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Simultaneous stretch forming and deep drawing in axisymmetrical sheet forming
Journal of Materials Processing Technology 97 (2000) 82±87
Simultaneous stretch forming and deep drawing in axisymmetrical sheet forming T.C. Lim, S. Ramakrishna*, H.M. Shang Department of Mechanical and Production Engineering, National University of Singapore, 10 Kent Ridge Crescent, 119260 Singapore Received 27 May 1999
Abstract From the product point of view, the onset of a visible neck during sheet metal forming sets the limit to which a blank material can be formed. In any forming process, the blank material is subjected to deformations that are mixtures of typical stretch forming and typical deep drawing, interlaced in an intricated manner with the progression of forming. Through varying forming parameters such as blank size, tool pro®le and blank-holding force, this paper explores the complex relationship between the mixture and the overall formability of sheet metal. # 2000 Elsevier Science S.A. All rights reserved. Keywords: Deep drawing; Stretch forming; Formability; Optimization
1. Introduction
2. Theory and formulation
Deep drawing [1] and stretch forming [2] are two extreme processes in sheet metal forming. In deep drawing, the entire blank is pushed through a die to produce a complete cup with an essentially unchanged thickness. Stretch forming is performed purely through stretching of the blank material within the die throat in the absence of draw-in of the ¯ange. The prevention of draw-in is achieved by the use of a beaded or serrated blank-holder. It is noticeable, however, that increasing the amount of stretching within the die throat during deep drawing (by increasing the blank-holding load, for instance) enables the formed cup to possess a larger surface area, resulting in a greater cup height for equal amounts of draw-in [3,4]. In stretch forming, introducing ¯ange draw-in allows more material for stretching and therefore a greater polar height is achieved at fracture [5,6]. In other words, the interaction of drawing and stretching of a blank allows a change in the maximum attainable cup height. The choice of proper blank size, blank-holder force and tool pro®le enables one to perform optimization, such as the maximization of cup height, by means of simultaneous stretching and draw-in.
In general sheet forming, the blank material is stretched biaxially with simultaneous drawing-in of the ¯ange, resulting in an increase in surface area with subsequent thinning. Thus, the area strain d is de®ned as
* Corresponding author. Tel.: 65-844-2216; fax: 65-773-3537 E-mail address: [email protected] (S. Ramakrishna)
d
dA ; A
(1)
where A is the current product area. The amount of stretching at any stage of forming is thus ln
A ; Aob
(2)
where Aob is the undeformed blank area. The incremental draw-in of the flange is defined as the loss in projected base area d ÿ
dAproj ; Aproj
(3)
where Aproj is the projected cup base area. At any deformation stage, the overall draw-in is Rob 2ln ; (4) Rob0
0924-0136/00/$ ± see front matter # 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 9 9 ) 0 0 3 3 7 - 4
T.C. Lim et al. / Journal of Materials Processing Technology 97 (2000) 82±87
83
Table 1 Sheet forming classification according to the X-factor Stretch-drawing spectrum
X-factor
Amount of stretching
Amount of drawing
Stretching relative to drawing
Overall thickness change
Sheet forming classification
Case 1 Case 2
X<0 X0
<0 0
>0 >0
< <
Negative stretching Pure deep drawing
Case Case Case Case
01 X!1
>0 >0 >0 >0
>0 >0 >0 0
< > >
Thickening No overall thickness change Thinning Thinning Thinning Thinning
3 4 5 6
where Rob and Rob' are the rim radius of the undeformed and deformed blank, respectively. The ratio of stretching to draw-in is obtained easily [4±6] as X
:
(5)
As shown in Table 1, axisymmetrical sheet forming processes may be classi®ed into six categories in accordance with the X-parameter. The values of draw-in ( ) and stretching () can be measured from the rim radius and the product area. The product area (A) may be calculated theoretically based upon the cup height, the rim radius and the tool pro®le. As shown in Fig. 1, the product area is sub-divided into two portions, the ¯ange area and the shell area A Aflange Ashell ;
