Some questions of the contact interaction theory in two-roll modules

The study is devoted to the analysis of contact stresses distribution in a two-roll module. The results of the study of the theory of contact interaction in two-roll modules are presented: an analytical description of the curve shape of the roll contact and determination of the relationship between tangential and normal stresses. Mathematical models of the roll contact curves and the friction stress models in a generalized two-roll module are determined. The obtained models are general in the sense that they are applicable for partial cases of interaction in two-roll modules. Formulas are found for calculating the neutral angles of two-roll modules. Dependencies between the forces acting in the rolls and the stresses distributed under the influence of these forces are stated. It is revealed that these dependences do not change with a change in the angle of supply of the material layer to the line of centers and the angle of inclination of the upper roll relative to the vertical.


Introduction
Roller machines are widely used in many branches of industry. The main operating element of roll machines is a roll pair. The roll pair and the processed material together compose a two-roll module.
The main task of the theory of contact interaction of two-roll modules is the modeling of contact (normal and tangential) forces, that is, obtaining analytical dependencies that describe the patterns of distribution of these forces. When modeling the contact forces, the main factors are the models of friction stresses that take into account the effect of friction in the contact zone of the rolls and link tangential and normal stresses, as well as the models of the rolls contact curves that describe the shapes of these curves.
In two-roll modules, the forces applied to the roll axis are transmitted to the material being processed along the contact curves of the rolls. In two-roll modules with rolls having elastic coatings, mutual strain of the contacting bodies occurs. In this case, the contact curves have complex configurations and depend mainly on the geometrical and strain characteristics of the roll surface layers and the material being processed. The exact shapes of the roll contact curves in this case have not yet been clarified. In this regard, the problem of contact interaction (for example, analysis of contact stresses distribution) is solved by preliminary selecting the formula for the roll contact curve.
Usually an arc of a circle, an ellipse and a parabola is applied, described by formula  cos 1 e a r   [1], or simple mechanical models defined by equations R r  and   cos cos R r  [2].
There are many mathematical models of friction stresses obtained by theoretical, experimental or experimental-theoretical methods. In the study of contact interaction in two-roll modules, the dry friction model (Amonton-Coulomb law) [3,4,5] [7], experimental data are needed on the extent of the adhesion zone l , on the friction stress at the beginning of the adhesion zone 0 t , and the strip thickness in the neutral section 0 h . Therefore, the models of friction stresses currently used in the theory of contact interaction of two-roll modules are considered approximate. For this reason, the theoretical curves of contact stresses distribution obtained are considered approximate, and they do not correspond to the experimental distribution diagrams [8].
It follows from the foregoing that obtaining theoretical curves of contact stress distribution corresponding to experimental diagrams is currently impossible due to the lack of correct models of roll contact curves and friction stress models.
Two-roll modules belong to the main operating elements of the roll machine or perform auxiliary functions. In this regard, many of them are asymmetric. Moreover, quite often several types of asymmetry are realized simultaneously, for example, two types of geometrical asymmetry -different diameters and the tilt of the material layer relative to the horizontal.
In this regard, to systematize the studies of contact interaction, first of all, a generalized scheme for the interaction of a roll pair with the processed material was selected, that is, a generalized model of two-roll modules was developed [10] based on the analysis of functional structures and classifications of roll modules [9], it served as the object of study of contact phenomena.
In this two-roll module, the rolls are positioned relative to the vertical by tilting to the right at an angle  , have unequal diameters ) ( The aim of the study is a mathematical modeling of the roll contact curves and friction stresses, i.e. an analytical description of the curve shape of the roll contact and determination of the relationship between shear and normal stresses in the generalized two-roll module under consideration.

Modeling the curve shape of the roll contact
We study the first problem, i.e. modeling the contact curves of the rolls.
First, consider the interaction of the material layer with the lower roll. The shape of the contact curve is analyzed in polar coordinates. This curve (curve A 1 A 2 ) consists of two sections Figure 1. Interaction scheme in a two-roll module MSTU 2020 Journal of Physics: Conference Series 1546 (2020) 012132 Contact interaction in two-roll modules with rolls having elastic coatings can be considered by analogy with the rolling of an elastic wheel over deformable soil [11]. In the theory of wheel rolling, the analytical definition of the contact line is associated with the analysis of the ratio of the strain rates of interacting bodies [12]. In various studies, this ratio is considered constant [12,13]. Then the ratio of wheel and soil strains is equal to the ratio of their strain rates [13].
To describe the curve shape of the roll contact, similar to the theory of wheel rolling, the ratio of strains of surface layers of the rolls and the material layer is considered to be equal to the ratio of their strain rates.
At each point of the roll contact curve, the strain in contacting bodies occurs along the n -n line perpendicular to the roll contact curve. Then at point B 1 of section A 1 K we have (figure 1) In view of expressions (2) To simplify research, expression (4) Geometrical analysis shows that the transition from formula (4) to formula (5) means replacing the straight line  Summarizing equations (5) and (6), we find the equations of the lower roll contact curve By analogy with the definition of the equation of the contact curve of the lower roll, we determine the equation of the contact curve of the upper roll where , The systems of equations (7) and (8) describe the shapes of the roll contact curves in the two-roll module under consideration. In this case, the influence of geometrical characteristics of one of the rolls on the shape of the contact curve of another one determines the contact angles, and the strain characteristics of the material layer and both rolls are completely included into the rate ratio. In the two-roll module under consideration, the upper roll is free. It can be a driving one. In this case, the two-roll module has a transmission mechanism between the rolls. The gripping conditions in a tworoll module with one driving roll differ from the gripping conditions with two driving rolls. The gripping conditions in the two roll module determine the contact angles. Therefore, two-roll modules with one driving roll and with two driving rolls have different contact angles [14]. In this regard, we can say that the influence of the transmission mechanism on the curve shape of contact rolls is also determined by the contact angles.
All partial cases of the two-roll module under consideration and the corresponding partial types of the system of equations (7) and (8) are analyzed. It was revealed that the formulas obtained are general in the sense that they describe all partial cases of the interaction of the processed material with the pairs of rolls in two-roll modules. Therefore, they can be used in modeling friction stresses.

Friction stress modeling
Now proceed to the study of the second problem, i.e. to friction stresses modeling.
First, we analyze the stress state of the contact interaction of the material layer and the lower roll along the contact curve 12 11 A A . In the steady-state interaction process, the lower roll is affected by: pressure force of the clamping devices Considering the lower roll in equilibrium under applied forces, we obtain    (12) From the force scheme of the compression zone ( figure 2) Since the considered process is a steady-state process, assume that where  1 C is a constant quantity.
Hence we have 0 2 1 Formula connecting the tangential and normal stresses at the points of recovery zone of the lower roll is obtained similarly. It has the form 12  These conditions lead to equality 12 11 C C  . Then from expressions (14) and (15) (17) and (18) we obtain the model of friction stresses for the lower driving roll In the two-roll module under consideration, the upper roll is free. In this case, the forces 2 F  and 2 T  acting on the upper roll change direction [9]. Therefore, the quantities j t 2 ) 2 , 1 (  j and 2 F have opposite signs in the formulas of system (19). In this regard, the model of friction stresses for the upper roll has the form The systems of equations (19) and (20) determine the models of friction stresses in the considered two-roll module. They show that the models of friction stresses in two-roll modules are independent of the feed inclination of the material layer to the center line and of the upper roll inclination relative to the vertical. An analysis of these models shows that they describe stress models for all partial cases of the two-roll module under consideration.