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On springback of double-curved autobody panels
International Journal of Mechanical Sciences 43 (2001) 5}37
On springback of double-curved autobody panels Nader Asna"* Gra( nges Technology, SE-612 81 Finspa ng, Sweden Received 14 September 1998; received in revised form 19 October 1999; accepted 23 November 1999
Abstract The springback of double curved autobody panels is studied theoretically and experimentally. Both steel and aluminium sheets are included in this investigation. The obtained results show that the springback is decreased with increasing binder force, increasing curvature, increasing sheet thickness and decreasing yield strength. This paper comprises also a discussion on the plastic strains and their in#uence on the springback. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Springback; Double curved; Panel; Autobody; Sheet metal forming; Aluminium; Steel
1. Introduction Better shape accuracy on automobile outer panels has been required and discussed in many years. It is, therefore, essential to clarify the factors governing geometrical surface defects on pressed panels and to "nd methods for improvement. These geometrical surface defects are initiated by buckling or springback due to elastic recovery [1]. In this study, only the springback of double-curved panels is at the focus. There are chie#y two factors which initiate springback on curved panels [1]: (I) The non-uniform distribution of the stress during forming. (II) The plastic bending moment released upon unloading. This paper deals only with factor II. For a discussion on factor I, the reader is referred to Refs. [1,2]. The concept used in this study is, for simplicity, described "rst for single-curved panels. In the next sections, this concept is applied to double-curved panels.
* Tel.: #46-122-830-39; fax: #46-122-124-87. E-mail address: nader.asna"@techno.graenges.se (N. Asna"). 0020-7403/01/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 0 3 ( 9 9 ) 0 0 1 0 1 - 0
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Nomenclature Subscripts 1 and 2 refer to paths 1 and 2, respectively. a &a b &b C E K &K K C l &l M &M M & M M & M n q R &R N N RH & RH N N t ¹ &¹ ¹ & ¹ ¹ & ¹
shift in the position of the neutral axes extent of the elastic region a constant which varies with the strain ratio Young's modulus strength coe$cient strength coe$cient half-width of the punch moment per unit width elastic portion of M & M , respectively plastic portion of M & M , respectively strain-hardening exponent punch pressure minor & major punch radius, respectively radii of the centre "bre sheet thickness tension elastic portion of ¹ & ¹ , respectively plastic portion of ¹ & ¹ , respectively
Greek and mixed letters a b c &c *h & *h e &e e & e e & e e & e e C k l p & p p & p p W p C ¹ &¹ B B ¹ &¹ U U
stress ratio strain ratio half of the sectorial angle springback total strains elastic portion of e & e , respectively strains at yielding plastic portion of e & e , respectively e!ective plastic strain coe$cient of friction Poisson's ratio elastic stresses stresses in the plastic region yield strength ("R ) N e!ective stress tension at the inlet of the die pro"le radius tension at the outlet of the die pro"le radius
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Fig. 1. The in#uence of tension on the stress and strain distribution and on the position of the neutral axis in pressing with a single-curved punch [3,4].
Consider a blank formed by a single-curved punch in two-steps, Fig. 1. In the "rst step, the blank is made to adopt the punch shape by application of a moment, M. In the second step, a tension, ¹, is applied. Assumed that M and ¹ act at the mid-thickness. The strain and stress distributions in the encircled zone in Fig. 1 vary with the magnitude of ¹.
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When the moment M'0 and ¹"0, the neutral axis is assumed to coincide with the center line, Fig. 1(a). In this case, the strains and the stresses above the centerline are tensile and equal to the strains and the stresses below the centerline which are compressive, Fig. 1(a). Applying a tension results in a shift, a, in the position of the neutral axis and enlarges the zone which is subject to tensile strains and stresses, Fig. 1(b). As the tension is increased, a grows, M decreases, and the portion of the cross section deformed elastically shifts towards the inner surface, Figs. 1(b)}(d). When the tension is su$ciently high, the neutral axis disappears from the cross section and both the inner and outer surfaces are deformed plastically in tension, Fig. 1(e). At this stage the plastic bending moment M"M P0. Fig. 1(e) exhibits an important case * bending under tension with entirely plastic cross section [3,4]. Unloading the tension, ¹, in Fig. 1 results in the sheet sliding around the punch without any change of shape, whilst unloading the moment, M, will change the radius of curvature from R to N R , Fig. 2 [5]. It can be shown that for forming with a single-curved punch [5]: Q 1 1 1 12M 12M (1!l) C" C " ! " , (1) * R R R Et E t N Q in which E is the Young's modulus, t the sheet thickness, l the Poisson's ratio, and M "M C the plastic bending moment applied before unloading. It can, furthermore, be shown that, Fig. 2 and [5]
*(1/R) 2*h/l,
(2)
in which *h is the springback in terms of height di!erence at the punch edge and l"half-width of the curved panel (Fig. 2). Eqs. (1) and (2) mean that the springback, measured as *(1/R) or *h, varies with the applied bending moment before unloading M"M . C The applied tension, ¹, and the bending moment, M, are, as shown in Fig. 1, uniquely related. If the sheet material is elastic perfectly plastic, one can plot the non-dimensional diagram of M/M C versus ¹/>t shown in Fig. 3. In Fig. 3, > is the yield strength, t the sheet thickness, and >t the yield tension. Fig. 3 shows that once the sheet starts to become plastic, the bending moment decreases very rapidly towards zero at which the sheet is fully plastic. Eqs. (1) and (2) show how this fall in moment a!ects the springback, Fig. 3. Thus, if the tension is su$ciently high to deform the entire sheet cross section plastically, then on unloading there is theoretically no springback and the sheet conforms exactly to the punch curvature.
Fig. 2. Unloading the moment, M, results in a curvature change.
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Fig. 3. The relationship between the bending moment M and tension ¹ [6].
Fig. 4. The moments and tensions acting along one of the two symmetry planes.
In this study, the concept described above is used to study the springback of double-curved panels.
2. Theoretical treatment The in#uence of the applied tension on the bending moment and, thereby, the springback was shown in Figs. 1 and 3. The magnitude of this tension is determined by the restraining force acting in the #ange area, Fig. 4. This restraining force is in turn dependent upon the applied blank holder force (binder force), the shape of the drawbeads (if used), friction (both binder-sheet friction and punch-sheet friction), and pro"le radii (both punch and binder). The larger the restraining force, the
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Fig. 5. The strain and stress distribution in a double-curved sheet element.
larger are the tensions acting on the double-curved surface, Fig. 4. The larger these tensions are, the smaller will be the springback. The theoretical analysis presented in this paper only concerns the tensions and moments acting on the double-curved panel surface. In other words, this paper does not comprise any mathematical expressions for the relationship between ¹ (¹ ), ¹ (¹ ), ¹ (¹ ) and ¹ (¹ ). U U U U B B See Fig. 4! Those interested in such expressions are referred to Ref. [2]. Note that Fig. 4 exhibits solely the cross section along one of the two symmetry planes on a double-curved panel. The analytical model constructed below comprises both of these planes. Compare Fig. 4 with Fig. 5. Fig. 5 exhibits the strain and stress distribution in a small element of the double-curved panel surface. Note in this "gure that neutral axis lies at z"!a along the 1-direction (path 1) and at z"!a along the 2-direction (path 2). The strain, e, consists of a portion, which is associated with extension of the middle "bre, and a portion, which has to do with the bending. One can therefore write [6] e "(z#a )/RH N
(3)
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Fig. 6. The assumed material behaviour.
and e "(z#a )/RH , N where (Fig. 5) RH "R #t/2 N N
(4)
(5)
and RH "R #t/2. N N Utilizing Eqs. (3) and (4), the strain ratio on the double-curved panel surface is found
(6)
e (7) b " "[(z#a )/RH ]/[(z#a )/RH ]. N N e In any sheet element considered on the panel surface (along the 1- or 2-direction), continuity demands that (see also Fig. 5) a "a , where Eq. (7) becomes
(8)
b "e /e "RH /RH . N N Assume that
(9)
E plane stress is prevailing (p O0, p O0 and p "p "0), Fig. 5, X E the stress}strain relationship in Fig. 6 describes the material behaviour. That is p"Ee,
(10)
where p is the stress in the elastic region, e the strain in the elastic region, and E the Young's modulus, and p"K ' (e)L, C C C
(11)
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in the plastic region, where p is the e!ective stress, e the e!ective plastic strain, K the strength C C C coe$cient, and n the strain-hardening exponent. The point at which Eq. (10) intersects Eq. (11) * the yield point * is characterized by (Fig. 6) p "K (ln 1.002)L"Ee. (12) C Note in Fig. 6 that the de"nition of the e!ective plastic strain, e , does not comply with the C scienti"cally correct terminology (according to which the elastic portion of the total strain is actually larger than that indicated in Fig. 6). The de"nition used in Fig. 6 leads to an overestimation of the magnitude of e . Since the magnitude of e obtained experimentally along C C paths 1 and 2, Fig. 5, is not expected to be greater than +0.05 (+5%), and since the de"nition above facilitates the analytical treatment of the springback, the terminology shown in Fig. 6 is used. E the strain ratio b , given by Eq. (9), is unchanged in both the elastic and the plastic regimes during the deformation. This means that the loading is proportional, i.e., the sheet metal is proportionally stretched and bent without any reversed loading caused by drawing and bending followed by stretching. Based on the assumptions above, an analytical model is constructed for the tensions and moments acting on, the strain gradients on, and the springback of the double-curved panel surface. This analytical model is accounted for in the appendix. Here, the key equations are used to describe how the component and tool design can bene"t from such an analytical model. 2.1. Tensions, moments, strain gradients and springback Applied at the middle surface, the total tension in the 1-direction is (Fig. 5) (13) ¹ "¹ #¹ , where ¹ is the tension caused by elastic stresses and ¹ the contribution of the plastic stresses to the total tension. These tension are given by (1#lb ) E ¹ " (1!l) 2RH N
Cp RH t W N ! a ! 2 E
(14)
and
K RH t/2#a Cp L> ! W . ¹ " N (n#1) RH E N In the 2-direction (Fig. 5)
(15)
¹ "¹ #¹ , where (1#l/b ) E ¹ " (1!l) 2RH N
(16)
b Cp RH t W N ! a ! 2 E
(17)
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and
K RH t/2#a b Cp L> ! W . (18) ¹ " N (n#1) RH E N Acting about the mid-thickness, the total moment/unit width in the 1-direction is, Fig. 5, M "M #M , (19) where M and M are the elastic and plastic portions respectively of the total moment. These moments are given by (1#lb ) E M " (1!l) RH N
Cp RH Cp RH a a t a t W N W N ! # # ! E 3E 6 2 24 8
(20)
and
K RH t/2#a Cp L> K RH Cp RH ! W M " N ! N a ! W N (n#2) RH (n#1) E E N In the 2-direction (Fig. 5)
t/2#a Cp L> ! W RH E N
M "M #M , where
(21)
(22)
(1#l/b ) E M " (1!l) RH N
a t a t b Cp RH b Cp RH a W N W N ! # # ! 2 24 8 E 3E 6
(23)
and
K (RH ) t/2#a b Cp L> K RH b Cp RH ! W M " N ! N a ! W N (n#2) E E RH (n#1) N t/2#a b Cp L> ! W . (24) RH E N Using Fig. 7 and the equilibrium equations in the 1- and 2-directions, the strain gradients over the panel surface are obtained
and
de /d "k (1#jb )/[n/e !(1#b )]
(25)
de 1 1 n "k 1# ! 1# . (26) d
jb b e The distributions of the tensions acting on the panel surface are derived with the aid of Figs. 7 and 8 and
¹ "¹ e\I >H@ A \( U
(27)
¹ "¹ e\I > H@ A \( . U
(28)
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Fig. 7. Forces acting on an element of the double-curved sheet (bending moments not considered).