(6)
where the flange area (Aflange) is ÿ 2 Aflange R2ob ÿ Rpt Rs
(7)
with Rpt and Rs being the cup wall radius and the die shoulder radius, respectively. The shell area (Ashell) consists of the flat top punch contact area; the punch nose contact area, the cup wall area and the die shoulder contact area. The product shape, based upon the tool dimensions, is shown in Fig. 1, which figure shows also that the cup wall is bounded by the punch and the die throat, enabling the cup wall radius to be assumed as Rpt Rp Rt =2:
(8)
For the case where the cup height is greater than or equal to the summation of the punch nose and die shoulder radii, it
Dominant in deep drawing Balanced drawing and stretching Dominant in stretch forming Pure stretch forming
may be assumed that the cup wall is vertical. This assumption allows the shell area to be summed up as A00shell 2Rpt H Rpt ÿRn 2 ÿ2 Rn Rs Rpt ÿRn Rs ; (9) where Rn is the punch nose radius Eq. (9) is accurate due to the small clearance between the punch and die shoulder, which enables the formation of the vertical cup wall. For the case where the cup height is less than the summation of the punch nose and die shoulder radii, the cup wall is inclined. A similar derivation of cup-wall inclination by Qin et. al. [6] and Manabe et. al. [7] reveals much complication in the shell area expression. A simple and yet accurate procedure in obtaining the shell area during the inclined cup wall stage is proposed here. The shell area is hereforth approximated to the form of a second-order polynomial as A0shell C0 C1 H C2 H 2 : (10) Con®ning the assumed equation to three boundary conditions (i) at H 0, A0shell Rpt Rs 2 :
(11)
(ii) at H (RnRs), A00shell :
A0shell
(12)
(iii) at H (RnRs), dA0shell dA00shell ; dH dH
(13)
and solving simultaneously the three resulting equations from Eqs. (11)±(13) gives the coefficients in Eq. (10) as ÿ 2 (14) C0 Rpt Rs ; C1 2Rpt 2
Rpt ÿRn 2 ÿ Rpt Rs 2 Rn Rs
2ÿ4 Rpt ÿRn Rs ; and C2 Fig. 1. Tool and workpiece geometry.
Rpt Rs 2 ÿ Rpt ÿRn 2 Rn Rs 2
(15)
Rpt ÿRn Rs 2ÿ : R n Rs (16)
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T.C. Lim et al. / Journal of Materials Processing Technology 97 (2000) 82±87
for H < Rn, where C0 R2pt ;
(20)
C1 2ÿ6 Rpt ÿRn
(21)
and C2 4ÿ Fig. 2. Various iso-height curves at Rob 75 mm and Rn 23 mm; H* H/Rpt.
During sheet forming, it is clear that the tool (the punch and die size and the pro®le) and the workpiece dimensions are ®xed. It is then possible to plot a family of iso-height curves for ®xed tool and workpiece geometries. Each of the iso-height curves represent equal cup height for various combinations of stretching and draw-in. Figs. 2±4 show iso-height curves for variation in punch penetration, blank size and punch nose radius, respectively. For the case where the die shoulder is small in comparison to the punch radius, as in most blank holders, one may assume a zero die shoulder radius to give a simpli®ed model. The ¯ange and shell areas are thus reduced to Aflange R2ob0 ÿR2pt ;
(17)
A00shell 2Rpt H Rpt ÿRn 2 ÿ2Rn Rpt ÿRn
(18)
for H Rn, and A0shell C0 C1 H C2 H 2
(19)
Rpt ÿ3: Rn
(22)
Substitution of Eq. (17) and (18) or Eq. (19) into Eqs. (6) and (2) gives the relationship between stretching, draw-in and punch penetration and geometric parameters as Rpt ÿRn H ln eÿ 2ÿ6 R2ob 2 Rpt H (23) 4ÿ ÿ3 Rn Rob for H < Rn and ( ln e
ÿ
2 ) 2Rpt H Rpt Rn Rn 2 ÿ4 2 3ÿ Rob Rob Rob (24)
for H Rn. The simpli®ed model is accurate only when the die shoulder radius is small in comparison to the punch radius. In addition to the simpli®ed model a linearized model is also proposed based upon pure stretch forming (X!1) and pure deep drawing (X 0). For the pure stretch forming operation, the amount of stretching is that with zero draw-in pure 0 ;
(25)
whilst pure deep drawing is characterized by a zero amount of stretching pure 0 :
(26)
Upon observation of Figs. 2±4, a linearized model is proposed as pure ÿ pure = pure ; where Fig. 3. Iso-height curves at H* 1.5 and Rn 23 mm for various blank sizes.
pure ln
(
Ashell
(27)
)
Rpt Rs 2
when fully drawn, and ( ) Rpt Rs 2 Ashell 1ÿ : pure ln Rob R2ob
(28)
(29)
A summary of the procedure for obtaining the amount of stretching is as follows:
*
Fig. 4. Iso-height curves at H 1.5 and Rob 75 mm for various punch profiles.
