Force equilibrium at the blank-holder and punch The punch force

q B q B O A B F C T 1B T 1B sin q B Figure 4.7 Diagram showing the relation between punch force and side-wall tension.

4.2.8 Tension distribution over the section

It is now possible to determine the tension at each point along a strip as illustrated in Figure 4.8. If the strain at the mid-point ε 1O is known or specified, the centre-line tension T 1O can be calculated from Equation 4.8. As the sheet between O and B is sliding outwards against an opposing friction force from B to O, the tension in the sheet will increase. The angle of wrap θ B can be determined from the punch depth h and the tool geometry. The tension at B can be found from Equation 4.13b, i.e. T 1B = T 1O expμ.θ B O q B A B D F G B E B C a C O A B D E F G T 1 s b Figure 4.8 Distribution of tension forces across the sheet in a draw die. In the side-wall, between B and C, the sheet is not in contact with the tooling and the tension is constant, i.e. T 1C = T 1B . If the surface of the sheet under the blank-holder is horizontal as shown, the angle turned through between C and D will be the same as θ B and hence the tension at D, and also at E, will be equal to that at the centre-line, i.e. T 1D = T 1E = T 1O . From E to F, the tension falls to zero as indicated in Section 4.2.6. 52 Mechanics of Sheet Metal Forming 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