Fig. 8. Distribution of the tensions is derived with the aid of symbols given in this "gure.
Eqs. (1) and (2) and Fig. 9 are used to derive expressions for the springback along paths 1 and 2 (the 1- and 2-direction Fig. 5) 6l M (1!l) *h " (1#lb )Et
(29)
6l M (1!l) . *h " (1#(l/b )Et
(30)
and
2.2. Discussion Eqs. (13) and (16) show the expressions derived for the tensions acting along paths 1 and 2 on a double-curved panel (Fig. 5). The second term in these expressions indicates the contribution of the plastic stresses to the total tension, Eqs. (15) and (18). Therefore, (t/2#a )/RH !Cp /E"0 N W
(31)
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Fig. 9. Eqs. (29) and (30) calculate *h and *h .
and (t/2#a )/RH !b Cp /E"0, (32) N W should give the magnitude of a and a at the yield point. Utilizing Eq. (9) and rewriting Eqs. (31) and (32), one obtains a "a "Cp RH /E!t/2. W N Normalizing Eq. (33) yields 2a 2a 2Cp RH " " W N !1, t t tE
(33)
(34)
in which RH and C are de"ned by Eqs. (5) and (A.9) in the appendix, respectively. N In this investigation, two di!erent punches will be used. One of these punches * punch No. 1 * has the radii R "665 mm and R "2000 mm. Both punches are 230 mm wide and N N 400 mm long, Fig. 5. Four di!erent sheet materials will, furthermore, be used * steel (FeP04), and the aluminium qualities 5182, 6016 and 6111.
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N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37 Table 1 The position of the neutral axis, a /(t/2), at the yield point for the materials in this investigation, when pressed with punch No. 1. See also Fig. 5 and Eqs. (33) and (34) Material
t (mm)
E (MPa)
l
C
b
p RH W N (MPa) (mm)
a "a (mm)
a /(t/2)" a /(t/2)
Steel (FeP04) 5182-O
0.8 1.2 2.0 1.2 2.0 1.2 2.0
205 000 71 000 71 000 70 000 70 000 70 000 70 000
0.30 0.33 0.33 0.33 0.33 0.33 0.33
0.952 0.921 0.921 0.921 0.921 0.921 0.921
0.3326 0.3327 0.3328 0.3327 0.3328 0.3327 0.3328
172 149 144 95 113 167 175
0.1314 0.6870 0.2445 0.2323 !0.0094 0.8631 0.5340
0.3285 1.1450 0.2445 0.3872 !0.0094 1.4385 0.5340
6016-T4 6111-T4
665.4 665.6 666 665.6 666 665.6 666
Table 1 shows the position of the neutral surface at the yield point for these sheet materials, when pressed with punch No. 1. The values of a ("a ) and a /(t/2) ("a /(t/2)) in Table 1 are calculated by Eqs. (33) and (34), respectively. Note in Table 1 that a ("a ) and a /(t/2) ("a /(t/2)) are negative for the 2-mm thick sheet of material 6016. Utilizing Eqs. (31) and (32), one can "nd that if t/(2RH )'Cp /E, (35) N W then a ("a ) will be negative at the &elastic limit'. This implies that yielding may occur during the "rst loading step * application of the bending moments * if the yield strength is small, E is large, minor punch radius is small and/or the sheet thickness is large. Fig. 10 displays the moments and tensions acting on an arbitrary element (along path 1 or 2) of the double-curved surface versus the shift in the position of the neutral surface. The "gure is based on Eqs. (13), (16), (19) and (22) and concern 2-mm thick 6016 and 0.8-mm thick FeP04 (steel). For 2-mm thick 6016 steel, plastic deformation is, as shown in Fig. 10, initiated (actually proceeds; see Table 1) and the moments fall in magnitude as soon as any tensions are applied. To initiate plastic deformation (and a reduction in moments) as 0.8-mm thick FeP04 is considered, ¹ and ¹ must, however, be increased to approximately 40 and 22 N/mm, respectively and 2a /t ("2a /t) must be moved to 0.33. See Fig. 10. Fig. 11 displays the springback versus the shift in the position of the neutral surface at the panel walls. The "gure is based on Eqs. (29) and (30). Fig. 11 shows, in other words, how much the neutral surface must be moved to initiate plastic deformation and how fast the springback is reduced after yielding. When pressed with punch No. 1, 1.2-mm thick 6111 should display a severe springback, Fig. 11. The larger the restraining force in the #ange area (for instance, the larger the binder force) the larger are the tensions acting on the double-curved surface (Fig. 4). The larger these tensions, the larger is the shift in the position of the neutral surface, and the larger this shift, the smaller is the springback. Assume now that a 2-mm thick sheet of material 6016 is pressed with punch No. 1 and that the obtained shift in the position of the neutral surface is a "a "t/2"1 at the centre of this panel. Figs. 12 and 13, which are based on Eqs. (13), (16), (19), (22), (27)}(30), show the variation of the shift
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Fig. 10. Moments and tensions acting on an arbitrary element (along path 1 or 2, Fig. 5) of the double-curved panel surface versus the shift in the position of the neutral surface. The "gure is based on Eqs. (13), (16), (19) and (22). See also Table 1.
in the position of the neutral axes, the tensions, the moments and the springback over the face of this panel (along paths 1 and 2). a , a , ¹ , and ¹ are smallest at the panel centre and increase towards the panel walls, whilst M and M are largest at the panel centre and decrease towards the panel walls. See Figs. 12 and 13. The important case of bending under tension with entirely plastic cross section was discussed in chapter 1. It was shown in Fig. 1(e) that the bending moment and, thereby, the springback are very
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Fig. 11. Springback versus the shift in the position of the neutral surface at the panel wall: (a) path 1 (the "gure is based on Eq. (29)) and (b) path 2 (the "gure is based on Eq. (30)).
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Fig. 12. The variation of the shift in the position of the neutral axes and the tensions over the panel face (paths 1 and 2). The "gure is valid for 2-mm thick 6016-T4 pressed with punch No. 1. It is assumed that a "a "t/2"1 at the panel centre. The "gure is based on Eqs. (13), (16), (27) and (28). The coe$cient of friction k "0.1. Concerning the other parameter values, see Table 1.
small, if the tension is so high that the entire cross section becomes plastic. It is of great interest to the upcoming experimental analysis to calculate the emerging plastic strains, when the entire cross section becomes plastic. The "rst terms in Eqs. (13) and (16) indicate the elastic portion of the tensions acting on the double-curved surface. Therefore, (Cp RH /E)!(a !t/2)"0 W N
(36)
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Fig. 13. The variation of the moments and the springback over the panel face (paths 1 and 2). The "gure is valid for 2-mm thick 6016-T4 pressed with punch No. 1. It is assumed that a "a "t/2"1 at the panel centre. The "gure is based on Eqs. (13), (16), (19), (22), (27)}(30). The coe$cient of friction k "0.1. For the other parameter values, see Table 1. See also Fig. 12.
and (b Cp RH /E)!(a !t/2)"0, (37) W N should give the position of the neutral axis, when the tensions just have become entirely plastic. Rewriting Eqs. (36) and (37), one obtains a "a "Cp RH /E#t/2. (38) W N At z"t/2, Eqs. (A.20) and (A.24) in the appendix give the plastic strains on the panel surface e "e !eW "(t/2#a )/RH !Cp /E, N W
(39)
t/2#a b Cp ! W. e "e !e " RH E N
(40)
and
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Fig. 14. Plastic strains, which emerge on the panel surface, if the sheet materials in this study are pressed with punch No. 1, and if the entire cross section just have become plastic. The "gure is based on Eqs. (41)}(42).
Combining Eqs. (9) and (38)}(40) gives e "t/RH N
(41)
and e "t/RH . N Combining Eqs. (41) and (A.15) in the appendix yields the e!ective plastic strain
(42)
e"(4(1#b #b )/3t/RH . (43) C N Eqs. (41)}(43) yield the plastic strain along the 1-direction, the plastic strain along the 2-direction and the e!ective plastic strain in any sheet element along path 1 or 2 (Fig. 5), if the entire cross section of this element just have become plastic. Using these equations, the strains displayed in Fig. 14 are obtained for the sheet materials in this study, Table 1. Figs. 14 shows that the plastic strains, which emerge theoretically when the entire cross section just have become plastic, are very small * (1%. The word theoretically is italicized, since it is assumed that the loading occurs in two steps * the bending moments are applied "rst, after which the tensions are exerted. It is, furthermore, assumed that plastic deformation is caused by the tensions. In practice, the tensions and the bending moments are exerted at the same time and interact during the forming. The strain gradients along paths 1 and 2, the panel centre being the origin, should also be discussed. Substituting e and e plotted in Fig. 14 into Eqs. (25) and (26), respectively, one can see that the strain gradients along paths 1 and 2 become small and positive. These gradients become small, particularly since the ratios n/e and n/e in Eqs. (25) and (26) are large (for the n-values, see Table 2). Note that these gradients are small, even if e+5%, which is a practically C more reasonable value.
3. Materials The mechanical properties of the materials used in this investigation are displayed in Table 2, where t is the sheet thickness, R the yield strength, R the ultimate tensile strength, A the N K
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Table 2 Mechanical properties of the materials used in this investigation Material
t (mm)
R N (MPa)
R K (MPa)
A (%)
r
n
K C (MPa)
E (MPa)
l
Steel (FeP04) 5182-O-EDT
0.8 1.2 2.0 1.2 2.0 1.2 2.0
172 149 144 95 113 167 175
306 284 285 190 219 301 298
44 27 30 29 31 28 27
1.69 0.72 0.83 0.67 0.65 0.60 0.56
0.23 0.30 0.27 0.24 0.24 0.26 0.24
527 498 498 344 344 520 520
205000 71000 71000 70000 70000 70000 70000
0.30 0.33 0.33 0.33 0.33 0.33 0.33
6016-T4-EDT 6111-T4-V65S-H90
R , R , A , r and n for these materials are extracted from Ref. [7]. N K The gauge length was 80 mm. See Ref. [8]. See Ref. [9].
