ASYMMETRICAL COLD STRIP ROLLING. A NEW ANALYTICAL APPROACH

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1 ASYMMETICAL COLD STIP OLLING. A NEW ANALYTICAL APPOACH ODICA IOAN * In this pape is given a solution of asymmetical stip olling poblem using a Bingham type constitutive equation and the petubation method. olling pessue distibution, olling foce, olling toque, font tension, position of neutal points, which ae affected by vaious olling conditions ae analyzed. The influence of the olling speed on the whole pocess was studied using a Bingham type constitutive equation.. INTODUCTION A unifom plastic model of symmetical stip olling in which peipheal velocity and adius of the uppe oll ae equal to those of the lowe oll was poposed by Oowan []. ecently, the symmetical stip olling using a Bingham type constitutive equation and the petubation method was studied by N. Şandu and G. Camenschi [, 3]. Asymmetical olling pocess was used to manufactue plates and sheets. The asymmety is due to the olls with diffeent speeds o with diffeent diametes. This pocess impoves poductivity of the olling opeation-the olling foce, pessue, toque is educing, and the popeties of the stip suface ae impoved. An analytical solution fo asymmetical stip olling pocess was poposed by Y. M. Hwang and G.Y. Tzou, using the slab method [4] and the olling pessue, olling foce and toque wee obtained. In this pape is given a solution of asymmetical stip olling poblem using the method pesented in [, 3].. SOLUTION OF THE POBLEM In ode to solve the poblem of asymmetical stip olling the following assumptions wee made: a the oll is igid; the stip being olled is Bingham type viscoplastic, incompessible mateial; b defomation is plain stain; c the pocess takes place with high speed; * Univesitatea Spiu Haet Bucueşti. ev. oum. Sci. Techn. Mec. Appl., Tome 54, N o, P. 34, Bucaest, 9

2 odica Ioan d the fictional facto between the oll and the mateial is constant ove the ac of contact; e the flow diection of the stip at the entance and the exit of the oll bite ae hoizontal. ω O δ P P S O θ S α α P P x δ O ω Fig. The geomety of asymmetical cold olling. Fig. is a schematic illustation of the asymmetical stip olling pocess. The viscoplastic defomation egion is bounded by two singula sufaces S and S. This zone is divided in thee egions by the sufaces SN and SN passing though the neutal points N fo the uppe oll, and N fo the lowe oll. The position of the neutal points N, and N and the equation of the sufaces SN and SN will be detemined. Assuming that the angle of stip and the uppe oll between the contact point at the entance in the defomation zone and the contact point at the exit fom it (α, also the adius of the uppe oll (δ ae known, we get the following conditions in ode that the pocess to take place δ = ( ( δsin α δsin α δ sin α, (

3 3 Asymmetical cold stip olling. A new analytical appoach 3 δ α = acsin sin α δ, which we have fom the geomety of the figue. Equations of the poblem ae a Cauchy equation divt =ρ d v, ( dt b continuity equation Id =, (3 c defomation ate tenso equation T ( d Bingham type constitutive equation d= v v, (4 t I k = p η d, IIt > k, (5 II d In ( (5, t is Cauchy stess tenso, p pessue, η, k mateial constants, d defomation ate tenso, II d the second invaiant of defomation ate tenso, II t the second invaiant of tension tenso deviato, v velocity vecto. Witing equations ( (5 in pola coodinates and tuning them into non dimensional fom by means of the elations =, v = v v, v v v θ =, p p η v = θ. (6 The following dimensionless combinations ae put into evidence Bingham s numbe Bg = k η v, and e = ρ v eynolds numbe. η We shall suppose that e <<, theefoe the inetial tems ae neglected and extending the method used in [] fo the symmetical stip olling, we get (, v k i i O( Bg Ψi θ = Ψ ϕ θ η, (7 whee bi ei Ψi ( θ = aiθ sin θ di cosθ, ( 8

