Worked example punch stretching

and the current thickness is t = t exp −ε t = 0.9 × exp −0.387 = 0.61 mm The stress ratio, from Equation 2.14, is α = 2 × 0.69 + 1 2 + 0.69 = 0.89 The effective strain from Equation 2.19 is ε = 43 1 + 0.69 + 0.69 2 0.229 = 0.389 The effective stress is σ = 700 × 0.389 0.2 = 580 MPa From Equation 2.18, the meridional stress is σ φ = 580 1 − 0.89 + 0.89 2 = 611 MPa The meridional tension is, T φ = 611 × 10 6 × 0.61 × 10 −3 = 373 kNm The semi-angle subtended by the tangent circle is given by sin φ = 2850 The punch force is F = 2πrT φ sin φ = 2π × 28 × 10 3 × 373 2850 = 37 kN

9.3 Effect of punch shape and friction

For punches that are not hemispherical, the strain distribution depends on the punch shape. Two cases are shown in Figure 9.7; the amount of thinning, given by the magnitude of the thickness strain |ε t |, is greatest where the curvature of the profile is greatest. Friction also affects the strain distribution and the greatest punch displacement. On the left of each diagram, the frictionless case is shown; on the right, the case where there is friction between the punch and the sheet is illustrated. In the flat-bottomed punch, Figure 9.7a, friction is confined to the punch corner radius. This prevents the tension across the face of the punch from increasing sufficiently to stretch the material over the face of the punch. The maximum depth that can be formed is therefore less with friction than without it. With the pointed punch, Figure 9.7b, the reverse effect occurs. Without friction, strain is concentrated near the nose and the depth before failure is limited. The effect of friction is to reduce the tension at the nose and spread the strain over a greater area; this permits greater depth of forming. Stretching over a punch is often a pre-form operation, where material is redistributed to obtain favourable conditions for the final stage. In the second stage, the shape of the punch is defined by the part design, but for the preform punch there is some flexibility in 134 Mechanics of Sheet Metal Forming r a b r With friction With friction −e t −e t m m Figure 9.7 Strain distributions for a a flat-bottomed punch, and b a pointed punch, without and with friction. the shape and the tool designer needs to consider both curvature and friction in arriving at a suitable tool.

9.4 Exercises

Ex. 9.1 In the example in Section 9.1.3, the extensometer initially rests on a circle of 50 mm diameter on the flat sheet. The initial sheet thickness is 1.2 mm. At some instant in the test, the pressure is 6.4 MPa, the spherometer measures a vertical distance of 3 mm and the extensometer indicates that the circle has grown to a diameter of 61 mm. Determine the effective stress and strain at this instant. [Ans: 620 MPa, 0.4] Ex. 9.2 Determine the principal radius of curvature in the meridional direction in the unsupported sheet adjacent to the tangent point in the example given in the example in Section 9.2.1. [Ans: −56 mm] Ex. 9.3 Using an approximate analysis, determine the pressure versus bulge height char- acteristic for the operation shown in Figure 9.1. A disk of 1.2 mm thickness is clamped around a circle of 100 mm diameter and bulged to a height of 45 mm. The stress–strain curve is σ = 350ε 0.18 MPa. Determine the effective strain at maximum pressure. [Ans: 0.4] Stretching circular shells 135