Fig. 15. True stress versus true strain for the studied sheet materials. The data concerning 5182-O, 6016-T4 and 6111-T4 are extracted from the tensile data provided by Ref. [7] and concern the 1.2-mm thick sheets of these materials.
total elongation (gauge length"50 mm), r the normal anisotropy, n the strain-hardening exponent, K the strength coe$cient, and E"modulus of elasticity. C Fig. 15 displays true stress versus true strain obtained by tensile testing of the studied sheet materials along the rolling direction. Note that Fig. 15 concerns the 1.2 mm-thick sheets of the aluminium grades in Table 2.
4. Experimental procedure Two di!erent punches were used, the dimensions of them being shown in Fig. 16. The drawbeads used were semi-circular in cross section, the radius being 8 mm. Sheets with three di!erent thicknesses were pressed in this study (Table 2). Prior to pressing the 2-mm thick sheets (and after pressing the 0.8 and 1.2-mm thick sheets), the lower binder was
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23
Fig. 16. The punches used in this investigation.
Fig. 17. The panel shape after the trimming and the sites at which the panel shape were measured.
machined in order to keep the ratios clearance/thickness and lower binder proxle radius/thickness unchanged. Rectangular blanks (400 mm;570 mm) were pressed with both punches. For strain measurement, a quadrilateral grid was etched on each blank prior to pressing. Varying the binder force varied the restraining force in the #ange area. In each case, the maximum utilized binder force was slightly lower than that resulting in fracture. The shapes of the punches used (Fig. 16), were "rst determined in a coordinate-measuring machine. Each panel was then trimmed by laser cutting so that only 5 mm of the panel wall was left at its corners (Fig. 17). After this trimming, the panel shape was measured in the same coordinatemeasuring machine. The panel shape was then compared to the shape of the punch, with which the panel was pressed, and z-deviation from the punch shape at each measuring site was determined.
5. Results Fig. 18 displays the z-deviation from the punch shape (springback) versus the distance from the panel centre. This "gure concerns 2 mm thick AA6111 pressed with punch No. 1. All three panels in Fig. 18 were pressed with drawbeads * db in the legends.
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Fig. 18. z-deviation from the punch shape along path 1 (top "gure), *h , and along path *h , versus the distance from the panel centre. Material"6111. Sheet thickness"2.0 mm. Punch No. 1 was used. BHF in the legend"applied blank holding force in metric tonnes (;10 kN). db in the legend"drawbeads were used.
As exhibited in Fig. 18, the springback is reduced with increasing blank holding force * BHF in the legends. In this case, the maximum applied BHF was 55 metric tonnes (;10 kN), Fig. 18. Using a slightly higher BHF would lead to fracture at the panel corners. 55 t lies, in other words, just beneath the fracture limit. Fig. 19 shows the e!ective plastic strain measured along paths 1 and 2 versus the distance from the panel centre. Since the strains are very small, the measurement errors may be relatively large. It was predicted in the previous sections that the strain gradients along paths 1 and 2 would be positive and small. This prediction can be considered as veri"ed, even taking the possible measurement errors mentioned above into account (Fig. 19). Similar results (as those shown in Figs. 18 and 19) were obtained for the panels pressed in the other sheet materials in this study. Concerning these results, the reader is referred to Ref. [2]. It was predicted in the previous sections that the springback is reduced with increasing shift in the position of the neutral surface. The larger the restraining force (BHF), the larger is this shift. The larger the BHF, the smaller is, in other words, the springback. This prediction is veri"ed by Fig. 18.
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Fig. 19. E!ective plastic strain along paths 1 and 2 versus the distance from the panel centre. Material"6111. Sheet thickness"2.0 mm. Punch No. 1 was used. BHF in the legend"applied blank holding force in metric tonnes (;10 kN). db in the legend"drawbeads were used.
If the mean value of the z-deviations at the panel walls (*h and *h ), Fig. 18, are normalized by the crown height (h and h ) and plotted versus the applied BHF, Figs. 20 and 21 are obtained. Fig. 20 displays the variation of *h /h and *h /h with BHF obtained on the panels pressed with punch No. 1. As shown in Fig. 20, *h /h and *h /h decrease with increasing BHF. Note that the maximum BHF plotted for each sheet material is the highest attainable. For instance, for 1.2-mm thick AA6111, 30 t is the highest attainable BHF. Applying a slightly higher BHF leads to fracture at the panel corners. The larger the restraining force (the higher the BHF), the larger is the shift in the position of the neutral surface. Therefore, Fig. 20 can be compared with Fig. 11. Since the same BHF leads to di!erent 2a /t and 2a /t, when applied to the di!erent sheets, the x-axis of Figs. 11 and 20 are not completely compatible. Fig. 20 shows, in fact, di!erent segments of Fig. 11. Note too that the sheets materials, for which 2a /t and 2a t are small at the yield point, exhibit a small springback at relatively moderate BHF levels. Compare Fig. 20 with Table 1 or Fig. 11.
26
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
Fig. 20. Normalized mean value of the springback at the panel walls, *h /h (path 1; top "gure) and *h /h (path 2), versus the applied blank holding force. The "gures concern the panels pressed with punch No. 1. Drawbeads were used.
Fig. 21 shows the variation of *h /h and *h /h with BHF obtained on the panels pressed with punch No. 2. The springback is reduced signi"cantly when punch No. 2 is used. Compare Figs. 20 and 21. Punch No. 2 has smaller radii than punch No. 1, Fig. 16. This means, in turn, that the shift in the position of the neutral surface at the yield point is smaller, when punch No. 2 is used, Eq. (34).
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
27
Fig. 21. Normalized mean value of the springback at the panel walls *h /h (path 1; top "gure) and *h /h (path 2), versus the applied blank holding force. The "gures concern the panels pressed with punch No. 2. Drawbeads were used.
Theoretically, plastic deformation may, furthermore, be initiated during the "rst loading step * application of the bending moments, Eq. (35). All this indicates that plastic deformation is initiated at smaller BHF levels, when punch No. 2 is used. Initiation of plastic deformation plays a signi"cant role in springback reduction. *h /h and *h /h are therefore smaller, as punch No. 2 is utilized. See Figs. 20 and 21.
28
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
Fig. 22. Normalized mean value of the springback at the panel walls, *h /h , versus the e!ective plastic strain at the panel centre. The "gures concern the panels pressed with punches No. 1 and No. 2 and drawbeads.
It was shown in the previous sections that the emerging plastic strains on the double-curved surface may be very small, even if the entire cross section is plastic (a condition which results in a very small springback), Eqs. (41)}(42) and Fig. 14. The strain level on a panel that exhibits a small springback may, therefore, di!er only slightly from that attained on a panel (pressed in the same material and thickness and with the same punch) which shows a large springback. Recall too that the strain gradients along paths 1 and 2 are positive and small.
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
29
Fig. 23. The panel shape after the second trimming and the sites at which the panel shape were measured.
This prediction is veri"ed by the experimental data presented in Figs. 18 and 19. It is also veri"ed by Fig. 22, which displays *h /h at the panel walls versus the e!ective plastic strain at the panel centre. The di!erence in strain level can, as exhibited in Fig. 22, be very small, although the di!erence in springback is large or very large. See, for instance, the values obtained for 1.2-mm thick AA6111 (Fig. 22). The springback reduces with increasing binder force. Increasing the binder force leads to larger strains. However, as the springback is minimized, increasing the binder force leads only to larger strains. See, for instance, the values obtained for the steel sheet (Fig. 22). Therefore, it is improper (as commonly done) to evaluate the springback of di!erent sheet materials by comparing the deviations obtained on panels pressed (in di!erent sheet materials) to the same strain level at the centre.
6. Discussion It was shown that the tensions acting on the double-curved surface of the panel interact with the restraining forces developed in the #ange area. The higher the restraining forces, the higher the tensions acting on the double-curved surface of the panel. The tensions acting on the double-curved surface a!ect the shift in the position of the neutral axis, a and a (a "a in the same sheet element). The higher the tensions, the larger is a ("a ). The larger a ("a ) is, the smaller will be the moments acting on the double-curved surface and, "nally, the smaller the moments, the smaller will the springback be. It was, in other words, shown that the higher the restraining forces developed in the #ange area, the smaller the springback. The presented experimental data verify this prediction. The restraining forces can be increased by increasing the blank holding force and/or by using a more `severea drawbead shape, etc. In this study, the restraining forces were increased by increasing the binder force. It was also shown that the shift in the position of the neutral axis at the yield point (elastic limit), theoretically determined by a "a "Cp RH /E!t/2, W N
(33)
30
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
should play a signi"cant role in minimization (practical elimination) of the springback. The experimental results presented in Figs. 20 and 21 verify this prediction. In case a "a "0, also the tensions acting on the double curved surface ¹ "¹ "0 at the elastic limit. All this signi"es that if the sheet material and the punch shape are selected in such a fashion that R /E"p /E"t/(2CRH ), N W N
(44)
the springback can be minimized (practically eliminated) at moderate restraining force levels. This is an important conclusion, particularly for those practical cases, in which the restraining force is created solely by the aid of drawbeads. It is commonly held that sheet aluminium exhibits a larger springback than steel sheet in industrial applications. Eq. (44) shows that the springback can be minimized with an appropriate punch design and sheet material (particularly yield strength and sheet thickness) selection, regardless of the material category. Applying a higher binder force reduces the springback. However, the experimental results show that increasing the binder force is not a su$cient measure, as far as the minimization (practical elimination) of springback is concerned. A higher binder force may lead to fracture (at the panel corners), before the springback is minimized. The punch corner radii must, therefore, be increased (in order to avoid fracture), so that the springback can be minimized by applying higher binder forces (increasing the restraining forces). See for instance, 1.2-mm thick AA6111, which cannot withstand higher binder forces than 30 t, Fig. 20. The panels pressed in this study were actually trimmed once more so that solely the doublecurved surface was left, Fig. 23. The panel shapes after this second trimming were compared to the punch shapes. The results obtained after the second trimming comply with those attained after the "rst. Those interested in the results obtained after this second trimming are referred to Ref. [2]. Note, once again, that the theoretical analysis presented in this paper only concerns the tensions and moments acting on the double-curved panel surface. To prevent this presentation from taking unreasonable proportions, this paper does not comprise any mathematical expressions for the relationship between ¹ (¹ ), ¹ (¹ ) ¹ (¹ ) and ¹ (¹ ). See Fig. 4! Those inter U U U U B B ested in such expressions are referred to Ref. [2]. Therefore, a quantitative comparison between predicted and measured springback cannot be conducted here. Those interested in such a comparison are referred to Ref. [2].
