The hydrostatic bulging test

r a a b s f s q e f , e q e f e q Figure 9.2 a Membrane stresses on the spherical shell. b Distribution of membrane strains at some stage in the bulging process. As indicated, this process is used as a test for mechanical properties. The strain at the pole can be measured by a thickness gauge, or by measuring in some way the expansion of a circle at the pole; this information, together with measurement of the curvature of the pole and the current bulging pressure, permits calculation of the effective stress–strain curve from Equations 9.1 to 9.3. If the material is anisotropic, the stress–strain curve obtained shows the properties in the through-thickness direction. It may be shown that the stress state at the pole is equivalent to hydrostatic tension plus uniaxial compression in the through-thickness direction and it is assumed that the hydrostatic stress will not influence yielding. The most important reason for using this test is that quite large strains can be obtained before failure even in materials having very little strain-hardening. Following Section 5.4.1, and as shown in Figure 9.3, the membrane strain at failure in biaxial stress, ε ∗ φ = ε ∗ θ , is greater than the strain at necking in the tensile test. Also, from Equation 9.2, the effective strain is twice the membrane strain in biaxial tension. Necking in tensile test Bulge test e q e f e ∗ q , e ∗ f n Figure 9.3 Forming limit diagram for a low-strain-hardening material showing the end points in the tensile and bulge tests.

9.1.2 An approximate model of bulging a circular diaphragm

An approximate model of the process can be obtained if it is assumed that the surface is spherical, that the membrane strains are equal everywhere and not just at the pole and that the thickness is uniform. The area of the deformed surface in Figure 9.1 is 2πρh and equating volumes before and after deformation gives π a 2 t = 2πρht 130 Mechanics of Sheet Metal Forming 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