An experimental study on some formability evaluation methods in negative incremental forming

Journal of Materials Processing Technology 186 (2007) 45–53

An experimental study on some formability evaluation methods in negative incremental forming G. Hussain a,∗ , L. Gao a , N.U. Dar b a

College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China b College of Mechanical Engineering, University of Engineering & Technology, Taxila, Pakistan Received 20 April 2006; received in revised form 25 November 2006; accepted 2 December 2006

Abstract In single point incremental forming (SPIF), the final thickness of a deformed sheet can be predicted by the sine law. Therefore, the formability in SPIF can be expressed as the maximum wall angle (θ max ) that a sheet would endure without fracturing. In the present study, two tests were carried out in order to evaluate the formability of an aluminum sheet. In the first test, conical frustums and square pyramids, a set of each, were produced by systematically varying the wall angle in order to investigate θ max . In the second test, four conical frustums, each having varying wall angle, designed by revolving different curved lines were formed to fracture. The results revealed that the value of θ max obtained from the former test was smaller than those obtained from the latter one. Moreover, a variation among the values of θ max obtained from the parts of the second test was also found. Since the first test shows the minimum possible value of θ max , it should be employed in combination with the second test so as to minimize the number of experiments required. © 2006 Elsevier B.V. All rights reserved. Keywords: Negative incremental forming; Formability tests; Maximum wall angle; Sine law

1. Introduction In the recently past years, many sheet-metal-forming techniques have been under study in order to develop novel forming processes, such as laser forming, water assisted forming and single point incremental forming (SPIF), characterized by high flexibility. These techniques are characterized by the possibility of being easily adopted to produce small production batches using low cost tooling. Among these innovative processes [1–7], SPIF has attained a great attention. In the simplest form of the SPIF process, the final component shape is determined by the relative movement of a tool with respect to the blank rather than the die shape. The process is usually carried out on CNC machines where it is possible to assign and control the tool motion according to the fixed paths [4–7]. The process has two variants: negative incremental forming and positive incremental forming [8] (Fig. 1). In the former variant, the blank is backed with a die that increases the probability to produce parts with sharp corners in contrast with the latter one. ∗

Corresponding author. Tel.: +86 136 7516 1625. E-mail address: gh [email protected] (G. Hussain).

0924-0136/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2006.12.005

Like other sheet-metal-forming processes, SPIF also suffers of some forming defects for instance spring back and bending close to the clamped zone [9]. However, these defects can be minimized by optimizing the process parameters. The production rate of the novel process is not as high as those of existing ones. Nevertheless, some other outstanding features, such as flexibility and low cost tooling, make it feasible to manufacture parts in small batches for various applications [10–13], which are listed below: • It is a very economical process for rapid prototyping; • The method creates large regions of homogenous deformation and avoids the large stress and strain gradients. Due to this fact, a specimen formed by the process is considered to be more reliable to calibrate a void nucleation model than the tensile specimens; • Finally, it is capable to manufacture a variety of irregularshaped components and highly customized medical products. In short, due to the inherent advantages and flexibility, the process offers the possibility to implement a powerful alternative.

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Nomenclature Terms Generatrix a curve whose motion generates a surface or a solid Revolved surface a surface generated by the motion of generatrix Transition point a point, which is located closest to fracture, on which the theoretical and experimental values of thickness of a part are found to be in accordance Relations t = t0 sin α = t0 cos θ the sine law Notations a, b, e, k constants used in the equation of circle, ellipse, parabola and exponential function C(xc ,yc ) transition point (Fig. 4d) on a part, whose wall angle varies D(xd ,yd ) fracture point (Fig. 4d) on a part whose wall angle varies hp the depth of a part measured to an arbitrary point P(x,y) P(x,y) an arbitrary point on a surface (Fig. 4b) P1 (x1 ,y1 ), P2 (x2 ,y2 ) the initial and end points of a generatrix or a surface (Fig. 4b) R radius of a circular arc t thickness of a formed part t0 thickness of a blank Greek symbols α half-apex angle of a part θ wall/slope angle of a part θp wall or slope angle on an arbitrary point on a part θ 1 ,θ 2 the wall/slope angles on the initial and end points of a part or a generatrix, respectively

