On the critical radius in sheet bending

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1 On the critical radius in sheet bending Kals, J.A.G.; Veenstra, P.C. Published: 1/1/1974 Dcument Versin Publisher s PDF, als knwn as Versin f Recrd (includes final page, issue and vlume numbers) Please check the dcument versin f this publicatin: A submitted manuscript is the authr's versin f the article upn submissin and befre peer-review. There can be imprtant differences between the submitted versin and the fficial published versin f recrd. Peple interested in the research are advised t cntact the authr fr the final versin f the publicatin, r visit the DOI t the publisher's website. The final authr versin and the galley prf are versins f the publicatin after peer review. The final published versin features the final layut f the paper including the vlume, issue and page numbers. Link t publicatin Citatin fr published versin (APA): Kals, J. A. G., & Veenstra, P. C. (1974). On the critical radius in sheet bending. (TH Eindhven. Afd. Werktuigbuwkunde, Labratrium vr mechanische technlgie en werkplaatstechniek : WT rapprten; Vl. WT33). Eindhven: Technische Hgeschl Eindhven. General rights Cpyright and mral rights fr the publicatins made accessible in the public prtal are retained by the authrs and/r ther cpyright wners and it is a cnditin f accessing publicatins that users recgnise and abide by the legal requirements assciated with these rights. Users may dwnlad and print ne cpy f any publicatin frm the public prtal fr the purpse f private study r research. Yu may nt further distribute the material r use it fr any prfit-making activity r cmmercial gain Yu may freely distribute the URL identifying the publicatin in the public prtal? Take dwn plicy If yu believe that this dcument breaches cpyright please cntact us prviding details, and we will remve access t the wrk immediately and investigate yur claim. Dwnlad date: 5. Dec. 218

2 ON THE CRITICAL RADIUS IN SHEET BENDING J.A.G. Kals P.C. Veenstra presented t XXIV General Assembly f CIRP Tky EINDHOVEN UNIVERSITY OF TECHNOLOGY DEPARNT OF MECHANICAL ENGINEERING DIVISION OF PRODUCTION ENGINEERING reprt nr. PT 33

3 ON THE CRITICAL RADIUS IN SHEET BENDING by J.A.G. Kals and P.C. Veenstra INTRODUCTION Actually it is very difficult t find a suitable experimental defrming limit fr sheet bending peratins. In many cases literature nly prvides data fr particular sheet materials and in general the criterin cnnected with the data is either nt very well defined r nt mentined at all. Different values might be due t different criteria n which the experiments are based. The defrming limit culd be defined t be the amunt f strain which starts a remarkable granulatin in the uter surface f the bending zne. In anther case a visible rientatin f this granulatin r the start f surface cracks might be mre relevant. In general, hwever, it is very difficult t carry ut experiments based n direct visual bservatin. In this paper a simple theretical mdel is prpsed, which prved t be rather useful in a number f practical applicatins. It is based n the cnsideratin that lcal strain effects like lcalized granulatin, necking, cracks etc. in a mre r less unifrmly stressed area arise frm strain peaks which must have been preceded by a lcal instability. Since plastic instability can be calculated frm tensile test data, it is pssible t derive a safe upper limit fr the bending radius which can serve as a guideline fr tl and prduct design. LOCAL SURFACE INSTABILITY Starting frm the cnsideratin that a peak strain will ccur at the uter surface f the bending zne fr hmgeneus materials, we have t deal with a plane stress situatin in a thin surface layer : (I)

4 - 2 - where 1 s the bending stress. Then the stress-strain relatins are de: J d). = 2"" ] (2-i) da (2) de: 2 = 2"" ] (2i-J) de: 3 d)' = - '2 1 (I +i) Nw, by subsfituting the strain increments given by Equatins (2) int the general expressin fr the effective strain increment it is eas ily shwn that (3) de = - 2 de: 3 I (de: 3 J+i ) where (4) I = i2 - i + 1 Cnsider a small (unifrmly strained) surface element (5) da r= = - de: 3 when cnstancy f vlume is assumed.

