# Improvement of Numerical Modelling Considering Plane Strain Material Characterization with an Elliptic Hydraulic Bulge Test

^{*}

## Abstract

**:**

## 1. Introduction

_{2}= 0 is not possible. Flores et al. [5] and Wagoner [6] came up against this challenge by defining a dependence of the two principal stresses with the material anisotropy that is quantified in uniaxial tensile tests. Wagoner referred to the mean vertical anisotropy which can be ascribed to the theory of Hill [4] with a factor m = 2. In contrast to this, Flores et al. described a relationship to the material anisotropy in transverse direction. Furthermore, for the model of Flores et al., a maximum tolerable transverse contraction of 0.02 true strain was assumed. Within this boundary, the area under plane strain conditions could be determined for each time step. Investigations by Kuwabara [7] with a modified cruciform tensile test proved that the relationship of the second principal stress to the material anisotropy overestimates this stress component and leads to a higher stress.

## 2. Experimental Setup and Methodology

#### 2.1. Materials and Specimen Preparation

_{0}= 1.0 mm. The specimens used for the elliptical hydraulic bulge test are circular with a diameter of 395.0 mm. Two notches under 0° 45° and 90° according to rolling direction are used for identification purposes and for a robust alignment in the testing setup, respectively. All specimens are extracted by laser cutting (TruLaser Cell 7020, Trumpf GmbH + Co. KG, Ditzingen, Germany).

#### 2.2. Methodology

#### 2.2.1. Experimental Setup

_{1}= 200 mm and d

_{2}= 75 mm with a die radius of 28 mm. The resulting aspect ratio is 2.67 for the ellipsoid. The introduced elliptical geometry is numerically optimized for failure under plane strain conditions in the center of the specimen and optimized to a reduced bending strain overlap which would reduce the formability and causes failure in the die radius. To measure the hydraulic pressure for the determination of the stress, a pressure sensor with 40 MPa is integrated. The local strain distribution during the forming process is detected by a three-dimensional optical strain measurement system (ARAMIS, GOM GmbH, Braunschweig, Germany) with a camera resolution of five megapixels. The testing device is able to perform material testing with a constant strain rate by controlling the forming speed during the test with the help of the optical measurement system. Therefore, the volume flow of the hydraulic medium is regulated with an online strain measurement. This information is used to control the speed of the hydraulic pressure transducer (Pregler GmbH Co. KG, Deggendorf, Germany).

#### 2.2.2. Experimental Procedure

_{1}of 30 mm in x direction and 5 mm in y direction. For the determination of the curvature r

_{2}in y direction, an evaluation area of 30 mm in y direction and 5 mm in x direction are used. Figure 2 illustrates the evaluation areas for both curvatures.

#### 2.2.3. Determination of Flow Curves under Plane Strain

_{1}and r

_{2}, the pressure and the resulting total strain tensor in the pole center, the resulting principal stress components can be determined for each measured time increment in Equations (1) and (2) with the shell theory [23].

_{2}(t) is the determined radius and t(t) refers to the actual sheet thickness. The same counts for Equation (2). Here, additionally, the radius r

_{1}(t), which corresponds to the curvature in x direction, is needed. The sheet thickness in the pole can be calculated with the measured strain data (see Equation (3)).

_{0}is needed. The true plastic strain in sheet thickness direction ε

_{3,pl}(t) is calculated in Equation (4).

_{1,pl}(t) and y direction ε

_{2,pl}(t). To determine the true plastic strain components, the elastic strain has to be reduced from the measured total strain in both directions. Therefore, the elastic strain in both principal directions has to be calculated, (see Equations (5) and (6)).

_{1}to the elastic strain in ε

_{2}direction and vice versa, the Poisson ratio ν is needed. The plastic strain ratio can be determined with Equations (7) and (8).

_{1,total}and ε

_{2,total}in Equations (7) and (8) are measured with the optical strain measurement device.

#### 2.2.4. Verification of the Proposed Approach

## 3. Results of Plane Strain Hydraulic Bulge Tests

## 4. Process Verification and Validation

#### 4.1. Verification of the Evaluation Area for the Curvature Determination

_{y}= 68.5 mm. For the curvature in x direction, a resulting radius r

_{x}= 370 mm can be measured. For this evaluation, the best fit cylinder can be determined with a standard deviation of 0.057 mm for the distance between the cylinder and the surface of the specimen. With respect to the magnitude of the curvature, the standard deviations of 0.01 mm and 0.057 mm are insignificant. Hence, the evaluation area can be used to determine the resulting radii of the curvatures with sufficient accuracy.