Step 1: Measurement of cup height and rim radius; Step 2: Compute Aflange from Eq. (7); Step 3: If H (Rn Rs)
T.C. Lim et al. / Journal of Materials Processing Technology 97 (2000) 82±87
85
Table 2 Present test combinations
Fig. 5. Comparison of iso-height curves at H* 1, Rob 75 mm and Rn 23 mm for the newly-proposed model (P), the geometricallysimplified model (S) and the linearized model (L).
Rob (mm)
Rn (mm)
F (kN)
H (mm)
75 75 75 70 75 75 75 75 75 80
23 23 23 23 9 23 23 23 37.5 23
60 60 60 60 60 30 60 90 60 60
12.66 21.10 29.00 59.20 26.08 59.22 37.60 31.24 45.40 33.88
Ashell A00shell from Eq. (9); Else Ashell A0shell from Eq. (10); Step 4: Compute A and from Eqs. (6) and (2), respectively. Good correlation between the newly developed (detailed) stretching model with the simpli®ed and linearized models is shown in Fig. 5 The simpli®ed and linearized models offer quick reference to the relationship between stretching, drawin, tool dimensions, blank sizes and depth of punch penetration. In this paper, only the detailed model for stretching is used for comparison with experimental data.
and 90 kN were applied. Table 2 shows all of the test combinations in the present investigation. A ®xed punch speed of 16 mm/min was used in all the tests. The end-point of forming is determined as follows: when fracture occurs, a cracking sound is emitted and the punch penetration is then stopped. The fractured cup height, Hfrac,is the limiting cup height, Hmax. When fracture does not occur even until a cup is fully drawn, then the full-cup height, Hfull, becomes the limiting cup height. Based on material incompressibility, the meridional arc length between one circular grid and its concentric neighbour is
3. Experimental technique
ds
A Hille 20/40 ton Universal Sheet Metal Testing Machine was used. The punch tools used are of the following dimensions: punch radius Rp 37.5 mm; punch nose radius Rn 9, 23 and 37.5 mm; die throat radius Rt 40 mm; and die shoulder radius Rs 6 mm. The sheet material used for the investigation was commercial brass of 1.26 mm thickness. All specimens were tested in the as-received condition. The blank size used are speci®ed by the blank radius Rob 70, 75 and 80 mm. In the preparation of the circular specimens, large brass sheets were ®rst sheared into square blanks. Blanks with deep scratches were discarded so as to ensure that the cup height at fracture are not dominated by existing thickness irregularities. After the preparation of circular blanks, concentric grids were plotted on one side of the blank using the QBASIC program and the Roland Plotter 1100. The circular grids are of radial intervals of 2 mm, these grids enabling the measurement of product area. A ``Pye-Unicam'' 2-D travelling microscope was used to measure the radial distance of each grid after deformation. The thickness at each grid was obtained from the measurement of the ®nal thickness using an ``Ono Sokki'' digital gauge and sensor. To obtain the effects of process parameters on the formability of brass sheets, three major parameters were chosen. A punch radius of Rp 37.5 mm with nose radii Rn 9, 23 and 37.5 mm were used whilst blank-holder forces of 30, 60
Piece-wise summation of circular ring areas from the pole, i 0, to the outermost printed grid, i (Nÿ1), on the product gives the experimental product area
r0 t 0 dr0 : rt
Aexpt 2t0 dr0
(30)
N ÿ1 X r0 i r0 i1 ; ti ti1 i0
(31)
where r0 and dr0 are the radius of ith circular grid and the radial grid interval, respectively (before blank deformation), whilst ti and t0 are the thickness at the ith grid after deformation and the blank thickness before deformation, respectively. 4. Results and discussion Fig. 6(a) and (b) show the relationship between stretching and draw-in as an axisymmetrical sheet forming characteristic during punch penetration. Fig. 7(a), Fig. 8(a) and Fig. 9(a) show the amount of stretching simultaneous with draw-in at the limit of forming for various blank-holder forces, tool pro®les and blank sizes, respectively. The limiting cup heights and their X-factors are depicted correspondingly in Fig. 7(b), Fig. 8(b) and Fig. 9(b). Fig. 7(a) shows that in the case of fracture, the release in blank-holder force delays the occurrence of necking so that the increase in punch penetration is due to more draw-in of the ¯ange material than stretching of existing shell material.
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T.C. Lim et al. / Journal of Materials Processing Technology 97 (2000) 82±87
Fig. 6. (a) Experimental (^) and theoretical (&) stretching versus drawin for punch penetration at Rob 75 mm, Rn 23 mm and F 60 kN. (b) Experimental (^) and theoretical (&) X-factor for increasing punch penetration at Rob 75 mm, Rn 23 mm and F 60 kN
Fig. 8. (a) Experimental (^) and theoretical (&) stretching versus drawin for Rob 75 mm, Rn 9, 16, and 37.5 mm, F 60 kN. (b) Experimental (^) and theoretical (&) X-factor versus limiting cup height for Rob 75 mm, Rn 9, 16 and 37.5 mm, F 60 kN.
