An approximate model of bulging a circular diaphragm

from which t = t a 2 2ρh 9.4 From Equation 9.2, for a material obeying the stress–strain law σ = Kε n , the membrane stress is σ φ = σ = K ln t t n From Equation 9.3, the pressure to bulge the diaphragm is p = 2σ φ t ρ = 4σ t h a 2 1 1 + ha 2 2 9.5 using ρ = a 2 + h 2 2h 9.6 In bulging a diaphragm, the pressure may reach a maximum dp = 0 and then bulging continues under a falling pressure gradient. Rupture will occur when the strain at the pole reaches the forming limit as shown in Figure 9.3. This provides an example of different instabilities in processes, as discussed in Section 5.1. If the diaphragm is considered as a load-carrying structure, then the maximum pressure point constitutes instability and failure. If it is a metal forming process, instability is when necking and tearing occur at the forming limit curve, which, as mentioned, is usually beyond the maximum pressure point. As discussed in Section 5.1, most metal forming processes are displacement controlled, rather than load controlled, and local necking usually governs the end-point.

9.1.3 Worked example the hydrostatic bulging test

Equipment designed to obtain an effective stress strain curve by bulging a circular diaphragm with hydrostatic pressure is shown, in part, in Figure 9.4. An extensometer measures the current diameter D of a small circle near the pole of original diameter D and a spherometer measures the height of the pole above this circle h. The current pressure is p and the original thickness of the sheet is t . Assuming that within this circle a state of uniform biaxial tension exists and that the shape is spherical, obtain relations for the effective stress and strain at this instant. p D h Figure 9.4 Small region at the pole in a hydraulic bulge test. Stretching circular shells 131 Solution. The hoop strain around a circle of diameter D is, ε θ = ln π D π D = ln D D Assuming ε θ = ε φ , from Equation 9.1, ε = 2ε θ = 2 ln DD and t = t D D 2 Note that this is a different relation from Equation 9.4 in the approximate model, Section 9.1.2. In the approximate model, the whole of the diaphragm is considered and the dimension a is fixed. With the extensometer, the gauge circle of diameter D increases during deformation. An alternative relation for strain can be obtained assuming that the volume of the spherical cap is 2πρht, and that this originally was of volume π D 2 t 4. This leads to a very similar result to that above. The radius of curvature of the surface, from Equation 9.6, for a = D2, is ρ = h D h 2 + 4 8 and from Equation 9.1 and 9.5, σ = pρ 2t = p 16 h t D h 2 + 4 D D 2

9.2 Stretching over a hemispherical punch

If a disc is clamped at the edge and stretched by a hemispherical punch, as shown in Figure 9.5, the tension in the sheet will increase with punch displacement. If there is no friction between the sheet and the punch, the greatest strain will be at the pole and the strain distribution will be similar to that in Figure 9.2. Failure would be anticipated by tearing at the pole. In practice, it is very difficult to achieve near-frictionless conditions and the effect of friction is investigated here. p m × p T f T f + d Tf R df r r 2 r 1 r Figure 9.5 Stretching a circular blank with a hemispherical punch. 132 Mechanics of Sheet Metal Forming