Thickness change during bending Friction between the points A and B

or T B = T A exp μ π 2 10.24 It is assumed that the tension is less than the yielding tension, so there is no change of thickness during this part of the process.

10.5.4 Unbending at B

During unbending at B there will be an increase in tension and a decrease in thickness as at A above and these can be determined from relations similar to Equations 10.20 and 10.21 using the tension T B instead of T . The final tension in the process in Figure 10.7 is T f = T B + T B .

10.5.5 Worked example drawing over a radius

Sheet is drawn over a radius under plane strain conditions as shown in Figure 10.7. The initial thickness is 2 mm and the tool radius is 8 mm i.e. ρ = 9 mm. The material has a constant plane strain flow stress of 300 MPa. The back tension is 250 kNm and the friction coefficient is 0.08. Assuming an efficiency factor of 0.8, determine by approximate methods the thickness of the sheet and the tension at exit. Solution. The yield tension at the first bend is T y = 300 × 10 6 × 2 × 10 −3 = 600 kN m From Equation 10.2, the increment in tension at the first bend is T = 300 × 10 6 2 × 10 −3 2 4 × 0.8 × 9 × 10 −3 1 + 250 600 2 = 49 kN m The change in thickness at the first bend is t 2 = 2 2 × 9 250 600 = 0.046mm i.e. t = 0.092mm and t = 1.91mm The tension after the first bend is 250 + 49 = 299 kNm. The tension after the sheet has moved over the curved surface, from Equation 10.24, is T = 299 exp 0.08 × π2 = 339 kN m The yield tension at the unbend is T y = 300 × 10 6 × 1.91 × 10 −3 = 573 kN m Combined bending and tension of sheet 149 The increase in tension as the sheet straightens is T = 300 × 10 6 1.91 × 10 −3 2 4 × 0.8 × 9 × 10 −3 1 + 339 573 2 = 51 kN m and the exit tension is 299 + 51 = 350 kNm. The change in thickness at the unbend is t t = − 1.91 2 × 9 339 573 = −0.0628 and the thickness increment is −0.0628 × 1.91 = −0.12, and the final thickness is 1.91 − 0.12 = 1.79 mm.

10.6 Draw-beads

In draw die forming as in Chapter 4, draw-beads are used to generate tension in the sheet. A draw-bead is illustrated schematically in Figure 10.10. The sheet enters at the left with a small tension T , and then undergoes a series of bending or unbending processes marked by the broken lines in the diagram. At each bending or unbending point, following Equation 10.20, the tension will increase, by T = 1 4η T y ρt 1 + T T y 2 and the thickness decrease, as indicated in Equation 10.21, by t t = − 1 2ρt T T y Between the points where the curvature changes, there will be an increase in tension, T f , due to friction, of T + T f = T exp μθ T T f h t n q r Figure 10.10 A draw-bead used to increase the tension in a sheet from T to T f as it passes through from left to right in a draw die. 150 Mechanics of Sheet Metal Forming