Fig. 7. (a) Experimental (^) and theoretical (&) stretching versus drawin for Rob 75 mm, Rn 23 mm, F 30, 60 and 90 kN. (b) Experimental (^) and theoretical (&) X-factor versus limiting cup height for Rob 75 mm, Rn 23 mm, F 30, 60 and 90 kN.
Fig. 9. (a) Experimental (^) and theoretical (&) stretching versus drawin for Rob 70, 75 and 80 mm, Rn 23 mm, F 60 kN. (b) Experimental (^) and theoretical (&) X-factor versus limiting cup height for Rob 70, 75 and 80 mm, Rn 23 mm, F 60 kN.
Delay in necking accounts for a greater depth of punch penetration as shown in Fig. 7(b), thereby implying that where excessive stretching occurs, reducing the blankholder force increases the cup-height limit. On the other hand, it is widely understood that the height of a pure deepdrawn cup can be increased through the introduction of stretching by means of increased blank-holder force. This brings one to conclude that an optimal blank holder force exist which gives maximum cup height.
Fig. 8(a) shows the fracture locus for various punch pro®le in the stretching versus draw-in plane. The use of a sharp punch nose (as in Rn 9 mm) introduces large plastic bending of the blank at the contact region at the punch nose, thereby limiting the amount of stretching at the punch ¯at top contact area. Sharp bending at the punch nose contact induces necking during early stage of punch penetration, resulting in low limiting cup height. Hemispherical punch (Rn 37.5 mm) on the other hand facilitates stretching
T.C. Lim et al. / Journal of Materials Processing Technology 97 (2000) 82±87
throughout the punch contact such that a large amount of stretching is attained, resulting in large polar height (Fig. 8(b)) even at moderate draw-in, as shown in Fig. 8(a). Fig. 9(a) presents a linear correlation between stretching and draw-in. Large blank size (Rob 80 mm) inhibits ¯ow of ¯ange into the shell portion, resulting in large amount of stretching to facilitate punch penetration. The excessive stretching induced, sets the limit in draw-in (Fig. 9(a)) and cup height (Fig. 9(b)). Reduction of blank size decreases the resistance in ¯ange draw-in, which in turn enhances draw-in and lowers stretching during punch penetration. Lowering in stretching delays the occurrence of necking such that the limiting cup height is improved. For pure deep drawing process, decrease in blank size lowers the limiting cup height not only due to absence of stretching but also due to limitation in ¯ange material. This suggests that blank size can be optimized to produce maximum achievable cup height. 5. Conclusions It is widely known that limitation in cup height for a pure deep-drawing process is set by the blank area whilst that for pure stretch forming is set by severe thinning. The attainment of maximum cup height is shown to be possible through the interplay of stretching and draw-in by means of the proper choice in forming conditions. The use of the limiting cup height curve in an X-factor versus cup height plane allows a die designer to select optimal forming conditions in terms of blank size, tool pro®le and clamping force.
87
The actual amount of stretching and draw-in are lost when the X-parameter is used: the introduction of the stretching versus draw-in plane provides a better alternative. The location of a point in the stretching versus draw-in curve gives insight into the amount of stretching, the amount of draw-in and the cup height. The X-factor is represented by the slope at a point on the plane subtending to the origin. Comparison between a series of iso-height curves (Fig. 2, for example) with the limiting locus in the stretching versus draw-in plane shows the relationship between the forming conditions and the corresponding limiting cup height. A cup forming manufacturer may control the amount of thinning (stretching) and ®nal rim radius (draw-in) of a formed cup of certain height through the guidance of an iso-height curve in the stretching versus draw-in diagram. References [1] S.Y. Chung, H.W. Swift, Proc. IMechE. 165 (1951) 199±211. [2] B. Kaftonoglu, J.M. Alexander, J. Inst. Metals. 90 (1961±1962) 457± 470 [3] P.K. Lee, C.Y. Choi, T.C. Hsu, Trans ASME B 94 (1973) 925± 930. [4] S. Qin, H.M. Shang, C.J. Tay, J.X. Mo, J. Mater. Proc Techol. 59 (1996) 386±390. [5] H.M. Shang, S. Qin, J. Mo, J. Mater. Proc. Technol. 62 (1996) 454± 457. [6] S. Qin, H.M. Shang, C.J. Tay, J. Mo, J. Mater. Proc. Technol. 63 (1997) 117±122. [7] K. Manabe, S. Yoshihara, H. Nishimura, Proc. ASME Dynamics Systems and Control Division 58 (1996) 175±181.
