UDK 621.7:621.89 Original scientific article/Izvirni znanstveni članek ISSN 1580-2949 MTAEC9, 43(1)23(2009) SCHEMES OF METAL-WORKING PROCESSES AND THE RELATED TRIBOLOGICAL EQUATIONS OF FLUID MECHANICS SHEME PROCESOV PREDELAVE KOVIN IN TRIBOLOŠKE ENAČBE MEHANIKE FLUIDOV ZANJE Dušan Ćurčija, Ilija Mamuzic University of Zagreb, Faculty of Metallurgy Sisak, Aleja narodnih heroja 3, 44000 Sisak, Croatia mamuzic@simet.hr Prejem rokopisa — received: 2007-10-15; sprejem za objavo - accepted for publication: 2008-10-02 We present a survey of the most frequently used equations for applications in cold drawing and rolling, for smooth and rough surfaces, for the effect of lubricant inertial forces, for more advanced theoretical solutions as well as the equations for cold drawing with a solid lubricant and the combination solid lubricant-emulsion. The compression processes are described using cylindrical coordinates. Also, the basic equations for the flowing of the lubricant on inclined planes related to screw rolling and for the forming of metals with fluids are given. Key words: lubrication, metal forming Reynolds differential equation, Monte-Carlo method Podan je pregled najbolj pogosto uporabljenih enačb pri hladnem valjanju in vlečenju, za gladke in hrapave površine, za vpliv vztrajnostnih sil, za bolj napredne teoretične rešitve in enačb za vlečenje s trdnim mazivom in kombinacijo trdo mazivo -emulzija. Tlačni procesi so opisani s cilindričnimi koordinatami. Podane so tudi osnovne enačbe za tok maziva na nagnjeni površini, ki se nanašajo na navojno valjanje in za oblikovanje kovine s fluidi. Ključne beside: mazanje, preoblikovanje kovin, Reynoldova diferencialna enačba, metoda Monte Carlo 1 INTRODUCTION The investigations and development of modern plastic working technology covers the following topics: physical modelling and simulation, computer simulation and characteristics and the behaviour of the material during processing. The gradients representing the changes of temperature and mechanical stresses are greater for a greater per-pass (partial) deformation. In Figure 1 the compression force F, the heat flow H, the deformation direction D and the rolling direction K for simple rolling Figure 1: Directions of the gradients in a physical simulation of metal rolling1 Slika 1: Smeri gradientov pri fizikalni simulaciji valjanja kovin1 Materiali in tehnologije / Materials and technology 43 (2009) 1, 23-30 are shown. The physical simulation, as a laboratory representation of the process, is based on the law of similarity and allows only a limited extrapolation. The simulation of the rolling process occurs by applying the principles of viscoplasticity and the use of analytical solutions increases with the rapid development of modern theoretical and experimental methods for the investigation of the plastic deformation of metals. A torsional plastometer was applied with success for the determination of the rolling force (Figure 2) and the obtained data can be applied for the correction of the Figure 2: Torsionmeter2 Slika 2: Merilec torzije2 23 D. ĆURČIJA, I. MAMUZIĆ: SCHEMES OF METAL-WORKING PROCESSES AND THE RELATED TRIBOLOGICAL Figure 3: Measuring the surface stresses on a sheet's surface Slika 3: Merjenje povr{inskih napetosti na traku calibration as well as the regulation of the rolling gap for continuous rolling stands. In Figure 3 a scheme is given for measuring the stresses on the sheet surface with a stressmeter3. 