# On the Use of Advanced Friction Models for the Simulation of an Industrial Stamping Process including the Analysis of Material and Lubricant Fluctuations

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Sheet Material

#### 2.2. Mechanical Characterisation

#### 2.3. Topography Analysis

^{2}). The surface topography parameters on the primary surface (S-F) were computed using SensoMap Premium 7.4 metrology software for data post-processing in accordance with the ISO 25178 standard [27].

- Average roughness (Sa). The arithmetic mean of the absolute value of the height within the surface. It is the most commonly used parameter to assess surface topography, together with the two-dimensional roughness parameter Ra.
- Root mean square roughness (Sq). A measurement of the asymmetry of the surface deviation about the mean plane.
- Maximum height (Sz). The sum of the maximum value of the surface peak height and the maximum value of the surface valley within the defined area.
- Skewness (Ssk): The degree of bias of the roughness shape, i.e., the asperity of the surface. A positive skewness gives rise to a surface with more peaks or asperities, whereas a negative skewness leads to more valleys.

#### 2.4. Experimental Measurements

#### 2.5. Strip Drawing Tests

_{N}) is applied by moving the upper die using a servomotor coupled to a mechanical jack. Subsequently, the strip is pulled between dies at a constant sliding velocity (v). Both tangential (F

_{T}) and normal (F

_{N}) forces are recorded during the test. F

_{T}is the sum of interface friction forces between the sheet and the upper and lower dies. Therefore, the CoF (µ in Equation (1)) can be calculated with the Coulomb friction law (Equation (1)):

^{3}at 50° and a viscosity of 19.6 mm

^{2}/s at 40 °C. A micropipette was used to apply the specified amount of lubricant in each test. Three repetitions per sliding velocity and contact pressure were tested to ensure representative results.

^{2}of lubricant was applied for all contact pressures and a sliding velocity of 10 mm/s. Then, the effect of the amount of lubricant on the friction coefficient was tested with amounts ranging from 0.5 g/m

^{2}to 4 g/m

^{2}for the same sliding velocity and various pressures. Finally, three lubrication levels were tested based on production measurements: 1 g/m

^{2}, 2.4 g/m

^{2}(average lubrication), and 4 g/m

^{2}(maximum measured). These results were used to develop the friction models that generated the inputs of the numerical model. The results were also used to reproduce the distribution of lubricant in each zone and on each side of the sheet.

#### 2.6. Tribological Models

- Constant. In most industrial simulations, a Coulomb constant model is used for steel deep drawing. In this work, the typical constant value of 0.15 was applied so as to compare with the remaining models.
- Pressure and velocity dependent (P-v dependent). Several studies have demonstrated the influence of contact pressure and sliding velocity on the friction coefficient [4,5,8,10]. In this work, a pressure- and velocity-dependent friction model to calculate the effective coefficient of friction ${\mu}_{eff}$ was assumed (Equation (2)) following the potential distribution of Filzek [4]:$${\mu}_{eff}=\mu {\left(\frac{p}{{p}_{ref}}\right)}^{e-1}-a\xb7\mathrm{l}\mathrm{n}\frac{max\left({\nu}_{rel},{\nu}_{ref}\right)}{{\nu}_{ref}}$$
^{2}. The maximum, minimum, and mean lubricant amount values were selected to analyze the differences in the tribological models for these boundary values. For simulation, a P-v-dependent single model with a medium level of lubricant was defined, as AutoForm is not able to create different models for each side of the sheet for this tribological model type. Therefore, a simulation model with a P-v-dependent 2.4 g/m^{2}model was implemented. The average between the two faces was considered for each measured zone, and it was assumed that the P-v-dependent model for the average amount of lubricant was an appropriate approximation. - TriboForm with lubrication zones. In this case, TriboForm
^{®}software was used to create a contact-pressure- and sliding-velocity-dependent model. This model also considers material elasto-plasticity and tool roughness. AutoForm has a TriboForm plug-in that permits implementation of this complex model. It can also define different amounts of lubricant in various zones and sides of the sheet by means of lubrication spots. Nine lubrication zones were defined for each zone and side of the sheet to replicate industrial conditions. These amounts of lubricant were measured in the industrial precut. Three tribological models were defined for three lubricant levels based on the industrial measurements: 1, 2.4, and 4 g/m^{2}. As for the P-v-dependent models, the maximum, minimum, and mean values were selected to analyze the differences between models. To feed the simulation model, the TriboForm 1 g/m^{2}model was used as the basis. Additional amounts of lubricant were applied to the different zones and sides of the sheet based on experimental measurements with lubrication spots.

