Worked example the hydrostatic bulging test

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 The distribution of strain, ε φ , from the centre to the clamped edge is shown in Figure 9.6a. Because of the frictional contact stress μp, the maximum tension and strain is at some distance from the pole and this is where failure by splitting is expected. The strain along a meridian is plotted in the forming limit diagram in Figure 9.6b. The sheet tends to thin near B and then tear around a circle at B, where B is an intermediate point between the pole, A, and the edge, C. Frictionless r With friction A B With friction B e f e q e f C A C FLC Frictionless a b Figure 9.6 a Strain distribution for stretching a circular blank and b plot of the strains along a meridian for stretching with and without friction at the punch face. Outside the contact region, the surface pressure is zero and from Equation 7.3 the principal curvatures are connected by T θ ρ 1 = − T φ ρ 2 As both T θ and T φ are tensile positive in punch stretching, it follows that the principal curvatures are of opposite sign and the surface is of negative Gaussian curvature as is shown in Figure 9.5.

9.2.1 Worked example punch stretching

In stretching a sheet over a hemispherical punch as shown in Figure 9.5, the punch diameter is 100 mm and the initial sheet thickness is 0.9 mm. The tangent point dividing the contact from the non-contact region is at a radius of 28 mm. Grid circles on the sheet initially of 3.5 mm diameter are measured at the tangent point; along the meridional direction the major axis of the deformed circle is 4.4 mm and the minor axis in the hoop direction is 4.1 mm. The material has a stress strain relation of σ = 700ε 0.2 MPa. Determine the punch force. Solution. The major and minor strains and the strain ratio are ε φ = ln 4.43.5 = 0.229; ε θ = ln 4.13.5 = 0.158 and β = 0.1580.229 = 0.69 The thickness strain is ε t = − 1 + 0.69 × 0.229 = −0.387 Stretching circular shells 133