Strain and thickness distribution

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 c If in the condition shown, the edge of the sheet just comes to the point F , estimate approximately the initial semi blank width. d If, in the position shown, the side wall is about to split, estimate the punch force P and the strain at the centre ε 1 . Solutions 1. Tension at the centreline T O = σ 1 t = Kε n 1O t exp −ε 1O = 750 × 10 6 × 0.8 × 0.03 0.23 exp −0.03 = 260kNm 2. Arc length sin θ = a − R P R f − R P = 0.33 − 0.01 2.8 − 0.01 , θ = 0.115rad = 6.6 ◦ arc OA = R f − R p θ = 2.8 − 0.01 × 0.115 = 0.321m arc AB = R P π 2 − θ = 0.01 × π 2 − θ = 0.015m CB = 0.028 Arc CD = π2 × 0.01 = 0.015m DF = 0.080m ∴ arc length OF = 0.460m 3. Tensions Using Equation 4.13b T A = T O expμθ = 260 exp0.1 × 0.115 = 263kNm T B = T O expμπ2 = 263 exp0.1 × 3.142 = 304kNm T C = T B T C = T D expμπ2 = T D exp 0.1 × 3.142 = 304kNm T D = T E = 308 expμπ2 = 260kNm 4. Blankholder force From Equation 4.14, we have T D = 2μB, ∴B = T D 2μ = 260 2 × 0.1 = 1300kN 5. Strains T A = Kε n 1A t exp −ε 1A = 600ε 1A exp −ε 1A = 263 ∴ by trial and error, ε 1A = 0.032 Simplified stamping analysis 55