5 - 3 - With Equatin (5), Equatin (3) can be written as fllws (6) de = 21 da T+T A Fr a cnstant value f i we have a straight strain path and Equatin (6) can be integrated t (7) - 21 In A = l+i A wherea is the initial value f A. Using the strain hardening equatin accrding t Ludwik : _ )n (8) cr = c ( + where C dentes the characteristic flw stress and n is the strain hardening expnent, the wrk dne per unit vlume will be given by the equatin (9) dw = cr de = C(e+e:)n de s Cnsidering Equatins (6) and (7), this will give the result (1) If after a certain amunt f strain, the dissipatin f wrk per unit surface increaseshws,a maximum value the relevant surface element becmes a weakened spt, whilst the surrunding surface elements are strengthening further. Thus we see, that the stability limit is fund by (I 1) d da In ther wrds

6 - 4 - (12) 1 AI' c + = n n A + 2I E where c indicates the critical value f the surface element. Fr i = cnstant we find, cnsidering Equatins (2), (3) and (6), after integratin (13) Curves representing these equatins are shwn in Fig. O. Actually Equatins (13) are defining the frming limit diagram fr the sheet material n the base f lcal instability, which is the first cnditin. fr necking. In the case f simple sheet bending (i.e. bending in ne plane) the surrunding surface material is nt preventing the instable spt frm strngly increasing its strain. BENDABILITY CRITERION In rder t make a first step twards a slutin f the practical prblem the displacement f the neutral surface f bending is neglected. The difference in the result is nt very imprtant fr materials with a lw degree f bendability. Further it is assumed that the resulting tensin inthebending zne is zer and that bending takes place under plane-strain cnditins (the average transverse strain being zer during the bending). S Using the last f the Equatins (13) we find that i = i. Substitutin f this value int the first f Equatins (13) leads t ( 15) E = n - Ic Z As usual it is assumed that plane sectins remain plane during the

7 - 5 - bending. The distributin f elngatin (r engineering strain) f fibres acrss the sheet is therefre linear. The bending strain in any fibre LS In (1 + Z) p where P is the radius t the central surface and y is the distance f the fibre frm the central surface. S the bending strain in the uter surface f the bending zne is (16) E = In (1 +!-) L 2p as a first apprximatin. Thus, using Equatin (15) and Equatin (16) prvides the bending gemetry cnnected with an instable surface layer (17) Pc -- = s An easy practical apprximatin is achieved fr =, S s and ex I+x.. (18) Pc S The curves representing Equatins (17) and (18) are shwn in Fig. 1. The left part f the diagram is f practical imprtance and shws a neglectable difference between the curves. BENDING IN SUCCESSIVE STEPS Fr the sake f simplicity the fllwing calculatins are based n Equatin (18). Frm Fig. 1 it becmes clear thatsheet materials with a small n-value difficulties can arise when a relatively sharp bending edge is wanted. Apart frm chsing a mre ductile sheet material, a sharp edge may be achieved by bending in steps applying an amnealing peratin in between the steps.

8 - 6 - The develpment f a practical guideline fr the planning f the steps may start frm the assumptin, that the ductility f the material can be restred cmpletely by annealing. S the increase in bending strain in the sheet surface (19) I = In (1+ z!-) - In (1+ 2 s ) Pj Pj-l during any bending step j is restricted t the value given by Equatin (15). S fr =,s=s and ex IX l+x we btain the simple law (2) () c. 2n + (n+l) which enables a successive calculatin f the critical bending radius by each step using the value f the initial radius. Frm this frmula Equatin (21) can be derived, if pwers f n are neglected.! (21) (f) = J c. -1 2n (j + n 1: a) a= This apprximate frmula gives the critical values f the bending radius directly fr each step_ The curves representing Equatin (2) are shwn in Fig. 2. DISCUSSION Startingfrm an analysis f lcal plastic instability under biaxial stress Equatin (2) is btained. This is a very imprtant result, since it allws us t determine a safe upper limit fr the first bending peratin f sheet and useful indicatin fr planning the fllwing steps. Since instability cannt be bserved, a direct experimental verificatin was impssible. Hwever, frm a number f practical applicatins the criterin prved t be a very useful ne. In additin t this the criterin develped is in agreement with a number