Several studies have been performed with emphasis on assessing the formability in this forming method [14–17]. There are two ways to express formability: an FLC that presents the limiting strains [14,15], and θ max that is the maximum wall angle (θ max ) without fracture of the sheet metal [16,17]. In the current work, formability has been expressed as θ max . Young [16] produced a collection of conical samples with a variety of wall angles in order to investigate θ max for an aluminum sheet. In order to avoid forming of a large number of samples required, as reported in [16], a new formability test was devised in the previous work [17]. In this test, a conical surface, which was designed by revolving a circular arc, having varying wall angle was formed to fracture. In the present investigations, both the tests [16,17] were employed to test the formability of an aluminum sheet. However, the variety of shapes formed in the previously devised test (see [17] for detail) was increased from one to four in order to study the effect of the specimen shape on formability. For this purpose, conical frustums each with vary-

ing wall angle were modeled by revolving various curves, which were the segments of circular, parabolic, elliptical and exponential curves, and formed to facture by negative incremental forming. The results were quantitatively analyzed within – and between – the specimens of the tests. Based on the analysis, some conclusions have been drawn, and a methodology to determine the minimum possible value of the formability parameter ‘θ max ’ has been proposed. 2. The experimental set-up and the process parameters The material used in the current study was an aluminum sheet with 0.91 mm thickness. Tensile tests were performed in order to determine the mechanical properties of the sheet, given as follows: elastic modulus = 72 GPa, yield strength = 94 MPa, ultimate tensile strength = 177 MPa, elongation = 21%, and modulus of rigidity = 27 GPa. The commercial CAD/CAM software ‘UG NX-3’ was utilized for designing and tool path generation. The spiral tool path [4–7] was used to control the tool motion. The blank having the size of (140 mm × 140 mm) was held at the periphery by a blank holder as presented in Fig. 2, and a CNC milling machine tool was employed to deform the blank. A HSS tool with the hemispherical end of 8 mm diameter shaped the part at the room temperature. The forming tool traveled along the closed-loop path at the horizontal feed rate of 2500 mm/min and the vertical feed of 0.15 mm/revolution. Both the tests were conducted under the same forming conditions. 3. Formability evaluation The formability of the aluminum sheet was evaluated by employing two methods, which are described as follows. 3.1. Test-1 In SPIF, the final thickness of a part depends on the wall angle (slope angle θ has been referred as wall angle) [16–18]. Therefore, in an attempt to form a new material by SPIF, the initial testing begins with a search for the maximum wall angle that the metal would withstand without fracturing. The maximum wall angle ‘θ max ’ was investigated by producing the collections of conical frustums and truncated square pyramids, a few of which are shown in Fig. 3. The deformation angle was varied in steps, as detailed in Table 1, until ‘θ max ’ was determined. In all of the parts, the base and bottom dimensions were kept constant, which are given as follows: major diameter of frustum = 68 mm, minor diameter of frustum = 30 mm, side length of the base of pyramid = 68 mm, and side length of the bottom of pyramid = 30 mm. 3.2. Test-2 3.2.1. A brief introduction of the test In the Test-1, a large number of parts have to be produced in order to evaluate the exact formability of a sheet. This laborious and costly testing method can be replaced by a novel one, as

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Fig. 1. Variants of single point incremental forming [8]: (a) negative incremental forming and (b) positive incremental forming.

Table 1 Systematic investigation of θ max by employing the Test-1 Part type

θ a (◦ )

Outcome

Conical frustums

50 55 60 65 70 68 66 66.5 66.3

Formed Formed Formed Formed Formed Formed Formed Fracture Fracture

66.5 66.3 66

Fracture Fracture Formed

Truncated square pyramids a

Wall/slope angle of part.

proposed in [17], which is capable to determine the formability at low cost and reduced processing time. This innovative method makes use of a curved-line-generatrix1 to generate a revolved surface whose wall angle varies. The CAD model of such a surface has been shown in Fig. 4a–b. The surface is supposed to fracture on a point D(xd ,yd ) before reaching the designed depth, for the wall angle on the end point of the surface is kept 90◦ normally (see the sine law in [17,18]). Fig. 4c shows a part formed in the test conducted in the previous work [17] in order to determine the formability of an aluminum sheet, and Fig. 4d represents the thickness profile of the formed part. C(xc ,yc ) is a transition point on which the actual and theoretical thickness values of the part are found to be in accordance. Hence, according to [17], the wall angle on this point is regarded as θ max .

1

Generatrix is a curve whose motion generates a surface or a solid.

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G. Hussain et al. / Journal of Materials Processing Technology 186 (2007) 45–53 Table 2 The design data of the parts formed in the Test-2 Part’s generatrix

Circular curve Elliptical curve Exponential curve Parabolic curve a b c d e

Fig. 2. The experimental set-up.

