# A Numerical Procedure Based on Orowan’s Theory for Predicting the Behavior of the Cold Rolling Mill Process in Full Film Lubrication

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Plastic Deformation Model

- -
- ϕ (rad) is the rolling angle
- -
- θ is an angular variable varying between zero and ϕ
- -
- x is the horizontal coordinate on the rolling axis
- -
- y is the vertical coordinate perpendicular to the rolling axis
- -
- ϕ
_{0}(rad) is the rolling angle corresponding to the beginning of the inlet zone - -
- ϕ
_{iw}(rad) is the rolling angle corresponding to the end of the inlet zone and the beginning of the work zone - -
- ϕ
_{wo}(rad) is the rolling angle corresponding to the end of the work zone and the beginning of the outlet zone - -
- ϕ
_{3}(rad) is the rolling angle corresponding to the end of the outlet zone - -
- ω (rad/s) is the rolling speed
- -
- S
_{i}, S_{o}(m) are the strip thicknesses before and after the rolling mill process, Figure 1 - -
- R (mm) is the radius of the rolls
- -
- A, A′ (m
^{2}) are the inner and outer cylindrical sections used in the Orowan’s theory - -
- F(ϕ) (N) is the horizontal force on the strip, Figure 2
- -
- p(ϕ) (Pa) is the pressure acting on the strip, Figure 2
- -
- $\overline{\mathsf{\tau}}$ (ϕ) (Pa) is the tangential stress on the strip planar surface, Figure 2
- -
- τ(ϕ) (Pa) is the tangential stress acting on the strip cylindrical section, Figure 2
- -
- t (ϕ) (Pa) is the radial pressure on the strip cylindrical section, Figures 2 and 3
- -
- F
_{τ}(ϕ) and F_{t}(ϕ) (N) are the two components of the horizontal force F in the tangential and radial directions, respectively, Figures 2 and 3 - -
- $w\left(\mathsf{\varphi},\mathrm{a}\right)$ is the $\mathrm{Orowan}\u2019\mathrm{s}\text{}\mathrm{function}$, Figure 4
- -
- σ
_{s}(MPa) is the material yield stress - -
- y
_{1}(ϕ) (m) is the function expressing the profile of the strip measured from the longitudinal axis and its semi-thickness, Figures 2 and 5 - -
- y
_{2}(ϕ) (m) is the function expressing the profile of the fluid film measured from the longitudinal axis and the semi-thickness of the strip and the fluid film together, Figure 5 - -
- h
_{0}(ϕ) (m) is the function expressing the semi-thickness of the fluid film at the beginning of the inlet zone, Figure 5 - -
- h(ϕ) (m) is the function expressing the semi-thickness of the fluid film
- -
- $\overrightarrow{v}=\text{}u\text{}\hat{\mathrm{i}}+v\text{}\hat{\mathrm{j}}$ (m/s) is the velocity field with the horizontal component u and the vertical component v
- -
- u
_{1}(ϕ) (m/s) is the horizontal velocity at y_{1}(ϕ) - -
- u
_{2}(ϕ) (m/s) is the horizontal velocity at y_{2}(ϕ) - -
- u
_{10}(m/s) is the horizontal velocity at y_{1}(ϕ_{0}) - -
- u
_{13}(m/s) is the horizontal velocity at y_{1}(ϕ_{3}) - -
- v
_{1}(ϕ) (m/s) is the vertical velocity at y_{1}(ϕ) - -
- v
_{2}(ϕ) (m/s) is the vertical velocity at y_{2}(ϕ) - -
- μ (Pa∙s) is the lubricant viscosity
- -
- μ
_{0}(Pa∙s) is the lubricant viscosity in standard conditions - -
- γ (Pa
^{−1}), α (°C^{−1}) are the two Barus constants - -
- T (°C) is the temperature
- -
- f is the friction coefficient
- -
- a is Orowan’s parameter
- -
- Q (m
^{3}/s) is the lubricant flow rate

_{s}is the material yield stress, and θ is an angular variable varying between zero and ϕ.

_{1}(ϕ) is the function expressing the profile of the strip measured from the longitudinal axis and its semi-thickness (Figure 2 and Figure 5).

#### Considerations about w

## 3. Global Model

- -
- the material is rigid and perfectly plastic;
- -
- strain-hardening effects are negligible;
- -
- surfaces are considered perfectly smooth (the roughness of contact surfaces being neglected);
- -
- strip width is much larger than strip thickness, thus the problem can be considered in two dimensions;
- -
- lubricant is uncompressible;
- -
- lubricant flow is laminar;
- -
- lubricating area is much smaller than the roll circumference;
- -
- lubricant film thickness is much smaller than material thickness reduction;
- -
- inertia forces are negligible;
- -
- rolls are rigid.

#### 3.1. Viscosity

^{−8}Pa

^{−1}and α = 0.0163989 °C

^{−1}.

#### 3.2. Model Equations and Domains

_{0}represents the angle at the beginning of the inlet zone, y

_{1}(ϕ) is the function expressing the profile of the strip measured from the longitudinal axis and its semi-thickness, and y

_{2}(ϕ) is the function expressing the profile of the fluid film measured from the longitudinal axis and the semi-thickness of the strip and the fluid film together; ${y}_{1}$ and ${y}_{2}$ are given by the following:

_{0}is the function expressing the semi-thickness of the fluid film at the beginning of the inlet zone and S

_{i}is the metal sheet thickness (Figure 5).