4 4 odica Ioan 4 and with B i Ei ϕi θ = Aiθ sin θ Di cos θ Ki( θ cos θ Ki( θ sin θ dk θ = fi ( θ sin θ, dθ dk θ = fi ( θ cos θ, dθ (9 dψ i dψ i dfi ( θ d d = θ dθ dθ dθ dψ d Ψ dψ d Ψ d 4 d d 4 θ θ θ dθ i i i i. ( Velocity vecto components ae given by dψi k i i v v = ϕ θ O dθ η vi k θ = ϕ i θ O η ( Bg ( Bg The physical components of the stess tenso ae i vη dψ i ηv t = a d i ci θ dψ i d Fi ( θ C ln k θ ϕ θ A O Bg F ai i θ. i i i,, ( ( ( θ i vη ηv F C tθθ = a c k A ϕ θ O i i ln Bg i i i i ai,

5 5 Asymmetical cold stip olling. A new analytical appoach 5 with v t = k ϕ θ O i θ dψ i η d Ψ i dθ Bg d i θ Fi ( θ F i i d Ψi dψ θ = dθ 4 dθ. (3 Denoting by X, Y the stess esultants in diections Ox, Oy, acting on a θ α, α, we have suface with the equation = ( θ,, α ( θ cos ( θ θθ sin d, X = t t θ t t θ θ α α ( θ sin ( θ θθ cos d. Y = t t θ t t θ θ α (4 So, we get X j sin α sin α = a 4ηv α α j ( α ( α b j cosαsin α cosαsinα cosαcosα cosαcosα ej cj ( ( α sinα ( α sinα α α ( α α Bg Aj ( α sinα ln α sinα ln (5 ( j B sin sin j C sin sin α α α α 4 α α α α ( α Ej ( ( α α ( α α α I cos cos cos j

6 6 odica Ioan 6 ( sin F sin F j j a α α α α α α j whee α K α K α, ( jcos jsin I j α aj bjcost-ejsin t = d t, (6 F α j ( t with j =,, 3 placed in egions,, 3 of the defomation zone and the vetical stess esultants is Y cosα cosα = 4η α α j a v b j sin αsinα sinαsinα j ( α ( α sinαcos α sinαcos α ej cj ( ( α cosα ( α cos α α α ( α α Bg -Aj ( α cosα ln α cosα ln ( j B cos cos j C cos cos α α α α 4 α α α α (7 ( α E j ( ( α α ( α α α I sin sin sin j ( cos F cos F j j a α α α α α α j α K α K α, j =,. ( jsin jcos ( N We compute the stess esultants acting on sufaces θ =α, ( α, ( α and we obtain

7 7 Asymmetical cold stip olling. A new analytical appoach 7 and ( α T = tθ θ=α d = vηψ α δsinα N ( α N k ϕ α α Ψ α δ α N F sin ( α N = tθθ θ=α d = vηa δsin α N ( α N ηv δsinα F α C c ( N k ϕ α N a sin sin δ α δ N α δsinα A ln N ln ( (. (8 (9 3 N. In the same way we compute T, 3 T, T, T, 3 T, N, 3 N, N, N and 3. THE DISCONTINUITY SUFACES Consideing sufaces S and S as singula sufaces fo the velocity and stess field, the dynamical compatibility conditions ae v =, and we shall have [ ] n [ t n ] = kl k ( Ψi, θ vjjsinθ= Ci, i =,. ( We impose that the discontinuity suface S passes though the point P, fo θ=α and we get the equation of suface S b e v a θ α ( sin θ sinα ( cos θ cos α (