7. Conclusions The following conclusions apply: E Increasing the binder force, increasing the curvature, increasing the sheet thickness and/or decreasing the yield strength reduces the springback. E The shift in the position of neutral surface at the yield point a "a "Cp RH /E!t/2, W N
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
31
plays a signi"cant role in minimization (practical elimination) of the springback. The smaller a and a at the yield point, the smaller the springback at moderate binder force (or restraining force) levels. If a and a are large at the yield point, more gentle corner and pro"le radii are necessary to minimize the springback. These radii must be gentle so that high binder (restraining) forces can be applied without risking fracture at the panel corners. E If the sheet material and the punch shape are selected in such a fashion that R /E"p /E"t/(2CRH ), N W N the springback can be minimized (practically eliminated) at moderate restraining force levels. This is an important conclusion, particularly for those practical cases, in which the restraining force is created solely by the aid of drawbeads. E Consider two panels pressed in the same sheet material and with the same punch but with di!erent binder forces. The di!erence in strain level between these two panels can be very small, although the di!erence in springback is large or very large (between the same two panels). Therefore, it is improper (as commonly done) to evaluate the springback of di!erent sheet materials by comparing the deviations obtained on panels pressed (in di!erent sheet materials) to the same strain level at the centre.
Acknowledgements The present investigation was carried out, whilst the author was with the Swedish Institute for Metals Research, and funded by Volvo Car Corporation/Go( teborg and the National Swedish Agency for Industrial and Technical Development (Nutek), which are gratefully acknowledged.
Appendix A In the elastic region (Fig. 6), Hooke's law (plane stress, p "0) can be applied [4], X p "E(e #le )/(1!l) and
(A.1)
p "E(e #le )/(1!l), (A.2) in which p "stress in the 1-direction (Fig. 5), p "stress in the 2-direction (Fig. 5), e "strain in the 1-direction, e "strain in the 2-direction and l"Poissons's ratio. Combining Eqs. (A.1) and (A.2) with Eqs. (3), (4) and (9) in the main text, we obtain p "[(1#lb )/(1!l)]E(z#a )/RH N
(A.3)
and
p "
l 1# b
(1!l) E(z#a )/RH . N
(A.4)
32
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Rewriting Hooke's law (plane stress, p "0) in terms of strains yields X e "(p !lp )/E and
(A.5)
e "(p !lp )/E. (A.6) Combining Eqs. (9) and (12) in the main text and (A.3), (A.5) and (A.6) with von Mises yield criterion (see Eq. (A.18)), we obtain eW "Cp /E W
(A.7)
and eW "b Cp /E, W in which
(A.8)
(1!l) b #l b #l \ C" (A.9) 1! # (1#lb ) 1#lb 1#lb and eW and eW are the yield strains in the 1- and the 2-directions, respectively (Figs. 5 and 6). At the yield point (Figs. 5 and 6), z"b !a
(A.10)
and z"b !a , which combined with Eqs. (3) and (4) give eW "b /RH N
(A.11)
(A.12)
and eW "b /RH . (A.13) N Combining Eqs. (A.7) and (A.8) with Eqs. (A.12) and (A.13), respectively and utilizing Eq. (9), we obtain b "b "CRH p /E. (A.14) N W In the plastic region in Fig. 6, von Mises yield criterion (plane stress, p "p "0, e O0) can be X utilized. The e!ective plastic strain, e, can then be denoted as [6] C 4 e"e (1#b #b ), (A.15) C 3 N
where e is the plastic strain in the 1-direction (Fig. 5) and b is the strain ratio, Eq. (9). Assume that p "a p .
(A.16)
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
The stress ratio, a , in Eq. (A.16) is related to the strain ratio, b , Eq. (9) by [6] a "(2b #1)/(2#b ). The e!ective stress, p, can therefore be denoted as [6] C p"p (1!a #a, C where p "the stress in the 1-direction (Fig. 5). Substituting Eqs. (A.15) and (A.18) in Eq. (11), one obtains p "K C
[(4/3(1#b #b )]L (e )L. (1!a #a
33
(A.17)
(A.18)
(A.19)
Note that z#a Cp ! W, e "e !eW " (A.20) RH E N in which e is the total strain in the 1-direction given by Eq. (3) and eW the yield strain in the 1-direction given by Eq. (A.7). Substituting Eq. (A.20) into Eq. (A.19) yields p "K
z#a Cp L ! W , RH E N
(A.21)
where [(4/3(1#b#b)]L . (1!a #a Combining Eqs. (9), (A.16) and (A.19) yields K "K C
p "[K a /(1!a #a] C
4 (1#b #b) 3
(A.22)
L b (e )L.
(A.23)
Note that (A.24) e "e !eW "(z#a )/RH !b Cp /E, N W in which e is the total strain in the 2-direction given by Eq. (4) and eW the yield strain in the 2-direction given by Eq. (A.8). Combining Eqs. (A.23) and (A.24), we obtain p "K [(z#a )/RH !b Cp /E]L, N W where K a [(4(1#b #b)/3]L K " C . bL (1!a #a
(A.25)
(A.26)
34
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
A.1. Tensions and moments Applied at the middle surface, the total tension in the 1-direction is ¹ "¹ #¹ , (A.27) where ¹ is the tension caused by elastic stresses and ¹ the contribution of the plastic stresses to the total tension. In the elastic region, Fig. 5(a), the tension will be
@ \?
R
p dz. (A.28) \R Combining Eqs. (A.3), (A.14) and (A.28) and integrating, we obtain Eq. (14) in the main text. The contribution of the plastic stresses to the total tension is ¹ "
¹ "
p dz.
(A.29)
@ \? Combining Eqs. (A.14) and (A.21) with Eq. (A.29) and integrating yields Eq. (15) in the main text. A derivation analogous to the above yields the tension in the 2-direction, Fig. 5(b). See Eqs. (16)}(18) in the main text. Acting about the mid-thickness, the total moment per unit width in the 1-direction is M "M #M , (A.30) where M and M are the elastic and plastic portions, respectively, of the total moment. In the elastic region (Fig. 5(a))
@ \?
R
p z dz. \R Combining Eqs. (A.3), (A.14) and (A.31) and integrating yield Eq. (20) in the main text. The contribution of the plastic stresses to the total moment is M "
(A.31)
p z dz. (A.32) @ \? Combining Eqs. (A.14) and (A.21) with Eq. (A.32) and integrating yield Eq. (21), which is shown in the main text. A derivation analogous to the above yields the moment in the 2-direction, Fig. 5(b). See Eqs. (22)}(24) in the main text. M "
A.2. Strain gradients and distribution of tensions over the panel surface Let us "rst denote the tension ratio, j, as j"¹ /¹ , in which ¹ and ¹ are the tensions given by Eqs. (13) and (16), respectively.
(A.33)
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
35
Consider an element of the double-curved sheet and the forces acting on it (bending moments not considered), Fig. 7. Equilibrium in the radial direction in this "gure requires ¹ /RH #¹ /RH "q. (A.34) N N Assuming Coulomb friction and that the friction coe$cient, k , is constant during the forming, the equilibrium of forces in the 1-direction in Fig. 7 yields d¹ "k qRH d . N Combining Eqs. (A.34) and (A.35), one obtains
(A.35)
d¹ "k (¹ /RH #¹ /RH )RH d . N N N Substituting Eq. (A.33) into Eq. (A.36) gives
(A.36)
d¹ /¹ "k (1/RH #j/RH )RH d . N N N It is known that
(A.37)
¹ "p t and, hence, one can write
(A.38)
d¹ /¹ "dp /p #dt/t. It is, furthermore, known that
(A.39)
p "K (e )L, why one can write dp /p "n de /e . The thickness component in Eq. (A.39) can be denoted as dt/t"de "!(de #de ). R Substituting Eq. (9) into Eq. (A.41) leads to dt/t"de "!(1#b ) de . R Substituting Eqs. (A.40) and (A.42) into Eq. (A.39) results in
(A.40)
(A.41)
(A.42)
(A.43) d¹ /¹ "[n/e !(1#b )] de . Combining Eqs. (9) and (A.37) with Eq. (A.43), we obtain Eq. (25) in the main text. A derivation analogous to the above yields Eq. (26). See the main text. Eqs. (25) and (26) give the strain gradients over the panel surface. Let us now look at the distribution of the tensions over the panel surface, starting with the tension along path 1. Combining Eqs. (9) and (A.37), we obtain d¹ /¹ "k (1#jb ) d . Integrating this equation with the symbols given in Fig. 8(a) yields
2U d¹ A " k (1#jb ) d , ¹ 2 (
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N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
which becomes ln(¹ /¹ )"k (1#jb )(c ! ). (A.44) U Eq. (A.44) can be rewritten as Eq. (27), which is shown in the main text. Note that c in Fig. 8(a) and Eq. (27) is known, since both l and R (RH ) are known. N N A derivation analogous to the above gives Eq. (28), which can be seen in the main text. Note that c in Fig. 8(b) and Eq. (28) is known, since both l and R (RH ) are known. N N Note, "nally, that it is possible to attain a better description of the relationship between ¹ and ¹ and between ¹ and ¹ than those given by Eqs. (27) and (28) by taking the in#uence of the U U punch pro"le radius into consideration. For simplicity, Eqs. (27) and (28) are preferred here. A.3. Springback Combining Eqs. (1) and (2) yields *h "6l M /(E t) C
(A.45)
and *h "6l M /(E t), (A.46) C in which M "M de"ned by Eq. (19) and M "M given by Eq. (22) in the main text. Utilizing C C Eqs. (9), (A.3)}(A.4), (A.7)}(A.8), (A.30) and (A.31), one can easily show that E "(1#lb )E/(1!l)
(A.47)
and
l E/(1!l). (A.48) E " 1# b Substituting Eq. (A.47) into Eq. (A.45) and Eq. (A.48) into Eq. (A.46) yield Eqs. (29) and (30). These latter equations are given in the main text. Note that (Fig. 9) sin "l /R N
(A.49)
sin "l /R . N
(A.50)
and
References [1] Yoshida K et al. Analysis and control of buckling behaviour due to elastic recovery in press forming. Proceedings of the 13th Biennial Congress of the International Deep Drawing Research Group, IDDRG, Melbourne, Australia, February 20}24, 1984. p. 73}84. [2] Asna" N. Mechanics of sheet metal forming, Ph.D. Thesis, LTU-DT-1997/6, Lulea University of Technology, Sweden, 1997. p. BI1}3.
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37
[3] Nagpal V et al. ICAM mathematical modelling of sheet metal formability indices and sheet metal forming processes. Technical Report AFML-TR-79-4168, AFML/LTC, WPAFB, OH 45433, USA, 1979. [4] Mielnik EM. Metalworking science and engineering. New York, U.S.A.: McGraw-Hill, 1991. p. 34, 743}51. [5] Duncan JL et al. Die forming approximations for aluminium sheet. Sheet Metal Industries 1978. p. 1015}25. [6] Marciniak Z et al. Mechanics of sheet metal forming. London: Edward Arnold, 1992. p. 17, 68}85, 143}5. [7] Draft "nal report, Centre de Recherches Metallurgiques (CRM) contribution to the low weight vehicle programme properties of aluminium alloys for body structure. Brite-EuRam project 5656, Contract No. BRE2-CT92-0266. [8] BjoK rnfot S. Springback in forming of high strength steel sheet. Lulea University of Technology, Technical Report 1983:60T p. 15 (in Swedish). [9] Adler T et al. MATEDS-the aluminium database from SkanAluminium. SkanAluminium and Royal Institute of Technology, Stockholm, 1993.