#### 4.2. Strain Distribution Analysis

#### 4.3. Verification of the Proposed Stress Calculation

#### 4.4. Process Validation with Notched Tensile Tests

## 5. Improvement of Sheet Thickness Distribution Prognosis in Forming Simulations

#### 5.1. Enhanced Material Modelling with the Yield Criterion Yld2000-2d

_{pl,eq}= 0.05 because of the high standard deviation in HBT at the beginning of plastic deformation.

_{1}and σ

_{2}, which are discussed above, are used. The methodology for the optimization is presented in Figure 13. The root mean squared error between the experimentally determined point under plane strain and the identified curve is calculated first. Then, the root mean squared error is minimized by changing the yield locus exponent. This is suitable because the shape of the curvature changes with a variation of this parameter. The higher the yield locus exponent, the straighter the curvature of the yield surface. In contrast, a lower m value leads to a round contour. With this procedure, the yield locus exponent is adapted to the experimental data. The resulting parameters are given in Table 4 for all three materials, AA5182, DC06 and DP600, whereby a yield locus exponent of m = 5.16 results for AA5182. The two steel grades allocate a value of m = 5.59 for the DC06 and m = 3.42 for the DP600. The determined yield locus exponents have no metal physical motivation, but are based on experimental data, which are not valid for the yield locus exponents given in the literature.

#### 5.2. Improvement of Numerical Simulation

#### 5.2.1. AA5182

#### 5.2.2. DC06

_{pl}= 0.05, leads to a rising difference between real and modelled material behavior with proceeding plasticization. The investigated cup has a drawing depth of 50.0 mm; therefore, a high plastic deformation can be assumed.

