Determination of Constitutive Relationships of Tubular Materials at Various Strain Rates Using Hydro-Bulging Experiments

  • 40 Accesses


Liquid impact forming (LIF) is a rapid tube hydroforming technique. The deformation behaviour of a metal tube may be different in LIF from that in tube hydroforming. A constitutive equation or equivalently an equivalent stress-strain relationship is generally used to describe the deformation behaviour of metals. The purpose of this work is to model the deformation behaviour of tubular materials in LIF using the Johnson-Cook (JC) structural model. LIF hydro-bulging experiments combined with the analytical approach based on the membrane theory and the force equilibrium equation were used to determine the model coefficients A, B, C, and n in the equations for SS304 stainless steel tubular materials. Finite element (FE) simulations of hydro-bulging under various impact velocities were carried out to validate the resultant JC model. The relationship between the strain rates and impact velocities was determined, the bulging heights between the equivalent stress-strain curves at different impact velocities were analysed, and the bulging heights obtained by FE simulations and experimental results were compared. The results show that the proposed approach using the JC model is suitable to define the stress-strain behaviour of tubular materials in LIF.

This is a preview of subscription content, log in to check access.

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10


v (mm/s):

Impact velocity, Fig. 1.

Fa (N):

Impact loading from a pusher, driven by a punch press ram, Fig. 1.

P (MPa):

hydraulic pressure, Fig. 1.

A (MPa):

Johnson-Cook structural model coefficient, equation (1).

B (MPa):

Johnson-Cook structural model coefficient, equation (1).

n (−):

Hardening exponent in Johnson-Cook structural model, equation (1).

σe (−):

Equivalent stress, equation (1).

εe (−):

Equivalent strain, equation (1).

\( \dot{\varepsilon} \)(s−1):

Strain rate, equation (1).

\( \dot{\varepsilon_0} \)(s−1):

Reference strain rate, equation (1).

σx (MPa):

Longitudinal stress at the peak b of the bulged tube, Fig. 2.

σθ (MPa):

Circumferential stress at the peak b, Fig. 2.

σr (MPa):

Thickness stress at the peak b, Fig. 2.

εx (−):

Longitudinal strain at the peak b, Fig. 2.

εθ (−):

Circumferential strain at the peak b, Fig. 2.

ρθ (mm):

Circumferential radius at the peak b, Fig. 2.

r0 (mm):

Initial outer radius of the tube, Fig. 2.

h (mm):

Bulging height of the bulged tube, Fig. 2.

ρθ (mm):

Circumferential curvature radius of the bulged profile, Fig. 2.

ρx (mm):

Axial curvature radius of the bulged profile, Fig. 2.

t (mm):

Instantaneous wall thickness of the bulged tube, equation (4).

t0 (mm):

Initial wall thickness of tubular blanks, equation (6).

εt (−):

Radial thickness strain at the peak b, equation (6).

\( \dot{\varepsilon_{\mathrm{e}}} \) (s−1):

Equivalent strain rate, equation (10).

e (−):

Equivalent strain increment at a time step dt, equation (10).

dt (s):

A time step, equation (10).

l0 (mm):

Initial length of tubular blanks, Fig. 6.

lb (mm):

Bulge zone length, Fig. 6.

T (−):

Normalized total hydro-bulging time, Fig. 8.


  1. 1.

    Merklein M, Hofman M, Lechner M, Kuppert A (2014) A review on tailored blanks—production, applications and evaluation. J Mater Process Technol 214(2):151–164

  2. 2.

    Liu J, Liu X, Yang L, Liang H (2016) Investigation of tube hydroforming along with stamping of thin-walled tubes in square cross-section dies. P I Mech Eng B-J Eng 230(1):111–119

  3. 3.

    Rappel H, Beex LAA (2019) Estimating fibres' material parameter distributions from limited data with the help of Bayesian inference. Eur J Mech A-Solid 75:169–196

  4. 4.

    Rappel H, Beex LAA, Noels L, Bordas SPA (2019) Identifying elastoplastic parameters with Bayes' theorem considering output error, input error and model uncertainty. Probabilist Eng Mech 55:28–41

  5. 5.

    Rappel H, Beex LAA, Bordas SPA (2018) Bayesian inference to identify parameters in viscoelasticity. Mech Time-Depend Mat 22:221–258

  6. 6.

    Atkinson M (1996) Accurate determination of biaxial stress–strain relationships from hydraulic bulging tests of sheet metals. Int J Mech Sci 39(7):761–769

  7. 7.

    Sokolowski T, Gerke K, Ahmetoglu M (2000) Evaluation of tube formability and material characteristics: hydraulic bulge testing of tubes. J Mater Process Technol 98(1):34–40

  8. 8.

    Hwang Y-M, Lin Y, Altan T (2007) Evaluation of tubular materials by a hydraulic bulge test. Int J Mach Tool Manu 47(2):343–351

  9. 9.

    Bortot P, CerettiE GC (2008) The determination of flow stress of tubular material for hydroforming applications. J Mater Process Technol 203(1):318–388

  10. 10.

    XuY CLC, Tsien YC (2008) Prediction of work-hardening coefficient and exponential by adaptive inverse finite element method for tubular material. J Mater Process Technol 201:413–418

  11. 11.

    Liu J, Liu X, Yang L, Liang H (2013) Determination of flow stress of thin-walled tube based on digital speckle correlation method for hydroforming applications. Int J Adv Manuf Technol 69(1–4):439–450

  12. 12.

    Grolleau V, Gary G, Mohr D (2008) Biaxial testing of sheet materials at high strain rates using viscoelastic bars. Exp Mech 48(3):293–306

  13. 13.

    Huang C, Liu J, Zhong Y et al (2014) Exploring liquid impact forming technology of the thin-walled tubes. Appl Mech Mater 633-634:841–844

  14. 14.

    Hajializadeh F, Mashhadi MM (2015) Investigation and numerical analysis of impulsive hydroforming of aluminium 6061-T6 tube. J Manuf Process 20:257–273

  15. 15.

    Yang L, Wang N, Jia H (2016) Determination of material parameters of welded tube via digital image correlation and reverse engineering technology. Mater Manuf Process 31(3):328–334

  16. 16.

    Lianfa Y, Cheng G (2008) Determination of stress–strain relationship of tubular material with hydraulic bulge test. Thin-Walled Struct 46(2):147–154

  17. 17.

    Boudeau N, Malécot P (2012) A simplified analytical model for post-processing experimental results from tube bulging test: theory, experimentations, simulations. Int J Mech Sci 65(1):1–11

  18. 18.

    Tang Z, Liang J, Xiao Z et al (2010) Three-dimensional digital image correlation system for deformation measurement in experimental mechanics. Opt Eng 49(10):1–9

  19. 19.

    Koç M, BillurE CÖN (2011) An experimental study on the comparative assessment of hydraulic bulge test analysis methods. Mater Des 32(1):272–281

Download references


This work was financially supported by the National Natural Science Foundation of China (51564007), Natural Science Foundation of Guangxi Province (2017GXNSFAA198133), and GUET Excellent Graduate Thesis Program under Grant (16YJPYSS01).

Author information

Correspondence to L. Yang.

Ethics declarations

Conflict of Interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yang, L., Fu, Y., He, Y. et al. Determination of Constitutive Relationships of Tubular Materials at Various Strain Rates Using Hydro-Bulging Experiments. Exp Tech 44, 127–136 (2020).

Download citation


  • Liquid impact forming
  • Hydro-bulging
  • Tubes
  • FE simulation
  • Constitutive equation
  • Strain rates