On springback of double-curved autobody panels Nader Asna"* Gra( nges Technology, SE-612 81 Finspa ng, Sweden Received 14 September 1998; received in revised form 19 October 1999; accepted 23 November 1999
Abstract The springback of double curved autobody panels is studied theoretically and experimentally. Both steel and aluminium sheets are included in this investigation. The obtained results show that the springback is decreased with increasing binder force, increasing curvature, increasing sheet thickness and decreasing yield strength. This paper comprises also a discussion on the plastic strains and their in#uence on the springback. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Springback; Double curved; Panel; Autobody; Sheet metal forming; Aluminium; Steel
1. Introduction Better shape accuracy on automobile outer panels has been required and discussed in many years. It is, therefore, essential to clarify the factors governing geometrical surface defects on pressed panels and to "nd methods for improvement. These geometrical surface defects are initiated by buckling or springback due to elastic recovery [1]. In this study, only the springback of double-curved panels is at the focus. There are chie#y two factors which initiate springback on curved panels [1]: (I) The non-uniform distribution of the stress during forming. (II) The plastic bending moment released upon unloading. This paper deals only with factor II. For a discussion on factor I, the reader is referred to Refs. [1,2]. The concept used in this study is, for simplicity, described "rst for single-curved panels. In the next sections, this concept is applied to double-curved panels.
* Tel.: #46-122-830-39; fax: #46-122-124-87. E-mail address: nader.asna"@techno.graenges.se (N. Asna"). 0020-7403/01/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 0 3 ( 9 9 ) 0 0 1 0 1 - 0
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Nomenclature Subscripts 1 and 2 refer to paths 1 and 2, respectively. a &a b &b C E K &K K C l &l M &M M & M M & M n q R &R N N RH & RH N N t ¹ &¹ ¹ & ¹ ¹ & ¹
shift in the position of the neutral axes extent of the elastic region a constant which varies with the strain ratio Young's modulus strength coe$cient strength coe$cient half-width of the punch moment per unit width elastic portion of M & M , respectively plastic portion of M & M , respectively strain-hardening exponent punch pressure minor & major punch radius, respectively radii of the centre "bre sheet thickness tension elastic portion of ¹ & ¹ , respectively plastic portion of ¹ & ¹ , respectively
Greek and mixed letters a b c &c *h & *h e &e e & e e & e e & e e C k l p & p p & p p W p C ¹ &¹ B B ¹ &¹ U U
stress ratio strain ratio half of the sectorial angle springback total strains elastic portion of e & e , respectively strains at yielding plastic portion of e & e , respectively e!ective plastic strain coe$cient of friction Poisson's ratio elastic stresses stresses in the plastic region yield strength ("R ) N e!ective stress tension at the inlet of the die pro"le radius tension at the outlet of the die pro"le radius
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
7
Fig. 1. The in#uence of tension on the stress and strain distribution and on the position of the neutral axis in pressing with a single-curved punch [3,4].
Consider a blank formed by a single-curved punch in two-steps, Fig. 1. In the "rst step, the blank is made to adopt the punch shape by application of a moment, M. In the second step, a tension, ¹, is applied. Assumed that M and ¹ act at the mid-thickness. The strain and stress distributions in the encircled zone in Fig. 1 vary with the magnitude of ¹.
8
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When the moment M'0 and ¹"0, the neutral axis is assumed to coincide with the center line, Fig. 1(a). In this case, the strains and the stresses above the centerline are tensile and equal to the strains and the stresses below the centerline which are compressive, Fig. 1(a). Applying a tension results in a shift, a, in the position of the neutral axis and enlarges the zone which is subject to tensile strains and stresses, Fig. 1(b). As the tension is increased, a grows, M decreases, and the portion of the cross section deformed elastically shifts towards the inner surface, Figs. 1(b)}(d). When the tension is su$ciently high, the neutral axis disappears from the cross section and both the inner and outer surfaces are deformed plastically in tension, Fig. 1(e). At this stage the plastic bending moment M"M P0. Fig. 1(e) exhibits an important case * bending under tension with entirely plastic cross section [3,4]. Unloading the tension, ¹, in Fig. 1 results in the sheet sliding around the punch without any change of shape, whilst unloading the moment, M, will change the radius of curvature from R to N R , Fig. 2 [5]. It can be shown that for forming with a single-curved punch [5]: Q 1 1 1 12M 12M (1!l) C" C " ! " , (1) * R R R Et E t N Q in which E is the Young's modulus, t the sheet thickness, l the Poisson's ratio, and M "M C the plastic bending moment applied before unloading. It can, furthermore, be shown that, Fig. 2 and [5]
*(1/R) 2*h/l,
(2)
in which *h is the springback in terms of height di!erence at the punch edge and l"half-width of the curved panel (Fig. 2). Eqs. (1) and (2) mean that the springback, measured as *(1/R) or *h, varies with the applied bending moment before unloading M"M . C The applied tension, ¹, and the bending moment, M, are, as shown in Fig. 1, uniquely related. If the sheet material is elastic perfectly plastic, one can plot the non-dimensional diagram of M/M C versus ¹/>t shown in Fig. 3. In Fig. 3, > is the yield strength, t the sheet thickness, and >t the yield tension. Fig. 3 shows that once the sheet starts to become plastic, the bending moment decreases very rapidly towards zero at which the sheet is fully plastic. Eqs. (1) and (2) show how this fall in moment a!ects the springback, Fig. 3. Thus, if the tension is su$ciently high to deform the entire sheet cross section plastically, then on unloading there is theoretically no springback and the sheet conforms exactly to the punch curvature.
Fig. 2. Unloading the moment, M, results in a curvature change.
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
9
Fig. 3. The relationship between the bending moment M and tension ¹ [6].
Fig. 4. The moments and tensions acting along one of the two symmetry planes.
In this study, the concept described above is used to study the springback of double-curved panels.
2. Theoretical treatment The in#uence of the applied tension on the bending moment and, thereby, the springback was shown in Figs. 1 and 3. The magnitude of this tension is determined by the restraining force acting in the #ange area, Fig. 4. This restraining force is in turn dependent upon the applied blank holder force (binder force), the shape of the drawbeads (if used), friction (both binder-sheet friction and punch-sheet friction), and pro"le radii (both punch and binder). The larger the restraining force, the
10
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
Fig. 5. The strain and stress distribution in a double-curved sheet element.
larger are the tensions acting on the double-curved surface, Fig. 4. The larger these tensions are, the smaller will be the springback. The theoretical analysis presented in this paper only concerns the tensions and moments acting on the double-curved panel surface. In other words, this paper does not comprise any mathematical expressions for the relationship between ¹ (¹ ), ¹ (¹ ), ¹ (¹ ) and ¹ (¹ ). U U U U B B See Fig. 4! Those interested in such expressions are referred to Ref. [2]. Note that Fig. 4 exhibits solely the cross section along one of the two symmetry planes on a double-curved panel. The analytical model constructed below comprises both of these planes. Compare Fig. 4 with Fig. 5. Fig. 5 exhibits the strain and stress distribution in a small element of the double-curved panel surface. Note in this "gure that neutral axis lies at z"!a along the 1-direction (path 1) and at z"!a along the 2-direction (path 2). The strain, e, consists of a portion, which is associated with extension of the middle "bre, and a portion, which has to do with the bending. One can therefore write [6] e "(z#a )/RH N
(3)
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
11
Fig. 6. The assumed material behaviour.
and e "(z#a )/RH , N where (Fig. 5) RH "R #t/2 N N
(4)
(5)
and RH "R #t/2. N N Utilizing Eqs. (3) and (4), the strain ratio on the double-curved panel surface is found
(6)
e (7) b " "[(z#a )/RH ]/[(z#a )/RH ]. N N e In any sheet element considered on the panel surface (along the 1- or 2-direction), continuity demands that (see also Fig. 5) a "a , where Eq. (7) becomes
(8)
b "e /e "RH /RH . N N Assume that
(9)
E plane stress is prevailing (p O0, p O0 and p "p "0), Fig. 5, X E the stress}strain relationship in Fig. 6 describes the material behaviour. That is p"Ee,
(10)
where p is the stress in the elastic region, e the strain in the elastic region, and E the Young's modulus, and p"K ' (e)L, C C C
(11)
12
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
in the plastic region, where p is the e!ective stress, e the e!ective plastic strain, K the strength C C C coe$cient, and n the strain-hardening exponent. The point at which Eq. (10) intersects Eq. (11) * the yield point * is characterized by (Fig. 6) p "K (ln 1.002)L"Ee. (12) C Note in Fig. 6 that the de"nition of the e!ective plastic strain, e , does not comply with the C scienti"cally correct terminology (according to which the elastic portion of the total strain is actually larger than that indicated in Fig. 6). The de"nition used in Fig. 6 leads to an overestimation of the magnitude of e . Since the magnitude of e obtained experimentally along C C paths 1 and 2, Fig. 5, is not expected to be greater than +0.05 (+5%), and since the de"nition above facilitates the analytical treatment of the springback, the terminology shown in Fig. 6 is used. E the strain ratio b , given by Eq. (9), is unchanged in both the elastic and the plastic regimes during the deformation. This means that the loading is proportional, i.e., the sheet metal is proportionally stretched and bent without any reversed loading caused by drawing and bending followed by stretching. Based on the assumptions above, an analytical model is constructed for the tensions and moments acting on, the strain gradients on, and the springback of the double-curved panel surface. This analytical model is accounted for in the appendix. Here, the key equations are used to describe how the component and tool design can bene"t from such an analytical model. 2.1. Tensions, moments, strain gradients and springback Applied at the middle surface, the total tension in the 1-direction is (Fig. 5) (13) ¹ "¹ #¹ , where ¹ is the tension caused by elastic stresses and ¹ the contribution of the plastic stresses to the total tension. These tension are given by (1#lb ) E ¹ " (1!l) 2RH N
Cp RH t W N ! a ! 2 E
(14)
and
K RH t/2#a Cp L> ! W . ¹ " N (n#1) RH E N In the 2-direction (Fig. 5)
(15)
¹ "¹ #¹ , where (1#l/b ) E ¹ " (1!l) 2RH N
(16)
b Cp RH t W N ! a ! 2 E
(17)
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
13
and
K RH t/2#a b Cp L> ! W . (18) ¹ " N (n#1) RH E N Acting about the mid-thickness, the total moment/unit width in the 1-direction is, Fig. 5, M "M #M , (19) where M and M are the elastic and plastic portions respectively of the total moment. These moments are given by (1#lb ) E M " (1!l) RH N
Cp RH Cp RH a a t a t W N W N ! # # ! E 3E 6 2 24 8
(20)
and
K RH t/2#a Cp L> K RH Cp RH ! W M " N ! N a ! W N (n#2) RH (n#1) E E N In the 2-direction (Fig. 5)
t/2#a Cp L> ! W RH E N
M "M #M , where
(21)
(22)
(1#l/b ) E M " (1!l) RH N
a t a t b Cp RH b Cp RH a W N W N ! # # ! 2 24 8 E 3E 6
(23)
and
K (RH ) t/2#a b Cp L> K RH b Cp RH ! W M " N ! N a ! W N (n#2) E E RH (n#1) N t/2#a b Cp L> ! W . (24) RH E N Using Fig. 7 and the equilibrium equations in the 1- and 2-directions, the strain gradients over the panel surface are obtained
and
de /d "k (1#jb )/[n/e !(1#b )]
(25)
de 1 1 n "k 1# ! 1# . (26) d
jb b e The distributions of the tensions acting on the panel surface are derived with the aid of Figs. 7 and 8 and
¹ "¹ e\I >H@ A \( U
(27)
¹ "¹ e\I > H@ A \( . U
(28)
14
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
Fig. 7. Forces acting on an element of the double-curved sheet (bending moments not considered).
