Modelling pressure rolling of asymmetric rolling process

The paper presents a comparative analysis between experimental results and modelling in order to interpret the value of the contact pressure on the asymmetric longitudinal rolling. It is also intended action and the different behaviour of upper cylinder compared to the lower cylinder action in situations when both are driven, or only one operates. In the modelling will be presented on the basis of boundary conditions imposed rolling pressure variation in the degree of reduction and also re size arc length of contact. Determining a curve is also important to determine the locus of points which characterize symmetry conditions partial rolling process between unequal diameters cylinders.


Establishing engineering mathematical models for the rolling process
This paper is part of the modern trends of gradual replacement of the physical experiments with numerical ones in the design and optimisation of the technological processes of plastic deformation by rolling. By its nature, this paper requires a multidisciplinary approach, starting with the deep understanding of the theory of complex phenomena of metallic materials plastic deformation by rolling, and continuing with the modern techniques of processing and interpretation of the experimentally obtained data. [1] The length of the contact arc between the metallic material and the rolls is one of the decisive parameters influencing the strength, temperature and kinematic conditions of the rolling process. To determine the length of the contact arc for symmetrical and asymmetrical cold rolling, in addition to the roll reduction and radius, we must also know the elastic deformations of the rolls and the strip undergoing the rolling process, values that depend on the strength of the rolling process.

The model used for the contact arc length
This problem can be solved by using the rolling theory equations, as well as the elasticity theory equations, based on the well-known models and methods for calculating the length of the contact arc for cold rolling.
Using the Hertz's concept (relationship 1), a number of models have been obtained, as following: a) The Hitchcock's model, which takes into account the elastic compression of the rolls: Rthe radius of the working rolls; Δhthe absolute reduction; the mean normal contact stress; the Poisson's ratio; the elastic modulus of the material which the rolls are made of Relationship 2 does not take into account the elastic deformations of the strip undergoing the rolling process, and the normal stress diagram is adopted in the shape of semi ellipse. b) The Roberts model takes into account the contact between the rolls and an ideal rigid plan: -The normal force acting per unit length of the strip; c) Ţelicov and Grişcov proposed the following relationship to calculate the increase of the contact arc length by taking into account the asymmetry of the normal stresses diagram: (6) This relationship (6) helps reducing the errors between the calculated values and the experimental ones. d) The Ţelicov's model introduces also the influence of elastic deformations of the strip: The mean normal contact stress, the contact arc length and its increase along the plane passing through the centres of the rolls are calculated by taking into account the elastic deformations of the rolls and the strip; e) Dinnik shows in his model that the determination of the contact arc length increase x l without taking into account the length of the strip and the diagram symmetry for normal stresses is irrational, proposing x l to be determined using the relationship: a -Coefficient considering the asymmetry of the normal stresses diagrams: h l -Thickness of the strip after rolling For determining the value of x lc , Dinnik recommends the Relationship 3.
f) The relationship proposed by Cepurchin determines the contact arc length taking into account only the elastic compression of the rolls, as follows: √ (11) c -Shape coefficient of the normal stresses diagram g) Based on the previous model, Poluhin obtained the dependence for determining the length of the contact arc by taking into account the elastic deformations of the rollers and strip. Therefore: √ ( ) (12) Where:  [1] Since the elastic deformations of the strips are neglected in all the proposed models, we are going to analyse the plastic contact portion between the strip and the rolls to determine the mean contact pressure.
Thus, there are a number of models proposed in the literature, which are distinguished by simplicity and compactness, and are obtained considering a constant value of the resistance to deformation for the mean values of the respective parameters. The models proposed by these authors, without taking into account the influence of elastic deformations of the rolls, are shown in the following table:

Bland and Ford
. where: For the convenience of further analysis of the model, we replace the "p m " value with the dimensionless parameter "n", which is the stress state coefficient: And dependencies: for determining: : With consideration of relations, models of the Table 1, take the form Dependencies can be simplified to some extent, if adopted: After entering in we obtain:  The relationships confirm the fact that the stress state coefficient "n", found in all the analysed models, finally represents a function of three dimensionless parameters: 1the coefficient of friction; 2the ratio between the radius of the working rolls and the thickness of the bar at the entrance into the deformation zone; 3the relative reduction, which is also the criterion of similarity;

The experimental plant and testing method
The research for this theme purpose have been made on a 170 mm reversing two-high rolling mill, created and installed in the no conventional technologies and plastic deformation laboratory of the Engineering Faculty from Hunedoara [1][2][3]. Figure 1, [1] shows a general view of the installation, which has been built by the author of this paper and which, compared to all the other mills, allows a simultaneous registering of the symmetric and asymmetric rolling process parameters, which are the rolling force (F d and F a ), the side efforts (X d and X a ), the pressure on the surface of contact with the superior roll (p ms ) and the with inferior one (p mi ), the real lengths of the springs of contact with the superior roll (l s ) and with the inferior one (l i ). 5device for registering the real length of the contact springs; 6 -bearing 7superior roll bearing; 8-rolling force (F) detector; 9side force (X) detector Since the intended purpose was to study the qualitative aspect of phenomena related to the asymmetric longitudinal rolling, for eliminating the inevitable influence of the iron oxides (scale) on the process, the tests were carried out on samples of aluminum and copper, with the following dimensions [4]: In this paper, by analyses and calculations, we are going to use the experimentally obtained values for pressure, and the actual contact lengths of the arcs with the upper and lower roll, for equal diameter and different diameter. Figure 2 shows the dependence of the real length of the contact arcwith the upper roll (smaller) and the lower roll (bigger)versus the reduction, when rolling the samples whose thicknesses are h 0 =12; 6; 2 and 1 mm [4].

