The stribeck curve in cold flat rolling

Abstract

Cold flat rolling of lubricated steel strips was studied. The three-component system of rolling - the mill, the rolled strip and their interface - was analyzed using a two-step mathematical model, a 1D model and flat rolling experiments. The utilized experimental data have been taken from McConnell and Lenard [1]. The objective of the study was the examination of the interactions of the three components and the development of the Stribeck curve. In the first part of the work, a two-dimensional finite element model was used in which the elastic deformation of the work roll and the elastic–plastic deformation of the strip were considered. Special attention was paid to the events at the lubricated interface where an upgraded version of Levanov’s model was utilized. The model was developed to analyze the local variables at the roll-strip interface. The parameters of the model were determined in an iterative manner, minimizing the differences of the measured and computed roll force and torque. Thus, the relative velocity between the roll and the strip, the roll pressure, the interfacial shear stress and the temperature were obtained. The effect of the temperature on the material parameters of the constitutive equations was also taken into account. Another model was then employed to consider the effects of the local variables on the lubricant’s viscosity. These were then used to obtain the local values of the Sommerfeld number, which in turn led to its average value and to the traditional shape of the Stribeck curve. Further, a simple 1D model of the flat rolling process was also tested for its ability to lead to the Stribeck curve. It was concluded that while both advanced and simple models allowed the development of the curve, removing the simplifying assumptions yielded a more reliable plot.

Appendix: Models and formula mentioned in introduction

The friction factor

is given as the ratio of the interfacial shear stress (τ) and the yield strength (k) in pure shear of the softer material in the contact:

$$ \begin{array}{ll}m=\tau /k\hfill & 0\le m\le 1\hfill \end{array} $$

(9)

For the perfect lubrication m = 0 and the sticking conditions occur for m = 1.

Amonton and Coulomb frictional model

The friction stress (τ) is directly proportional to local normal pressure p.

$$ \tau =\mu p $$

(10)

Wahnheim-Bay’s frictional model

The general frictional model is based on the slip-line theory and the nominal friction stress (τ _n ) is given by the following:

$$ {\tau}_n=m\alpha k $$

(11)

where m is the friction factor, α is the ratio between real and apparent area of contact. The model indicates that Amonton Coulomb’s law is valid whereas the friction stress approaches a constant value at high normal pressures.

Hitchkock’s formula

The modified roll radius (R’) caused by elastic deformation can be expressed by Hitchcock’s formula:

$$ {R}^{\prime }=R\left[1+\frac{16\left(1-{v}^2\right)}{\pi E}\frac{F_r}{b\varDelta h}\right] $$

(12)

where R is the radius of the unloaded work roll, b is the width of rolled strip, Δh = h ₀ -h ₁ and h ₀ , h ₁ are the initial and final thickness of strip and F _r is the rolling force.