Constitutive Modeling of the High-Temperature Flow Behavior of α-Ti Alloy Tube

  • Yanli Lin
  • Kun Zhang
  • Zhubin He
  • Xiaobo Fan
  • Yongda Yan
  • Shijian Yuan


In the hot metal gas forming process, the deformation conditions, such as temperature, strain rate and deformation degree, are often prominently changed. The understanding of the flow behavior of α-Ti seamless tubes over a relatively wide range of temperatures and strain rates is important. In this study, the stress–strain curves in the temperature range of 973-1123 K and the initial strain rate range of 0.0004-0.4 s−1 were measured by isothermal tensile tests to conduct a constitutive analysis and a deformation behavior analysis. The results show that the flow stress decreases with the decrease in the strain rate and the increase of the deformation temperature. The Fields–Backofen model and Fields–Backofen–Zhang model were used to describe the stress–strain curves. The Fields–Backofen–Zhang model shows better predictability on the flow stress than the Fields–Backofen model, but there exists a large deviation in the deformation condition of 0.4 s−1. A modified Fields–Backofen–Zhang model is proposed, in which a strain rate term is introduced. This modified Fields–Backofen–Zhang model gives a more accurate description of the flow stress variation under hot forming conditions with a higher strain rate up to 0.4 s−1. Accordingly, it is reasonable to adopt the modified Fields–Backofen–Zhang model for the hot forming process which is likely to reach a higher strain rate, such as 0.4 s−1.


constitutive model Fields–Backofen model flow behavior hot metal gas forming statistical analysis α-Ti seamless tube 



This study was financially supported by National Key R&D Program of China (2017YFB0304400, 2017YFB0306300), the National Natural Science Foundation of China (Nos. 51575131, 51405102), the program for Changjiang Scholars and Innovative Research Team in University (No. IRT1229). The authors would like to take this opportunity to express their sincere appreciation to the funds.