2 FLUID MECHANICS The concept of the boundary layer (Figure 4) was proposed by Prandl in 1904. The thickness of the fluid layer (5x) can differ significantly from the flowing line 2. The layer has, however, a constant flow velocity. Below the laminar part of the layer 3, the flowing velocity (v) decreases and on the solid surface the fluid is at a standstill, f is the boundary laminary layer, w is the transition area and K is the turbulent part of the boundary layer 1. The representation in Figure 3 shifts the Navier-Stokes and Reynolds equations in the domain of velocity. The use of emulsions for the plastic working of metals led to a significant lowering of production costs and to savings with expensive natural oils. In Figure 5 the equilibrium is shown for the surface tension of a drop of light liquid on the surface of a heavier liquid: °12 = a13 cos02 + a23 cos0j (1) For 02 --> 0 the adhesion work (W) is calculated using the Jung-Dupre equation Figure 5: Equilibrium of the surface tension a for a light liquid (3) on a heavier liquid (1) in air 25 Slika 5: Ravnotežje povr{inske napetosti a za lahko tekočino (3) na težji tekočini (1) in na zraku (3)5 W = a23 (1 + cos0j) (2) Investigations of the use of equation (2) in metallurgy were carried out by Ju. P. Abdulov6. In a fluid mechanical metallurgical investigation different equations are used for the flat (Figure 5) and for the inclined plane (Figure 6). The case of lubrication of a surface with vertical movement is met, also (Figure 7). In this work we will examine the fluid friction (friction with hydrodynamic lubricant), for which Newton's law is applied: F = z S v / h (3) where F is the friction force, z is the flowing capacity, S is the sliding surface, V is the velocity of the relative transfer, and h is the thickness of the lubricant layer. For a description of the case of liquid friction in the plastic deformation of metals the Nady equation is used. Let us start with an analysis on the basis of Figure 8. The sheet of thickness h is covered with the lubricant layer £(x) ahead of the section entering the deformation zone, for a wedge-shaped rolling gap a with the gap Figure 4: Boundary layer for the flow of fluid on the flat plane4 Slika 4: Mejni sloj za tok maziva na ravni povr{ini4 Figure 6: Shaping of a drop of liquid on an inclined plane7 where x, y are the Descartes coordinates; u,v are the corresponding flowing velocities; t is the time; g is the acceleration due to gravity; and ß is the angle of inclination Slika 6: Oblikovanje kapljice tekočine na nagnjeni povr{ini x, y Descartesove koordinate; u,v hitrosti pretokov; t čas; g konstanta gravitacije; ß kot nagiba 24 Materiali in tehnologije / Materials and technology 43 (2009) 1, 23-30 D. ĆURČIJA, I. MAMUZIĆ: SCHEMES OF METAL-WORKING PROCESSES AND THE RELATED TRIBOLOGICAL ... Figure 7: The dragging of fluid on a vertically moving metal surface8 where hs is the thickness of the fluid layer on the stagnation line of a metal surface moving with the constant velocity of U, h o is the the thickness of the dragged fluid layer on the metal sheet, and v is the sheet velocity Slika 7: Vle~enje maziva na povr{ino s pokon~nim gibanjem8 hs - debelina sloja maziva na mirujo~i to~ki metalne povr{ine, ki se premika s stalno hitrostjo U, h o - debeline sloja maziva, ki ga vle~e kovinska povr{ina, v - hitrost traka angle a, sheet velocity vo, rolls velocity vr and rolls radius R. The Reynolds' differential equations of fluid mechanics describing the representation in Figure 8 are: dp/dx = ôMvo + vR) / e2(x) - HßQI £3(x) (4) t = ^(vR - v0) / e(x) - (e(x)/2) dp/dx (5) For x = 0 we have e(x) = £o in the entering section of the deformation zone. For the change of pressure gradient dp/dx = 0 the tangential stress in the lubricant layer (t) is: t = ß(vr - Vo) / e(x) (6) In equation (6), attributed to Nady, p is the lubricant dynamic viscosity, Q is the lubricant flow, e(x) is the lubricant layer thickness ahead of the section entering the deformation zone and dp/dx is the pressure gradient in the lubricant. 