3.2.2. Design of parts In the previous investigation [17], only one shape was designed by revolving an arc of a circle. The selection of curve to create a revolved surface may affect the test results. Due to this reason, three more surfaces were designed in the present study. In order to model these surfaces, three curve-segments from circle, ellipse and parabola and one segment from an expo-

Part design θ 1 a (◦ )

θ 2 b (◦ )

hc (mm)

D1 d (mm)

D2 e (mm)

40 40 40 40

80 80 80 80

68.15 72 122.5 157.5

110 110 110 110

31.36 23.2 14 13.3

Wall/slope angle on the initial point of part. Wall/slope angle on the end point of part. Designed depth of part. Major diameter of part. Minor diameter of part.

nential curve were selected as generatrices. The included angle of all the curves was kept constant, i.e., 40◦ , which is evident from the design data given in Fig. 5a–d. Because the sheet was not expected to be deformed to a large angle, curves with 80◦ end slope were preferred in order to reduce the blank size and forming time. The base of each revolved surface was 110 mm in diameter (Table 2), while the design of the surfaces varied in the bottom sizes and the depth values due to different slope distribution along the generatrices (see Fig. 5). 3.2.3. Mathematical formulation of the wall angle and thickness As the wall angle of each specimen in the Test-2 varies, the derivation of mathematical expressions to compute the wall angle and thickness on an arbitrary point on the spec-

Fig. 3. Some parts formed in the Test-1: (a) representative conical frustums and (b) representative square pyramids.

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Fig. 4. Illustration of the Test-2 [17]: (a) the generatrix and revolved surface (b), terminology, (c) the formed part, and (d) thickness profile of the formed part.

imen is needed. The procedure adopted to derive these expressions has been given in detail in the previous paper [17]; for ready reference, however, it is re-written briefly as below: In SPIF, metal deformation obeys the sine law [16–18], which is expressed mathematically as follows: t = t0 sin α = t0 sin(90 − θ) = t0 cos θ

(1)

where t is the final thickness, t0 the blank thickness, α the halfapex angle and θ is the wall/slope angle. 

Let a circular curve P1 P2 having an included angle of θ be selected as a generatrix to generate a surface whose wall angle varies Fig. 4a–b. D(xd ,yd ) is a point on which failure may occur during forming. Say P(xp ,yp ) is an arbitrary point on the surface. The thickness tp and the wall angle θ p on this point can be computed by using

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Fig. 5. Design of the generatrices: (a) circular curve, (b) elliptical curve, (c) exponential curve, and (d) parabolic curve.

the following relations:

3. For the surface designed by revolving the exponential curve (Fig. 5c):

1. For the surface designed by revolving the circular curve (Fig. 5a): y  y − h  p 1 p θp = cos−1 = (2a) R R tp = t0

yp t0 = (y1 − hp ) R R

(2b)

where y1 − hp = yp , R is the radius of the circular curve. 2. For the surface designed by revolving the elliptical curve (Fig. 5b):    −b −1 2 2 θp = tan (3a) b − (y1 − hp ) a(y1 − hp )  

 −b tp = t0 cos tan−1 (3b) b2 − (y1 − hp )2 a(y1 − hp ) where y1 − hp = yp , a and b are constants.

θp = tan−1 {a(hp + y1 )}

(4a)

tp = t0 cos[tan−1 {a(hp + y1 )}]

(4b)

where hp + y1 = yp , a is a constant. 4. For the surface designed by revolving the parabolic curve (Fig. 5d): θp = tan−1 {2



k(hp + y1 )}

tp = t0 cos[tan−1 {2



k(hp + y1 )}]

(5a) (5b)

where hp + y1 = yp , k is a constant. In Eqs. (2)–(5), t0 , R, a, b, e, k and y1 are known values, while the depth hp to a point P(x,y) is measured from the part under study.

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Fig. 6. . The parts formed in the Test-2 designed by revolving: (a) the circular curve, (b) the elliptical curve, (c) the exponential curve, and (d) the parabolic curve.

3.2.4. Procedure of formability evaluation The test specimens as per the design data given in Table 2 were formed under the same forming conditions employed in the Test1. Each part continued to form until fracture occurred (Fig. 6), and the machine tool was manually stopped upon occurrence of the fracture.

In order to determine the transition point ‘c(xc ,yc )’ (see Fig. 4d), a number of points in an increment of 0.5 mm were scribed on each part close to the fracture. For this purpose, the part was held at the periphery, and the points were scribed with a depth gauge as illustrated in Fig. 7a. After marking points, each part was cut into quarters so as to facilitate measurement of the

Fig. 7. Set-up for: (a) scribing points and measuring depth, and (b) measuring thickness.

