Hole expansion Drawing Applications of the simple theory

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 r i r T f T f T r i r a c b T q − T T f r i T r T q r Figure 7.8 a Drawing of a circular shell. b Location of tensions on the yield locus. c Distribution of tensions in the shell. The equilibrium equation is then dT φ dr + T r = 0 7.9 Integrating and substituting the boundary condition that T φ = 0 at r = r , we obtain, T φ = T ln r r and T θ = T φ − T = −T 1 − ln r r 7.10 This stress distribution is illustrated in Figure 7.8c. It may be seen from Figure 7.8b that the maximum value for the meridional tension at the inner radius of the drawn shell is when T φ = T . Substituting this in Equation 7.10 gives that the maximum size shell that can be drawn is when ln r r i = 1 or r r i = e = 2.72 = LDR 7.11 This so-called Limiting Drawing Ratio LDR given by the simple analysis is very approximate and actual values in the range of 2.0–2.2 are usually observed.

7.5.3 Nosing and flaring of tube

The end of a tube may be ‘nosed’ or ‘necked’ by pushing into a converging die as shown in Figure 7.9a or ‘flared’ using a cone punch as shown in Figure 7.9b. In nosing, the tensions lie in the third quadrant of the yield locus, Figure 7.9c, and it may be shown that, for the boundary condition T φ = 0 at r i the tensions are 114 Mechanics of Sheet Metal Forming r i r i r i r i r i r r r r r r T f T f T f T f T f T f T q T q T q T q a c b d e f − T − T − T r r T r i − T T Figure 7.9 a Nosing and b flaring the end of a tube. Location of the tensions on the yield locus