Simultaneous stretch forming and deep drawing in axisymmetrical sheet forming T.C. Lim, S. Ramakrishna*, H.M. Shang Department of Mechanical and Production Engineering, National University of Singapore, 10 Kent Ridge Crescent, 119260 Singapore Received 27 May 1999
Abstract From the product point of view, the onset of a visible neck during sheet metal forming sets the limit to which a blank material can be formed. In any forming process, the blank material is subjected to deformations that are mixtures of typical stretch forming and typical deep drawing, interlaced in an intricated manner with the progression of forming. Through varying forming parameters such as blank size, tool pro®le and blank-holding force, this paper explores the complex relationship between the mixture and the overall formability of sheet metal. # 2000 Elsevier Science S.A. All rights reserved. Keywords: Deep drawing; Stretch forming; Formability; Optimization
1. Introduction
2. Theory and formulation
Deep drawing [1] and stretch forming [2] are two extreme processes in sheet metal forming. In deep drawing, the entire blank is pushed through a die to produce a complete cup with an essentially unchanged thickness. Stretch forming is performed purely through stretching of the blank material within the die throat in the absence of draw-in of the ¯ange. The prevention of draw-in is achieved by the use of a beaded or serrated blank-holder. It is noticeable, however, that increasing the amount of stretching within the die throat during deep drawing (by increasing the blank-holding load, for instance) enables the formed cup to possess a larger surface area, resulting in a greater cup height for equal amounts of draw-in [3,4]. In stretch forming, introducing ¯ange draw-in allows more material for stretching and therefore a greater polar height is achieved at fracture [5,6]. In other words, the interaction of drawing and stretching of a blank allows a change in the maximum attainable cup height. The choice of proper blank size, blank-holder force and tool pro®le enables one to perform optimization, such as the maximization of cup height, by means of simultaneous stretching and draw-in.
In general sheet forming, the blank material is stretched biaxially with simultaneous drawing-in of the ¯ange, resulting in an increase in surface area with subsequent thinning. Thus, the area strain d is de®ned as
* Corresponding author. Tel.: 65-844-2216; fax: 65-773-3537 E-mail address: [email protected] (S. Ramakrishna)
d
dA ; A
(1)
where A is the current product area. The amount of stretching at any stage of forming is thus ln
A ; Aob
(2)
where Aob is the undeformed blank area. The incremental draw-in of the flange is defined as the loss in projected base area d ÿ
dAproj ; Aproj
(3)
where Aproj is the projected cup base area. At any deformation stage, the overall draw-in is Rob 2ln ; (4) Rob0
0924-0136/00/$ ± see front matter # 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 9 9 ) 0 0 3 3 7 - 4
T.C. Lim et al. / Journal of Materials Processing Technology 97 (2000) 82±87
83
Table 1 Sheet forming classification according to the X-factor Stretch-drawing spectrum
X-factor
Amount of stretching
Amount of drawing
Stretching relative to drawing
Overall thickness change
Sheet forming classification
Case 1 Case 2
X<0 X0
<0 0
>0 >0
< <
Negative stretching Pure deep drawing
Case Case Case Case
0
>0 >0 >0 >0
>0 >0 >0 0
< > >
Thickening No overall thickness change Thinning Thinning Thinning Thinning
3 4 5 6
where Rob and Rob' are the rim radius of the undeformed and deformed blank, respectively. The ratio of stretching to draw-in is obtained easily [4±6] as X
:
(5)
As shown in Table 1, axisymmetrical sheet forming processes may be classi®ed into six categories in accordance with the X-parameter. The values of draw-in ( ) and stretching () can be measured from the rim radius and the product area. The product area (A) may be calculated theoretically based upon the cup height, the rim radius and the tool pro®le. As shown in Fig. 1, the product area is sub-divided into two portions, the ¯ange area and the shell area A Aflange Ashell ;
(6)
where the flange area (Aflange) is ÿ 2 Aflange R2ob ÿ Rpt Rs
(7)