_{1}(ϕ

_{0}).

_{1}(ϕ

_{wo}) and y

_{1}(ϕ

_{3}).

## 4. Results and Discussion

- -
- R = 0.255 m
- -
- ω = 60.13 rad/s
- -
- μ
_{0}= 5 Pa∙s - -
- S
_{i}= 12 × 10^{−4}m - -
- u
_{10}= 11.7 m/s - -
- σ
_{s}= 730 MPa - -
- Q = 2.74 × 10
^{−4}m^{3}/s - -
- γ = 10
^{−8}Pa^{−1}

_{iw}. ϕ

_{iw}represents the value of the angle at the end of the inlet zone and beginning of the work zone.

_{o}) coming out from the rolling mill plant could be measured. All the tests were performed using the same working parameters and the same materials. In particular, the material used for the tests was a martensitic aging steel used for aeronautic and automotive applications, with a yield stress of 730 MPa and initial metal sheet thickness of 12 × 10

^{−4}m. The rolling mill plant had rolls with a radius of 0.255 m and the basic working parameters used for the tests were as follows: a roll angular velocity of 60.13 rad/s, an initial horizontal velocity of the metal sheet of 11.7 m/s and an environmental temperature of 20 °C. The oil/water emulsion used as a lubricant had a value of μ

_{0}equal to 5 Pa∙s, a flow rate of 2.74 × 10

^{−4}m

^{3}/s, and an outlet temperature of 40 °C.

_{o}considered in this work is the mean value resulting after 10 tests in the same environmental conditions, as already mentioned. Table 1 shows the results of the experimental tests. Therefore, the mean value of S

_{o}was compared with the ones predicted by the model proposed in this work and the one based on the slab analysis, as Table 2 summarizes.

_{i}= 1.240 mm to S

_{o}= 0.898 mm. The model using Orowan’s theory predicts S

_{o}= 0.882 mm, while the model based on the slab analysis predicts S

_{o}= 0.934 mm. This means that the real thickness reduction is equal to 27.6% and the thickness reduction predicted by Orowan’s theory and slab analysis are equal to 28.9% and 24.7%, respectively.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Tension state (radial pressure t and tangential stress $\mathsf{\tau}$) in a generic point of the element section.

**Figure 4.**Function $w\left(\mathsf{\varphi},\mathrm{a}\right)$. Red curves are obtained by varying ϕ between 0° and 30°; the blue curve represents the polynomial approximation evaluated by Equation (10).

**Figure 6.**Diagram p*-ϕ*: comparison of results obtained by Orowan’s theory (black curve) and the slab analysis (red curve).

**Figure 7.**Diagram p*-ϕ*: comparison of results obtained by Orowan’s theory with different models of viscosity.

**Figure 8.**Film shapes obtained by Orowan’s theory with different temperatures, where y

_{1}(ϕ*) and y

_{2}(ϕ*) are the profiles of the strip and of the fluid film from the longitudinal axis, respectively.

Test Number | Measured Value of S_{o} (Plant) |
---|---|

1 | 0.898 mm |

2 | 0.899 mm |

3 | 0.898 mm |

4 | 0.899 mm |

5 | 0.898 mm |

6 | 0.899 mm |

7 | 0.897 mm |

8 | 0.899 mm |

9 | 0.897 mm |

10 | 0.896 mm |

**Table 2.**Summary of results and comparison between the final thickness of the sheet: measured, evaluated with Orowan’s theory, and the slab analysis models.

Variable | Measured Mean Value (Plant) | Orowan’s Model | Slab Analysis |
---|---|---|---|

S_{o} | 0.898 mm | 0.882 mm | 0.934 mm |

**Table 3.**Summary of results and comparison between the final thickness: measured and evaluated with Orowan’s theory for different temperatures.

Variable | Measured Mean Value (Plant) | T = 40 °C | T = 140 °C |
---|---|---|---|

S_{o} | 0.898 mm | 0.882 mm | 0.889 mm |

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**MDPI and ACS Style**

Valigi, M.C.; Malvezzi, M.; Logozzo, S.
A Numerical Procedure Based on Orowan’s Theory for Predicting the Behavior of the Cold Rolling Mill Process in Full Film Lubrication. *Lubricants* **2020**, *8*, 2.
https://doi.org/10.3390/lubricants8010002

**AMA Style**

Valigi MC, Malvezzi M, Logozzo S.
A Numerical Procedure Based on Orowan’s Theory for Predicting the Behavior of the Cold Rolling Mill Process in Full Film Lubrication. *Lubricants*. 2020; 8(1):2.
https://doi.org/10.3390/lubricants8010002

**Chicago/Turabian Style**

Valigi, Maria Cristina, Monica Malvezzi, and Silvia Logozzo.
2020. "A Numerical Procedure Based on Orowan’s Theory for Predicting the Behavior of the Cold Rolling Mill Process in Full Film Lubrication" *Lubricants* 8, no. 1: 2.
https://doi.org/10.3390/lubricants8010002