Abstract
In the present study, the mechanical model of section element in tube bulging area was established through the stress conditions of hydraulic bulging. Based on the mechanical model, the equivalent stress and equivalent strain equations of the section element in the bulging area were derived, respectively. Combined with the experimental data, the equivalent stress and strain of the section element in the bulging area were fitted by polynomial, and the plastic hardening model of thin-walled tube under pulsating loading was obtained. In order to verify the model precision, the plastic hardening models obtained from pulsating hydroforming and non-pulsating hydroforming were taken as the material model by finite element simulation respectively. The experimental results were compared with the simulation results which showed that as a pulsating hydraulic loading material parameters, the model established in this paper had higher precision.
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Abbreviations
- P (Mpa):
-
Pulsating fluid pressure (Fig. 1)
- T (s−1):
-
Pulsation cycle (Fig. 1)
- L (mm):
-
Length of metal thin-walled tube (Fig. 2(a); Eq. (3); Table 1)
- I (mm):
-
Length of tube bulging area (Fig. 2(a); Eq. (3); Table 1)
- i (s):
-
Time i in the process of tube bulging (Fig. 2(a))
- hi (mm):
-
Tube bulging height at time i (Fig. 2(a))
- Pi (Mpa):
-
Instantaneous internal pressure (Fig. 2(a))
- ra (mm):
-
Initial radius of thin-walled tube (Fig. 2(a); Eq. (3); Table 1)
- t0 (mm):
-
Initial wall thickness of thin-walled tube (Fig. 2(a); Table 1)
- σaz (Mpa):
-
Axial stress at point a (Fig. 2(b); Eqs. (7), (10), (11) and (14)-(16))
- σbz (Mpa):
-
Axial stress at point b (Fig. 2(b); Eqs. (7), (10), (11) and (14)-(16))
- ρaz (mm):
-
Axial radius of curvature at point a (Fig. 2(b); Eq. (15))
- ρaz (mm):
-
Axial radius of curvature at point b (Fig. 2(b); Eq. (15))
- ρaθ (mm):
-
Circumferential radius of curvature at point a (Fig. 2(b); Eqs. (14)-(16))
- ρbθ (mm):
-
Circumferential radius of curvature at point a (Fig. 2(b); Eqs. (14)-(16))
- ta (mm):
-
Wall thickness of thin-walled tube at point a (Fig. 2(b); Eqs. (7), (8), (10)-(12) and (17))
- tb (mm):
-
Wall thickness of thin-walled tube at point b (Fig. 2(b); Eqs. (7), (10)-(12) and (17))
- tp (mm):
-
Wall thickness of thin-walled tube under hydraulic pressure P (Fig. 2(b); Eqs. (7), (12), (17), (20) and (21))
- dla (mm):
-
The length of the circumferential arc belong to the circle passing point a before and after bulging (Fig. 2(b); Eqs. (7), (8) and (18))
- dlb (mm):
-
The length of the circumferential arc belong to the circle passing point b before and after bulging (Fig. 2(b); Eqs. (7)-(9) and (18))
- dforming (mm):
-
The length of the tube bulging profile from point a to point b (Fig. 2(b); Eqs. (7), (10), (11) and (13))
- θ (°):
-
The angle between the section of the unit body and the tube axis (Fig. 2(b); Eqs. (7), (10) and (11))
- φ (°):
-
The angle between the straight line passing through point a and b and the horizontal coordinate axis (Fig. 2(b); Eq. (19))
- oa (—):
-
Center of curvature at point a (Fig. 2(b))
- ob (—):
-
Center of curvature at point b (Fig. 2(b))
- σθ (Mpa):
-
Circumferential stress (Fig. 2(c); Eqs. (20) and (21))
- Ψa (°):
-
The angle between the tangent of point a on the bulging profile and the horizontal coordinate axis (Fig. 2(d); Eqs. (7), (10) and (11))
- Ψa (°):
-
The angle between the tangent of point b on the bulging profile and the horizontal coordinate axis (Fig. 2(d); Eqs. (7), (10) and (11))