9 - 7 - f recmmandatins in literature. Fr example: Oehler {I} mentins a value p 2.5 s fr steel. Since the average n-value fr steel is c.2 Equatin (18) prvides the same value. Hwever, the strain-hardening expnent is nt ften mentined in literature, therefre a cmparisn is difficult. Mrever a first investigatin has been carried ut in rder t check the theretical bending criterin in a mediate way. Fr that purpse three different steel sheets with a thickness f 12. mm and a plished surface have been subjected t bending peratins in a V-tl. By watching changes f the surface appearance by eye very carefully the values f the bending radius have been measured fr beginning mattness, fr the start f perceptible directinality in the surface appearance and. finally fr visible crack initatin. The results f these tests are represented in Fig. 3. The bservatins cme up t expectatins : mattness is nt cnnected with instability, but directinality f the surface structure and crack' frmatin are bviusly preceded by lcal instability. With this the practical significance f the theretically develped criterin is prvisinally established, althugh the relatin between this and different practical criteria has t be investigated in a mre systematical way. Particularly, the effect f sheet thickness is f practical imprtance. Actually, practical values {2} f the bending radius exceed the minimum values given by the criterin very ften. But the demands made n prducts differ cnsiderably and generally are nt defined very well. Especially in the case f dynamic lads r chemical treatment f the prduct after bending, the instability criterin seems t meet the needs in a prper way. Als fr thick sheet materials, where the strain gradient in nrmal directin has a smaller value in cnnectin with surface instability, it might be wise t restrict the bending radius accrding t the limit f surface instability. It is neither pssible nr desirable t discuss all the practical bending prcesses here. It is nly suggested that sme bending prblems can be analysed in a mre systematical way. Especially in the case f additinal drawing stresses in the plane f the sheet, it is necessary t take the shift f the neutral plane f bending int accunt. It can be calculated by equating the ttal tensin t the difference between the bending tensin and cmpressin.

10 OEHLER/KAISER Sahnitt- Stanz- und Ziehwerkzeuge. Springer Verlag, 1957, p. 189, W.P. ROMANOWSKI Handbek vr de mderne Stansteahniek. Kluwer, p. 77 (translated frm Handbuch der Stanzereitechnik, VEB Verlag Technik, Berlin).

11 r l» >... lis...,.... l» 1 lis... 9 l» ::s - lis > lis C w -C.- lis......, / / / / cri tical va I ue f smallest st rai n 1 (Etc/n) 2 fig. Lcal instability limit fr stretching f sheet materia I PT

12 ... 1 VI I-.- '" "'... ::J 8 ) t- c: "' c: Q).. 6 v ' \- V Q).4 >... Q) t... r- 2 1, 1 " \ \\ \ K,... ', Eq( I \.,,, "' fig.l r- _ IEq strain - hardening expnent n r (n - t ) Surface instability bundary in shee t bend i ng PT

13 ... 1r--r r r r-; r : Q..U m c -u c Q)...c r v Q) >... Q) 2 Hr r L-.f.2.3 strain hardening expnent n.4.5 fig. 2 Surface instability limit fr suctessive bending steps (EO:O) PT

14 p'!t.rial ) :12mm ind. n mattness pis direct cracks a.o S 7.25 b c r sheet thickness s:: 12 mm ther. surface instability --- visual surface rughness (mattness) visual di rectinal ity in rughness A crack frmatin 6 1#1... 1#1 :;I.. "Q 1'\1 r:a c.- "Q C I».Q I», II I I I I I, I I Q11b I I O..3 strain hardening expnent n fig.3 Experimental frmability criteria in sheet bending PT

