Approximate models of forming axisymmetric shells

boundary will therefore be a central force along the axis, as shown in Figure 7.5. The magnitude of the axial force is, Z = T φ 2π r sin φ 7.5 where the subscript zero refers to the boundary conditions. • At any instant during forming, all elements of the shell are assumed to be deforming plastically. • The sheet obeys a plane stress Tresca yield condition and, as mentioned, strain-hardens in such a way that the product of flow stress and thickness remains constant, i.e. σ f t = T = constant. f Z T f r Figure 7.5 Boundary force conditions for a shell.

7.5 Applications of the simple theory

7.5.1 Hole expansion

We consider a circular blank stretched over a domed punch as shown in Figure 7.6a. At any instant, there is a circular hole at the centre of radius r i , and a meridional tension is applied at the outer radius r . At the edge of the hole, the meridional tension must be zero and a state of uniaxial tension in the circumferential direction would exist. We expect that the meridional tension would become more tensile towards the outer edge and in the yield locus, the tensions would fall in the first quadrant of the diagram as illustrated in Figure 7.6b. T f T q r i r i r r T f T a b Figure 7.6 a Hole expansion process with the sheet stretched over an axisymmetric punch. b Region on the tension yield locus for this process. 112 Mechanics of Sheet Metal Forming As seen from Figure 7.6b, the tensions in the sheet are T φ and T θ = T . The equilibrium equation, Equation 7.4, is dT φ dr − T − T φ r = 0 7.6 which on integrating and substitution of the boundary condition, T φ = 0 at r = r i , gives T φ = T 1 − r i r 7.7 The tension distribution given by Equation 7.7 is illustrated in Figure 7.7. T f T q r i r r T T Figure 7.7 Stress distribution for hole expansion in a circular blank. For a circular blank without a central hole, the stress state at the pole is, by symmetry, that T φ = T θ . From Figure 7.6b it is seen that this can only occur when both are equal to the yield tension T . The equilibrium equation is then dT φ dr = 0 i.e. the merdional tension does not change with radius and hence the stress distribution is uniform and T φ = T θ = T This relation is useful for determining the punch load. If, instead of being stretched over a punch, the sheet is clamped around the edge and bulged by hydrostatic pressure, the hoop strain around the edge will be zero and the hoop tension at the outer edge will be less than the meridional stress. In this case, the simple model does not predict the tension distribution well; at the edge, the strain state must be plane strain and from the flow rule, Equation 2.13c, we predict that T θ = T φ 2.

7.5.2 Drawing

If a circular blank is drawn into a circular die as shown in Figure 7.8a, we may anticipate that the meridional tension will be tensile positive at the throat and zero at the outer edge. As any circumferential line will shrink during drawing, the hoop tensions are likely to be negative or compressive. The tensions will therefore lie in the second quadrant of the yield locus as shown in Figure 7.8b where T φ − T θ = T 7.8 Simplified analysis of circular shells 113