Reverse bending Elastic unloading and springback

6.7 Small radius bends

6.7.1 Strain distribution

In the previous sections, the strain in bending was assumed to be a linear function of the distance from the middle surface. If the radius of the bend is approximately the same as the sheet thickness, a more refined analysis is necessary. In Figure 6.20, a length l of sheet of thickness t is bent under plane strain and constant thickness conditions to a middle surface radius of ρ. In the deformed shape, the length of the middle surface is l a = ρθ A l B C D y t 2 y A′ B′ C′ D′ q R r r b r t Figure 6.20 Element of sheet bent to a small radius bend. and the volume of the deformed element is, θ 2 R 2 − R 2 1 where R = ρ + t2 is the radius of the outer surface and R = ρ − t2 is the radius of the inner surface. As the volume of the element remains constant, we obtain from the above l t 1 = θ 2 R 2 − R 2 1 = θρt i.e. l = θρ 6.37 If the length of the middle surface in the deformed condition is l a then, l a = ρθ = l 6.38 i.e. the length of the middle surface does not change. In the previous simple analysis, it was assumed that a fibre at some distance y from the middle surface remained at that distance during bending. In small radius bends, this is 96 Mechanics of Sheet Metal Forming not the case. We consider a fibre y from the middle surface in the undeformed state as shown in Figure 6.20. Equating the shaded volumes shown, θ 2 r 2 − ρ − t 2 2 = y + t 2 l 6.39 Given that, θ = lρ = l ρ , the radius of the fibre initially at y is r = 2ρy + ρ 2 + t 2 4 6.40 The length of the fibre, A ′ B ′ , is l = rθ, and substituting θ = l ρ we obtain l l = 1 + 2y ρ + t 2ρ 2 6.41 The change in fibre length on bending is shown in Figure 6.21a. Fibres initially above the middle surface will always increase in length. Fibres below the middle surface will decrease in length initially, but may then increase. The minimum length is denoted by the point B in Figures 6.21a and b. The minimum length is found by differentiating Equation 6.41 with respect to curvature 1ρ and equating to zero. From this, the strain in a fibre begins to reverse when t ρ = − 4y t 6.42 Substituting in Equation 6.41, the minimum length of such a fibre is l l = 1 − 2y t 2 6.43 and the radius of the fibre at B that has reached its minimum length is r b = ρ 2 − t 2 2 = RR 6.44 The importance of this strain reversal is that it must be taken into account in determining the effective strain in a material element at a particular distance from the middle surface. If there is no reversal, the effective strain can be determined approximately from the initial and final lengths of the fibre. If there is a reversal, the effective strain integral should be integrated along the whole deformation path. In Figure 6.21c, the bold curve on the right shows the effective strain determined from the above analysis. The broken line is that determined from the simple, large bend ratio analysis. There is a significant difference due to two factors: • appropriate integration of the effective strain; and • the non-linear distribution of strain derived from Equation 6.41. The curves in Figure 6.21c are calculated for a bend ratio ρt = 23. At the inner surface, y = −t2, the strain calculated from Equation 6.41 is ε 1 = −1.4. In the simple analysis ε 1 = y ρ = −t 2ρ = −0.75 Bending of sheet 97