# Physical Modelling and Numerical Simulation of the Deep Drawing Process of a Box-Shaped Product Focused on Material Limits Determination

^{1}

^{2}

^{*}

## Abstract

**:**

_{m}≥ 1.47 and an average strain hardening exponent of n

_{m}≥ 0.23 must be used for the deep drawing of the bathtub.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Press-Die-Pressing System

#### 2.2. Material

_{0}= 0.5 mm was used. From the point of view of formability, the material properties are shown in Table 3. These were measured according to the following standards: mechanical properties by STN EN ISO 6892-1, the normal anisotropy ratio by STN EN ISO 10113, and the strain hardening exponent by STN EN ISO 10275 using the TIRAtest 2300 testing machine (TIRA Maschinenbau GmbH, Rauenstein, Germany) controlled by a PC.

#### 2.3. Numerical Simulation Model

#### 2.3.1. Material Hardening Model

_{0}is the pre-strain, and φ

_{pl}is the plastic strain.

#### 2.3.2. Material Yield Locus

_{Y}is the equi-biaxial yield stress.

_{0}, r

_{45}, r

_{90}(i.e., plastic strain ratios), and these were measured using tensile tests performed on the specimens taken at 0°, 45°, and 90° to the rolling direction (see Table 3).

_{p0.2}for each Lankford’s coefficient r

_{α}and either the Hill 90 coefficient m or the equi-biaxial yield stress σ

_{y}. An equi-biaxial yield stress of σ

_{y}= 220 MPa was used for the calculations, which was obtained from a hydraulic bulge test using the HYDROTEST device. Uniaxial yield values R

_{p0.2}and Lankford’s coefficients for the rolling directions of 15°, 30°, 60°, and 75° were additionally tested according to standards shown previously (Table 5). The values of the Hill 90 yield locus constants α, β, γ, and m are shown in Table 6.

#### 2.3.3. Failure Criteria

_{1}when φ

_{2}= 0 was calculated as follows: [30,35]

_{0}is the material thickness, and n is the strain hardening exponent. The left side of the FLC curve was calculated from Equation (11) and the right side from Equation (12).

#### 2.3.4. Boundary Conditions

^{−1}, and the punch speed was set to 5 m·s

^{−1}. In the z direction, both increased linearly from zero to the final value to prevent dynamic effects at contact. This was defined as the imposed velocity by the curve [30]. Contact conditions were defined by the friction coefficient and the Coulomb laws defined the friction. Water-based lubricant is used when real bathtubs are produced, so the friction coefficient was set to a constant value of 0.09 due to the friction between the steel sheet and tool steel [28,36]. The blankholder force was set to 340 kN when the bathtub model pressing free of wrinkles and fracture was drawn.

## 3. Results

_{simul}is the thickness evaluated from the numerical simulation, Th

_{real}is the thickness measured on the physical model, and Th

_{nom}is the nominal steel sheet thickness. The results are shown in Table 8 and in Figure 8. Based on the evaluation using the minimal thickness criterion, the best combination seems to be the Hill 48 yield locus and the Krupkowski material hardening law. The values of thickness at both critical regions were the closest to the thicknesses measured on the bathtub model pressing. The graph on Figure 8 shows the deviation of thickness evaluated from the numerical simulation and the thickness measured on the bathtub model pressing for each combination material hardening law/yield locus.

_{45}where the lowest value was measured. The results comply with those of [37], because the plastic strain ratio expresses the steel sheet’s resistance to thinning. Thus, greater thinning of the thickness appears in the direction of the lowest plastic strain ratio.

## 4. Discussion

_{m}≥ 1.47 and n

_{m}≥ 0.23 to obtain bathtub model pressings free of fracture. To improve the deep drawing process of the bathtub model pressing or the real bathtub in production, it is recommended that these values are higher. This is because the plastic strain ratio improves the steel sheet’s resistance to thinning of the thickness, while the strain hardening exponent unifies the strain distribution during deformation and prevents necking [37].

## 5. Conclusions

- The Hill 48 and Hill 90 yield locus mathematical models and the Hollomon and Krupkowski hardening law mathematical models for cold rolled low carbon aluminum-killed steel for enameling were determined from tensile tests at angles of 0°, 45°, and 90° to the rolling direction and bulge tests. Experimental Kosmalt 190 steel with a thickness of a
_{0}= 0.5 mm showed extra deep drawing quality with r_{m}= 1.57 and n_{m}= 0.226. - In all numerical simulations and physical experiments, the bathtub model pressing was drawn free of fracture and wrinkles when simulated at the same blankholder force (340 kN) and friction (0.09) values. Keeler-Brazier’s mathematic model was used to define the forming limit curve and to determine material fracture in numerical simulations.
- The best yield locus/hardening law combination appeared to be Hill 48/Krupkowski. This was determined by comparing the wall thicknesses of model pressing in selected sections after simulations and physical experiments. The deviations at the local minima were 0.7% and −1.0% in section A-A (longitudinal) and 2.4% in section B-B (corner). The course of relative thickness change evaluated from numerical simulations and experimental measurements showed good conformity.
- The material’s anisotropy limits were found to be r
_{m}= 1.47 and n_{m}= 0.23 when the model pressing free of fracture was drawn in a numerical simulation. Virtual materials were defined from experimentally measured values of the plastic strain ratio.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Experimental drawing of the die: (

**a**) overview; (

**b**) details of drawbeads and grooves; (

**c**) dimensions of the drawbeads [mm].

**Figure 3.**Microstructure and texture of the Kosmalt 190 material in the rolling direction: (

**a**) microstructure; (

**b**) texture.

**Figure 4.**Simulation model and blank: (

**a**) meshed components of the drawing die; (

**b**) shape and dimensions of the blank.

**Figure 6.**Section of the bathtub model used to measure the thickness (the rolling direction 0° was on the longitudinal axis).