with Rpt and Rs being the cup wall radius and the die shoulder radius, respectively. The shell area (Ashell) consists of the flat top punch contact area; the punch nose contact area, the cup wall area and the die shoulder contact area. The product shape, based upon the tool dimensions, is shown in Fig. 1, which figure shows also that the cup wall is bounded by the punch and the die throat, enabling the cup wall radius to be assumed as Rpt Rp Rt =2:
(8)
For the case where the cup height is greater than or equal to the summation of the punch nose and die shoulder radii, it
Dominant in deep drawing Balanced drawing and stretching Dominant in stretch forming Pure stretch forming
may be assumed that the cup wall is vertical. This assumption allows the shell area to be summed up as A00shell 2Rpt H Rpt ÿRn 2 ÿ2 Rn Rs Rpt ÿRn Rs ; (9) where Rn is the punch nose radius Eq. (9) is accurate due to the small clearance between the punch and die shoulder, which enables the formation of the vertical cup wall. For the case where the cup height is less than the summation of the punch nose and die shoulder radii, the cup wall is inclined. A similar derivation of cup-wall inclination by Qin et. al. [6] and Manabe et. al. [7] reveals much complication in the shell area expression. A simple and yet accurate procedure in obtaining the shell area during the inclined cup wall stage is proposed here. The shell area is hereforth approximated to the form of a second-order polynomial as A0shell C0 C1 H C2 H 2 : (10) Con®ning the assumed equation to three boundary conditions (i) at H 0, A0shell Rpt Rs 2 :
(11)
(ii) at H (RnRs), A00shell :
A0shell
(12)
(iii) at H (RnRs), dA0shell dA00shell ; dH dH
(13)
and solving simultaneously the three resulting equations from Eqs. (11)±(13) gives the coefficients in Eq. (10) as ÿ 2 (14) C0 Rpt Rs ; C1 2Rpt 2
Rpt ÿRn 2 ÿ Rpt Rs 2 Rn Rs
2ÿ4 Rpt ÿRn Rs ; and C2 Fig. 1. Tool and workpiece geometry.
Rpt Rs 2 ÿ Rpt ÿRn 2 Rn Rs 2
(15)
Rpt ÿRn Rs 2ÿ : R n Rs (16)
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T.C. Lim et al. / Journal of Materials Processing Technology 97 (2000) 82±87
for H < Rn, where C0 R2pt ;
(20)
C1 2ÿ6 Rpt ÿRn
(21)
and C2 4ÿ Fig. 2. Various iso-height curves at Rob 75 mm and Rn 23 mm; H* H/Rpt.
During sheet forming, it is clear that the tool (the punch and die size and the pro®le) and the workpiece dimensions are ®xed. It is then possible to plot a family of iso-height curves for ®xed tool and workpiece geometries. Each of the iso-height curves represent equal cup height for various combinations of stretching and draw-in. Figs. 2±4 show iso-height curves for variation in punch penetration, blank size and punch nose radius, respectively. For the case where the die shoulder is small in comparison to the punch radius, as in most blank holders, one may assume a zero die shoulder radius to give a simpli®ed model. The ¯ange and shell areas are thus reduced to Aflange R2ob0 ÿR2pt ;
(17)
A00shell 2Rpt H Rpt ÿRn 2 ÿ2Rn Rpt ÿRn
(18)
for H Rn, and A0shell C0 C1 H C2 H 2
(19)
Rpt ÿ3: Rn
(22)
Substitution of Eq. (17) and (18) or Eq. (19) into Eqs. (6) and (2) gives the relationship between stretching, draw-in and punch penetration and geometric parameters as Rpt ÿRn H ln eÿ 2ÿ6 R2ob 2 Rpt H (23) 4ÿ ÿ3 Rn Rob for H < Rn and ( ln e
ÿ
2 ) 2Rpt H Rpt Rn Rn 2 ÿ4 2 3ÿ Rob Rob Rob (24)
for H Rn. The simpli®ed model is accurate only when the die shoulder radius is small in comparison to the punch radius. In addition to the simpli®ed model a linearized model is also proposed based upon pure stretch forming (X!1) and pure deep drawing (X 0). For the pure stretch forming operation, the amount of stretching is that with zero draw-in pure 0 ;
(25)
whilst pure deep drawing is characterized by a zero amount of stretching pure 0 :
(26)
Upon observation of Figs. 2±4, a linearized model is proposed as pure ÿ pure = pure ; where Fig. 3. Iso-height curves at H* 1.5 and Rn 23 mm for various blank sizes.
pure ln
(
Ashell
(27)
)
Rpt Rs 2
when fully drawn, and ( ) Rpt Rs 2 Ashell 1ÿ : pure ln Rob R2ob
(28)
(29)
A summary of the procedure for obtaining the amount of stretching is as follows:
*
Fig. 4. Iso-height curves at H 1.5 and Rob 75 mm for various punch profiles.
Step 1: Measurement of cup height and rim radius; Step 2: Compute Aflange from Eq. (7); Step 3: If H (Rn Rs)
T.C. Lim et al. / Journal of Materials Processing Technology 97 (2000) 82±87
85
Table 2 Present test combinations
Fig. 5. Comparison of iso-height curves at H* 1, Rob 75 mm and Rn 23 mm for the newly-proposed model (P), the geometricallysimplified model (S) and the linearized model (L).