#### 2.7. Simulation Set-Up

## 3. Results

#### 3.1. Tensile Tests and FLDs

_{p 0.2%}), ultimate strength (R

_{m}) Lankford coefficients at 0°, 45°, and 90° (r

_{0}, r

_{45}, and r

_{90}), and hardening exponent (n). Similar values were observed between batches.

#### 3.2. Topography Analysis

#### 3.3. Strip Drawing Tests

^{2}and 4 g/m

^{2}of lubricant at a contact pressure of 15 MPa. This decrease in the CoF with contact pressure followed the same trend for all levels of lubricant. The difference between the CoF of the minimum and maximum amounts of lubricant also increased slightly as the contact pressure rose.

^{2}(Figure 7a), 2.4 g/m

^{2}(Figure 7b), and 4 g/m

^{2}(Figure 7c). The results show that both contact pressure and sliding velocity have a significant influence on the CoF. The decreasing trend of the CoF with contact pressure is more pronounced from 2 to 10 MPa.

^{2}(Figure 7a) and 4 g/m

^{2}(Figure 7c), a slight increase in the CoF can be observed at the end of the test, although there was no galling in the die blocks. For the minimum lubrication, the CoF ranges from 0.1208 (at 2 MPa and 10 mm/s) to 0.054 (at 25 MPa and 150 mm/s). At the maximum level of lubricant, the CoF ranges from 0.1140 (at 2 MPa and 10 mm/s) to 0.047 (at 25 MPa and 150 mm/s).

#### 3.4. Tribological Models

- Pressure and velocity dependent (P-v dependent). Figure 8 plots the three P-v-dependent models developed for the minimum (Figure 8a), mean (Figure 8b), and maximum (Figure 8c) amounts of lubricant. As observed in the experimental tests, the CoF presents higher values for lower sliding velocities and lower pressures. The boundaries of all the P-v-dependent models are well defined for small contact pressures (2 MPa), whereas for higher contact pressures there are some disparities with respect to the experimental results. The P-v-dependent models for 1 and 4 g/m
^{2}report a slight overestimation of the CoF for high contact pressures and sliding velocities. As an example, the P-v-dependent model for 4 g/m^{2}presents a small disparity for a sliding velocity of 150 mm/s, in that it estimates a CoF of 0.55 versus an experimental value of 0.047. The increase in the CoF for 25 MPa and 10 mm/s in 1 and 4 g/m^{2}lubrication levels cannot be replicated by the P-v-dependent models, which show a maximum deviation of 18% with respect to the experimental value for the 1 g/m^{2}P-v-dependent model. The coefficients of the P-v-dependent friction models (Equation (2)) were calculated with least squares methodology using Microsoft Excel and are presented in Table 4. The P-v-dependent model for 2.4 g/m^{2}reported the lowest error, with an RMSE of 0.04 (Table 5) and model boundaries in strong agreement with the experimental results.

- TriboForm with lubrication zones. These models were composed of a cast iron tooling with an average roughness of Sa = 0.65 µm and sheet material with a roughness of Sa = 1.4 μm. The nine lubrication zones defined for each side of the sheet are illustrated in Figure 9. Note that the lubrication scope in the TriboForm library created ranges from 0.5 to 3 g/m
^{2}. Hence, zones with amounts of lubricant higher than 3 g/m^{2}were limited to a maximum of 3 g/m^{2}.

^{2}. The higher RMSE values from the 1 and 4 g/m

^{2}models were due to a singularity in the CoF value for a contact pressure of 25 MPa and sliding velocity of 10 mm/s, among other factors. As for R-squared, all models presented similarly high values. For 2.4 g/m

^{2}, the highest R-squared was found for both the P-v-dependent and TriboForm models.

^{2}TriboForm models presented the lowest friction coefficient for all contact pressures. In the case of sliding velocities lower than 10 mm/s, all the TriboForm models reported a higher CoF than the P-v-dependent models for the same contact pressures. For example, the TriboForm predicted CoF reached a value of 0.146 for 1 MPa, 1 mm/s, and 1 g/m

^{2}, which is 11.81% higher than the P-v-dependent model for the same conditions. For sliding velocities higher than 150 mm/s and high contact pressures (40 MPa), there are also differences between the two models. For example, for 300 mm/s a global minimum CoF of 0.025 was obtained in the TriboForm model of 4 g/m

^{2}, versus a CoF value of 0.043 in the P-v-dependent model of 4 g/m

^{2}. With respect to the constant model (CoF 0.15), substantial differences were observed. For high contact pressures and sliding velocities (e.g., 40 MPa and 300 mm/s), the P-v-dependent model estimated a CoF 64.67% lower than the constant model. Under the same conditions, the TriboForm model estimated a CoF 76.95% lower than the constant model. The TriboForm model for 1 g/m

^{2}estimated a CoF value of 0.419 at 40 MPa and 300 mm/s, which was 40.13% higher than the TriboForm model for 4 g/m

^{2}(with a CoF of 0.029).