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Table 3 The data of critical points scribed on the parts formed in the Test-2 Part’s generatrix

Circular curve Elliptical curve Exponential curve Parabolic curve a b c d e f g

Depth of part measured to a point

Actual and analytical thickness of part on point D(xd ,yd )

Actual and analytical thickness of part on point C(xc ,yc )

hr a (mm)

hd b (mm)

hc c (mm)

td(th) d (mm)

td(exp) e (mm)

tc(th) f (mm)

tc(exp) g (mm)

53.47 59.34 58.39 46.34

50.29 56 54.5 40.5

45.44 52.6 48.5 32.5

0.299 0.291 0.2873 0.29

0.26 0.255 0.258 0.268

0.337 0.3186 0.309 0.317

0.34 0.32 0.31 0.316

Reached or formed depth of part. Depth of part measured to the point D(xd ,yd ). Depth of part measured to the point C(xc ,yc ). Analytical thickness of part on the point D(xd ,yd ). Actual thickness of part on the point D(xd ,yd ). Analytical thickness of part on the point C(xc ,yc ). Actual thickness of part on the point C(xc ,yc ).

actual thickness with a dial gauge indicator (Fig. 7b). The analytical thickness values on the points were computed by putting the known depth ‘hp ’ values in the respective Eqs. (2a), (3a), (4a) and (5a) depending on the part under investigation. Although several points were scribed on each part, the data of the critical points, i.e., transition points and failure points, only have been given in Table 3. It is obvious that the actual thickness value on a failure point ‘D(xd ,yd )’, which is closest to the fracture, is smaller than the predicted one. While, the actual and predicted thickness values on a transition point are found to be in good agreement. Therefore, the wall angle on a transition point was regarded as θ max , and, for each part, was calculated using one of the relevant equations, i.e., Eqs. (2a), (3a), (4a) or (5a). 4. Discussion Fig. 8a presents the values of θ max obtained from the parts formed in the Test-1. Both the parts – cones and pyramids –

Fig. 8. Formability values of the sheet: (a) by the Test-1 and (b) by the Test-2.

show the same formability value, i.e., 66◦ . However, the results of the depth measurement of the pyramid and cone formed with 66.5◦ wall angle showed that the former was 2 mm shorter than the latter. This finding, which is similar to the one found by Shim and Park [15], indicates that the deformation at a corner is greater than that along a side. But, from a practical perspective, no remarkable effect was noticed. The results (Fig. 8b) obtained from the Test-2 shows a trend of increasing value θ max , revealing that the formability depends on the shape produced in the test. In fact, slope distribution along the depth varies from specimen to specimen, thus causing a variation in the location of the transition point whose data have been presented in Table 3. It means that the formability of the sheet depends on the slope distribution along the depth of the test specimen. Even the smallest value of θ max shown by the Test-2 is bigger than that shown by the Test-1 (Fig. 8). This finding strengthens the one discussed in the preceding paragraph. In this context, an explanation based on the methodology as adopted by Iseki [19] in his analysis of the deformation force may be given as follows. Say a part having varying wall angle and a part having constant wall angle are under the forming process (Fig. 9). Let F is the resultant deformation force on an arbitrary point N, which is an instantaneous position of the forming tool, under equilibrium conditions. As the force distribution in SPIF is very complex, it is analyzed on one point and the reaction forces are neglected

Fig. 9. An analysis of the deformation force: (a) in a part with varying wall angle, and (b) in a part with constant wall angle.

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for simplification. The part’s segment on which the force component F sin θ acts is longer in the part whose wall angle does not vary than in the part whose wall angle varies, i.e., MN  LN. Although the frictional force between the tool and sheet may partially nullify the effect of F sin θ, the possibility that the force affects the metal in the vicinity of the contact area with the tool can not be ruled out. Therefore, the affected vicinity might be larger in the part having constant wall angle than in the part having varying wall angle. Due to this reason, an earlier fracture occurs in the Test-1 than in the Test-2. 5. Concluding remarks In the present study, two tests were conducted to evaluate the formability of an aluminum sheet in negative incremental forming. In the first test, collections of cones and pyramids were produced by varying the deformation angle in small steps in order to investigate the maximum wall angle‘θ max ’ without fracture of the sheet. In the second test, four conical pyramids, each having varying wall angle, designed by revolving different curved lines were formed to fracture. The results of the tests were analyzed quantitatively, and some observations were made that are given as below: (1) In the first test, during searching θ max , some parts having sharp corners, i.e., square pyramids, fractured at the corners, which shows that a corner undergoes greater deformation than a straight side for the same tool path. However, from a practical standpoint, this fact does not affect the value of θ max remarkably. (2) In the second test, the value of θ max varies from specimen to specimen, showing that the formability of a sheet depends on the curvature of the generatrix of the test specimen. (3) The formability values shown by the second test are higher than that shown by the first test. This is due the fact that the wall angle of each specimen in the second test varies, while the specimens of the first test have constant wall angles. Although either of the aforementioned tests can be employed to investigate θ max , a safe formability value is always required to limit the design specifications of components so as to avoid sheet fracture during forming. On the other hand, the first test in which a large number of parts have to be formed is costly and time consuming. Therefore, both the tests should be employed in combination in order to acquire the minimum possible value of θ max . A new testing procedure is proposed as follows: (1) Form a part having varying wall angle, and determine a value of θ max , say it is θ max(ref) ; (2) Form either a cone or a pyramid with the constant wall angle of θ max(ref) ; and form few more parts by decreasing wall angle until the sheet would withstand a maximum wall angle without fracturing.