Fig. 8. Distribution of the tensions is derived with the aid of symbols given in this "gure.
Eqs. (1) and (2) and Fig. 9 are used to derive expressions for the springback along paths 1 and 2 (the 1- and 2-direction Fig. 5) 6l M (1!l) *h " (1#lb )Et
(29)
6l M (1!l) . *h " (1#(l/b )Et
(30)
and
2.2. Discussion Eqs. (13) and (16) show the expressions derived for the tensions acting along paths 1 and 2 on a double-curved panel (Fig. 5). The second term in these expressions indicates the contribution of the plastic stresses to the total tension, Eqs. (15) and (18). Therefore, (t/2#a )/RH !Cp /E"0 N W
(31)
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
15
Fig. 9. Eqs. (29) and (30) calculate *h and *h .
and (t/2#a )/RH !b Cp /E"0, (32) N W should give the magnitude of a and a at the yield point. Utilizing Eq. (9) and rewriting Eqs. (31) and (32), one obtains a "a "Cp RH /E!t/2. W N Normalizing Eq. (33) yields 2a 2a 2Cp RH " " W N !1, t t tE
(33)
(34)
in which RH and C are de"ned by Eqs. (5) and (A.9) in the appendix, respectively. N In this investigation, two di!erent punches will be used. One of these punches * punch No. 1 * has the radii R "665 mm and R "2000 mm. Both punches are 230 mm wide and N N 400 mm long, Fig. 5. Four di!erent sheet materials will, furthermore, be used * steel (FeP04), and the aluminium qualities 5182, 6016 and 6111.
16
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37 Table 1 The position of the neutral axis, a /(t/2), at the yield point for the materials in this investigation, when pressed with punch No. 1. See also Fig. 5 and Eqs. (33) and (34) Material
t (mm)
E (MPa)
l
C
b
p RH W N (MPa) (mm)
a "a (mm)
a /(t/2)" a /(t/2)
Steel (FeP04) 5182-O
0.8 1.2 2.0 1.2 2.0 1.2 2.0
205 000 71 000 71 000 70 000 70 000 70 000 70 000
0.30 0.33 0.33 0.33 0.33 0.33 0.33
0.952 0.921 0.921 0.921 0.921 0.921 0.921
0.3326 0.3327 0.3328 0.3327 0.3328 0.3327 0.3328
172 149 144 95 113 167 175
0.1314 0.6870 0.2445 0.2323 !0.0094 0.8631 0.5340
0.3285 1.1450 0.2445 0.3872 !0.0094 1.4385 0.5340
6016-T4 6111-T4
665.4 665.6 666 665.6 666 665.6 666
Table 1 shows the position of the neutral surface at the yield point for these sheet materials, when pressed with punch No. 1. The values of a ("a ) and a /(t/2) ("a /(t/2)) in Table 1 are calculated by Eqs. (33) and (34), respectively. Note in Table 1 that a ("a ) and a /(t/2) ("a /(t/2)) are negative for the 2-mm thick sheet of material 6016. Utilizing Eqs. (31) and (32), one can "nd that if t/(2RH )'Cp /E, (35) N W then a ("a ) will be negative at the &elastic limit'. This implies that yielding may occur during the "rst loading step * application of the bending moments * if the yield strength is small, E is large, minor punch radius is small and/or the sheet thickness is large. Fig. 10 displays the moments and tensions acting on an arbitrary element (along path 1 or 2) of the double-curved surface versus the shift in the position of the neutral surface. The "gure is based on Eqs. (13), (16), (19) and (22) and concern 2-mm thick 6016 and 0.8-mm thick FeP04 (steel). For 2-mm thick 6016 steel, plastic deformation is, as shown in Fig. 10, initiated (actually proceeds; see Table 1) and the moments fall in magnitude as soon as any tensions are applied. To initiate plastic deformation (and a reduction in moments) as 0.8-mm thick FeP04 is considered, ¹ and ¹ must, however, be increased to approximately 40 and 22 N/mm, respectively and 2a /t ("2a /t) must be moved to 0.33. See Fig. 10. Fig. 11 displays the springback versus the shift in the position of the neutral surface at the panel walls. The "gure is based on Eqs. (29) and (30). Fig. 11 shows, in other words, how much the neutral surface must be moved to initiate plastic deformation and how fast the springback is reduced after yielding. When pressed with punch No. 1, 1.2-mm thick 6111 should display a severe springback, Fig. 11. The larger the restraining force in the #ange area (for instance, the larger the binder force) the larger are the tensions acting on the double-curved surface (Fig. 4). The larger these tensions, the larger is the shift in the position of the neutral surface, and the larger this shift, the smaller is the springback. Assume now that a 2-mm thick sheet of material 6016 is pressed with punch No. 1 and that the obtained shift in the position of the neutral surface is a "a "t/2"1 at the centre of this panel. Figs. 12 and 13, which are based on Eqs. (13), (16), (19), (22), (27)}(30), show the variation of the shift
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
17
Fig. 10. Moments and tensions acting on an arbitrary element (along path 1 or 2, Fig. 5) of the double-curved panel surface versus the shift in the position of the neutral surface. The "gure is based on Eqs. (13), (16), (19) and (22). See also Table 1.
in the position of the neutral axes, the tensions, the moments and the springback over the face of this panel (along paths 1 and 2). a , a , ¹ , and ¹ are smallest at the panel centre and increase towards the panel walls, whilst M and M are largest at the panel centre and decrease towards the panel walls. See Figs. 12 and 13. The important case of bending under tension with entirely plastic cross section was discussed in chapter 1. It was shown in Fig. 1(e) that the bending moment and, thereby, the springback are very
18
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
Fig. 11. Springback versus the shift in the position of the neutral surface at the panel wall: (a) path 1 (the "gure is based on Eq. (29)) and (b) path 2 (the "gure is based on Eq. (30)).
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
19
Fig. 12. The variation of the shift in the position of the neutral axes and the tensions over the panel face (paths 1 and 2). The "gure is valid for 2-mm thick 6016-T4 pressed with punch No. 1. It is assumed that a "a "t/2"1 at the panel centre. The "gure is based on Eqs. (13), (16), (27) and (28). The coe$cient of friction k "0.1. Concerning the other parameter values, see Table 1.
small, if the tension is so high that the entire cross section becomes plastic. It is of great interest to the upcoming experimental analysis to calculate the emerging plastic strains, when the entire cross section becomes plastic. The "rst terms in Eqs. (13) and (16) indicate the elastic portion of the tensions acting on the double-curved surface. Therefore, (Cp RH /E)!(a !t/2)"0 W N
(36)
20
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Fig. 13. The variation of the moments and the springback over the panel face (paths 1 and 2). The "gure is valid for 2-mm thick 6016-T4 pressed with punch No. 1. It is assumed that a "a "t/2"1 at the panel centre. The "gure is based on Eqs. (13), (16), (19), (22), (27)}(30). The coe$cient of friction k "0.1. For the other parameter values, see Table 1. See also Fig. 12.
and (b Cp RH /E)!(a !t/2)"0, (37) W N should give the position of the neutral axis, when the tensions just have become entirely plastic. Rewriting Eqs. (36) and (37), one obtains a "a "Cp RH /E#t/2. (38) W N At z"t/2, Eqs. (A.20) and (A.24) in the appendix give the plastic strains on the panel surface e "e !eW "(t/2#a )/RH !Cp /E, N W
(39)
t/2#a b Cp ! W. e "e !e " RH E N
(40)
and
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
21
Fig. 14. Plastic strains, which emerge on the panel surface, if the sheet materials in this study are pressed with punch No. 1, and if the entire cross section just have become plastic. The "gure is based on Eqs. (41)}(42).
Combining Eqs. (9) and (38)}(40) gives e "t/RH N
(41)
and e "t/RH . N Combining Eqs. (41) and (A.15) in the appendix yields the e!ective plastic strain
(42)
e"(4(1#b #b )/3t/RH . (43) C N Eqs. (41)}(43) yield the plastic strain along the 1-direction, the plastic strain along the 2-direction and the e!ective plastic strain in any sheet element along path 1 or 2 (Fig. 5), if the entire cross section of this element just have become plastic. Using these equations, the strains displayed in Fig. 14 are obtained for the sheet materials in this study, Table 1. Figs. 14 shows that the plastic strains, which emerge theoretically when the entire cross section just have become plastic, are very small * (1%. The word theoretically is italicized, since it is assumed that the loading occurs in two steps * the bending moments are applied "rst, after which the tensions are exerted. It is, furthermore, assumed that plastic deformation is caused by the tensions. In practice, the tensions and the bending moments are exerted at the same time and interact during the forming. The strain gradients along paths 1 and 2, the panel centre being the origin, should also be discussed. Substituting e and e plotted in Fig. 14 into Eqs. (25) and (26), respectively, one can see that the strain gradients along paths 1 and 2 become small and positive. These gradients become small, particularly since the ratios n/e and n/e in Eqs. (25) and (26) are large (for the n-values, see Table 2). Note that these gradients are small, even if e+5%, which is a practically C more reasonable value.
3. Materials The mechanical properties of the materials used in this investigation are displayed in Table 2, where t is the sheet thickness, R the yield strength, R the ultimate tensile strength, A the N K
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Table 2 Mechanical properties of the materials used in this investigation Material
t (mm)
R N (MPa)
R K (MPa)
A (%)
r
n
K C (MPa)
E (MPa)
l
Steel (FeP04) 5182-O-EDT
0.8 1.2 2.0 1.2 2.0 1.2 2.0
172 149 144 95 113 167 175
306 284 285 190 219 301 298
44 27 30 29 31 28 27
1.69 0.72 0.83 0.67 0.65 0.60 0.56
0.23 0.30 0.27 0.24 0.24 0.26 0.24
527 498 498 344 344 520 520
205000 71000 71000 70000 70000 70000 70000
0.30 0.33 0.33 0.33 0.33 0.33 0.33
6016-T4-EDT 6111-T4-V65S-H90
R , R , A , r and n for these materials are extracted from Ref. [7]. N K The gauge length was 80 mm. See Ref. [8]. See Ref. [9].