8 8 odica Ioan 8 k ϕ ( θ v ( θ sin θ ( α sin α =. η We impose that the discontinuity suface S passes though the point P, fo θ=α and we get the equation of suface S b e v aθ α ( sin θ sin α ( cos θ cosα k ϕ ( θ v ( θ sin θ ( α sin α =. η (3 On sufaces SN Imposing that the equation of suface and S we have satisfied[ v ] =. N N S suface passes though S N in the fom n ( N, N α α, we will obtain k ϕ ( θ ϕ ( θ =. v Ψ θ Ψ α Ψ3 θ Ψ3 α η N Simila, we get the equation of suface 3 S N k ϕ ( θ ϕ ( θ =. v Ψ θ Ψ α Ψ3 θ Ψ3 α η N 3 (4 (5 4. BOUNDAY CONDITIONS The following conditions will be used: a vi θ (, α =, vi ( θ, α =, i =,3 ; b the discontinuity suface S passes though the point P (also, S passes though the point P ; c we assume that X I =, whee X I = X and X ( α = ( α ( α = ( α II = X ( α = ( α ( α = ( α d the following fiction conditions ae imposed θ=α m II t θ = t θ=α, t θ θ=α = m II t θ=α, 3 3

9 9 Asymmetical cold stip olling. A new analytical appoach 9 θ θ=α = m II θ=α, t t θ θ= α = m II θ= α, t t obtain 3 3 θ θ= α = m II θ= α, t t e on sufaces S N ( N and N θ θ= α = m II θ= α ; t t S we have [ ] v =, f v ( α, α = ω δ, n X ( N, X 3 N =, v α α = ω δ. Fom condition a we have ϕ ( α =, espectively i i X X 3 N = ; ϕ α = and we A iα Bi sin α Di Ei cos α = [ Kicos α Kicos α ], A iα Bi sin α Di Ei cosα =, i =,3. (6 Imposing that the discontinuity suface S passes though the point P (, = α θ= α, we get v b e ( sin sin a α α α α ( cos α cos α v α sin α α sin α =. Using the continuity equation v= v and sin (7 α α α sin α =, (8 we will have b e a( α α ( sin α sin α ( cos α cosα =. (9 In a simila way, imposing that the discontinuity suface S passes though the point P, we get b e a( α α ( sin α sin α ( cos α cos α =. (3 Fom the fiction condition θ=α m II t θ = t θ=α, we obtain whee so b sin α e cosα = m F α, (3 a b ab cosα a e sinα = F α, (3

10 odica Ioan m ( bsin α ecosα = m a b e cos α sin α and with the notation m m = γ, (34 we get bsin α ecos α = γ ( a bcos α esin α. (35 We also obtain that with and ( ( α ( α Ψ (33 m ϕ α m ϕ α =, (36 F Ψ m = F ( α α. (37 eplacing (37 in (36, we have ϕ ( α = γϕ( α. (38 Using the expessions of ϕ ( α and ϕ ( α, we get A B ( cos sin E ( sin cos γ γ α α γ α α = = f( α 4 K( γsin α cosα K( sin α γcosα. ( Fom fiction condition t θ θ=α = m II t θ=α, we have Aγ B γ cosα sinα E γ sinα cosα = = b sin α e cos α =γ ( a b cos α e sin α Fom the expessions of ϕ ( α and (4 ϕ α = γ ϕ α. (4 3 ϕ α we get 3 A γ B γ cosα sinα E γ sinα cosα = = f3( α 4 K3( cosα γsinα K3( sinα γcosα. (4 In a simila way, using that θ θ=α = m II t t θ=α, we obtain

11 Asymmetical cold stip olling. A new analytical appoach bsinα ecosα =γ a bcosα esinα. (43 So, we have bisin α eicos α = γ ( ai bicos α eisin α, i =,3, ϕ i( α = γϕi( α, i =,3 (44 and also b sin α e cos α = γ a b cos α e sin α i i i i i ϕi α = γ ϕ α Fom the condition which means fom whee N N i ( N,, i =,3., i =,3 v α α = ω δ, we get the equation fo N N ( α N (45 k ϕ ( α v ( α Ψ ( α η = ω δ, (46 Bg ω δ ϕ α α α Ψ α =, (47 N N v ωδ ωδ 4Bg v v ϕ α Ψ α Bgϕ ( α α = ( N, Simila, fom condition. (48 v α α = ω δ, we get the equation fo Bg ω δ ϕ α α α Ψ α =, (49 3 N N 3 v

