DISTRIBUTION NORMAL CONTACT STRESSES IN THE ROLL GAP AT A CONSTANT SHEAR STRESS

Transcription

1 Acta Metallurgica Slovaca, Vol., 05, No., p DISTRIBUTION NORMAL CONTACT STRESSES IN THE ROLL GAP AT A CONSTANT SHEAR STRESS Rudol Pernis ), Tibor Kvačkaj )* ) Považská Bystrica, Slovakia ) Technical University o Košice, Faculty o Metallurgy, Košice, Slovakia Received: Accepted: * Corresponding author: e-ail: tibor.kvackaj@tuke.sk, Tel.: , Letna 9,04 00 Košice, Slovakia Abstract Dierential equation describing stress state in the rolling gap was derived irst tie by Karan. Since the solution o the dierential equation is not easy any authors try to sipliy o entry conditions. Soe authors have replaced the circular arch o the contact zone o rolls by straight line, polygonal curve or parabola or sipliications o solution. These sipliications allow to obtain analytical solution dierential equation but with acceptation soe inaccuracy o the inal results. Another solution o the dierential equation was ocused on the substitution o the analytical solution by the nuerical solution, but should also expect soe uncertainty o the inal results. A ore sophisticated solution was given by Gubkin is based on deining a constant shear stress and the approxiation o the circular arch through the straight line. Gubkin or analytic solution o dierential equation used one constant that includes the riction coeicient and second constant which is including the geoetry o the rolling gap. The contribution o this paper is an original analytical solution o the dierential equation based on the description o the contact arc by the equation o a circle. The proposed solution or the calculation o noral stress distribution is described by two constants. The irst constant is describing the geoetry o the rolling gap and the second describes riction coeicient. The inal solution o dierential equation is su o two independent unctions involving the shear stress as a variable value. The proposed solution does not consider with aterial work hardening during processing. Keywords: theory o rolling, constant shear stress, dierential equations, noral stress, relative stress Introduction The process o lengthwise rolling can be described as the action o active orces onto rolling direction with the consideration o equilibriu conditions or the eleent. Theory o lengthwise rolling process was presented or the irst tie by von Karan [] in 95. This theory described the equilibriu conditions o the eleent in the rolling gap by two-diensional dierential equation. The geoetrical characterization o the dierential eleent is represented in Fig.. The horizontal projection o all the orces acting on the eleent ust be in equilibriu state. On the base o the su o the horizontal active orces was derived dierential equation o contact stresses or two diensional deoration in the consideration o the orward and the backward slip zones. The procedure or derivation o the two-diensional dierential equation can be ound in the classical literature o rolling Počta [], Avitzur [3], Hensel and Spittel [4] and DOI 0.776/as.vi.549 p-issn e-issn

2 Acta Metallurgica Slovaca, Vol., 05, No., p Fig. Deterination o geoetric relationships Mieik [5]. More recent literature are the publications Hajduk and Konvičný [6], Kollerová et al [7] and Pernis [8]. The stress state in rolling gap describes dierential equation: dσ x dx σn x dy y dx τ y 0 (.) where: - plus sign (+) is or backward slip zone - inus sign () is or orward slip zone - n - noral contact stress on rolls - - shear stress between rolls and rolling aterial - x, y - coordinates o the cylinder touching the rolled aterial The ollowing Tresca condition o plasticity is used in eq.(): n x σ a (.) where: - y n axial principal stress (vertical direction) - x inial principal stress (horizontal direction) - a - stress which represents real deoration resistance [9 ] The shear stress can be deterined ro on the base two ollowing assuptions: - the shear stress is varied and also is proportional to the noral contact stress on rolls: n (3.) - the shear stress is constant and also is proportional to the real deoration resistance: a (4.) Substituting eq. () and eq.(4) to the eq.() is obtained the ollowing or o two-diensional dierential equation with a single variable that is stress: dσ n dx σa y dy σa dx y 0 (5.) DOI 0.776/as.vi.549 p-issn e-issn

3 Acta Metallurgica Slovaca, Vol., 05, No., p The geoetry o the roll which is in contact with the rolling aterial can be described by the coordinates x and y wherein the variable y is a unction o the coordinate x. The eq.(5) is characterized by two constants: actual deoration resistance and riction coeicient. Assuing that the aterial during the rolling does not exhibit work hardening it can be the dierential equation divided by the actual deoration resistance a and used new variable which is relative noral contact stress σ : a n n n (6.) where: - is a siga unction Substituting eq. (6) to the eq.(5) is obtained orula in which is eliinating the actual deoration resistance a : dσ n dx dy 0 (7.) y dx Using this equation (eq.(7)) the solution becoes independent o the properties o the rolled aterial. Solution o dierential equation according to Gubkin The analytical solutions o eq.() are based on certain assuptions and ainly sipliications. Gubkin [3] approxiated the circular arc with a parabola and also by a straight line as shown in Fig.. h h y x (8.) l d dy h (9.) dx l d Fig. The Contact Arc according to Gubkin Substituting eq. (9) to the eq.(7) is obtained ollowing or: DOI 0.776/as.vi.549 p-issn e-issn