Fig. 15. True stress versus true strain for the studied sheet materials. The data concerning 5182-O, 6016-T4 and 6111-T4 are extracted from the tensile data provided by Ref. [7] and concern the 1.2-mm thick sheets of these materials.
total elongation (gauge length"50 mm), r the normal anisotropy, n the strain-hardening exponent, K the strength coe$cient, and E"modulus of elasticity. C Fig. 15 displays true stress versus true strain obtained by tensile testing of the studied sheet materials along the rolling direction. Note that Fig. 15 concerns the 1.2 mm-thick sheets of the aluminium grades in Table 2.
4. Experimental procedure Two di!erent punches were used, the dimensions of them being shown in Fig. 16. The drawbeads used were semi-circular in cross section, the radius being 8 mm. Sheets with three di!erent thicknesses were pressed in this study (Table 2). Prior to pressing the 2-mm thick sheets (and after pressing the 0.8 and 1.2-mm thick sheets), the lower binder was
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
23
Fig. 16. The punches used in this investigation.
Fig. 17. The panel shape after the trimming and the sites at which the panel shape were measured.
machined in order to keep the ratios clearance/thickness and lower binder proxle radius/thickness unchanged. Rectangular blanks (400 mm;570 mm) were pressed with both punches. For strain measurement, a quadrilateral grid was etched on each blank prior to pressing. Varying the binder force varied the restraining force in the #ange area. In each case, the maximum utilized binder force was slightly lower than that resulting in fracture. The shapes of the punches used (Fig. 16), were "rst determined in a coordinate-measuring machine. Each panel was then trimmed by laser cutting so that only 5 mm of the panel wall was left at its corners (Fig. 17). After this trimming, the panel shape was measured in the same coordinatemeasuring machine. The panel shape was then compared to the shape of the punch, with which the panel was pressed, and z-deviation from the punch shape at each measuring site was determined.
5. Results Fig. 18 displays the z-deviation from the punch shape (springback) versus the distance from the panel centre. This "gure concerns 2 mm thick AA6111 pressed with punch No. 1. All three panels in Fig. 18 were pressed with drawbeads * db in the legends.
24
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Fig. 18. z-deviation from the punch shape along path 1 (top "gure), *h , and along path *h , versus the distance from the panel centre. Material"6111. Sheet thickness"2.0 mm. Punch No. 1 was used. BHF in the legend"applied blank holding force in metric tonnes (;10 kN). db in the legend"drawbeads were used.
As exhibited in Fig. 18, the springback is reduced with increasing blank holding force * BHF in the legends. In this case, the maximum applied BHF was 55 metric tonnes (;10 kN), Fig. 18. Using a slightly higher BHF would lead to fracture at the panel corners. 55 t lies, in other words, just beneath the fracture limit. Fig. 19 shows the e!ective plastic strain measured along paths 1 and 2 versus the distance from the panel centre. Since the strains are very small, the measurement errors may be relatively large. It was predicted in the previous sections that the strain gradients along paths 1 and 2 would be positive and small. This prediction can be considered as veri"ed, even taking the possible measurement errors mentioned above into account (Fig. 19). Similar results (as those shown in Figs. 18 and 19) were obtained for the panels pressed in the other sheet materials in this study. Concerning these results, the reader is referred to Ref. [2]. It was predicted in the previous sections that the springback is reduced with increasing shift in the position of the neutral surface. The larger the restraining force (BHF), the larger is this shift. The larger the BHF, the smaller is, in other words, the springback. This prediction is veri"ed by Fig. 18.
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
25
Fig. 19. E!ective plastic strain along paths 1 and 2 versus the distance from the panel centre. Material"6111. Sheet thickness"2.0 mm. Punch No. 1 was used. BHF in the legend"applied blank holding force in metric tonnes (;10 kN). db in the legend"drawbeads were used.
If the mean value of the z-deviations at the panel walls (*h and *h ), Fig. 18, are normalized by the crown height (h and h ) and plotted versus the applied BHF, Figs. 20 and 21 are obtained. Fig. 20 displays the variation of *h /h and *h /h with BHF obtained on the panels pressed with punch No. 1. As shown in Fig. 20, *h /h and *h /h decrease with increasing BHF. Note that the maximum BHF plotted for each sheet material is the highest attainable. For instance, for 1.2-mm thick AA6111, 30 t is the highest attainable BHF. Applying a slightly higher BHF leads to fracture at the panel corners. The larger the restraining force (the higher the BHF), the larger is the shift in the position of the neutral surface. Therefore, Fig. 20 can be compared with Fig. 11. Since the same BHF leads to di!erent 2a /t and 2a /t, when applied to the di!erent sheets, the x-axis of Figs. 11 and 20 are not completely compatible. Fig. 20 shows, in fact, di!erent segments of Fig. 11. Note too that the sheets materials, for which 2a /t and 2a t are small at the yield point, exhibit a small springback at relatively moderate BHF levels. Compare Fig. 20 with Table 1 or Fig. 11.
26
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
Fig. 20. Normalized mean value of the springback at the panel walls, *h /h (path 1; top "gure) and *h /h (path 2), versus the applied blank holding force. The "gures concern the panels pressed with punch No. 1. Drawbeads were used.
Fig. 21 shows the variation of *h /h and *h /h with BHF obtained on the panels pressed with punch No. 2. The springback is reduced signi"cantly when punch No. 2 is used. Compare Figs. 20 and 21. Punch No. 2 has smaller radii than punch No. 1, Fig. 16. This means, in turn, that the shift in the position of the neutral surface at the yield point is smaller, when punch No. 2 is used, Eq. (34).
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
27
Fig. 21. Normalized mean value of the springback at the panel walls *h /h (path 1; top "gure) and *h /h (path 2), versus the applied blank holding force. The "gures concern the panels pressed with punch No. 2. Drawbeads were used.
Theoretically, plastic deformation may, furthermore, be initiated during the "rst loading step * application of the bending moments, Eq. (35). All this indicates that plastic deformation is initiated at smaller BHF levels, when punch No. 2 is used. Initiation of plastic deformation plays a signi"cant role in springback reduction. *h /h and *h /h are therefore smaller, as punch No. 2 is utilized. See Figs. 20 and 21.
28
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
Fig. 22. Normalized mean value of the springback at the panel walls, *h /h , versus the e!ective plastic strain at the panel centre. The "gures concern the panels pressed with punches No. 1 and No. 2 and drawbeads.
It was shown in the previous sections that the emerging plastic strains on the double-curved surface may be very small, even if the entire cross section is plastic (a condition which results in a very small springback), Eqs. (41)}(42) and Fig. 14. The strain level on a panel that exhibits a small springback may, therefore, di!er only slightly from that attained on a panel (pressed in the same material and thickness and with the same punch) which shows a large springback. Recall too that the strain gradients along paths 1 and 2 are positive and small.
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
29
Fig. 23. The panel shape after the second trimming and the sites at which the panel shape were measured.
This prediction is veri"ed by the experimental data presented in Figs. 18 and 19. It is also veri"ed by Fig. 22, which displays *h /h at the panel walls versus the e!ective plastic strain at the panel centre. The di!erence in strain level can, as exhibited in Fig. 22, be very small, although the di!erence in springback is large or very large. See, for instance, the values obtained for 1.2-mm thick AA6111 (Fig. 22). The springback reduces with increasing binder force. Increasing the binder force leads to larger strains. However, as the springback is minimized, increasing the binder force leads only to larger strains. See, for instance, the values obtained for the steel sheet (Fig. 22). Therefore, it is improper (as commonly done) to evaluate the springback of di!erent sheet materials by comparing the deviations obtained on panels pressed (in di!erent sheet materials) to the same strain level at the centre.
6. Discussion It was shown that the tensions acting on the double-curved surface of the panel interact with the restraining forces developed in the #ange area. The higher the restraining forces, the higher the tensions acting on the double-curved surface of the panel. The tensions acting on the double-curved surface a!ect the shift in the position of the neutral axis, a and a (a "a in the same sheet element). The higher the tensions, the larger is a ("a ). The larger a ("a ) is, the smaller will be the moments acting on the double-curved surface and, "nally, the smaller the moments, the smaller will the springback be. It was, in other words, shown that the higher the restraining forces developed in the #ange area, the smaller the springback. The presented experimental data verify this prediction. The restraining forces can be increased by increasing the blank holding force and/or by using a more `severea drawbead shape, etc. In this study, the restraining forces were increased by increasing the binder force. It was also shown that the shift in the position of the neutral axis at the yield point (elastic limit), theoretically determined by a "a "Cp RH /E!t/2, W N
(33)
30
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
should play a signi"cant role in minimization (practical elimination) of the springback. The experimental results presented in Figs. 20 and 21 verify this prediction. In case a "a "0, also the tensions acting on the double curved surface ¹ "¹ "0 at the elastic limit. All this signi"es that if the sheet material and the punch shape are selected in such a fashion that R /E"p /E"t/(2CRH ), N W N
(44)
the springback can be minimized (practically eliminated) at moderate restraining force levels. This is an important conclusion, particularly for those practical cases, in which the restraining force is created solely by the aid of drawbeads. It is commonly held that sheet aluminium exhibits a larger springback than steel sheet in industrial applications. Eq. (44) shows that the springback can be minimized with an appropriate punch design and sheet material (particularly yield strength and sheet thickness) selection, regardless of the material category. Applying a higher binder force reduces the springback. However, the experimental results show that increasing the binder force is not a su$cient measure, as far as the minimization (practical elimination) of springback is concerned. A higher binder force may lead to fracture (at the panel corners), before the springback is minimized. The punch corner radii must, therefore, be increased (in order to avoid fracture), so that the springback can be minimized by applying higher binder forces (increasing the restraining forces). See for instance, 1.2-mm thick AA6111, which cannot withstand higher binder forces than 30 t, Fig. 20. The panels pressed in this study were actually trimmed once more so that solely the doublecurved surface was left, Fig. 23. The panel shapes after this second trimming were compared to the punch shapes. The results obtained after the second trimming comply with those attained after the "rst. Those interested in the results obtained after this second trimming are referred to Ref. [2]. Note, once again, that the theoretical analysis presented in this paper only concerns the tensions and moments acting on the double-curved panel surface. To prevent this presentation from taking unreasonable proportions, this paper does not comprise any mathematical expressions for the relationship between ¹ (¹ ), ¹ (¹ ) ¹ (¹ ) and ¹ (¹ ). See Fig. 4! Those inter U U U U B B ested in such expressions are referred to Ref. [2]. Therefore, a quantitative comparison between predicted and measured springback cannot be conducted here. Those interested in such a comparison are referred to Ref. [2].
7. Conclusions The following conclusions apply: E Increasing the binder force, increasing the curvature, increasing the sheet thickness and/or decreasing the yield strength reduces the springback. E The shift in the position of neutral surface at the yield point a "a "Cp RH /E!t/2, W N
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
31
plays a signi"cant role in minimization (practical elimination) of the springback. The smaller a and a at the yield point, the smaller the springback at moderate binder force (or restraining force) levels. If a and a are large at the yield point, more gentle corner and pro"le radii are necessary to minimize the springback. These radii must be gentle so that high binder (restraining) forces can be applied without risking fracture at the panel corners. E If the sheet material and the punch shape are selected in such a fashion that R /E"p /E"t/(2CRH ), N W N the springback can be minimized (practically eliminated) at moderate restraining force levels. This is an important conclusion, particularly for those practical cases, in which the restraining force is created solely by the aid of drawbeads. E Consider two panels pressed in the same sheet material and with the same punch but with di!erent binder forces. The di!erence in strain level between these two panels can be very small, although the di!erence in springback is large or very large (between the same two panels). Therefore, it is improper (as commonly done) to evaluate the springback of di!erent sheet materials by comparing the deviations obtained on panels pressed (in di!erent sheet materials) to the same strain level at the centre.