15 On the Critical Radius in Sheet Bending I. A. G. KaJs (2), and P. C. Veenstra (1) PTMO Intrductin It is very difficult t find fr sheet bending peratins a suitable experimental defrming limit. In many cases literature prvides data nly fr particular sheet materials and, in general, the criterin cnnected with the data is either nt very well defined r nt mentined at all. Different values might be due t different criteria n which the experiments are based. The defrming limit culd be defined as the amunt f strain which starts a nticeable granulatin in the uter surfac f the bending zne. In anther case a visible rientatin f this granulatin r the start f surface cracks might be mre relevant. In general, hwever, it is very difficult t carry ut experiments based n direct visual bservatin. In this paper a simple theretical mdel is prpsed, which has prved rather useful in a number f practical applicatins. It is based n the cnsideratin that lcal strain effects such as lcalized granulatin, necking, cracks, etc. in a mre r less unifrmly stressed area arise frm strain peaks which must have been preceded by a lcal instability. Since plastic instability can be calculated frm tensile test data, it is pssible t derive a safe upper limit fr the bending radius which can serve as a guideline fr tl and prduct design. Leal Surface Instability Starting frm the cnsideratin that a peak strain will ccur at the uter surface f the bending zne fr hmgeneus materials, we have t deal with a plain stress situatin in a thin surface layer: where O! is the bending stress. Then the stress-strain relatins are del = A a 1 (2-1) da de2 = a 1 (21-1) da - a 1 (1+1) Nw, by substituting the strain increments given by Equatins (2) int the general expressin fr the effective strain increment (1) (2) With Equatin (5), Equatin (3) can be written as fllws: dt da. r Fr a cnstant value f i we have a straight strain path and Equatin (6) can be integrated t L In L 1+1 A where A is the initial value f A. Using the strain hardening equatin accrding t Ludwik: (6) (7) (j C Cf+E)n (8) where C dentes the characteristic flw stress and n is the strain hardening expnent, the wrk dne per unit vlume will be given by the equatin Cnsidering Equatins (6) and (7), this will give the result: (9) (1) If after a certain amunt f strain, the dissipatin f wrk per unit surface increase shws a maximum value the relevant surface element becmes a weakened spt, whilst the surrunding surface elements are strengthened further. Thus the stability limit is fund by (11) In ther wrds A E n n A 21 (12) where c indicates the critical value f the surface element. Fr i = cnstant we find, cnsidering Equatins (2), (3) and (6), after integratin 2-1 ( _ 1+1 E ) 1+1 n 21! (13) E = (n _ 1+1 E ) 2c 21 it is easily shwn that where 2 de di=-.::::...: (3) Actually Equatins (13) define the frming limit diagram fr the sheet material n the base f lcal instability, which is the first cnditin fr necking. In the case f simple sheet bending (i.e. bending in ne plane) the surrunding surface material des nt prevent the instable spt frm strngly increasing its strain. Cnsider a small (unifrmly strained) surface element A=X1X2, then (4) Beadability Criteria In rder t make a first step twards a slutin f the practical prblem the displacement f the neutral surface f bending is neglected. The difference in the result is nt very imprtant fr materials with a lw degree f bendability. Further it is assumed that the resulting tensin in the bending zne is zer and that bending takes place under plane-strain cnditins (the average transverse strain being zer during the bending). S (14) and. when cnstancy f vlume is assumed. Using the last f the Equatins (13) we find that i == l/z. Substitutin f this value int the first f Equatins (13) leads t (15) Annals f the CIRP VL