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As discussed in Section 5, the metal in the vicinity of the contact area with the tool is affected in SPIF. Further research is required in order to investigate the relationship among the formability, the deformation force, the curvature of generatrix and the vicinity affected. References [1] M. Otsu, K. Osakada, M. Fujii, Controlled laser forming of sheet metal with shape measurement and using database, in: Proceedings of the Metal Forming 2000, Rotterdam, 2000, p. 433. [2] B. Jurisevic, K.C. Heiniger, K. Kuzman, M. Junkar, Incremental sheet metal forming with a high-speed water jet, in: Proceedings of the IDDRG Congress, Bled Slovenia, May 11–15, 2003, pp. 139–148. [3] L. Filice, L. Fratini, New trends in sheet metal stamping processes, in: Proceedings of the PRIME Conference, 2001, pp. 143–148. [4] S. Matsubara, Incremental backward bulge forming of a sheet metal with a hemispherical head tool, J. JSTP 35 (1994) 1311. [5] J. Jeswiet, Incremental single point forming, in: Proceedings of the Conference of North American Manufacturing Research Institution (NAMRI), 2001, pp. MF01–MF246. [6] T. Maki, Sheet Fluid Forming and Sheet Dieless NC Forming, Amino Corporation, Japan, 2005. [7] J. Kopac, Z. Kampus, Incremental sheet metal forming on CNC milling machine-tool, J. Mater. Process. Technol. 162/163 (2005) 622– 628. [8] J.J. Park, Y.H. Kim, Fundamental studies on the incremental sheet metal forming technique, J. Mater. Process. Technol. 40 (2003) 447–453. [9] G. Ambrogio, I. Costantino, L. De Napoli, L. Filice, L. Fratini, M. Muzzupappa, Influence of some relevant process parameters on the dimensional accuracy in incremental forming: a numerical and experimental investigation, J. Mater. Process. Technol. 153/154 (2004) 501–507. [10] W.B. Livers, A.K. Pilkey, D.J. Lloyd, Using incremental forming to calibrate a void nucleation model for automotive aluminum sheet alloys, Acta Mater. 52 (2004) 300–3007. [11] D. Leach, A.J. Green, A.N. Bramley, A new incremental forming process for small batch and prototype parts, in: Proceedings of the Ninth International Conference on Sheet Metal, Leuven, 2001. [12] H. Amino, Y. Lu, S. Ozawa, K. Fukuda, T. Maki, Dieless NC forming of automotive service panels, in: Proceedings of the Conference on Advanced Techniques of Plasticity, 2002, pp. 1015–1020. [13] G. Ambrogio, L. De Napoli, L. Filice, G. Gagliardi, M. Muzzupappa, Application of incremental forming process for high customized medical product manufacturing, J. Mater. Process. Technol. 162/163 (2005) 156– 162. [14] H. Iseki, H. Kumon, Forming limit of incremental sheet metal stretch forming using spherical rollers, J. JSTP 35 (1994) 1336. [15] M.S. Shim, J.-J. Park, The formability of aluminum sheet in incremental forming, J. Mater. Process. Technol. 113 (2001) 654. [16] D. Young, Formability in computer numerically controlled single point incremental forming, Thesis, Queens University in conformity with the requirements for the degree of Master of Science, Canada, 2003. [17] G. Hussain, L. Gao, A novel method to test the thinning limit of sheet-metal in Negative Incremental Forming, Int. J. Mach. Tools Manuf., 2006 (URL: www.sciencedirect.com). [18] H.Y. Wei, L. Gao, S.G. Li, Investigation on thickness distribution of bulge type incrementally formed sheet metal part with irregular shapes, in: Proceedings of the IMCC, vol. 34, China, 2004, p. 147. [19] H. Iseki, An approximate deformation analysis and FEM analysis for the incremental bulging of sheet metal using spherical roller, J. Mater. Process. Technol. 111 (2001) 150.