Acknowledgements The present investigation was carried out, whilst the author was with the Swedish Institute for Metals Research, and funded by Volvo Car Corporation/Go( teborg and the National Swedish Agency for Industrial and Technical Development (Nutek), which are gratefully acknowledged.
Appendix A In the elastic region (Fig. 6), Hooke's law (plane stress, p "0) can be applied [4], X p "E(e #le )/(1!l) and
(A.1)
p "E(e #le )/(1!l), (A.2) in which p "stress in the 1-direction (Fig. 5), p "stress in the 2-direction (Fig. 5), e "strain in the 1-direction, e "strain in the 2-direction and l"Poissons's ratio. Combining Eqs. (A.1) and (A.2) with Eqs. (3), (4) and (9) in the main text, we obtain p "[(1#lb )/(1!l)]E(z#a )/RH N
(A.3)
and
p "
l 1# b
(1!l) E(z#a )/RH . N
(A.4)
32
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
Rewriting Hooke's law (plane stress, p "0) in terms of strains yields X e "(p !lp )/E and
(A.5)
e "(p !lp )/E. (A.6) Combining Eqs. (9) and (12) in the main text and (A.3), (A.5) and (A.6) with von Mises yield criterion (see Eq. (A.18)), we obtain eW "Cp /E W
(A.7)
and eW "b Cp /E, W in which
(A.8)
(1!l) b #l b #l \ C" (A.9) 1! # (1#lb ) 1#lb 1#lb and eW and eW are the yield strains in the 1- and the 2-directions, respectively (Figs. 5 and 6). At the yield point (Figs. 5 and 6), z"b !a
(A.10)
and z"b !a , which combined with Eqs. (3) and (4) give eW "b /RH N
(A.11)
(A.12)
and eW "b /RH . (A.13) N Combining Eqs. (A.7) and (A.8) with Eqs. (A.12) and (A.13), respectively and utilizing Eq. (9), we obtain b "b "CRH p /E. (A.14) N W In the plastic region in Fig. 6, von Mises yield criterion (plane stress, p "p "0, e O0) can be X utilized. The e!ective plastic strain, e, can then be denoted as [6] C 4 e"e (1#b #b ), (A.15) C 3 N
where e is the plastic strain in the 1-direction (Fig. 5) and b is the strain ratio, Eq. (9). Assume that p "a p .
(A.16)
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
The stress ratio, a , in Eq. (A.16) is related to the strain ratio, b , Eq. (9) by [6] a "(2b #1)/(2#b ). The e!ective stress, p, can therefore be denoted as [6] C p"p (1!a #a, C where p "the stress in the 1-direction (Fig. 5). Substituting Eqs. (A.15) and (A.18) in Eq. (11), one obtains p "K C
[(4/3(1#b #b )]L (e )L. (1!a #a
33
(A.17)
(A.18)
(A.19)
Note that z#a Cp ! W, e "e !eW " (A.20) RH E N in which e is the total strain in the 1-direction given by Eq. (3) and eW the yield strain in the 1-direction given by Eq. (A.7). Substituting Eq. (A.20) into Eq. (A.19) yields p "K
z#a Cp L ! W , RH E N
(A.21)
where [(4/3(1#b#b)]L . (1!a #a Combining Eqs. (9), (A.16) and (A.19) yields K "K C
p "[K a /(1!a #a] C
4 (1#b #b) 3
(A.22)
L b (e )L.
(A.23)
Note that (A.24) e "e !eW "(z#a )/RH !b Cp /E, N W in which e is the total strain in the 2-direction given by Eq. (4) and eW the yield strain in the 2-direction given by Eq. (A.8). Combining Eqs. (A.23) and (A.24), we obtain p "K [(z#a )/RH !b Cp /E]L, N W where K a [(4(1#b #b)/3]L K " C . bL (1!a #a
(A.25)
(A.26)
34
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
A.1. Tensions and moments Applied at the middle surface, the total tension in the 1-direction is ¹ "¹ #¹ , (A.27) where ¹ is the tension caused by elastic stresses and ¹ the contribution of the plastic stresses to the total tension. In the elastic region, Fig. 5(a), the tension will be
@ \?
R
p dz. (A.28) \R Combining Eqs. (A.3), (A.14) and (A.28) and integrating, we obtain Eq. (14) in the main text. The contribution of the plastic stresses to the total tension is ¹ "
¹ "
p dz.
(A.29)
@ \? Combining Eqs. (A.14) and (A.21) with Eq. (A.29) and integrating yields Eq. (15) in the main text. A derivation analogous to the above yields the tension in the 2-direction, Fig. 5(b). See Eqs. (16)}(18) in the main text. Acting about the mid-thickness, the total moment per unit width in the 1-direction is M "M #M , (A.30) where M and M are the elastic and plastic portions, respectively, of the total moment. In the elastic region (Fig. 5(a))
@ \?
R
p z dz. \R Combining Eqs. (A.3), (A.14) and (A.31) and integrating yield Eq. (20) in the main text. The contribution of the plastic stresses to the total moment is M "
(A.31)
p z dz. (A.32) @ \? Combining Eqs. (A.14) and (A.21) with Eq. (A.32) and integrating yield Eq. (21), which is shown in the main text. A derivation analogous to the above yields the moment in the 2-direction, Fig. 5(b). See Eqs. (22)}(24) in the main text. M "
A.2. Strain gradients and distribution of tensions over the panel surface Let us "rst denote the tension ratio, j, as j"¹ /¹ , in which ¹ and ¹ are the tensions given by Eqs. (13) and (16), respectively.
(A.33)
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
35
Consider an element of the double-curved sheet and the forces acting on it (bending moments not considered), Fig. 7. Equilibrium in the radial direction in this "gure requires ¹ /RH #¹ /RH "q. (A.34) N N Assuming Coulomb friction and that the friction coe$cient, k , is constant during the forming, the equilibrium of forces in the 1-direction in Fig. 7 yields d¹ "k qRH d . N Combining Eqs. (A.34) and (A.35), one obtains
(A.35)
d¹ "k (¹ /RH #¹ /RH )RH d . N N N Substituting Eq. (A.33) into Eq. (A.36) gives
(A.36)
d¹ /¹ "k (1/RH #j/RH )RH d . N N N It is known that
(A.37)
¹ "p t and, hence, one can write
(A.38)
d¹ /¹ "dp /p #dt/t. It is, furthermore, known that
(A.39)
p "K (e )L, why one can write dp /p "n de /e . The thickness component in Eq. (A.39) can be denoted as dt/t"de "!(de #de ). R Substituting Eq. (9) into Eq. (A.41) leads to dt/t"de "!(1#b ) de . R Substituting Eqs. (A.40) and (A.42) into Eq. (A.39) results in
(A.40)
(A.41)
(A.42)
(A.43) d¹ /¹ "[n/e !(1#b )] de . Combining Eqs. (9) and (A.37) with Eq. (A.43), we obtain Eq. (25) in the main text. A derivation analogous to the above yields Eq. (26). See the main text. Eqs. (25) and (26) give the strain gradients over the panel surface. Let us now look at the distribution of the tensions over the panel surface, starting with the tension along path 1. Combining Eqs. (9) and (A.37), we obtain d¹ /¹ "k (1#jb ) d . Integrating this equation with the symbols given in Fig. 8(a) yields
2U d¹ A " k (1#jb ) d , ¹ 2 (
36
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
which becomes ln(¹ /¹ )"k (1#jb )(c ! ). (A.44) U Eq. (A.44) can be rewritten as Eq. (27), which is shown in the main text. Note that c in Fig. 8(a) and Eq. (27) is known, since both l and R (RH ) are known. N N A derivation analogous to the above gives Eq. (28), which can be seen in the main text. Note that c in Fig. 8(b) and Eq. (28) is known, since both l and R (RH ) are known. N N Note, "nally, that it is possible to attain a better description of the relationship between ¹ and ¹ and between ¹ and ¹ than those given by Eqs. (27) and (28) by taking the in#uence of the U U punch pro"le radius into consideration. For simplicity, Eqs. (27) and (28) are preferred here. A.3. Springback Combining Eqs. (1) and (2) yields *h "6l M /(E t) C
(A.45)
and *h "6l M /(E t), (A.46) C in which M "M de"ned by Eq. (19) and M "M given by Eq. (22) in the main text. Utilizing C C Eqs. (9), (A.3)}(A.4), (A.7)}(A.8), (A.30) and (A.31), one can easily show that E "(1#lb )E/(1!l)
(A.47)
and
l E/(1!l). (A.48) E " 1# b Substituting Eq. (A.47) into Eq. (A.45) and Eq. (A.48) into Eq. (A.46) yield Eqs. (29) and (30). These latter equations are given in the main text. Note that (Fig. 9) sin "l /R N
(A.49)
sin "l /R . N
(A.50)
and
References [1] Yoshida K et al. Analysis and control of buckling behaviour due to elastic recovery in press forming. Proceedings of the 13th Biennial Congress of the International Deep Drawing Research Group, IDDRG, Melbourne, Australia, February 20}24, 1984. p. 73}84. [2] Asna" N. Mechanics of sheet metal forming, Ph.D. Thesis, LTU-DT-1997/6, Lulea University of Technology, Sweden, 1997. p. BI1}3.
N. Asnax / International Journal of Mechanical Sciences 43 (2001) 5}37
37
[3] Nagpal V et al. ICAM mathematical modelling of sheet metal formability indices and sheet metal forming processes. Technical Report AFML-TR-79-4168, AFML/LTC, WPAFB, OH 45433, USA, 1979. [4] Mielnik EM. Metalworking science and engineering. New York, U.S.A.: McGraw-Hill, 1991. p. 34, 743}51. [5] Duncan JL et al. Die forming approximations for aluminium sheet. Sheet Metal Industries 1978. p. 1015}25. [6] Marciniak Z et al. Mechanics of sheet metal forming. London: Edward Arnold, 1992. p. 17, 68}85, 143}5. [7] Draft "nal report, Centre de Recherches Metallurgiques (CRM) contribution to the low weight vehicle programme properties of aluminium alloys for body structure. Brite-EuRam project 5656, Contract No. BRE2-CT92-0266. [8] BjoK rnfot S. Springback in forming of high strength steel sheet. Lulea University of Technology, Technical Report 1983:60T p. 15 (in Swedish). [9] Adler T et al. MATEDS-the aluminium database from SkanAluminium. SkanAluminium and Royal Institute of Technology, Stockholm, 1993.