# A new analytical method for determination of the flow curve for high-strength sheet steels using the plane strain compression test

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## Abstract

A new analytical method to determine the effective tool width in contact with the sheet workpiece in the plane strain compression test, which changes during the test if a tool with a radius is used, is proposed. A detailed description of this method and the corresponding procedure of the flow curve determination for high-strength sheet steels are presented. The underpinning assumptions of the method are validated with the help of the FEA and the validation results are presented. Furthermore, with the help of the FEA, the main disadvantages of the plane strain compression test – strain inhomogeneity and possible tool misalignment – are investigated. It is shown that these disadvantages become negligible if a tool with a sufficiently large radius is used. The experimental validation of the proposed method was performed with the help of the uniaxial tensile test, the plane strain compression test and hydraulic bulge test on ten common high-strength and advanced high-strength sheet steels in the ultimate tensile strength range between 460 and 1260 MPa and the thickness range between 0.8 and 3.1 mm. The paper demonstrates that with the proposed analytical method for determination of the effective tool width, the plane strain compression test equipped with a tool with a sufficiently large radius becomes more appealing as a cost-efficient alternative to the hydraulic bulge test for the flow curve determination of high-strength sheet steels than it has been considered until now.

## Keywords

Plane strain compression test Flow curve AHSS## Nomenclature

*α*Angle between the free workpiece surface and the flat surface of the tool without a radius at the border of the ideal forming zone [°]

*α*_{R}Angle between the free workpiece surface constrained by the tool radius and the flat surface of the tool with a radius at the border of the ideal forming zone [°]

*α*_{C}Angle between the free workpiece surface unconstrained by the tool radius and the flat surface of the tool with a radius at the border of the ideal forming zone [°]

- \( \Delta {\overline{\varepsilon}}_{\mathrm{pl}} \)
Increment of the equivalent plastic strain [−]

- \( {\overline{\varepsilon}}^{\mathrm{vM}} \)
Equivalent strain according to von Mises [−]

- \( {\varepsilon}_{\mathrm{pl}}^{\mathrm{vM}} \)
Equivalent plastic strain according to von Mises [−]

- \( {\varepsilon}_{\mathrm{pl}}^{\mathrm{T}} \)
Equivalent plastic strain according to Tresca [−]

- \( {\overline{\varepsilon}}_{\mathrm{pl}} \)
Equivalent plastic strain according to von Mises [−]

- \( {\overline{\varepsilon}}_{\mathrm{pl}}^{\mathrm{max}} \)
Maximum equivalent plastic strain according to von Mises [−]

*ε*_{ps}Global true strain in the ideal plane strain stress state of the PSCT [−]

*ε*_{ps}^{ref}Reference global true strain in the plane strain stress state of the PSCT [−]

*ε*_{Rm}True strain at the uniform elongation of the UTT [−]

*ε*_{un}True strain converted from the plane strain stress state into the uniaxial stress state [−]

*μ*Coulomb friction coefficient [−]

*σ*_{ps}Global true stress in the ideal plane strain stress state of the PSCT [MPa]

*σ*_{ps}^{ref}Reference true stress in the plane strain stress state of the PSCT [MPa]

*σ*_{Rm}Global true stress at the uniform elongation of the UTT [MPa]

*σ*_{un}True stress under the uniaxial stress state [MPa]

- \( {\overline{\sigma}}^{\mathrm{T}} \)
Equivalent stress according to Tresca [MPa]

- \( {\overline{\sigma}}^{\mathrm{vM}} \)
Equivalent stress according to von Mises [MPa]

- \( \overline{\sigma} \)
Equivalent stress according to an arbitrary yield criterion [MPa]

*σ*_{x}Stress along the

*x*axis [MPa]*σ*_{y}Stress along the

*y*axis [MPa]*∆σ*Increment of the mean stress in the ideal forming zone [MPa]

*a*Tool width excluding the two radii [mm]

*a*_{w}Effective tool width [mm]

*b*Actual specimen width [mm]

*b*_{0}Initial specimen width [mm]

*b*_{f}Final specimen width [mm]

- \( {b}_{\mathrm{f}}^{\mathrm{max}} \)
Final maximum specimen width [mm]

*C*_{b}Specimen width spread coefficient [−]

*C*(*x*_{C};*y*_{C})Outermost contact point between the tool and the workpiece

*F*Force [N]

*f*_{ps}Fitting factor for the stress-state-dependent conversion of stresses and strains [−]

*h*Actual specimen thickness [mm]

*h*_{0}Initial specimen thickness [mm]

*h*_{f}Final specimen thickness [mm]

*k*Shear flow stress [MPa]

*l*_{0}Initial specimen length [mm]

*m*Spread exponent [−]

*R*Tool radius [mm]

*r*_{h}(*x*)Contour of the tool radius [−]

*v*_{x}Material speed along the

*x*axis at the border of the ideal forming zone [ms^{−1}]*v*_{y}(*y*)Material speed along the

*y*axis at the border of the ideal forming zone [ms^{−1}]*w*Tool width including the two radii [mm]

*W*_{∆}Work corresponding to the increment of the mean stress in the ideal forming zone [J]

*W*_{shear}Shear work [J]

*x*_{0}Half of the tool width excluding the two radii [mm]

*y*_{0}Half of the actual specimen thickness [mm]

- ETW
Effective tool width

- HBT
Hydraulic bulge test

- PSCT
Plane strain compression test

- UTT
Uniaxial tensile test

## Notes

### Acknowledgements

The hydraulic bulge test data were obtained via collaborations of Faurecia Autositze GmbH with Bilstein GmbH & Co. KG, Salzgitter Flachstahl GmbH, Tata Steel Europe Ltd., thyssenkrupp Steel Europe AG, voestalpine Stahl GmbH, whose support is acknowledged.

### Compliance with ethical standards

The hydraulic bulge test data, which was used as input data for this work, was obtained via previous collaborations of the Faurecia Autositze GmbH with Bilstein GmbH & Co. KG, Salzgitter Flachstahl GmbH, Tata Steel Europe Ltd., thyssenkrupp Steel Europe AG, voestalpine Stahl GmbH. The material testing in the frame of this work was performed at and funded by Faurecia Autositze GmbH, Stadthagen, Germany, a subsidiary of Faurecia S. E., Nanterre, France. Charles Chermette (participant of a V. I. E. program funded by Business France, Paris, France), Klaus Unruh and Ilya Peshekhodov have contributed to the work as employees of Faurecia Autositze GmbH, Stadthagen, Germany, a subsidiary of Faurecia S. E., Nanterre, France. Jérôme Chottin and Tudor Balan have contributed to the work as employees of Faurecia Sièges d’Automobiles, a subsidiary of Faurecia S. E., Nanterre, France. No further conflicts of interests are to declare.

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