Tension distribution over the section

The distribution of tension is shown in Figure 4.8b. If the punch face is only gently curved, the angle of wrap and the tension will only increase slowly with distance along the strip from O to A. At the corner radius, the tension increases rapidly and reaches a plateau in the side-wall. It then drops down due to friction at the die corner radius and falls to zero outside the clamping area. The blank-holder force required to generate this tension distribution is found from Equation 4.14. The higher the blank-holder force, the greater will be the strain over the punch face, however the process is limited by the strain in the side-wall. The tension here has a maximum value determined from Equations 4.8 and 4.10. If this maximum is reached, the side-wall will fail by splitting.

4.2.9 Strain and thickness distribution

The distribution of strain corresponding to the tension distribution in Figure 4.8 can be found from Equation 4.8. Writing this in the form ε + 2 √ 3ε 1 n exp −ε 1 = √ 3T 1 2Kt 4.16 shows that ε 1 must be found by a numerical solution. 30 300 250 20 10 0.1 0.2 0.3 0.4 0.5 A B C D E p T P MPa T kNm b 0.08 t 0.04 0.8 0.75 0.73 0.1 0.2 0.3 0.4 0.5 e 1 t mm e 1 A B C D E a Figure 4.9 Distribution of strain, thickness, pressure and tension over the sheet arc length. see Section 4.2.11. Simplified stamping analysis 53 The thickness is given by t = t exp −ε 1 4.17 The approximate current blank width b in Figure 4.3 can be found by equating volumes. It is sufficiently accurate to determine the thickness at the end of each zone, e.g. at O and A from Equation 4.17, and to calculate the volume in this segment as V ol. OA = R P θ A t O + t A 2 4.18 Summing all such volumes from O to F and subtracting from the initial volume gives the current volume between F and G and hence, as there is no change in thickness beyond F 1 F Gt = b t − O →F Vol. 4.19

4.2.10 Accuracy of the simple model

A two-dimensional model of a process that is in fact three-dimensional will obviously be an approximation. The magnitude of the errors will depend on the actual process and judgment must be exercised. An additional source of error is the effect of the sheet bending and unbending as it passes around the tool radii, particularly at the die corner radius, at C, in Figure 4.3. Two effects are important. Bending strains will cause work-hardening in the sheet and also, as shown later, bending and unbending under tension reduce the sheet thickness. Both these effects will reduce the strain at which the tension reaches a maximum and can lead to early failure in the side-wall.

4.2.11 Worked example 2D stamping

Drawing quality steel of 0.8 mm thickness is formed in a stamping dieas shown in Figure 4.3 but with vertical side walls. The plane strain stress–strain relation is σ 1 = 750ε 0.23 1 MPa In the two-dimensional plane strain model, the variables are: Semi punch width: a = 330 mm Punch face radius; R f = 2800 mm Corner radius: R p , R d = 10 mm Side wall length BC = 28 mm Land width: DE = 0 mm Clamp width: EF = 80 mm a Estimate the blankholder force per side, per unit width to achieve a strain ε 1 = 0.03 at the centre if the friction coefficient is 0.1. b If s is the arc length measured along the deformed sheet in the above condition, prepare diagrams in which the horizontal axes are each s, and the vertical axes are: i the membrane strain, ε 1 , ii sheet thickness iii the tension, T in kNm, and iv the contact pressure. 54 Mechanics of Sheet Metal Forming