Effect of punch shape and friction

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 10 Combined bending and tension of sheet

10.1 Introduction

In Chapter 6, the bending of sheet under a pure moment was studied. Here we investigate several situations in which both a tension and a moment are applied to the sheet. In the first instance, we consider an elastic, perfectly plastic sheet bent elastically over a former and then tension is applied. Combined tension and moment on a rigid, perfectly plastic sheet is then analysed and the case of sheet being dragged under tension over some die radius is studied. A similar nomenclature to that given in Section 6.2. is used; i.e. continuous sheet is subject to a force per unit width, or tension T applied at the mid-surface and moment per unit width of M. Bending occurs under a plane stress, plane strain state; the elastic modulus is E ′ as given in Equation 6.10, and the plane strain yield stress is S. It should be noted that because the analysis of plastic bending is non-linear, the order in which the tension and the moment are applied may influence the result.

10.2 Stretching and bending an elastic, perfectly plastic sheet

If the desired curvature of a sheet is less than the limiting elastic curvature, the sheet cannot be formed to shape simply by bending over a die block or former. It would either springback to the flat shape, or, if it was over-bent until it became partially plastic, the springback would be so great that the process would be difficult to control. In gently curved parts such as aircraft skin panels, the sheet is gripped on either side and pulled; it is then wrapped under tension over a former. Here we consider a case more akin to a stamping operation where the sheet is first curved elastically to the shape of the former and then tension is applied. This is illustrated in Figure 10.1, and a model is developed for a two-dimensional, frictionless case. M M T T r Figure 10.1 Bending and then stretching a sheet over a large radius of curvature former. 136