12 odica Ioan and fom (49 N ωδ ωδ 4Bgϕ3 α Ψ3 α v v Bgϕ 3 ( α α = Using that c X I =, we have cos cos a δ δ b α α δδ δ δ ctgαcos α ctgαcos α c δ δ e = and δ δsinα δ δsinα A sin α ln sin ln α ( ( δ δ B C E sin sin 4 α α δ sin I δ sin F δ α sin F a α α α α δ sinα( Kcosα Ksinα =. Fom condition e, X = X, we get = 3 N δ δ cos α cos α N δδ δ δ a b c e ctgαcos α ctgαcos α = δ δ δ δ cos α cos α N δδ δ δ a b c e3 ctgαcos α ctgα cos α δ δ and δ δ N sinα δ δ N sinα A sin α ln sin ln α ( ( δ δ B C E sin sin 4 α α. (5 (5 (5 (53 (54

13 3 Asymmetical cold stip olling. A new analytical appoach 3 δ sin I δ sin F δ α sin F a α α α α δ sin α( Kcos α Ksin α = δ δ N sinα δ δ N sinα = A3 sin α ln sin α ln ( ( δ δ B3 C3 E3 sin sin 4 α α δ sin 3 I δ sin F3 δ α sin F3 a α α α α 3 δ sin α( K3cos α K3sin α. Using that X = X, we obtain 3 N δ δ cos α cos α N δδ δ δ a b c e ctgαcos α ctgα cos α = δ δ δ δ cos α cos α N δδ δ δ = a b c e3 ctgαcos α ctgαcos α δ δ and δ δ N sinα δ δ N sinα A sin ln sin ln ( ( δ δ B C E sin sin 4 α α δ sin I δ sin F δ α sin F a α α α α δ sin α( Kcos α Ksin α = α α δ δ N sinα δ δ N sinα A3 sin ln sin ln ( ( = α α (56 (57

14 4 odica Ioan 4 δ δ B3 C3 E3 sin sin 4 α α δ sin 3 I δ sin F3 δ α sin F3 a α α α α 3 δ sinα( K3cosα K3sinα. whee so 5. THE OLLING TOQUES, OLLING STESS AND OLLING PESSUE We will detemine the olling toque M =δ T T T, (58 3 T =Ψ ( α v sin N η δ α Ψ α δsinα N Bg ϕ ( α F, α T 3 =Ψ3 ( α v sin N η δ α Ψ3 ( α δsinα N N Bg ϕ 3( α F3, α T =Ψ α v sin η δ α N Ψ ( α δsinα Bg ( N ϕ α F, α M T T3 T = v ηv δη Also, fo the olling toque of oll we have 3 Total olling toque will have the expession M = M M. (59. (6 M =δ T T T. (6 (6

15 5 Asymmetical cold stip olling. A new analytical appoach 5 Intoducing the stess II σ X x = (63 and σ = k 3, is the mean yield stess of the mateial we obtain the elative olling y stess the elation ( σ x δ δ cos α cos α = a b( Bg σ δδ δ δ y 3 e ctg αcos α ctg αcos α c ( δ δ sin A sin ln δ α δ α δ sinα C δ α sin ln B (64 δ E δ sin α δ sin α sin α I ( sin F sin F δ α α δ α α a 4 and the olling pessue δsin α( Kcosα Ksin α ( θ ti F θθ i = a Bg 4 ln i c i Ai ϕi θ C i. ηv ai (65 6. NUMEICAL ESULTS AND CONCLUSIONS Wee consideed diffeent cases: A. olls with diffeent diametes (geometical asymmety ; B. olls with same diametes, asymmety given by diffeent oll speed Wee studied diffeent chaacteistics of the asymmetical olling pocess.

