SIMULATION THE DETERMINATION OF LONGITUDINAL ROLLING STABILITY FOR DIFFERENT FRICTION MODELS

Summary. The results of experimental and theoretical researches of rolling processes indicate that the friction forces at the contact of specimen with rolls determine the very feasibility and stability of the process. The most widespread in describing friction in metal forming processes were models caused by the transfer of dependencies during the sliding of solids (Amonton-Coulomb law) and during plastic deformation of the contact layers of interacting bodies (Siebel's law). The question of comparing the results obtained is relevant, when only selected friction models differ under the task conditions. Researches of the influence the changes in contact conditions according to different friction models were carried out theoretically by integrating the T. Kármán equation and with the energy method. When comparing the results, their coherence with respect to the limit conditions of the process is established. With increasing reductions, the stability of the process changes, there are the same geometric limit values, upon reaching which slipping begins and stops the rolling. The change in friction indicators, as a result of a change in the contact interaction of the rolls with the specimen, leads to a change in the longitudinal stability of the process.

С ЕКЦІЯ ХІ. ЗАГ АЛ ЬНА М ЕХ А НІК А Т А М ЕХ АН ІЧ Н А І НЖЕН ЕР ІЯ is necessary to adequately describe the effect of friction in the zone of deformation to the process conditions. But while there is no uniform model of specimen contact interaction with rolls by rolling, different researchers use different models of friction during rolling, considering specific aspects of the process [3]. The actual question is the comparison of the results obtained by the numerical methods for the metal forming processes, when only the selected friction models differ.
2. Analysis of recent research and publications. Friction of contacting bodies is a complex physicochemical process, for which a sufficiently complete mathematic model has not yet been built, taking into account and describing with sufficient completeness its different nature aspects [4 -8]. Therefore, when solving problems of metal forming, we can limit ourselves to modeling the action of contact forces in the interaction of two or more bodies in the zone of deformation in quantitative terms.
To this date, it is customary to quantitatively associate friction stresses with force-power indicators that have greater certainty. The most common in describing friction in MF processes are the following models: • the first of them is due to the transfer of dependencies when sliding solids (dry friction, Amonton -Coulomb law) where  t are tangential stresses on the contact surface of the sample-tool; f is coefficient of friction; n p are normal stresses on the contact surface at the selected point; • the second model is due to the transfer of dependencies with predominantly plastic deformation of the contact layers of interacting bodies (Siebel's law) s t   =, (2) where  is the indicator of friction forces; s  is the yield strength to shear.
Experimental researches of contact interactions during the rolling show that both models find confirmation only in an integral sense throughout the deformation zone, while at some points of contact or cross sections within the this zone, significant deviations of the current ones indicators f or  are possible from average integral values [9,10]. If the mode of contact interaction is contacthydrodynamic, then it is necessary to use Newton's law of friction [11,12] with the doing of a separate research, but this mode is less common.
The quantitative certainty of these two models (1) and (2) and the experience of their use allows accumulating reference data, performing theoretical studies and conducting computer modeling in specific cases of MF processes. Therefore, these models are basic for taking into account the effect of friction during rolling process.
The above must be taken into account in researches of the impact the interactions in contact on the longitudinal stability then rolling sheet metal and profiles.
The method to estimate the limit capturing ability of rolls in a steady state [1], G based on the calculation of longitudinal normal stresses and forces arising in a metal that is plastically deformed. The initial data for their determination are diagrams of contact normal stresses, which can be obtained by solving the differential equation of T. Kármán using the friction model (1) that chosen for these conditions. Knowing the distribution of stresses in the contact zone, and using the plasticity equation, it is easy to calculate the longitudinal internal stresses, current longitudinal forces and their average integral value. With stable rolling, this force cannot be directed in the direction of rolling (to be positive), because it is a resistance force, a reactive force. If this force is compressive (negative), the process is stable and rolling in the zone of deformation is in a balanced state. It follows that rolling limit conditions occur when the force cp np Q  is zero value.
It was noticed that the loss of equilibrium of the strip can occur in the presence of a lagging zone. This is possible because with a further increase in the angle of capture of the forces pulling the metal into the rolls, it becomes insufficient to simultaneously overcome increase in ejected forces and balance the resulting In this analysis, the longitudinal stability of the strip in the zone of deformation, that is, rolling without slipping can be estimated by the value of the average integral longitudinal force of the plastically deformed metal.
In Fig.1 shows diagrams of the distribution of pressure (3) As can be seen from the distribution graph np x Q  , this force varies in the areas of the deformation zone of the strip, and at its boundaries it is zero value. It is proved [1] that the longitudinal stability of the rolling process is determined by the sign of the average integral value For negative values we have a stability of process, for positive valuesthe process is impossible, the limit of stability cp np 0 Q  = [14].
In accordance with the criteria used, it follows that when the horizontal force is cp np 0 Q   , the rolling process is stable, occurs without partial slipping. This corresponds to the results of exploratory experiments, which were conducted much earlier than theoretical analysis. 3. Main materials. When using the determination of contact friction stresses according to Siebel in the research the stability of the rolling process, it is necessary to proceed to energy methods for solving rolling problems [15,16].
The stability of the process by symmetrical rolling the strip of rectangular section in cylindrical rolls with equal diameter (flat deformation) is analyzed, and initially the influence of external zones is not taken into account (Fig. 2).
Let be v 1 is the magnitude of the exit specimen velocity from the deformation zone, then horizontal component of velocity in arbitrary section: The following are the normal components of the strain velocity tensor: When, using (5) and (7), is Taking into account the condition of constancy the volume of deformation zone and 2D metal deformation ( we get The vertical component of the flow velocity z v is found by integrating equality (10), provided that As a result, we get: Then we calculate the intensity of the deformation velocities: We find the power of internal resistance: The volume element in the deformation zone is dV (14) Or finally: When finding the surface integral, the formulas are used: where ков v is the sliding speed of the metal on the roll.
In the plane of the longitudinal section of the deformation zone (see Fig. 2), it is convenient to use the polar coordinate system. Then After using formulas (17), (18) and taking into account where γ is the neutral angle.
Further, according to the assumptions made, it is necessary to consider the balance of power during rolling, that is, the functionality ( ) ( ) If we consider Φ as a criterion for rolling stability, then a change in the sign of the quantity determines the limit transition to an unstable mode. The sign Φ ("plus" or "minus") is the same as that of a dimensionless quantity for which the analysis is simplified.
To verify the correctness of the statement, data from the monograph [1, p. It should be emphasized that, due to the action of internal forces in the deformation zone, the loss of longitudinal stability of the rolling process may also occur in the presence of a lead zone. Equality γ = 0 does not always reflect the limit conditions of deformation [1]. This circumstance must be taken into account when developing energy-saving technology by adjusting the rolling tension mode at continuous rolling mills.
In the research the effect of changing contact conditions on the stability of the rolling process using the indicator  , the known experimental data given in [1] are used. For rolling stepped samples with D = 195mm, h 1 = 4.2mm, ψ = 0.7 according to the results of the corresponding mathematical modeling when calculating the values of the power functional (20) using the MathCAD system, data were obtained (Table 1).