^{2}models, with the former presenting a marked change in the trend for these extreme values.

#### 3.5. Numerical Results

#### 3.6. Experimental Measurements

## 4. Discussion

^{2}(Figure 6b). Although our results indicated that the influence of the lubricant is strongest at higher contact pressures (Figure 6b), Filzek et al. [8] observed the opposite effect. This phenomenon could be attributed to hydrostatic and hydrodynamic fluid effects [36]. High levels of lubricant can lead to a mixed-layer lubrication regime, resulting in significant variability in CoF values for the same contact pressures and sliding velocities [37]. Contact area size can have also an effect on this variation In the CoF. Recklin et al. [38] observed variations in the effect of the amount of lubricant between the same tests with different contact areas due to its distribution over the contact area. Lubricant viscosity also can affect the results, as its low viscosity (19.6 mm

^{2}/s at 40 °C) might cause a lubricant sweep during the process.

^{2}. Lubricant in such quantities could cause wrinkles on the part, which may have led to zones with significant thickening.

## 5. Conclusions

- The use of complex tribological models with the friction coefficient as a function of contact pressure, sliding velocity, and amount of lubricant substantially improved the simulation accuracy compared with the constant model.
- The TriboForm with lubrication zones and P-v-dependent models predicted similar results; the TriboForm model predicted slightly more thickening.
- The constant model predicted a lower draw-in value than the P-v-dependent and TriboForm with lubrication zones models for all points measured. With respect to the experimental results, no clear trend was observed.
- For batches of DC06 mild steel, no significative differences in the friction coefficient were found. This suggests a variation in the roughness of the sheets as an incontrollable noise that is not batch dependent. Nonetheless, this did not significantly affect the friction coefficient results (less than 0.01).
- For the analyzed materials, there were no major differences in the values of the coefficient of friction based on the amount of lubricant applied. The variation in the amount of lubricant (from 0.5 to 4 g/m
^{2}) led to maximum CoF variation of 0.014.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Scheme of the strip drawing test (

**left**) and strip drawing test machine used for the tests (

**right**).

**Figure 2.**AutoForm Fe model set-up (

**left**) and final part (

**right**). In AutoForm set-up, from top to bottom: die, sheet and binder, and punch. Spacer blocks are not shown in the set-up.

**Figure 3.**(

**a**) True stress–strain curves in the rolling directions 0°, 45°, and 90°; (

**b**) Upper and lower Forming Limit Curves. In both cases the global upper and lower limit curves for each test are shown for the 5 batches.

**Figure 4.**Representative axonometric (

**above**) and two-dimensional (

**below**) projections of the surface textures corresponding to samples of the five batches: (

**a**) DC06_1; (

**b**) DC06_2; (

**c**) DC06_3; (

**d**) DC06_4; and (

**e**) DC06_5.

**Figure 5.**Height descriptor parameters of the five different batches of DC06 material: (

**a**) average roughness Sa; (

**b**) root mean square roughness Sq; (

**c**) maximum height Sz; and (

**d**) skewness Ssk.

**Figure 6.**CoF vs. contact pressure of (

**a**) the five the material batches analyzed (DC06_3) at 10 mm/s and 0.5 g/m

^{2}and (

**b**) one of the batches analyzed (DC06_3) for different amounts of lubricant at 10 mm/s.

**Figure 7.**CoF vs. contact pressure of the DC06_3 batch at different sliding velocities (10, 50, 100, and 150 mm/s): (

**a**) 1 g/m

^{2}; (

**b**) 2.4 g/m

^{2}; and (

**c**) 4 g/m

^{2}.

**Figure 8.**CoF vs. contact pressure of the DC06_3 batch at different sliding velocities (10, 50, 100, and 150 mm/s). P-v-dependent model and experimental data for different amounts of lubricant: (

**a**) 1 g/m

^{2}; (

**b**) 2.4 g/m

^{2}; and (

**c**) 4 g/m

^{2}.

**Figure 9.**Amount of lubricant (in g/m

^{2}) for the lubrication zones defined on each side of the part: (

**a**) top side and (

**b**) bottom side.

**Figure 10.**CoF vs. contact pressure of the DC06_3 batch at different sliding velocities (10, 50, 100, and 150 mm/s). TriboForm model and experimental data. (

**a**) 1 g/m

^{2}; (

**b**) 2.4 g/m

^{2}; and (

**c**) 4 g/m

^{2}.

**Figure 11.**A 3D plot of CoF vs. contact pressure and sliding velocity of constant, P-v-dependent, and TriboForm models. The dots represent experimental points from strip drawing tests.

