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 highstrength 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 highstrength and advanced highstrength 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 costefficient alternative to the hydraulic bulge test for the flow curve determination of highstrength sheet steels than it has been considered until now.
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Notes
 1.
The same convention is consistently applied to the calculation of the corresponding stress.
 2.
The curve was digitized and a factor ½ was applied since Becker [17] investigated a twice bigger geometry than here (R = 4 mm, h_{0}=6 mm and a = 6 mm).
Abbreviations
 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 stressstatedependent 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
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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.
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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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Appendices
Appendix 1: Shear correction of the analytical stress determination
The free surface boundary condition at the border of the ideal forming zone is only an approximation of reality. In reality, additional shear strains appear in the material at the exit of the tool due to the vertical movement of the material coming out of the tool. This acts as a constraint, which requires additional work to be performed. Ismar and Mahrenholtz [23] elaborated the theory of the corner correction to calculate the increased stress exerted by the tool to perform this additional work. Figure 2 represents the idealized material flow of this phenomenon in case of two symmetric tools with a vanishing radius of R = 0. The material inside the forming zone is supposed to move horizontally with a speed v_{x}. In the reality, the material in contact with the tools leaves the forming zone forming an angle α and obtains an additional vertical movement at a speed comprised between v^{tool 1} at the upper tool and v^{tool 2} at the lower tool. The angle α is then defined as the angle between the workpiece surface and the surface of the tool without a radius at the forming zone border. The vector equations of the velocity field lead to:
A linear profile of the vertical material speed is assumed (see Fig. 20), which gives Eq. (31).
The shear work within the isolated slab (W_{shear}), coming from the vertical movement of the material, is
The integral of the absolute function between –h/2 and h/2 is equal to the area of a square of a side h/2:
Therefore, W_{shear} can be written as a function of v_{x}, α and k as
In parallel, the work provided by the increased mean stress ∆σ at the left side of the isolated slab is
By the conservation of the energy, W_{shear}=W_{∆}, the following expression of ∆σ is obtained:
Adding the term ∆σ to the expression of the ideal solution of the stress exerted by the tool (Eq. (11)), which was obtained assuming no stress constraints at the border of the ideal forming zone, yields
Appendix 2: Procedure for determination of the flow curve for highstrength sheet steels with the help of the plane strain compression test and the proposed new analytical method
Equipment, workpiece and preliminary testing
Equipment
The equipment necessary to perform the test on modern highstrength sheet steels is a universal tensioncompression testing machine with a maximum load of at least 200 kN and a plane strain compression device with the two tools in contact with the workpiece shown in Figs. 1 and 4. The width of the tools should be chosen to satisfy the requirement on the relation between the tool width w and the workpiece thickness h of 2 < w/h < 4. The tools should have a sufficiently large radius to ensure homogenous deformation in the workpiece. In the present work, the tools with w = 4 mm and R = 0.5 mm were used. A larger tool radius of up to 2 mm is recommended. A good guiding should be included in the device to guide the upper tool with respect to the lower tool.
Workpiece
The initial thickness of the workpiece h_{0} should be consistent with the width of the tool w and satisfy the condition mentioned above. The initial width of the workpiece b_{0} is recommended to satisfy the condition of b_{0} ≥ 5w. The length of the workpiece l_{0} should ideally enable to perform enough repeated measurements on the same sample with a gap of about 10 mm between the two adjacent deformation areas. In the present work, the workpiece of b_{0} = 22 mm and l_{0} = 50 mm was used. Workpieces should be oriented with their longer side along the rolling direction for the plastic flow in the sheet plane to take place in the rolling direction. Specimen preparation can be realised by laser, water jet or electrical discharge machining to ensure good dimensional quality with no strain hardening at the edges. Alternatively, shear cutting of the specimen with subsequent polishing of cut edges to remove strain hardening can be performed.
Preliminary testing
Results of the uniaxial tensile test according to ISO 68921 must be provided for the same batch of the tested material. These results are necessary for conversion of the flow curve from the plain strain compression test to the uniaxial stress state.
Test preparation
Correction of the loaddisplacement curve
The tool displacement during the test is obtained in this work directly from the machine crosshead displacement. This displacement must be corrected for the machine compliance effect. Therefore, a correction forcedisplacement curve, which quantifies the compliance of the testing machine, is first determined via pressing the tool without sample until the maximum load of the testing machine. The correction curve for the setup testing machine and tool is expected to be well reproducible, therefore doing it after several tests is sufficient. However, the test to build the correction curve should be repeated, if the forcedisplacement curves recorded while performing the PSCT on specimens of the same material have a displacement offset to one another when repeating the test. Such an observation highlights displacements of the test setup which can be mimimised or avoided by loading the testing machine without a workpiece again until its maximum force. Alternatively, the displacement can be measured directly at the tool using tactile, optical or laserassisted measurement. In this case, determination of the correction forcedisplacement curve is not required.
Initial thickness and width of the workpiece
Thicknesses and widths of the workpiece in the areas where the tests are to be performed need to be measured and documented.
Lubrication
A good lubrication is essential. In this work it is recommended to apply a layer of grease and teflon on both surfaces of the specimen. It enables to minimize the friction between the tool and the work piece. Applying an extra layer of grease between the teflon and the tool was not found to be necessary.