#### 5.2.3. DP600

## 6. Summary and Outlook

## Author Contributions

## Conflicts of Interest

## References

- Hariharan, K.; Prakash, R.V.; Prasad, M.S.; Reddy, G.M. Evaluation of yield criteria for forming simulations based on residual stress measurement. Int. J. Mater. Form.
**2010**, 3, 291–297. [Google Scholar] [CrossRef] - Banabic, D. Sheet Metal Forming Processes—Constitutive Modelling and Numerical Simulation; Springer: Berlin, Germany, 2010. [Google Scholar]
- Xavier, M.D.; Plaut, R.L.; Schön, C.G. Uniaxial Near Plane Strain Tensile Tests Applied to the Determination of the FLC Formabillity Parameter. Mater. Res.
**2014**, 17, 982–986. [Google Scholar] [CrossRef] - Barlat, F.; Brem, J.C.; Joon, J.W.; Chung, K.; Dick, R.E.; Lege, D.J.; Porboghrat, F.; Choi, S.-H.; Chu, E. Plane stress yield function for aluminum alloy sheets—Part 1: Theory. Int. J. Plast.
**2003**, 19, 1297–1319. [Google Scholar] [CrossRef] - Flores, P.; Tuninetti, V.; Gilles, G.; Gonry, P.; Duchêne, L.; Habraken, A.-M. Accurate stress computation in plane strain tensile tests for sheet metal using experimental data. J. Mater. Process. Technol.
**2010**, 1772–1779. [Google Scholar] [CrossRef] - Wagoner, R.H. Maesurement and Analysis of Plane-Strain Work Hardening. Metall. Trans. A
**1980**, 11, 165–175. [Google Scholar] [CrossRef] - Kuwabara, T. Advances of Plasticity Experiments on Metal Sheets and Tubes and Their Applications to Constitutive Modeling. AIP Conf. Proc.
**2005**, 778, 20–39. [Google Scholar] [CrossRef] - Hecht, J.; Pinto, S.; Geiger, M. Determination of Mechanical Properties for the Hydroforming of Magnesium Sheets at Elevated Temperature. Adv. Mater. Res.
**2005**, 6–8, 779–786. [Google Scholar] [CrossRef] - Altan, T.; Palaniswamy, H.; Bortot, P.; Mirtsch, M.; Heidl, W.; Bechtold, A. Determination of sheet material properties using biaxial bulge tests. In Proceedings of the Second International Conference on Accuracy in Forming Technology, Chemnitz, Germany, 13–15 November 2006; pp. 79–92. [Google Scholar]
- Rees, D.W.A. Plastic flow in the elliptical bulge test. Int. J. Mech. Sci.
**1995**, 37, 373–389. [Google Scholar] [CrossRef] - Lazarescu, L.; Nicodim, I.P.; Comsa, D.-S.; Banabic, D. A procedure for the evaluation of flow stress of sheet metal by hydraulic bulge test using elliptical dies. Key Eng. Mater.
**2012**, 504, 107–112. [Google Scholar] [CrossRef] - Lazarescu, L.; Comsa, D.-S.; Nicodim, I.; Ciobanu, I.; Banabic, D. Characterization of plastic behaviour of sheet metals by hydraulic bulge test. Trans. Nonferr. Met. Soc. China
**2012**, 22, s275–s279. [Google Scholar] [CrossRef] - Shi, B.; Peng, Y.; Yang, C.; Pan, F.; Cheng, R.; Peng, Q. Loading path dependent distortional hardening of Mg alloys: Experimental investigation and constitutive modeling. Int. J. Plast.
**2017**, 90, 76–95. [Google Scholar] [CrossRef] - Safaei, M.; Lee, M.-G.; Zang, S.-I.; De Waele, W. An evolutionary anisotropic model for sheet metals based on non-associated flow rule approach. Comput. Mater. Sci.
**2014**, 81, 15–29. [Google Scholar] [CrossRef] - Manopulo, N.; List, J.; Hippke, H.; Hora, P. A Non-Associated Flow Rule Based on Yld2000-2d Model. In Proceedings of the 8th Forming Technology Forum Zurich 2015, Zurich, Switzerland, 29–30 June 2016. [Google Scholar]
- Zang, S.-I.; Lee, M.; Kim, J.H.; Safaei, M. A new representation of linear transformation tensor for the description of plastic subsequent anisotropy. AIP Conf. Proc.
**2013**, 1567, 508–511. [Google Scholar] [CrossRef] - Plunkett, B.; Cazacu, O.; Barlat, F. Orthotropic yield criteria for description of the anisotropy in tension and compression of sheet metals. Int. J. Plast.
**2008**, 24, 847–866. [Google Scholar] [CrossRef] - Kuwabara, T.; Mori, T.; Asano, M.; Hakoyama, T.; Barlat, F. Material modeling of 6016-O and 6016-T4 aluminum alloy sheets and application to hole expansion forming simulation. Int. J. Plast.
**2017**, 93, 164–186. [Google Scholar] [CrossRef] - Merklein, M.; Suttner, S.; Brosius, A. Charcaterisation of kinematic hardening and yield surface evolution from uniaxial to biaxial tension with continuous strain path change. CIRP Ann. Manuf. Technol.
**2014**, 14, 297–300. [Google Scholar] [CrossRef] - Suttner, S.; Merklein, M. Experimental and numerical investigation of a strain rate controlled hydraulic bulge test of sheet metal. J. Mater. Process. Technol.
**2016**, 235, 121–133. [Google Scholar] [CrossRef] - SEP 1240: 2006-07. Testing and Documentation Guideline for the Experimental Determination of Mechanical Properties of Steel Sheets for CAE-Calculations; National Standard; Beuth Verlag GmbH: Berlin, Germany, 2006. [Google Scholar]
- VDA 239-300: 2015-10. Experimental Determination of Mechanical Properties of Aluminium Sheets for CAE-Calculation Testing and Documentation; National Standard; Verband der Automobilindustrie e.V. (VDA): Berlin, Germany, 2015. [Google Scholar]
- Kollar, L.; Dulácska, E. Buckling of Shells for Engineers; John Wiley Sons: Budapest, Hungary, 1984; ISBN 978-0471903284. [Google Scholar]
- Jung, J.; Jun, S.; Lee, H.-S.; Kim, B.-M.; Lee, M.-G.; Kim, J.H. Anisotropic Hardening Behaviour and Springback of Advanced High-Strength Steels. Metals
**2017**, 7, 480. [Google Scholar] [CrossRef] - Pijlman, H.H.; Huetink, H.; Meinders, V.T.; Carleer, B.D.; Vegter, H. The Implementation of the Vegter yield Criterion and a physically based hardening rule in Finite Elements. In Proceedings of the 4th International World congres on Computational Mechanics (IACM ’98), Buenos Aires, Argentine, 29 June–2 July 1998. [Google Scholar]

**Figure 4.**Strain rate over equivalent strain (

**a**) and resulting curvature over time (

**b**) for elliptic hydraulic bulge test (HBT) of DC06.

**Figure 5.**True stress—true plastic strain curves with an elliptic hydraulic bulge test (HBT) according to rolling direction (RD), diagonal direction (DD) and transversal direction (TD) of AA5182.

**Figure 6.**Flow curves with an elliptic hydraulic bulge test (HBT) according to rolling direction (RD), diagonal direction (DD) and transversal direction (TD) for DC06.

**Figure 7.**True principle stress—true plastic strain curves with an elliptic hydraulic bulge test (HBT) according to rolling direction (RD) and transversal direction (TD) of DP600.

**Figure 9.**Evaluation area in elliptic hydraulic bulge test for determination of the curvature via best fit cylinders.

**Figure 10.**Resulting strain path in the pole of the elliptic specimen in contrast to the resulting forming limit curve (FLC) according to DIN EN ISO 12004-2 for DC06.