**Figure 12.**Simulation results of FLD diagrams of the part applying the three tribological models: (

**a**) constant (mu 0.15); (

**b**) P-v dependent; and (

**c**) TriboForm with lubrication zones. The legend represents the different strain states.

**Figure 13.**Thinning of the three tribological models analyzed: (

**a**) constant; (

**b**) P-v dependent, and (

**c**) TriboForm with lubrication zones. Negative values indicate more stretching (or more thinning) of the part with respect to reference thickness.

**Figure 14.**Thinning (in absolute value) of the three zones (1, 2 and 3) analyzed for the three tribological models. The value of maximum thinning of each zone highlighted was selected for the analysis.

**Figure 15.**Difference in draw-in prediction (in mm) for P-v-dependent and TriboForm with lubrication zones models with respect to the constant model (

**left**) for the eleven points analyzed (

**right**).

**Figure 16.**FLD contour diagrams of three zones of the industrial part. Tribological models (constant, P-v dependent and TriboForm) vs. experimental. (

**a**) Zone 1; (

**b**) Zone 2; and (

**c**) Zone 3.

**Figure 17.**Differences (in mm) between numerical draw-in of the three tribological models and experimental values for the 11 points analyzed.

**Figure 18.**Tangential velocity (in mm/s) for the three tribological models analyzed at the stage of maximum velocity: (

**a**) constant; (

**b**) P-v dependent; and (

**c**) TriboForm.

Factor Affecting CoF | Value |
---|---|

Contact pressure [MPa] | 2/5/10/15/25 |

Sliding velocity [mm/s] | 10/50/100/150 |

Amount of lubricant [g/m^{2}] | 0.5/1/2/2.4/3/4 |

Material | DC06 mild steel |

Sheet thickness | 0.64 mm |

Poisson ratio | 0.3 |

Young Modulus | 210 GPa |

Hardening model | Swift Hockett–Sherby |

Yield criteria | BBC2005 |

Blank holder | Force controlled (columns) |

Spacer blocks | From 0.5 to 0.9 mm |

Material | R_{p 0.2%} [MPa] | R_{m} [MPa] | r_{0} [-] | r_{45} [-] | r_{90} [-] | n [-] |
---|---|---|---|---|---|---|

DC06_1 | 156.9 | 294.5 | 2.110 | 1.837 | 2.560 | 0.247 |

DC06_2 | 140.0 | 280.9 | 2.060 | 1.877 | 2.820 | 0.248 |

DC06_3 | 147.4 | 293.3 | 2.043 | 1.680 | 2.493 | 0.243 |

DC06_4 | 151.4 | 296.5 | 2.050 | 1.910 | 2.640 | 0.250 |

DC06_5 | 152.4 | 292.9 | 2.013 | 1.647 | 2.420 | 0.240 |

Model | e | a | p_{ref} [MPa] | v_{ref} [mm/s] |
---|---|---|---|---|

P-v dependent 1 g/m^{2} | 0.8767 | 0.0116 | 1.9771 | 10 |

P-v dependent 2.4 g/m^{2} | 0.8817 | 0.0138 | 1.9901 | 10 |

P-v dependent 4 g/m^{2} | 0.8664 | 0.0108 | 1.9704 | 10 |

Model | Amount of Lubricant [g/m^{2}] | R-Squared [-] | RMSE [-] |
---|---|---|---|

P-v dependent | 1 | 0.895 | 0.006 |

2.4 | 0.963 | 0.004 | |

4 | 0.915 | 0.006 | |

TriboForm | 1 | 0.870 | 0.007 |

2.4 | 0.969 | 0.004 | |

4 | 0.941 | 0.005 |

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## Share and Cite

**MDPI and ACS Style**

Muñiz, L.; Trinidad, J.; Garcia, E.; Peinado, I.; Montes, N.; Galdos, L.
On the Use of Advanced Friction Models for the Simulation of an Industrial Stamping Process including the Analysis of Material and Lubricant Fluctuations. *Lubricants* **2023**, *11*, 193.
https://doi.org/10.3390/lubricants11050193

**AMA Style**

Muñiz L, Trinidad J, Garcia E, Peinado I, Montes N, Galdos L.
On the Use of Advanced Friction Models for the Simulation of an Industrial Stamping Process including the Analysis of Material and Lubricant Fluctuations. *Lubricants*. 2023; 11(5):193.
https://doi.org/10.3390/lubricants11050193

**Chicago/Turabian Style**

Muñiz, Laura, Javier Trinidad, Eduardo Garcia, Ivan Peinado, Nicolas Montes, and Lander Galdos.
2023. "On the Use of Advanced Friction Models for the Simulation of an Industrial Stamping Process including the Analysis of Material and Lubricant Fluctuations" *Lubricants* 11, no. 5: 193.
https://doi.org/10.3390/lubricants11050193