Springback in an elastic, perfectly plastic material

The ratio of the fully plastic moment to the limiting elastic moment has been shown to be M p M e = 3 2 Therefore combining the above with Equation 6.19, we obtain 1 ρ = − 3 2 1 ρ e = −3 S E ′ t 6.31 If the sheet has been unloaded from a curvature of 1ρ , the proportional change in curvature, from Equation 6.31 is 1ρ 1ρ = −3 S E ′ ρ t 6.32 or, from Equation 6.29, the change in bend angle is θ ≈ −3 S E ′ ρ t θ 6.33 Equation 6.33 is only approximate and applies to small differences in angle or curvature and to the case in which the sheet has been bent to a nearly fully plastic state. Nevertheless, the equation is very useful and indicates that springback is proportional to: • the ratio of flow stress to elastic modulus, SE ′ , which is small and often of the order of 11000; • the bend ratio ρ t ; • the bend angle. Thus springback will be large when thin high strength sheet is bent to a gentle curvature.

6.6.2 Residual stresses after unloading

When an elastic, perfectly plastic sheet is unloaded from a fully plastic state, it is shown above that the change in moment is M = −M p . Substituting in Equation 6.30 −St 2 4 t 3 12 = σ 1 max t 2 i.e. the change in stress at the outer fibre is σ 1 = − 3 2 S 6.34 Equation 6.34 supports the assumption that for the simple bending model given here, the unloading process is fully elastic. Thus the effect of unloading is equivalent to adding an elastic stress distribution of maximum value of –3S2 to the fully plastic stress state as shown in Figure 6.18. The residual stress distribution is shown on the right of Figure 6.18; this an is idealized repre- sentation arising from the simple model, but it does show that after unloading, the tension 94 Mechanics of Sheet Metal Forming ≡ S −3 S 2 M p M = 0 − M p − S 2 + Figure 6.18 Residual stress distribution after unloading from a fully plastic moment. side of the bend would have a significant compressive residual stress at the surface and there would be a residual tensile stress on the inner surface.

6.6.3 Reverse bending

If a sheet has been bent to a fully plastic state and unloaded, it is interesting to see what reverse bending is required to cause renewed plastic deformation. From Figure 6.16, it may be seen that the change in stress required at the outer fibre to just start yielding is −2S. Substituting in Equation 6.30, shows that the change in moment is M = − 2S t 2 t 3 12 = − St 2 3 6.35 The moment for reverse yielding is therefore M rev. = St 2 1 4 − 1 3 = − St 2 12 = − M e 2 6.36 Thus yielding starts at only half the initial yield moment as shown in Figure 6.19. This softening effect is important as there are a number of processes in which sheet goes through bend–unbend and reverse bend cycles. The actual softening is likely to be greater than that calculated above as most materials will also have some Bauschinger effect and yield at a reverse stress of magnitude less than S. M Moment Curvature M e = 4 M p = 1r 2 12 St 2 − − M p St 2 6 St 2 M e = − Figure 6.19 Reverse bending of an elastic, perfectly plastic sheet. Bending of sheet 95