Parameter | Real Bathtub | Bathtub Model |
---|---|---|

Geometry similarity (scale 1:5) | ||

Length [mm] | 1695 | 339 |

Width [mm] | 710 | 142 |

Height [mm] | 400 | 80 |

Wall to bottom radius [mm] (i.e., Punch radius [mm]) | 130 | 26 |

Wall to flange radius [mm] (i.e., Die radius [mm]) | 28 | 5.6 |

Mechanical similarity | ||

Press | Hydraulic Fritz Muller BZE 2000 | Hydraulic Fritz Muller BZE 100 |

Ram working velocity [mm·s^{−1}] | 25 | 15 |

Die and punch material | Cast steel | Cast steel |

Physical similarity | ||

Material | Enameling steel Kosmalt | Enameling steel Kosmalt |

Lubricant | Vantol S | Vantol S |

C | Mn | P | S | Al | N | Cu | Ni | Cr |
---|---|---|---|---|---|---|---|---|

0.030 | 0.140 | 0.009 | 0.008 | 0.042 | 0.003 | 0.014 | 0.015 | 0.013 |

Dir. [°] | R_{p0.2}[MPa] | Rm [MPa] | A_{80}[%] | r [–] | rm [–] | Δr [–] | n [–] | nm [–] | Δn [–] |
---|---|---|---|---|---|---|---|---|---|

0 | 158 ^{±0.9} | 280 ^{±1.3} | 45.5 ^{±0.3} | 1.58 ^{±0.036} | 0.226 ^{±0.002} | ||||

45 | 159 ^{±1.1} | 286 ^{±0.9} | 42.4 ^{±0.5} | 1.33 ^{±0.032} | 1.57 | 0.47 | 0.227 ^{±0.001} | 0.226 | –0.001 |

90 | 155 ^{±1,0} | 279 ^{±0.5} | 45.4 ^{±0.5} | 2.02 ^{±0.052} | 0.225 ^{±0.001} |

_{p0.2}—yield strength; Rm—ultimate tensile strength; A

_{80}—elongation (80 mm initial gage length); r—plastic strain ratio (r-value); r

_{m}—average r-value; Δr—planar anisotropy of r-value. n—strain hardening exponent (n-value); n

_{m}—average n-value; Δn—planar anisotropy of n-value.

Model | K [MPa] | n [–] | φ_{0} [–] |
---|---|---|---|

Hollomon | 496 | 0.226 | - |

Krupkowski | 505 | 0.248 | 0.00899 |

Direction | R_{p0.2} [MPa] | r [–] |
---|---|---|

15 | 166 ^{±1.1} | 1.56 ^{±0.037} |

30 | 166 ^{±1.3} | 1.46 ^{±0.041} |

60 | 168 ^{±0.9} | 2.03 ^{±0.029} |

75 | 165 ^{±1.2} | 2.14 ^{±0.040} |

α | β | γ | m |
---|---|---|---|

1.56158 | 1.19317 | 20.2109 | 3.02902 |

Simulation Number | Yield Locus/Hardening Law | Minimal Thickness [mm] | ||
---|---|---|---|---|

Section A-A | Section B-B | |||

C-E | G-H | D-E | ||

S1 | Hill 48/Hollomon | 0.421 | 0.330 | 0.380 |

S2 | Hill 48/Krupkowski | 0.417 | 0.363 | 0.375 |

S3 | Hill 90/Hollomon | 0.405 | 0.398 | 0.416 |

S4 | Hill 90/Krupkowski | 0.412 | 0.396 | 0.417 |

Experiment | 0.413 ± 0.006 | 0.368 ± 0.008 | 0.363 ± 0.006 |

Simulation Number | Section A-A | Section B-B | |
---|---|---|---|

C-E | G-H | D-E | |

S1 | 1.7% | –7.6% | 3.4% |

S2 | 0.7% | –1.0% | 2.4% |

S3 | –1.7% | 5.9% | 10.6% |

S4 | –0.3% | 5.7% | 10.8% |

Material | r_{0} | r_{45} | r_{90} | r_{m} | Result |
---|---|---|---|---|---|

Kosmalt 190 | 1.58 | 1.33 | 2.02 | 1.57 | Ok |

Virtual B | 1.48 | 1.28 | 1.82 | 1.47 | Necking |

Virtual C | 1.38 | 1.23 | 1.62 | 1.37 | Fracture |

_{m}= 0.226.

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**MDPI and ACS Style**

Tomáš, M.; Evin, E.; Kepič, J.; Hudák, J.
Physical Modelling and Numerical Simulation of the Deep Drawing Process of a Box-Shaped Product Focused on Material Limits Determination. *Metals* **2019**, *9*, 1058.
https://doi.org/10.3390/met9101058

**AMA Style**

Tomáš M, Evin E, Kepič J, Hudák J.
Physical Modelling and Numerical Simulation of the Deep Drawing Process of a Box-Shaped Product Focused on Material Limits Determination. *Metals*. 2019; 9(10):1058.
https://doi.org/10.3390/met9101058

**Chicago/Turabian Style**

Tomáš, Miroslav, Emil Evin, Ján Kepič, and Juraj Hudák.
2019. "Physical Modelling and Numerical Simulation of the Deep Drawing Process of a Box-Shaped Product Focused on Material Limits Determination" *Metals* 9, no. 10: 1058.
https://doi.org/10.3390/met9101058