Determination of optimum process parameters for wrinkle free products in deep drawing process

Journal of Materials Processing Technology 191 (2007) 51–54

Determination of optimum process parameters for wrinkle free products in deep drawing process Anupam Agrawal, N. Venkata Reddy ∗ , P.M. Dixit Department of Mechanical Engineering, Indian Institute of Technology, Kanpur, India

Abstract In the present work, an attempt is made to predict the minimum blank holding pressure required to avoid wrinkling in the flange region during axisymmetric deep drawing process. Thickness variation during the drawing is estimated using an upper bound analysis presented in this paper. The minimum blank holding pressure required to avoid wrinkling at each punch increment is obtained by equating the energy responsible for wrinkling to that which suppresses the wrinkles. The predictions of the developed model are validated with the published numerical and experimental results and are found to be in good agreement. Parametric study is then carried out to study the influence of some process variables on the blank holding pressure to avoid wrinkling. © 2007 Elsevier B.V. All rights reserved. Keywords: Deep drawing; Wrinkling; Blank holder pressure; Upper bound

1. Introduction Deep drawing process design involves, among the other things, determination of minimum blank holding pressure that is required to avoid wrinkles during the process. There have been many attempts [1–5] to obtain the minimum blank holding pressure that prevents wrinkling. Senior [1] has evaluated the energy terms that are responsible for causing wrinkles (due to membrane stresses) and suppressing wrinkles (due to buckling and blank holder pressure), considering the material to be isotropic with constant sheet thickness. Triantafyllidis and Needleman [2] studied the influence of blank holder stiffness on wrinkling behavior using elasto-plastic finite element analysis approach, modeling the flange as an annular plate subjected to axisymmetrical radial tension along its inner edge. They also incorporated the effect of transverse anisotropy on the wrinkling behavior. Yu and Johnson [3] studied the effect of blank holder pressure on the buckling behavior of the flange. In their work, strain energy of bending in plastic buckling is equated with the work done by the in-plane stresses (membrane stresses) to get the critical condition for wrinkling, considering the material to be non-hardening, isotropic with constant sheet thickness. Zeng and Mahdavian [4] studied the effect of temperature on the critical condition for wrinkling. Buckling moment due to ∗

Corresponding author. E-mail address: [email protected] (N.V. Reddy).

0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.03.050

the hoop stress, moment due to resistance to bending deflection, moment due to blank holder force and moment induced at the die radius were considered for predicting critical condition for wrinkling, assuming the material to be isotropic with constant sheet thickness. Wang and Cao [5] have done the analysis for square cup drawing based on energy conservation and plastic bending theory, modeling the flange as a plate under in-plane and lateral loading conditions. They also assumed constant sheet thickness. The above literature survey shows that various approaches used for finding the appropriate blank holding pressure have made assumptions that the sheet thickness does not change during the deformation or the material is isotropic and nonhardening. These assumptions are relaxed in the present work which addresses the problem of predicting the minimum blank holding pressure required to avoid wrinkling in the flange region during axisymmetric deep drawing process. The proposed method takes much less time compared to the finite element analysis, but predicts the final deformation quite accurately. 2. Formulation The formulation section has been divided into two parts; first the formulation for the thickness variation in the flange region is presented. Next part is on the analysis of prediction of number of wrinkles and determination of blank holding force required to avoid the wrinkles.

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A. Agrawal et al. / Journal of Materials Processing Technology 191 (2007) 51–54

Section 2. The volume flow rates at Section 3 and at section B inside zone-II are equated and using the condition that at Section 3 the normal velocity is equal to the punch velocity, we get the expression of velocity at any point in zone-II. The velocity at Section 2 is obtained as, u2 =

Fig. 1. Final cup shape displaying different zones.

2.1. Analysis for thickness variation The total deformation region is divided into different zones as shown in Fig. 1. Analysis of each zone is carried out by proposing a kinematically admissible velocity field, i.e. the velocity field satisfying the condition of normal velocity continuity and volume constancy. Thickness and velocity distribution in each zone is found out [6]. For analysis of flange region (zone-I), the volume flow rate at Section 2 is equated with the volume flow rate at section A (inside zone-I). In case of plastic deformation, the rate of volume change is zero. It can be expressed as, ε˙ ii = 0

(1)