**Figure 11.**Equivalent stress-strain curve for circular and elliptical HBT for AA5182, DC06 and DP600.

**Figure 12.**Flow curves of notched tensile tests in rolling direction (RD) and elliptic hydraulic bulge tests HBT in transversal direction (TD) for AA5182, DC06 and DP600.

**Figure 13.**Methodology for the mapping of the yield locus under plane strain conditions through variation of the yield locus exponent m.

**Figure 17.**Sheet thickness of cruciform cups of A5182 for different yield locus exponents in simulations and experimental result.

**Figure 18.**Sheet thickness progression in the plane strain area of the punch radius for simulations with different yield locus exponent and experimental measurement for AA5182.

**Figure 19.**Sheet thickness of cruciform cups of DC06 for different yield locus exponents in simulations and experimental result.

**Figure 20.**Sheet thickness progression in the plane strain area of the punch radius for simulations with different yield locus exponent and experimental measurement for DC06.

**Figure 21.**Sheet thickness of cruciform cups of DP600 for different yield locus exponents in simulations and experimental result.

**Figure 22.**Sheet thickness progression in the plane strain area of the punch radius for simulations with different yield locus exponent and experimental measurement for DP600.

**Table 1.**Mechanical properties of the materials AA5182-O, DC06 and DP600 derived from material characterization.

Material | Sheet Thickness t_{0} (mm) | Yield Stress Y_{0.2} (MPa) | Tensile Strength TS (MPa) |
---|---|---|---|

AA5182-O | 1.0 | 120–125 | 260–270 |

DC06 | 1.0 | 165–180 | 285–300 |

DP600 | 1.0 | 400–405 | 645–650 |

Material | Blank Holder Force (kn) | Drawing Depth (mm) |
---|---|---|

AA5182-O | 100 | 30 |

DC06 | 400 | 50 |

DP600 | 300 | 30 |

**Table 3.**Conventional identified parameters for the yield criterion Yld2000-2d for AA5182, DC06 and DP600.

Material | Alpha 1 | Alpha 2 | Alpha 3 | Alpha 4 | Alpha 5 | Alpha 6 | Alpha 7 | Alpha 8 | Yield Locus Exponent |
---|---|---|---|---|---|---|---|---|---|

AA5182 | 0.9937 | 0.9680 | 0.9650 | 1.0298 | 1.0169 | 1.0272 | 0.9732 | 1.0564 | 8 |

DC06 | 0.9946 | 1.1106 | 0.7804 | 0.8744 | 0.8911 | 0.6842 | 0.9960 | 1.0478 | 6 |

DP600 | 0.9570 | 0.9998 | 1.0879 | 1.0024 | 1.0146 | 0.9830 | 0.9661 | 0.9866 | 6 |

**Table 4.**Identified parameters for the yield criterion Yld2000-2d with experimental determined yield locus exponent under plane strain for AA5182, DC06 and DP600.

Material | Alpha 1 | Alpha 2 | Alpha 3 | Alpha 4 | Alpha 5 | Alpha 6 | Alpha 7 | Alpha 8 | Yield Locus Exponent |
---|---|---|---|---|---|---|---|---|---|

AA5182 | 1.0708 | 0.8466 | 0.8889 | 1.0383 | 1.0208 | 1.0717 | 0.9277 | 1.0954 | 5.16 |

DC06 | 0.9873 | 1.1261 | 0.8022 | 0.8696 | 0.8889 | 0.6747 | 1.0004 | 1.0442 | 5.59 |

DP600 | 1.1906 | 0.7409 | 0.8430 | 1.0001 | 1.0161 | 1.2006 | 0.9497 | 1.0293 | 3.42 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lenzen, M.; Merklein, M.
Improvement of Numerical Modelling Considering Plane Strain Material Characterization with an Elliptic Hydraulic Bulge Test. *J. Manuf. Mater. Process.* **2018**, *2*, 6.
https://doi.org/10.3390/jmmp2010006

**AMA Style**

Lenzen M, Merklein M.
Improvement of Numerical Modelling Considering Plane Strain Material Characterization with an Elliptic Hydraulic Bulge Test. *Journal of Manufacturing and Materials Processing*. 2018; 2(1):6.
https://doi.org/10.3390/jmmp2010006

**Chicago/Turabian Style**

Lenzen, Matthias, and Marion Merklein.
2018. "Improvement of Numerical Modelling Considering Plane Strain Material Characterization with an Elliptic Hydraulic Bulge Test" *Journal of Manufacturing and Materials Processing* 2, no. 1: 6.
https://doi.org/10.3390/jmmp2010006