Rob (mm)
Rn (mm)
F (kN)
H (mm)
75 75 75 70 75 75 75 75 75 80
23 23 23 23 9 23 23 23 37.5 23
60 60 60 60 60 30 60 90 60 60
12.66 21.10 29.00 59.20 26.08 59.22 37.60 31.24 45.40 33.88
Ashell A00shell from Eq. (9); Else Ashell A0shell from Eq. (10); Step 4: Compute A and from Eqs. (6) and (2), respectively. Good correlation between the newly developed (detailed) stretching model with the simpli®ed and linearized models is shown in Fig. 5 The simpli®ed and linearized models offer quick reference to the relationship between stretching, drawin, tool dimensions, blank sizes and depth of punch penetration. In this paper, only the detailed model for stretching is used for comparison with experimental data.
and 90 kN were applied. Table 2 shows all of the test combinations in the present investigation. A ®xed punch speed of 16 mm/min was used in all the tests. The end-point of forming is determined as follows: when fracture occurs, a cracking sound is emitted and the punch penetration is then stopped. The fractured cup height, Hfrac,is the limiting cup height, Hmax. When fracture does not occur even until a cup is fully drawn, then the full-cup height, Hfull, becomes the limiting cup height. Based on material incompressibility, the meridional arc length between one circular grid and its concentric neighbour is
3. Experimental technique
ds
A Hille 20/40 ton Universal Sheet Metal Testing Machine was used. The punch tools used are of the following dimensions: punch radius Rp 37.5 mm; punch nose radius Rn 9, 23 and 37.5 mm; die throat radius Rt 40 mm; and die shoulder radius Rs 6 mm. The sheet material used for the investigation was commercial brass of 1.26 mm thickness. All specimens were tested in the as-received condition. The blank size used are speci®ed by the blank radius Rob 70, 75 and 80 mm. In the preparation of the circular specimens, large brass sheets were ®rst sheared into square blanks. Blanks with deep scratches were discarded so as to ensure that the cup height at fracture are not dominated by existing thickness irregularities. After the preparation of circular blanks, concentric grids were plotted on one side of the blank using the QBASIC program and the Roland Plotter 1100. The circular grids are of radial intervals of 2 mm, these grids enabling the measurement of product area. A ``Pye-Unicam'' 2-D travelling microscope was used to measure the radial distance of each grid after deformation. The thickness at each grid was obtained from the measurement of the ®nal thickness using an ``Ono Sokki'' digital gauge and sensor. To obtain the effects of process parameters on the formability of brass sheets, three major parameters were chosen. A punch radius of Rp 37.5 mm with nose radii Rn 9, 23 and 37.5 mm were used whilst blank-holder forces of 30, 60
Piece-wise summation of circular ring areas from the pole, i 0, to the outermost printed grid, i (Nÿ1), on the product gives the experimental product area
r0 t 0 dr0 : rt
Aexpt 2t0 dr0
(30)
N ÿ1 X r0 i r0 i1 ; ti ti1 i0
(31)
where r0 and dr0 are the radius of ith circular grid and the radial grid interval, respectively (before blank deformation), whilst ti and t0 are the thickness at the ith grid after deformation and the blank thickness before deformation, respectively. 4. Results and discussion Fig. 6(a) and (b) show the relationship between stretching and draw-in as an axisymmetrical sheet forming characteristic during punch penetration. Fig. 7(a), Fig. 8(a) and Fig. 9(a) show the amount of stretching simultaneous with draw-in at the limit of forming for various blank-holder forces, tool pro®les and blank sizes, respectively. The limiting cup heights and their X-factors are depicted correspondingly in Fig. 7(b), Fig. 8(b) and Fig. 9(b). Fig. 7(a) shows that in the case of fracture, the release in blank-holder force delays the occurrence of necking so that the increase in punch penetration is due to more draw-in of the ¯ange material than stretching of existing shell material.
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Fig. 6. (a) Experimental (^) and theoretical (&) stretching versus drawin for punch penetration at Rob 75 mm, Rn 23 mm and F 60 kN. (b) Experimental (^) and theoretical (&) X-factor for increasing punch penetration at Rob 75 mm, Rn 23 mm and F 60 kN
Fig. 8. (a) Experimental (^) and theoretical (&) stretching versus drawin for Rob 75 mm, Rn 9, 16, and 37.5 mm, F 60 kN. (b) Experimental (^) and theoretical (&) X-factor versus limiting cup height for Rob 75 mm, Rn 9, 16 and 37.5 mm, F 60 kN.