16 6 odica Ioan 6 ( α t, θθ ηv.6.4. V /V = V /V =. olling pessue V /V =. V /V = Contact length (mm Fig. olling pessue fo vaious ole speed atios when =.5 mm, =.5 mm, α = 3, α =.477, δ = mm, δ = mm, Bg =.5, γ =., γ =.. We obseve that the olling pessue (Fig. in case A and Fig. 3 in case B inceases, with the deceasing of the oll speed atio. When the oll speed atio is the neutal points become a single one. The pocess is taking place in bette conditions when oll speed atio inceases. When the fiction facto atio inceases, the olling pessue inceases (Fig. 4. The olling pessue deceases with inceasing stip speed (Fig. 5. The olling pessue deceases with deceasing of the eduction (Fig. 6. The back foce inceases with the inceasing of the fiction facto; fo γ=.6 7, the back foce is zeo, which allows us to conclude that the pocess is taking place only by means of the fiction foces, without aplling a back foce (Fig. 7. The elative olling stess inceases with the inceasing of the fiction facto and deceases with the inceasing of the oll speed. The back foce deceases with the inceasing of the oll speed atio (Fig. 8. The back foce vetical esultant is small and inceases with the inceasing of the fiction facto (Fig. 9.

17 7 Asymmetical cold stip olling. A new analytical appoach V /V = V /V =. V /V =. V /V =.3 olling pessue Contact length (mm Fig. 3 olling pessue fo vaious ole speed atios when =.5 mm, =.75 mm, α = α =.36645, δ = δ = mm, Bg =.3, γ = γ =., V =.9v. olling pessue.5.5 γ /γ =3 γ /γ = Contact length (mm Fig. 4 olling pessue fo vaious fiction factos atios when =.3 mm, =.5 mm, α = 3, α =.38, δ = mm δ = 7 mm, Bg =.8, V =.65v, V =.8v, γ =..

18 8 odica Ioan 8 ( α t, θθ ηv olling pessue Bg=.8 Bg= Contact length (mm Fig. 5 olling pessue fo the stip speed when =.3 mm, =.5 mm, α = 3, α =.38, δ = mm, δ = 7 mm, V =.7v, V =.8v, γ =., γ =.. ( α t, θθ ηv olling pessue =4% =35% Contact length (mm Fig. 6 olling pessue fo vaious eductions when =.5 mm, α = 3, δ = mm, Bg =.5, V =.7v, V =.8v, γ =., γ =..

19 9 Asymmetical cold stip olling. A new analytical appoach 9 X II 4ηv γ Fig. 7 The back foce fo fiction facto when =.5 mm, =.75 mm, α = α =.36645, δ = δ = mm, Bg =.3, V =.6v, V =.9v. X II 4ηv V V Fig. 8 The back foce fo vaious ole speed atios when =.5 mm, =.5 mm, α = 3, α =.477, δ = mm, δ = mm, Bg =.5, γ =., γ =.. The back foce vetical esultant is small and emains constant with the inceasing of the oll speed atio (Fig.. The total olling toque M deceases with the inceasing of the oll speed atio (Fig.. The total olling toque M inceases with the inceasing of the fiction facto (Fig..

20 3 odica Ioan Y II 4ηv γ Fig. 9 The back foce vetical esultant fo fiction facto when =.5 mm, =.75 mm, α = α =.36645, δ = δ = mm, Bg =.3, V =.6v, V =.9v. Y II 4ηv V V Fig. The back foce vetical esultant fo vaious oll speed atios when =.5 mm, =.5 mm, α = 3, α =.477, δ = mm, δ = mm, Bg =.5, γ =., γ =. The olling pessue fo the lowe oll is smalle than the olling pessue fo uppe oll in defomation zone III. In zones I and II, they ae pactically, equal (Fig. 3. The neutal point fo uppe oll is moving towads the exit of the viscoplastic defomation zone, when the fiction facto inceases and is moving towads the entance in the viscoplastic defomation zone, when the olling speed inceases (Figs. 4, 5.