Discussions and technological interpretations
As can be seen in the figure, the curves are increasing differently, their points of intersection being found at different degrees of reduction, depending on the thickness of the rolled samples.
Up to the points of intersection (=30-35% for h 0 =12 and 6 mm), the length of the arcs afferent to the smaller-diameter roll (upper) is greater, and by further increasing the reduction, the actual lengths of the contact arcs afferent to the smaller-diameter roll become smaller. By overlapping the areas where the mean pressure is exerted by the upper roll (smaller) and the lower roll (bigger) for rolling the samples whose dimensions are h 0 =12 mm (Figure 3 and 4), we   For each thickness, the intersection points of the curves representing the mean pressure and the actual length of the contact arc are placed at the same value of the reduction, confirming the correctness of the experimentally obtained data [5].
For the samples whose dimensions are h 0 = 2 mm ( Figure 5 and 6), the intersection points of the curves l rs =f(, p ms ) and l ri =f(, p mi ) are moving considerably and correspond to the reduction ε=15-25%, and the length of the contact arcs afferent to the larger-diameter roll (lower) is greater, l i l s . It is obvious that, for reductions smaller than the mentioned range, l ri  l rs . The distribution of the pressure exerted by each roll, for the samples whose dimension is h 0 =2 mm, is shown in Figures 5 and 6. In this case, there is a total concordance with the results presented so far. Therefore, within the reduction range ε =10-25% and p ms = p mi .
With increasing degree of reduction, the mean pressure on the smaller-diameter roll (p ms ) has a growing value compared to the mean pressure exerted by the larger-diameter roll (p mi ).
At reductions smaller than 10-25%, similar to the lengths of the contact arcs, the inequality p mi p ms is the one which probably applies. The results obtained in the experimental research carried out when using rolls with unequal diameters are fully consistent with the laws of mechanics.
Indeed, for any difference between the diameters of the working rolls and any degree of reduction (ε) applied, F s = F i , or: Joining the intersection points l s and l i shown in Figure 2, for various thicknesses of the rolled samples, as well as the intersection points of the curves describing the mean pressure on the smaller roll (p ms ) and the bigger one (p mi ) in Figures 3, 4, 5 and 6, we obtain a parabola passing through the origin of the coordinate axes (Figure 7), which in general can be expressed by the equation: (44) where: yrepresents the mean pressure (or the lengths of the contact arcs, respectively); acoefficient taking into account the rolling conditions; Xcritical degree of deformation; nindex depending on the degree of asymmetry of the process. The obtained curve is in itself the geometrical locus of the points which characterise the partial symmetry conditions of the rolling process between unequal-diameter rolls when the following equalities occur, determining this symmetry: In this case, regardless of the ratio between the working diameters of the rolls, the metallic material comes out in a straight line.
The field located above this curve corresponds to the situation when, on the larger-diameter roll (lower), the pressure is higher and the contact arc length is lower, i.e.: In this case, all the rolled samples are bending upwards when coming out from the rolls (on the smaller-diameter roll).  The respective field is located under this curve, where the pressure exerted by the smaller-diameter roll (upper) is higher, and the contact arc is shorter: In this case, the rolled samples are bending downwards when coming out from the rolls (on the larger-diameter roll). Therefore, in the field located above the curve plotted in Figures 3, 4, 5 and 6, the pressures exerted by the larger-diameter roll are high, and in the field located under the curve, the pressures exerted by the smaller-diameter roll are much higher.
Obviously, when p mi p ms , the bigger is the difference between the working diameters of rolls and the bigger is the initial thickness of the sample, the larger is the field. Related thereto, we should mention that the mean pressure is decreasing with increasing the deformation unevenness.

Concluding remarks
The research results confirmed that the current views on the deformation theory, according to which, at asymmetrical rolling, the pressures exerted by the smaller-diameter roll are always higher, are flawed. To characterise the rolling process based on the results obtained from this research, we consider useful to introduce a coefficient able to characterise the degree of asymmetry of the process: Interpreting the dependencies presented in Figure 7, it results: If K a =1, this situation corresponds to the case when y=aX n , and characterises the adequate conditions required by a symmetrical process to occur between the unequal-diameter rolls.
The more differs the K a coefficient from the K a =1 value, the greater is the difference between the contact arc lengths and the mean pressures exerted by the unequal-diameter cylinders, and the more is increasing the degree of unevenness of the deformation.
At the asymmetrical process, when the rolls have unequal diameters, the p m value is much lower than in case of a symmetrical process under identical conditions of reduction. This phenomenon is fuelled by the denatured shape of the deformation zone, which creates favourable conditions for developing longitudinal tensile stresses, whose action over the bar during the rolling process can match the previous and subsequent stress applied, which considerably reduces the mean pressure exerted on the contact surfaces of the rolls.