Abstract
In this study, a comprehensive mathematical analysis of the bulge test is presented. The geometry of the bulge test and the stress state that prevails in this test are very similar to the expanding thin wall sphere. These similarities may lead many users to confuse the interpretation of incremental change in the radius of the sphere as radial strain and consequently invalid analysis of the instability conditions. Therefore, in this study, emphasis is placed on how to properly use the equations to obtain the strain stress curve by this method precisely. Assuming work hardening behavior to obey power law, based on stress state on the dome, the relationship between fluid pressure and the height of the dome is presented. The instability condition is then determined and compared with Hill's analysis. To confirm the mathematical analysis, the hydraulic machine equipped with a computer system for recording pressure data versus the height of the bulge was used. The bulge test is conducted on two typical sheet metals, copper and commercially pure aluminum. The measured fluid pressure during the deformation and the thickness distribution of the sample well confirmed the results of the analytical analysis.
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Abbreviations
- \(\sigma_{\theta }\), \(\sigma_{\varphi }\) and \(\sigma_{r}\) :
-
Hoop, meridian, and radial (thickness) stresses
- \(\varepsilon_{\theta }\), \(\varepsilon_{\varphi }\) and \(\varepsilon_{r}\) :
-
Hoop, meridian, and radial (thickness) strains
- \(\varepsilon_{t}\) :
-
The absolute value of thickness strain
- \(\overline{\sigma }\) :
-
Effective stress
- \(\overline{\varepsilon }\) :
-
Effective strain
- \(\overline{\varepsilon }_{{{\text{ins}}}}\) :
-
Effective strain at the onset of instability
- \(e_{\theta }\) :
-
Engineering hoop strain
- h :
-
Height of the dome
- a :
-
Base radius of the dome
- \(\rho\) :
-
The radius of the dome
- z :
-
The current height of any material point
- t 0 and t :
-
The initial and current thickness of the sheet
- n :
-
Work hardening exponent
- k :
-
Coefficient of strength
- p :
-
Oil pressure
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The authors appreciate Shiraz University for its financial support (Grant numbers of 99-GR-ENG-15) and research facilities used in this work.
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Ebrahimi, R., Lotfabad, F.R. A Comprehensive Mathematical Analysis on Achieving Stress–Strain Behavior at Large Strains in Bulge Test. Iran J Sci Technol Trans Mech Eng 46, 901–909 (2022). https://doi.org/10.1007/s40997-021-00451-3
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DOI: https://doi.org/10.1007/s40997-021-00451-3