- y (°):
-
The angle between the line connecting point a and point b on the bulging profile and the horizontal coordinate axis (Fig. 2(d); Eqs. (7), (10) and (11))
- σVz(f) (Mpa):
-
Axial stress at the bulging time i (Eqs. (1), (2), (20) and (21))
- ρθ(f) (mm):
-
Circumferential radius of curvature at the bulging time i (Eqs. (1), (2), (20) and (21))
- ti (mm):
-
Tube wall thickness at the bulging time i (Eqs. (1) and (2))
- Ff (N):
-
Friction between thin-walled tube and mold (Eqs. (1)-(3))
- Ψi (°):
-
The angle between the tangent of a point on the bulging profile and the horizontal coordinate axis (Eqs. (4), (5) and (19))
- f(z) (—):
-
Bulging profile curve function (Eqs. (5) and (6))
- ρz (mm):
-
Radius of axial curvature (Eqs. (6), (20) and (21))
- εz(j) (—):
-
Axial strain at any bulging time i (Eqs. (22) and (23))
- εθ(j) (—):
-
Circumferential strain at any bulging time (Eqs. (22) and (23))
- εt(j) (—):
-
Thickness strain at any bulging time i (Eqs. (22) and (23))
- dεz(j) (—):
-
The axial strain increment of tube at any bulging time i (Eqs. (23), (24) and (26))
- dεθ(j) (—):
-
The circumferential strain increment of tube at any bulging time i (Eqs. (23), (24) and (26))
- dεt(j) (—):
-
The thickness strain increment of tube at any bulging time i (Eqs. (23), (24) and (26))
- dεθ(i) (—):
-
The equivalent strain increment of tube at any bulging time i (Eqs. (24)-(26))
- εθ(i) (—):
-
The equivalent strain of tube at any bulging time i (Eqs. (25)-(27))
- σe(i) (Mpa):
-
Equivalent stress of tube material at any bulging time i (Eqs. (26) and (27))
- σz(i) (Mpa):
-
Axial stress of bulging tube at any bulging time i (Eqs. (26) and (27))
- σe(j) (Mpa):
-
Circumferential stress of bulging tube at any bulging time i (Eqs. (26) and (27))
- σt(i) (Mpa):
-
Thickness stress of bulging tube at any bulging time i (Eqs. (26) and (27))
- λ0, λ1, … (—):
-
The coefficients in the plastic hardening model of metal thin-walled tube (Eq. (27))
- σb (Mpa):
-
Tensile strength (Table 1)
- σs (Mpa):
-
Yield strength (Table 1)
- K(—):
-
The strength coefficient (Table 1)
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Acknowledgments
This work was supported by National Natural Science Foundation of China (51665018) and Youth Scientific Research Project of JXEDU (GJJ171265).
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Guolin Hu is a Ph.D. candidate of the School of Mechanical and Electrical Engineering, Jiangxi University of Science and Technology, Ganzhou, China. He received his master degree in Mechanical Manufacture and Automation from Guilin University of Electronic Technology. His research interests include machine manufacturing, metal forming and electromechanical control.
Chunrong Pan is a Professor of the School of Mechanical and Electrical Engineering, Jiangxi University of Science and Technology, Ganzhou, China. He received his Ph.D. degree in Mechanical Manufacture and Automation from Guangdong University of Technology. His research interests include machine manufacturing, optimization algorithm and electromechanical control.
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Hu, G., Pan, C. Investigation of the plastic hardening of metal thin-walled tube under pulsating hydraulic loading condition. J Mech Sci Technol 34, 4743–4751 (2020). https://doi.org/10.1007/s12206-020-1031-5
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DOI: https://doi.org/10.1007/s12206-020-1031-5