Testing
The specimen should be placed between the tools. The test speed should be chosen so that quasistatic deformation of the material can be ensured. The test should be stopped manually once the slope of the forcedisplacement curve starts to increase as shown in Fig. 21. For the next tests on the same material batch, it is recommended to stop the test so that the portion of curve after the slope increase is small.
The specimen is then to be moved to perform the second test, whereas a minimum distance between the deformation zones of at least 10 mm is to be considered. The following tests can be stopped automatically at the maximum force defined in the first test. More tests must be performed if high scattering in the curves is observed. Thereafter, a representative forcedisplacement curve as well as the correction curve are to be selected for postprocessing.
Measurements after the test
The final width and thickness corresponding to the selected test should be measured as illustrated in Fig. 22 and documented. For the final thickness h_{f}, it recommended to make an average of five measurements along the deformation zone. For determination of the final width b_{f}, the final maximum width b_{fmax} is to be measured and documented. An equivalent final width is estimated in the subsequent calculation.
Results analysis
To make an easy and robust flow curve elaboration from the UTT and the PSCT, an excel template including VBA macros can be implemented for the calculations, which requires only a few actions from the user. A description on how such a template can be realized is presented in the following.
Input data
The following input data need to be provided:
Thickness, width, and gauge length of the specimen from the uniaxial tensile test
The yield strength R_{p02} from the representative uniaxial tensile test
Initial width of the PSCT specimen at the test position
Final maximum width of the PSCT specimen at the test position
Tool width of the PSCT (4 mm in this work)
Tool radius of the PSCT (0.5 mm in this work)
Friction coefficient (0.03 in this work)
Results of the measurement of the sheet thickness before the PSCT
Results of the measurement of the sheet thickness after the PSCT
Forcedisplacement curve from the UTT
Forcedisplacement curve from the PSCT
Forcedisplacement correction curve from the PSCT (without the specimen)
Determination of the flow curve from the UTT
Determination of the flow curve from the UTT includes usual steps of determination of the engineering and true stresses and strains, determination of R_{p0.2} and R_{m},creation of the flow curve by selecting the true stress and true strain values between R_{p0.2} and R_{m} and substracting the elastic strain.
Determination of the corrected forcedisplacement curve from the PSCT
For determination of the corrected forcedisplacement curve from the PSCT, a search for the corresponding displacement of the tool in the forcedisplacement correction curve from the PSCT without the specimen for a number of force levels should be carried out. The displacement from the PSCT with the specimen should be then corrected by substracting the displacement from the PSCT without a specimen for the same force level.
Determination of the actual contact surface between the tool and the specimen in the PSCT
To determine the actual contact surface between the tool and the specimen, the final equivalent specimen width b_{f} is determined according to b_{f} = b_{f0} + (b_{fmax} – b_{f0}) × 0.77. The value 0.77 is recommended in this work based on results of the performed microscopy analysis of the specimen width spread. Determination of the spread coefficient C_{b} should be carried out according to Eq. (16). Finally, determination of the actual equivalent specimen width b is performed according to Eq. (15).
Determination of the effective tool width (ETW) is performed for two cases: α_{R} < α_{C} and α_{R} = α_{C} (Fig. 5). For the case of α_{R} < α_{C} (Fig. 5, left), at which the tool radius constrains the material flow out of the deformation zone, following steps need to be done for determination of the ETW:
Determination of x_{c} by Eq. (25)
Determination of y_{c} as h_{0}/2
Determination of ETW as 2x_{c}
Determination of α_{R} and α_{C} (Fig. 5) to check the condition α_{R} < α_{C}
For the case of α_{R} = α_{C} (Fig. 5, right), at which the tool radius does not constrain the material flow out of the deformation zone, following steps need to be done for determination of the ETW:
Determination of x_{c} by Eq. (28)
Determination of y_{c} by Eq. (29)
Determination of ETW as 2x_{c}
Determination of α_{R} and α_{C} (Fig. 5) to check the condition α_{R} = α_{C}
The actual ETW a_{w} is subsequently determined as the minimum of the ETW for the case of α_{R} < α_{C} and the ETW for the case of α_{R} = α_{C.}
Determination of the true strain in the PSCT
The true strain in the PSCT is determined under assumption of its homogeneous distribution under the contact surface as ε_{ps} = ln(h_{0}/h), where h is determined from the corrected forcedisplacement curve.
Determination of the true stress
The true stress in the PSCT is determined by σ_{ps} = F/(ba_{w}), where b is the actual specimen width or the length of the contact surface and w is the effective tool width.
Determination of the friction and shear corrected true stress
The true equivalent stress according to Tresca the material is exposed to during the PSCT \( {\overline{\sigma}}^{\mathrm{T}} \) is determined based on σ_{ps} using Eq. (12) by substituting a with a_{w}.
Determination of the combined flow curve from the UTT and PSCT
Determination of the combined flow curve consisting of the flow curve from the UTT and the flow curve from the PSCT converted to the uniaxial tension stress state using Eqs. (17)–(19) should be performed the way it is done for the combined flow curve consisting of the of the flow curve from the UTT and the flow curve from the hydraulic bulge test converted to the uniaxial tension stress state as described in ISO 16808.
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Chermette, C., Unruh, K., Peshekhodov, I. et al. A new analytical method for determination of the flow curve for highstrength sheet steels using the plane strain compression test. Int J Mater Form 13, 269–292 (2020). https://doi.org/10.1007/s12289019014854
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Keywords
 Plane strain compression test
 Flow curve
 AHSS