4 Acta Metallurgica Slovaca, Vol., 05, No., p dσ n ld h h ld dy 0 y (0.) and ater sipliication: dy dσ n (.) y where: ld (.) h By integration o the eq.() will be obtained the inscription or the backward slip zone: nb y CB (3.) and or the orward slip zone: nf y CF (4.) The integration constant C B or the backward slip zone and C F or the orward slip zone were speciied ro the boundary conditions o the aterial input and exit into resp. ro rolling gap. It is assued that the lengthwise rolling process is realized without orward and backward stretching orces, without aterial hardening during plastic deoration and without roll lattening. The vertical coordinates o points A and B according to Fig. are as ollows: point A: y=h /, point B: y=h 0 / and horizontal stress in these points is x =0. To apply σn σa ust be valid a condition o plasticity. Fro eq.(3) is deterined integration constant or the backward slip zone where is valid: σ nb a y=h 0 /: C B h0 (5.) Siilarly ro eq.(4) is deterined integration constant or the orward slip zone where is valid: σ nf a y=h / C F h (6.) The equation describing o the distribution o the relative noral contact stress or the orward slip zone can be written as ollows: y nb (7.) h 0 and the equation or the calculation o the relative noral contact stress or the backward slip zone can be written as ollows: y nf (8.) h DOI 0.776/as.vi.549 p-issn e-issn

5 Acta Metallurgica Slovaca, Vol., 05, No., p The coordinate y is described by eq.(8). To calculate the relative noral contact stress the coordinate y is replaced by the relative coordinate x/l d. y x h l (9.) 0 d or: y h x l d (0.) Substituting eq. (9) to the eq.(7) is obtained the ollowing or or the relative noral contact stress or the backward slip zone: nb x (.) ld Substituting eq. (0) to the eq.(8) is obtained the ollowing or or the relative noral contact stress or the orward slip zone: nf x (.) ld The visualization o the rolling equation eq.() and eq.() coordinate x/l d and constant shear stress is given in Fig. 3. in depend on the relative Fig. 3 The distribution o the relative noral contact stress in rolling gap at constant shear stress 3 New solution o dierential equation The ollowing part will be represented the new analytical solution o eq.() with condition constant shear stress. The analytical solution o the eq.() is based on description o circular arch by equation o the circle as is showing in Fig. 4. DOI 0.776/as.vi.549 p-issn e-issn

6 Acta Metallurgica Slovaca, Vol., 05, No., p Fig. 4 The description o contact arc by the circle The new or o eq.(7) ater odiication will be: dy dx dσ n 0 (3.) y y The solution o eq.(3) can be obtained by its integration [4-9]: dy dx d n y (4.) y σ While the let side o eq.(4) is siply integrated the right side consists ro two integrals which can be described by the unctions F (x) a F (x) and eq.(4) will take the ollowing or: C F x) F ( ) (5.) n ( x where C is integration constant. The unction dependence y=(x) in dierential eq. (3) is representing o the equation o the circle: x y y R, 0 y 0 h R (6., 7.) where: - R the roll radius - h exit thickness o rolling aterial I variable y is separated ro eq.(6) and is carried out the dierentiation o the eq.(8) then is obtained the ollowing orula: h x y R R x, dy dx R x (8., 9.) The deterination o ratio dy/y ro eq.(8), eq.(9) and substituting to the unction F (x) is obtained: x x dy dx F ( x) R x dx R (30.) y h R R x x x R R R DOI 0.776/as.vi.549 p-issn e-issn