Anisotropy is incorporated through the normal anisotropy ratio ¯ which is defined as the ratio of the width to thickness strain (R) rates. Substituting the values of each strain component in Eq. (1), one can get a differential equation for the radial velocity at section A. Using the boundary condition, uI = u2 at r = rd , the expression for thickness at any section within the flange is obtained as,  (1/R) r tI = t2 (2) rd where uI and tI are the radial velocity and thickness at any point within zone-I. For the analysis of die-arc region (zone-II), it is assumed that there will not be any thickness variation after the material crosses

(rd − ρd )t3 upunch rd tII

(3)

This value of u2 is used as a boundary condition for the zone-I analysis. For the analysis of wall (zone-III), punch-arc (zone-IV) and punch-bottom region (zone-V) it is assumed that once the material has reached the cup-wall after passing through the die-arc region, the deformation is negligible. Hence, no variation of thickness occurs; only stacking of the material takes place. 2.2. Wrinkling analysis For the prediction of number of wrinkles and blank holding pressure necessary to avoid wrinkling, first a suitable waveform is assumed based on geometrical and process conditions. The energy or moment caused by the forces corresponding to assumed deflection profile is determined for the chosen system. When the total energy or moments tending to restore equilibrium is greater than that due to the forces causing instability (occurrence of wrinkling), the system remains stable. Equating the two opposing energies or the moment values gives the critical condition for onset of wrinkling. The present analysis is based on the energy method. The factor responsible for wrinkling is the energy due to compressive hoop stress (Tp ). The bending energy (Up ), and the restraining energy provided by the blank holder (Es ) are the factors resisting the compressive instability (occurrence of wrinkling) of the flange. The condition for onset of wrinkling can be written as, T p = Es + U p

(4)

Fig. 2 shows the assumed waveform and the boundary conditions. Boundary condition for the wrinkled flange can be taken as, i.e. ω = 0 at r = rd and ω ≥ 0 for r0 ≥ r > rd . The waveform which satisfies the above mentioned boundary condition can be

Fig. 2. Assumed waveform and geometrical boundary conditions.

A. Agrawal et al. / Journal of Materials Processing Technology 191 (2007) 51–54

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expressed as ω = α(r − rd ){1 − cos(Nθ)}

(5)

where α = (Ω0 )/(2(r0 − rd )) General expression for the plastic bending energy of the circular flange in terms of the lateral deflection ω is given by [3], 2  2π  r0  2 D ∂ ω 1 ∂ω ∂2 ω ∂2 ω U p = + − D(1 − ν) 2 + 2 2 ∂r r ∂r ∂θ ∂r 0 rd 2    2 1 ∂2 ω 1 ∂ω 1 ∂ω 1 ∂2 ω × + 2 2 + D(1 − ν) − r ∂r r ∂θ r ∂r∂θ r 2 ∂θ × r dr dθ

(6)

where the buckling rigidity D for the plastic deformation can be expressed as D=

E0 t03 , 9

E0 =

4EEt 1/2 2 (E1/2 + Et )

,

Et =

dσ¯ d¯ε

(7)

Substituting the partial derivatives of Eq. (5) into Eq. (6) and considering N to be integer Eq. (6) reduces to,  r0  3πDα2 πα2 D(r − rd )2 4 πDα2 2 p U = + N N + 2r 2r 3 2 rd   1 (r − rd )2 4(r − rd ) dr (8) + × − r r3 r2 The work done by the membrane stresses is given by Yu and Johnson [3],       2  1 2π r0 ∂ω 1 ∂ω 2 p T = − σr t r dr dθ + σ␪ t 2 0 ∂r r ∂θ rd (9) Assuming σ t = σ bh and using the flow rule [9] dεr dε␪ = ¯ r − σ␪ ) + (σr − σt ) ¯ ␪ − σr ) + (σ␪ − σt ) R(σ R(σ =

dεt ¯ R(σt − σ␪ ) + (σt − σr )

¯ = 2.1, t0 = 0.5, Fig. 3. Validation of wrinkling model (Rp = 25, R0 = 50, R n = 0.345, copper).

where ωmax = 2α(r0 − rd ) is the maximum amplitude of deflection of the flange at r = r0 . While using a constant blank holding pressure, an allowance for the maximum wave amplitude is assumed. Senior [1] had concluded that without this assumed allowance the solution of the stability equations are of no use because the maximum amplitude and the number of wrinkles are interrelated. For the current modeling the allowance for the maximum wave amplitude (Ω0 ) is assumed to be 0.00017 mm [7]. Kawai [7] found that this value of Ω0 is more or less constant for material like aluminum, copper and brass under the condition of local wrinkling for deep drawing. Substitution of Eqs. (8), (9) and (11) into Eq. (4) and rearranging, one gets a function for blank holding force σ bh in terms of number of wrinkles N, for the given geometric and material parameters. Differentiating this function with N, one can obtain the minimum blank holding pressure required to suppress these wrinkles. From this blank holding pressure, the blank holding force can be calculated as Fbh = σbh π(r02 − rd2 )