16 As usual it is assumed that plane sectins remain plane during the bending. The distributin f elngatin (r engineering strain) f fibres acrss the sheet is therefre linear. The bending strain in any fibre is 1n (1 + ) where Q is the radius t the central surface and y is the distance f the fibre frm the central surface. S the bending strain in the uter surface f the bending zne is E. = 1n (1 + ) (16) 1 2p as a first apprximatin. Thus, using Equatin (15) and Equatin (16) prvides the bending gemetry cnnected with an instable surface layer: 1 (17) 2! exp (n- EO )-1! An easy practical apprximatin is achieved fr E and (18) The curves representing Equatins (17) and (18) are shwn in Fig. 1. The left part f the diagram is f practical imprtance and shws a neglectable difference between the curves. Bending in Successive Steps Fr the sake f simplicity the fllwing calculatins are based n Equatin (18). Frm Fig. 1 it becmes clear that fr sheet materials with a small n-value difficulties can arise when a relatively sharp bending edge is wanted. Apart frm chsing a mre ductile sheet material, a sharp edge maybe achieved by bending in steps, applying an annealing peratin in between the steps. The develpment f a practical guideline fr the planning f, the steps may start frm the assumptin that the ductility f the material can be restred cmpletely by annealing. S the increase in bending strain in the sheet surface b.e = 1n (1+ _8_) - 1n (1+ _8_) (19) 1 2pj 2pj_1 during any bending step j is restricted t the value given by Equatin (15). S frs =, s = S and e x 1 + x we btain the simple law ( :) '" 2n + (n+1) 8 J c Pj - 1 (2) which enables a successive calculatin f the critical bending radius by each step using the value f the initial radius. Frm this frmula Equatin (21) can be derived, if pwers f n are neglected. 8 j-1 ( p) =2n(j+n 1: a) (21) J c a= This apprximate frmula gives the critical values f the bending radius directly fr each step. The curves representing Equatin (2) are shwn in Fig. 2. Discussin Starting frm an analysis f lcal plastic instability under biaxial stress Equatin (2) is btained. This is a very imprtant result, since it allws us t determine a safe upper limit fr the first bending peratin f sheet and useful indicatin fr planning the fllwing steps. Since instability cannt be bserved, a direct experimental verificatin is impssible. Hwever, frm a number f practical applicatins the criterin prves t be a very useful ne. In additin t this the criterin develped is in agreement with a number f recmmendatins in literature. Fr example: Oehler [1] mentins a value Qe 2.5 s fr steel. Since the average n-value fr steel is.2, Equatin (18) prvides the same value. Hwever, the strain-hardening expnent is nt ften mentined in literature, therefre a cmparisn is difficult. Actually, practical values [2] f the bendingradius very ften exceed the minimum values given by the criterin. But the demands made n prducts differ cnsiderably and, generally, are nt defined very well. Especially in the case f dynamic lads r chemical treatment f the prduct after bending, the instability criterin seems t meet the needs in a prper way. Als fr thick sheet materials, where the strain gradient in nrmal directin has a smaller value in cnnectin with surface instability, it might be wise t restrict the bending radius accrding t the limit f surface instability. It is neither pssible nr desirable t discuss all the practical bending prcesses here. It is nly suggested that sme bending prblems can be analysed in a mre systematic way. Especially in the case f additinal drawing stresses in the plane f the sheet, it is necessary t take the shift f the neutral plane f bending int accunt. It can be calculated by equating the ttal tensin t the difference between the bending tensin and cmpressin. 1. Oehler/Kaiser, Schnitt-, Stanz- und Ziehwerkzeuge. Springer Verlag, 1957, p. 189, W. P. Rmanwski, Handbek vr de mderne Stanstechniek. Kluwer, p. 77 (translated frm Handbuch der Stanzereitechnik, VEB Verlag Technik, Berlin). '" c: "U c: c 6..;: v.. 4 > r- t " \ \, 1\ J r-z =:.:..::..: , , ' r ' t r Q.U t r 'tj c:.....c \ t t_ j U " > Qj \l--jY ::>..;;;;::---t t_ j strain - hardening expnent n Fig. 1. Surface instability bundary in sheet bending..4.5 r l n - t Y3) I L L strain hardening expnent n Fig. 2. Surface instability limit fr successive bending steps (e= ). 56

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