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

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Abstract

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.

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Abbreviations

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.

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Acknowledgements

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).

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Correspondence to L. Yang.

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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). https://doi.org/10.1007/s40799-019-00342-y

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Keywords

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