmecsci.2011.03.005.. The findings of this study contribute to a better understanding of the mechanical behavior of AHSS steels and provide insights for further advancements in modeling their response under monotonic and reversal loading conditions.

5. SUMMARY AND CONCLUSIONS

This study evaluated the performance of a user-defined combined hardening modeling method for advanced high-strength steels (AHSS) under monotonic and reversal loading conditions. The plastic behavior of TWIP980 and TRIP980 steels was investigated using a cyclic plasticity modeling approach, incorporating an isotropic von Mises yield criterion and a single-term Chaboche nonlinear kinematic hardening rule.

The results of the study demonstrate the following conclusions:

The findings of this study contribute to a better understanding of the mechanical behavior of TWIP980 and TRIP980 steels under different loading conditions. The model’s accuracy in capturing the Bauschinger effect and transient behavior demonstrates its potential for predicting the material response in AHSS applications. However, further improvements or alternative modeling approaches are required to effectively represent the permanent softening behavior observed in TWIP980 steel.

Future research efforts should focus on enhancing the modeling capabilities to capture the complex behavior of AHSS steels, including their permanent softening behavior. This will facilitate more accurate predictions of the material response and enable the reliable simulation of AHSS components in various industrial applications.