7 Acta Metallurgica Slovaca, Vol., 05, No., p where is a constant coprising the ollowing paraeters: h (3.) R Sipliication o eq.(30) can be obtained when a transoration ro the rectangular coordinate syste to polar coordinates is perored. Fro Fig. 5 is resulting the deterination o the polar K x; y. coordinate o point Fig. 5 Deinition o the position o point K[x; y] The transoration ro Cartesian coordinates to polar with can be ade as ollows: x R sin, resp. dx R d (3., 33.) Substituting eq. (3) and eq.(33) to eq. (30) will obtained new orula which is labeled as unction F (,) resulting ro the transoration o Cartesian coordinates previously labeled as a unction F (x) into polar coordinates : sin R d sin F (, ) d (34.) sin R sin To calculate the integral eq.(34) will be applied ollowing substitution: t, dt sin d (35., 36.) sin dt F (, ) d t (37.) t Graphical visualization o eq.(37) is shown in Fig. 6. Function F (,) throughout the project space shall take negative values. The next step is deinition o unction F ( x) ro eq.(5) as ollows: dx dx dx F ( x) R (38.) y y0 R x x R DOI 0.776/as.vi.549 p-issn e-issn

8 Acta Metallurgica Slovaca, Vol., 05, No., p and their transoration ro the Cartesian coordinates (F (x)) into the polar coordinates (F (, )): d F (, ) d. (39.) sin Fig. 6 Graphical visualization o unction F(, ) The ollowing substitutions are introduced or solution o the integral: z, z, z dz d (40.) z Substituting eq. (40) to eq. (39) will obtained orula: z F (, ), (4.) where: z a z dz a (4.) Next procedure or solving the integral eq.(4) consists in its decoposition into partial ractions: dz a dz F (, ). (43.) a z a a z and solution o eq.(43) will be as ollows: a z F (, ) z (44.) a a a a Reversing the introduction o the constant a into the eq.(44) and the use o substitution ro eq.(40) yields the orula: DOI 0.776/as.vi.549 p-issn e-issn

9 Acta Metallurgica Slovaca, Vol., 05, No., p. 3-4 DOI 0.776/as.vi.549 p-issn e-issn z z F ), ( (45.) ), ( F (46.) Graphical visualization o eq.(46) is shown in Fig. 7. Function F (,) throughout the project space shall take positive values. Fig. 7 Graphical visualization o unction F (, ) When the unctions F (, ) and F (, ) are substituted into eq.(5) then receives the ollowing orula: ), ( ), ( F F C n (47.) and will be obtained analytical solution or the relative noral contact stress in coplex or: C n (48.) where: C is an integration constant Equations describing the distribution o relative noral contact stress along rolling gap at constant shear stress will have the ollowing ors: - or backward slip zone: C B nb (49.) - or orward slip zone: C F nf (50.)

10 Acta Metallurgica Slovaca, Vol., 05, No., p. 3-4 DOI 0.776/as.vi.549 p-issn e-issn Rolling process is realized without orward and backward stretching orces, without aterial hardening during plastic deoration and without roll lattening. According to Fig. polar coordinate or point B is and or point A is =0. In these points horizontal stress is x =0. In order to peror a condition o plasticity or these points ust be valid a n σ σ. The integration constant C B or the backward slip zone shell be deterined ro the condition σ nb a and eq.(49)as ollows: C B (5.) Also integration constant C F or the orward slip zone shell be deterined ro the condition σ nf a and eq.(50)as ollows: C F (5.) Substituting equations describing o the integration constants eq.(5) and eq.(5) into eq.(49) and eq.(50) will obtained the inal analytical solutions or calculation o relative noral contact stress in backward and orward slip zones: nb (53.) nf (54.) The eq.(53) and eq.(54) can be written in short or as su o two unctions: ) (, F ) (, F n (55.) where: F (,φ) is depend only on geoetric constant and polar coordinate φ F (,φ) is depend on geoetric constant, polar coordinate φ and riction coeicient Fig. 8 The distribution o the relative noral contact stress in rolling gap with constant shear stress