Fig. 7. (a) Experimental (^) and theoretical (&) stretching versus drawin for Rob 75 mm, Rn 23 mm, F 30, 60 and 90 kN. (b) Experimental (^) and theoretical (&) X-factor versus limiting cup height for Rob 75 mm, Rn 23 mm, F 30, 60 and 90 kN.
Fig. 9. (a) Experimental (^) and theoretical (&) stretching versus drawin for Rob 70, 75 and 80 mm, Rn 23 mm, F 60 kN. (b) Experimental (^) and theoretical (&) X-factor versus limiting cup height for Rob 70, 75 and 80 mm, Rn 23 mm, F 60 kN.
Delay in necking accounts for a greater depth of punch penetration as shown in Fig. 7(b), thereby implying that where excessive stretching occurs, reducing the blankholder force increases the cup-height limit. On the other hand, it is widely understood that the height of a pure deepdrawn cup can be increased through the introduction of stretching by means of increased blank-holder force. This brings one to conclude that an optimal blank holder force exist which gives maximum cup height.
Fig. 8(a) shows the fracture locus for various punch pro®le in the stretching versus draw-in plane. The use of a sharp punch nose (as in Rn 9 mm) introduces large plastic bending of the blank at the contact region at the punch nose, thereby limiting the amount of stretching at the punch ¯at top contact area. Sharp bending at the punch nose contact induces necking during early stage of punch penetration, resulting in low limiting cup height. Hemispherical punch (Rn 37.5 mm) on the other hand facilitates stretching
T.C. Lim et al. / Journal of Materials Processing Technology 97 (2000) 82±87
throughout the punch contact such that a large amount of stretching is attained, resulting in large polar height (Fig. 8(b)) even at moderate draw-in, as shown in Fig. 8(a). Fig. 9(a) presents a linear correlation between stretching and draw-in. Large blank size (Rob 80 mm) inhibits ¯ow of ¯ange into the shell portion, resulting in large amount of stretching to facilitate punch penetration. The excessive stretching induced, sets the limit in draw-in (Fig. 9(a)) and cup height (Fig. 9(b)). Reduction of blank size decreases the resistance in ¯ange draw-in, which in turn enhances draw-in and lowers stretching during punch penetration. Lowering in stretching delays the occurrence of necking such that the limiting cup height is improved. For pure deep drawing process, decrease in blank size lowers the limiting cup height not only due to absence of stretching but also due to limitation in ¯ange material. This suggests that blank size can be optimized to produce maximum achievable cup height. 5. Conclusions It is widely known that limitation in cup height for a pure deep-drawing process is set by the blank area whilst that for pure stretch forming is set by severe thinning. The attainment of maximum cup height is shown to be possible through the interplay of stretching and draw-in by means of the proper choice in forming conditions. The use of the limiting cup height curve in an X-factor versus cup height plane allows a die designer to select optimal forming conditions in terms of blank size, tool pro®le and clamping force.
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The actual amount of stretching and draw-in are lost when the X-parameter is used: the introduction of the stretching versus draw-in plane provides a better alternative. The location of a point in the stretching versus draw-in curve gives insight into the amount of stretching, the amount of draw-in and the cup height. The X-factor is represented by the slope at a point on the plane subtending to the origin. Comparison between a series of iso-height curves (Fig. 2, for example) with the limiting locus in the stretching versus draw-in plane shows the relationship between the forming conditions and the corresponding limiting cup height. A cup forming manufacturer may control the amount of thinning (stretching) and ®nal rim radius (draw-in) of a formed cup of certain height through the guidance of an iso-height curve in the stretching versus draw-in diagram. References [1] S.Y. Chung, H.W. Swift, Proc. IMechE. 165 (1951) 199±211. [2] B. Kaftonoglu, J.M. Alexander, J. Inst. Metals. 90 (1961±1962) 457± 470 [3] P.K. Lee, C.Y. Choi, T.C. Hsu, Trans ASME B 94 (1973) 925± 930. [4] S. Qin, H.M. Shang, C.J. Tay, J.X. Mo, J. Mater. Proc Techol. 59 (1996) 386±390. [5] H.M. Shang, S. Qin, J. Mo, J. Mater. Proc. Technol. 62 (1996) 454± 457. [6] S. Qin, H.M. Shang, C.J. Tay, J. Mo, J. Mater. Proc. Technol. 63 (1997) 117±122. [7] K. Manabe, S. Yoshihara, H. Nishimura, Proc. ASME Dynamics Systems and Control Division 58 (1996) 175±181.