21 Asymmetical cold stip olling. A new analytical appoach 3 The neutal point fo lowe oll has an opposite vaiation then the neutal point fo uppe oll. M M M = δ δ olling toque (M - N-m/mm V V Fig. Vaiation of total olling toque with the oll speed atio when =.5 mm, =.5 mm, η = 3 Nmm - s, k = 4 Nmm -, v = 4 mms -, α = 3, α =.477, δ = mm, δ = mm, Bg =.5, γ =., γ =.. M M M = δ δ olling Toque (M - N-m/mm γ Fig. Vaiation of the elative olling toque vesus fiction facto when =.5 mm, =.75 mm, η = 3 Nmm - s, k = 4 Nmm -, v = 5 mms -, α = α =.36645, δ = δ = mm, Bg =.9, V =.6v, V =.9v. The esults obtained with petubation method ae in good ageement with the esults given in [4], whee the poblem was solved with slab method.

22 3 odica Ioan ( α tθθ, i ηv 7 6 Uppe ole olling pessue Lowe ole Contact length (mm Fig. 3 Vaiation of olling pessue fo uppe and lowe oll when X II =, =.5 mm, =.75 mm, α = α =.36645, δ = δ = mm, Bg =.9, V =.6v, V =.9v, γ = γ =.67. N (mm 6 4 Bg=.8 8 Bg= γ Fig. 4 Vaiation of the position of the neutal point with fiction facto fo diffeent speed atios when =.5 mm, =.75 mm, α = α =.36645, δ = δ = mm, Bg =.3, V =.6v, V =.9v.

23 3 Asymmetical cold stip olling. A new analytical appoach 33 N (mm Bg=.9 Bg= γ Fig. 5 Vaiation of the position of the neutal point with fiction facto fo diffeent speed atios when =.5 mm, =.75 mm, α = α =.36645, δ = δ = mm, Bg =.3, V =.6v, V =.9v. N (mm N N,,,3,4,5,6,7,8 γ Fig. 6 Vaiation of the position of the neutal points with fiction facto when =.5 mm, =.75 mm, α = α =.36645, δ = δ = mm, Bg =.3, V =.6v, V =.9v. The influence of olling speed on whole pocess was descibed using the Bingham numbe. Until now, using othe methods to solve the asymmetical olling poblem, this influence cannot be descibed. eceived on Mach 3, 5.

24 34 odica Ioan 4 EFEENCES. E. OOWAN, The Calculation of oll Pessue in Hot and Cold Flat olling, Poc. Inst. Mech. Eng., 5, N. ŞANDU, G. CAMENSCHI, Contibution to the Mathematical Appoach of the High-Speed Stip olling, Lucăile celei de-a XXV-a CNMS Supl. An. Şt. Univ. Ovidius Constanţa,. 3. N. ŞANDU, G. CAMENSCHI, Asymmetical Binay Stip Dawing, ev. oum. Math. Pues Appl., 46, -3,. 4. Y.M. HWANG, G.Z. TZOU, An Analytical Appoach to Asymmetical Cold Stip olling Using the Slab Method, JMEPEG,, 4, G. CAMENSCHI, N. ŞANDU, Viscoplastic Flow though Inclined Planes with Application to the Stip Dawing, Lett. Appl. Engng. Sci., 7, N. ŞANDU, G. CAMENSCHI, A Mathematical Model of the Stip Dawing Poblem, Bul. Ştiinţific al celei de a XXVI-a Confeinţe Naţionale de Mecanica Solidelo, Băila,. 7. G.Y. TZOU, M.N. HUANG, Study on Minimum Thickness of Asymmetical Cold PV olling of Sheet, Jounal of Mateials Pocessing Technology, 5, 3,. 8. G. CAMENSCHI, Intoducee în mecanica mediilo continue defomabile, Edit. Univesităţii Bucueşti,. 9.. IOAN, O soluţie a poblemei tageii benzilo folosind o teoemă de medie, Buletinul Ştiinţific al Univesităţii din Piteşti, Lucăile celei de-a XXVII-a Confeinţe de Mecanica Solidelo,, 7, 3... IOAN, Global and Local Methods fo Solving Stip Dawing Poblem, in: Topics in Applied Mechanics, vol. I, edited by Vetuia Chioiu and Tudo Sieteanu, Editua Academiei omâne, 3.