11 Acta Metallurgica Slovaca, Vol., 05, No., p Geoetric visualization o eq.(53) and eq.(54) is presented in Fig. 8. The curves represent the developent o the relative noral contact stress in rolling gap in the condition o constant shear stress in dependence to the relative coordinate x/l d. The paraeters are the relative thickness deoration and riction coeicient (=0,4). Maxial value o the relative noral contact stress in the rolling gap is observed at neutral point. Increasing o relative deoration o thickness has resulted in the growth o relative noral contact stress and shiting o neutral point in direction to point A i.e. towards to the exit plane o rolling gap. The presented solution is valid to the rolled aterial which does not working hardening during his processing. 4 Conclusion The analytical solution dierential equation describing stress state in the rolling gap with condition constant shear stress is given in this paper. The irst analytical solution o the dierential eq.() was entioned by the author Gubkin using a sipliied describtion o the circular arch o the contact zone by the straight line and later by the parabola. A constant shear stress was used as urther sipliy the author. However these sipliications have ipact to precision o the calculation o distribution o the noral stress in the rolling gap. The contribution o this paper is the new description o the circular arch o the contact zone by the equation o a circle. Approxiation o the circular arch by the equation o a circle causes a proble to obtain analytical solution o dierential eq.(). A new analytical approach or solution o this case is based on the transoration ro Cartesian coordinates ( x) to polar coordinates ( Description o distribution o the relative noral contact stress on rolls is represented by the su o two independent unctions n F F. The irst unction has a logarithic or F (, ) and describes the geoetry o the rolling gap and second F F F (, unction ) describes shear stress between the rolls and the rolling aterial. The second unction has character o the wrapping angle and is independent on riction coeicient. The new approach or solution o eq.() allows too obtained calculation or case when shear stress is not a constant. Acknowledgents This work was inancially supported by the VEGA /035/4 project and the project Technological preparation o electrotechnical steels with high pereability or electrodrives with higher eiciency which is supported by the Operational Progra Research and Developent ITMS , inanced through European Regional Developent Fund. Reerences [] Th. Kárán: Zeitschrit ür Angewandte Matheatik und Mechanik, Vol. 5, 95, No., p. 39-4, (in Geran) [] B. Počta: The Fundaentals o Metal Foring Theory, SNTL, Praha, 966, (in Czech) [3] B. Avitzur: Metal Foring: Processes and Analysis, McGraw-Hill, New York, 968 [4] A. Hensel, Th. Spittel: The Force and Energy Consuption in Metal Foring Processes, Deutscher Verlag ür Grundstoindustrie, Leipzig, 978, p. 58, (in Geran) [5] E.M. Mieik: Metalworking Science and Engineering, McGraw-Hill, New York, 99 [6] M. Hajduk, J. Konvičný: Power Requireents or Hot Rolled Steel, SNTL, Praha, 983, (in Czech) DOI 0.776/as.vi.549 p-issn e-issn

12 Acta Metallurgica Slovaca, Vol., 05, No., p [7] M. Kollerová et al.: The Rolling, ALFA, Bratislava, 99, (in Slovak) [8] R. Pernis, J. Kasala: The Theoretical Calculation o Rolling Pressure Miniu, In: Metal 0, Brno, CD-ROM (0), 0 [9] T. Kubina, J. Bořuta, I. Schindler: Acta Metallurgica Slovaca, Vol. 3, 007, No. 4, p [0] R. Pernis, J. Kasala, J. Bořuta: Kovové Materiály Metallic Materials, Vol. 48, 00, No., p [] T. Kvačkaj, I. Pokorný, M. Vlado: Acta Metallurgica Slovaca, Vol. 6, 000, No. 3, p [] R. Pernis: The Theory and Technology o Cups Production, TnUAD Trenčín, Trenčín, 009, (in Slovak) [3] S.I. Gubkin: Plastic Deoration o Metals, Metallurgizdat, Moscow, 960, (in Russian) [4] L. Matejíčka: Journal o Inequalities in Pure and Applied Matheatics. Vol. 9, 008, No. 3, Article 75 [5] I.S. Gradshteyn, I.M. Ryzhik: Table o Integrals, Series, and Products, Seventh edition, Acadeic Press is an iprint o Elsevier, London, 007 [6] L. Matejíčka: Matheatica Slovaca. Vol. 45, 995, No., p [7] A.P. Grudev: The Rolling Theory, Metallurgija, Moscow, 988, (in Russian) [8] S.E. Rokotjan: The Rolling Theory and The Metal Quality, Metallurgija, Moscow, 98, (in Russian) [9] V.B. Bachtinov: The Rolling Production, Metallurgija, Moscow, 987, (in Russian) Noenclature n noral contact stress [MPa] n relative noral contact stress [ ] nb relative noral contact stress (Backward slip zone) [ ] nf relative noral contact stress (Forward slip zone) [ ] siga unction (average relative noral contact stress) [ ] shear stress [MPa] x, y principal stress (, ) [MPa] a actual resistance to deoration [MPa] x, y rectangular coordinates [] R, polar coordinates [, rad] dx, dy coordinate dierentials x and y [ ] constant dierential equation [ ] gripping angle [rad] n neutral angle [rad] l d length o contact arc [] h 0, h thickness beore and ater deoration [] h n thickness in neutral section [] h av average thickness [] h absolute reduction [] relative reduction [ ] riction coeicient [ ] R radius o rollers [] C B integration constant (Backward slip zone) [ ] C F integration constant (Forward slip zone) [ ] DOI 0.776/as.vi.549 p-issn e-issn