3 TECHNOLOGY OF THE PLASTIC WORKING OF METALS This technology depends strongly on the quality of the technological lubricants used that: - diminish the contact friction, - remove the heat, cool the tool and diminish the wear, - diminish the deformation resistance and the deformation work, - diminish the sticking to the tool and keep the surface of the product clean. The basic groups examined in this work are: - liquid emulsions, - fats and compounds, - consistent lubricants, - transparent - glass lubricants, - powder lubricants, - metallic lubricants. The friction in cold deformation is, in principle, of the boundary type, and it is characterised with a great working pressure. The approaches in the development of the theory of friction are: - geometrical, with the friction coefficient p = tga, - molecular, with attraction based on a kinetical conception, - deformation, based on the deformation work for a determined volume, - a combination of different approaches. The first calculations for the lubricant layer were by Mizuno10. According to Figure 8, the thickness of the lubricant layer is: £0 = 3,^7^0 + vr) / a(1 - exp(-p0Y) (7) with g being the piezocoefficient lubricant viscosity, P0 the rolling pressure, a the rolling angle, ,0 the lubricant viscosity at atmospheric pressure, V0 and vr the working velocities of the tool and the rolling. Also, new solutions were suggested, for example, Perlina, Grudeva and Kolmogorova for the technology of the cold drawing of metals,11 according to Figure 9. The tube 3 moves with a velocity v0 through the matrix 1 with the entering gap y it is covered with the lubricant 2 of thickness e in the entrance section of the deformation zone. Figure 8: Cold rolling with a lubricant Slika 8: Hladno valjanje z mazanjem9 Figure 9: Cold drawing of metals with lubricant1 Slika 9: Hladno vle~enje z mazanjem11 Materiali in tehnologije / Materials and technology 43 (2009) 1, 23-30 25 D. ĆURČIJA, I. MAMUZIĆ: SCHEMES OF METAL-WORKING PROCESSES AND THE RELATED TRIBOLOGICAL Figure 10: Distribution of velocity for the drawing of tubes with fat lubricant12 Slika 10: Porazdelitev hitrosti pri vlečenju cevi z mastnim mazivom12 Figure 13: Uniform roughness of the rolls 1 and the sheet 2 surface1' Slika 13: Enakomerna hrapavost površine valja 1 in površine traka 2 1 production costs are lower and complex forms are achieved more easily than when using conventional deep drawing. The use of computers enabled us to also consider the surface roughness in the calculations (Figure 13) and the inertia of the lubricant with a great metal deformation velocity. In Table 1 are the differential equations for the average roughness. The development of mathematical calculations for Pilger rolling are in the initial phase because of the complexity of the working surface of the Pilger rolls (Figure 14). Figure 11: Pressing of metals13 Slika 11: Prešanje kovin13 The calculation is more complex for the case of combined liquid-solid lubrication. In Figure 10 the solid lubricant is the part 1 and the emulsion is the part 2. For the case in Figure 10 the maximum velocity of the emulsion is approximately 0.4 of the rolling velocity12. The pressing plastic deformation would be impossible without lubricant. The mathematical modelling in cylindrical coordinates is based on the scheme in Figure 11, with 1 being the round matrix, 2 the lubricant, 3 the mandrel, 4 the pressed metal and u the pressing velocity. The forming with hydraulic fluid at the pressure 2 occurs over the membrane 1 (Figure 12). The liquid can have the role of either the matrix or of the extractor. For this process, a smaller number of toolings is used, the Figure 14: Scheme of Pilger rolling15, where v is the rolling direction, w is the direction of the tube Dc movement, a is the gripping angle, w is the constant angle rotation of the Pilger rolls, gx is the angle of the neutral section, R0 and Rb are the radii of the rolls' calibers, dn/2 is the mandrel radius, Ssc is the thickness of the tube wall and Q is the direction of the material flow. Slika 14: Shema Pilgerjevega valjanja: v - smer valjanje, w - smer cevi, Dc smer gibanja, a - kot prijema, w - constantna kotna hitrost Pilgerjevih valjev, gx - kot nevtralnega prereza, R0 in Rb - polmera kalibrov valjev, dn/2 - polmer trna, Ssc - debelina stene cevi in Q -smer toka materiala Figure 12: Hydraulic forming of a sheet Slika 12: Hidravlično oblikovanje traka Figure 15: Elements of the Pilger stand: A is the roll, B is the mandrel16 Slika 15: Elementi Pilgerjevega ogrodja: A - valj, B - trn16 26 Materiali in tehnologije / Materials and technology 43 (2009) 1, 23-30 D. ĆURČIJA, I. MAMUZIĆ: SCHEMES OF METAL-WORKING PROCESSES AND THE RELATED TRIBOLOGICAL ... Table 1: The most frequently used differential equation of fluid mechanics applied for the description of the lubricant behaviour for different metal working processes Tabela 1: Najbolj pogosto uporabljene enačbe in mehanike loma za opis vedenja maziva pri različnih procesih preoblikovanja kovin Equation Figure 1. Smooth surface of the roll and of the metal worked (rolling, drawing, wire drawing) dp/dx = ßd2vx/dy2 ; dp/dy = 0 ; T = ß(vR - v0)/e(x) - (£(x)/2) dp/dx dp/dz = |â2vz/ây2 ; dv/dx + dv/dy + dv/dz = 0 where vz, vx are the corresponding lubricant velocities, e(x) is the lubricant-layer thickness, dp/dx is the pressure gradient, T is the tangential stress, ß is the lubricant dynamic viscosity, and vR and v0 are the working velocities for the tool and the material 14 11 2. Average roughness of the tool and of the material ^p/dx] = 6^(v0+ vr)-| \H(?(x0)\-[ • T1^3(x0)l } ] is the operator of the mathematical probability, £(x0) is the random lubricant layer thickness, depending on the roughness of the tool and of the material, £0 is the lubricant-layer thickness in the access section of the zone of metal deformation 3. Sheet oiling H3d3H/Cadz3 - (yCa)3/2( T/5)(H - 2Hs2/3) dH/dz + (3H - Hs - TH3) = 0 Hs = 3 - T2; H = 1 + a; dH/dz = -ca; d2H/dz2 = c2a; a = Aexp(-cz); z = x*/h* Ca = ßU/a; T = h0*(pg/ßU)1/2; y = a(v4g)-1/3p; H = h*/h0* 7 h0* is the ordinate of the free liquid surface, H is the dimensionless ordinate of the free surface, T is the dimensionless thickness h*/h0* of the layer dragged on the metal surface, U is the velocity of the sheet withdrawal, z is the coordinate, v is the kinematical viscosity, and a is the surface tension__ 4. Lubricant shaping on an inclined surface (inclined bending rolling) 'dp/'dx = ßd2u/dx2 + pgsi^ — âO/âx dp/dy = -pgcosß - âO/ây; dh/dt + ßdh/dx = v Shaping of a drop on the horizontal plane (cylindrical coordinates) 6 dpi d r = ßd 2vx!dx2 18 dp/dr is the cylindrical coordinate system, O is the potential of the diffusion forces resulting from the interaction of lubricant molecules and the metal surface O ~ 10-20 ((tga)2 — ct2))/h3; t is the flowing time for a drop of lubricant; v is the velocity__ 5. Metal pressing (extrusion), cylindrical coordinates (1/r)(â/â r(râvz/â r) = (1/ßt)äp/dz ßt= the lubricant viscosity, depending on the temperature and pressure according to the Barussa equation._ 6. Tube drawing with fat lubricant T1 = -T0 - dp/dx(h2 - y); T = T0 + K\ 70 I m-1 70 v = (h2 - y)c+1/(c + 1) • (1/K (dp/dx)) Ti, To are the critical tangential stresses at the boundary tool and worked piece, h is the gap height between the tool and the worked piece, y is the worked piece velocity, K, m, c are the rheological characteristics of the fat. 7. Effect of inertia and of the smooth tool and worked piece surfaces dp/dx = 6ß(v0 + vr)/£2(x) + Cm/e3(x) + p tg«(l6v02£2(x) - C12)/120e3(x) C1 = k/2 - (k2/4 + 2v0£0(8v0£0 + 3k)1/2; k = 120v/tga £(x) = £0 - ax + x2/2R - ax3/2R2 +... a is the gripping or drawing angle, R is the roll's radius, v0, vR are the working velocities for the tool and the worked piece, and v is the kinematical viscosity_ 8. New approaches to the mathematical modelling (rolling of metals) dk/d(y>/a) = 6W(Ah/£0)2(H0S - HHS)/H3HS)eMk; W = ß4v0 + vR)/orAha Hhs = Ah/2e0[(