(12)

(10)

The expressions for σ r and σ ␪ can be obtained in terms of σ bh . Substituting these expressions in Eq. (9), we get the expression for Tp in terms of σ bh , dεr , dε␪ and dεt . Strain increment values are obtained from the upper bound analysis of Section 2.1. The other energy term in Eq. (4) is the energy due to restraining forces provided by the blank holder (Es ). The restraining pressure on the blank can be applied either through a constant restraining pressure type blank holder or by a spring type blank holder. The former provides pressure which is independent of the flange deflection whereas the latter provides restraining force which is proportional to the flange deflection. Restraining energy due to blank holder (for constant restraining pressure type blank holder) can be calculated as Es = σbh (R20 − rd2 ){ωmax − t}

(11)

Fig. 4. Minimum blank holding pressure required to suppress wrinkles.

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A. Agrawal et al. / Journal of Materials Processing Technology 191 (2007) 51–54

Fig. 5. Effect of varying normal anisotropy required to suppress wrinkles.

3. Results and discussion To validate the upper bound model its predictions of the punch load and thickness variation versus punch stroke is compared with theoretical and experimental results of Dejmal et al. [8] using the same process conditions and are found to be in good agreement [6]. To validate the proposed wrinkling model, its predictions are compared with the published experimental and simulation results [1,3,7] (Figs. 3 and 4). The predictions of the present model are in very good agreement to the experimental results (Fig. 3), as compared to the simulation results of Senior [1]

and Yu and Johnson [3]. This can be attributed to the fact that present analysis considers the variation in thickness as well as the influence of anisotropy and strain hardening. Fig. 4 shows the comparison of minimum blank holding pressure required to suppress the wrinkles with the experimental results of Kawai [7] and it can be seen that they are in good agreement. It can be seen from Fig. 5 that number of wrinkles increases and the minimum blank holding pressure required decreases with normal anisotropy (Fig. 5). Whereas the number of wrinkles decreases and the minimum blank holding pressure required increases with initial sheet thickness (Fig. 6). The reason for the above variations (Fig. 5) can be explained as follows. The thickening of the flange increases with decrease in normal anisotropy, hence the energy required to further bend the sheet for the given number of wrinkles increases. To avoid further bending, pressure required to suppress wrinkles also increases. Also, as the initial sheet thickness increases the energy required to bend the sheet for the given number of wrinkles increases hence the number of wrinkles decreases and the minimum blank holding pressure required increases (Fig. 6). 4. Conclusions A model based on energy equilibrium method is presented to predict the minimum blank holding force required to prevent wrinkling. The predictions of the present model are in very good agreement (Figs. 4 and 5) with the experimental results [1,7], compared to published results [1,3]. References

Fig. 6. Effect of variation of initial sheet thickness on wrinkling.

[1] B.W. Senior, Flange wrinkling in deep-drawing operations, J. Mech. Phys. Solids 4 (1956) 235–246. [2] N. Triantafyllidis, A. Needleman, An analysis of wrinkling in the swift cup test, ASME J. Eng. Mater. Technol. 102 (1980) 241–248. [3] T.X. Yu, W. Johnson, The buckling of annular plates in relation to deep drawing process, Int. J. Mech. Sci. 24 (3) (1982) 175–188. [4] X.M. Zeng, S.M. Mahdavian, Critical conditions of wrinkling in deep drawing at elevated temperature, J. Mater. Proc. Technol. 84 (1998) 38–46. [5] X. Wang, J. Cao, An analytical prediction of flange wrinkling in sheet metal forming, J. Manuf. Processes 2 (2) (2000) 100–107. [6] A. Agrawal, N.V. Reddy, P.M. Dixit, Optimal blank shape prediction considering sheet thickness variation: an upper bound approach, J. Mater. Proc. Technol., submitted for publication. [7] N. Kawai, Bull. JSME 4–13 (1961) 169–192. [8] I. Dejmal, J. Tirosh, A. Shirizly, L. Rubinsky, On the optimal die curvature in deep drawing processes, Int. J. Mech. Sci. 44 (2002) 1245–1258. [9] R. Hill, The Mathematical Theory of Plasticity, Oxford University Press, London, 1964 (Chapter XII, pp. 317–340).