  1. 1.
    S. Novotny and M. Geiger, Process Design for Hydroforming of Lightweight Metal Sheets at Elevated Temperatures, J. Mater. Process. Technol., 2003, 138, p 594–599CrossRefGoogle Scholar
  2. 2.
    J.C. Benedyk, Hot Metal Gas Forming of Aluminum for Manufacturing Vehicle Structural Components, Light Metal Age, 2003, 61, p 18–19Google Scholar
  3. 3.
    A. Rusinek, J.A. Rodríguez-Martínez, and A. Arias, A Thermo-Viscoplastic Constitutive Model for FCC Metals with Application to OFHC Copper, Int. J. Mech. Sci., 2010, 52, p 120–135CrossRefGoogle Scholar
  4. 4.
    Y.C. Lin and X. Chen, A Critical Review of Experimental Results and Constitutive Descriptions for Metals and Alloys in Hot Working, Mater. Des., 2011, 32, p 1733–1759CrossRefGoogle Scholar
  5. 5.
    G.R. Johnson, and W.H. Cook. A Constitutive Model and Data for Metals Subjected to Large Strains, High Strain Rates and High Temperatures. in Proceedings of the 7th International Symposium on Ballistics, The Netherlands, 1983, pp. 541–543.Google Scholar
  6. 6.
    J. Li, F. Li, J. Cai, R. Wang, Z. Yuan, and G. Ji, Comparative Investigation on the Modified Zerilli–Armstrong Model and Arrhenius-type Model to Predict the Elevated-Temperature Flow Behaviour of 7050 Aluminium Alloy, Comput. Mater. Sci., 2013, 71, p 56–65CrossRefGoogle Scholar
  7. 7.
    L. Chen, G. Zhao, J. Yu, and W. Zhang, Constitutive Analysis of Homogenized 7005 Aluminum Alloy at Evaluated Temperature for Extrusion Process, Mater. Des., 2015, 66, p 129–136CrossRefGoogle Scholar
  8. 8.
    D.S. Fields and W.A. Bachofen, Determination of Strain Hardening Characteristics by Torsion Testing, ASTM, Proc. Am. Soc. Test. Mater., 1957, 57, p 1259–1272Google Scholar
  9. 9.
    Y. Sun, W.D. Zeng, Y.Q. Zhao, X.M. Zhang, Y. Shu, and Y.G. Zhou, Research on the Hot Deformation Behavior of Ti40 Alloy Using Processing Map, Mater. Sci. Eng. A, 2011, 528, p 1205–1211CrossRefGoogle Scholar
  10. 10.
    Y. Han, H. Wu, W. Zhang, D. Zou, G. Liu, and G. Qiao, Constitutive Equation and Dynamic Recrystallization Behavior of As-Cast 254SMO Super-Austenitic Stainless Steel, Mater. Des., 2015, 69, p 230–240CrossRefGoogle Scholar
  11. 11.
    H. Rastegari, M. Rakhshkhorshid, M.C. Somani, and D.A. Porter, Constitutive Modeling of Warm Deformation Flow Curves of an Eutectoid Steel, J. Mater. Eng. Perform., 2017, 26, p 2170–2178CrossRefGoogle Scholar
  12. 12.
    G.Z. Quan, W.Q. Lv, Y.P. Mao, Y.W. Zhang, and J. Zhou, Prediction of Flow Stress in a Wide Temperature Range Involving Phase Transformation for As-Cast Ti-6Al-2Zr-1Mo-1V Alloy by Artificial Neural Network, Mater. Des., 2013, 50, p 51–61CrossRefGoogle Scholar
  13. 13.
    Y. Han, G.J. Qiao, J.P. Sun, and D.N. Zou, A Comparative Study on Constitutive Relationship of As-Cast 904L Austenitic Stainless Steel During Hot Deformation Based on Arrhenius-Type and Artificial Neural Network Models, Comput. Mater. Sci., 2013, 67, p 93–103CrossRefGoogle Scholar
  14. 14.
    Y. Sun, W.H. Ye, and L.X. Hu, Constitutive Modeling of High-Temperature Flow Behavior of Al-0.62 Mg-0.73Si Aluminum Alloy, J. Mater. Eng. Perform., 2016, 25, p 1621–1630CrossRefGoogle Scholar
  15. 15.
    M. Zhou, Y.C. Lin, J. Deng, and Y. Jiang, Hot Tensile Deformation Behaviors and Constitutive Model of an Al-Zn-Mg-Cu Alloy, Mater. Des., 2014, 59, p 141–150CrossRefGoogle Scholar
  16. 16.
    J. Deng, Y.C. Lin, S. Li, J. Chen, and Y. Ding, Hot Tensile Deformation and Fracture Behaviors of AZ31 Magnesium Alloy, Mater. Des., 2013, 49, p 209–219CrossRefGoogle Scholar
  17. 17.
    P. Lin, Z.B. He, S.J. Yuan, and J. Shen, Tensile Deformation Behavior of Ti-22Al-25Nb Alloy at Elevated Temperatures, Mater. Sci. Eng. A, 2012, 556, p 617–624CrossRefGoogle Scholar
  18. 18.
    W.T. Jia, S. Xu, Q.C. Le, L. Fu, L.F. Ma, and Y. Tang, Modified Fields–Backofen Model for Constitutive Behavior of As-Cast AZ31B Magnesium Alloy During Hot Deformation, Mater. Des., 2016, 106, p 120–132CrossRefGoogle Scholar
  19. 19.
    Z.B. He, B.G. Teng, C.Y. Che, Z.B. Wang, K.L. Zheng, and S.J. Yuan, Mechanical Properties and Formability of TA2 Extruded Tube for Hot Metal Gas Forming at Elevated Temperature, Trans. Nonferrous Metals Soc. China, 2012, 22, p 479–484CrossRefGoogle Scholar
  20. 20.
    X.Z. Kai, C. Chen, X.F. Sun, C.M. Wang, and Y.T. Zhao, Hot Deformation Behavior and Optimization of Processing Parameters of a Typical High-Strength Al-Mg-Si Alloy, Mater. Des., 2016, 90, p 1151–1158CrossRefGoogle Scholar
  21. 21.
    L. Chen, G.Q. Zhao, and J.Q. Yu, Hot Deformation Behavior and Constitutive Modeling of Homogenized 6026 Aluminum Alloy, Mater. Des., 2015, 74, p 25–35CrossRefGoogle Scholar
  22. 22.
    F.S. Qu, Y.H. Zhou, L.Y. Zhang, Z.H. Wang, and J. Zhou, Research on Hot Deformation Behavior of Ti-5Al-5Mo-5V-1Cr-1Fe Alloy, Mater. Des., 2015, 69, p 153–162CrossRefGoogle Scholar
  23. 23.
    Z.H. Du, S.S. Jiang, and K.F. Zhang, The Hot Deformation Behavior and Processing Map of Ti-47.5Al-Cr-V Alloy, Mater. Des., 2015, 86, p 464–473CrossRefGoogle Scholar
  24. 24.
    Y.Q. Cheng, H. Zhang, Z.H. Chen, and K.F. Xian, Flow Stress Equation of AZ31 Magnesium Alloy Sheet During Warm Tensile Deformation, J. Mater. Process. Technol., 2008, 208, p 29–34CrossRefGoogle Scholar
  25. 25.
    X.H. Zhang, Z.S. Cui, and X.Y. Ruan, Warm Forging of Magnesium Alloys: the Formability and Flow Stress of AZ31B, J. Shanghai Jiao Tong Univ., 2003, 37, p 1874–1877Google Scholar

Copyright information

© ASM International 2018

Authors and Affiliations

  • Yanli Lin
    • 1
    • 2
  • Kun Zhang
    • 2
  • Zhubin He
    • 2
    • 3
  • Xiaobo Fan
    • 2
    • 3
  • Yongda Yan
    • 4
    • 5
  • Shijian Yuan
    • 2
  1. 1.School of Materials Science and EngineeringHarbin Institute of Technology at WeihaiWeihaiChina
  2. 2.School of Materials Science and EngineeringHarbin Institute of TechnologyHarbinChina
  3. 3.School of Mechanical EngineeringDalian University of TechnologyDalianChina
  4. 4.Key Laboratory of Micro-systems and Micro-structures Manufacturing of Ministry of EducationHarbin Institute of TechnologyHarbinChina
  5. 5.Center For Precision EngineeringHarbin Institute of TechnologyHarbinChina

Personalised recommendations