Geometry and strain in bending Plane strain bending

y e b e a 2 t 2 t Figure 6.3 Assumed strain distribution in bending. Following Equations 2.18b and 2.19c, for, β = 0, α = 12, we obtain σ 1 = 2 √ 3 σ f = S and ε 1 = √ 3 2 ε 6.6 where S is the plane strain flow stress. Equation 6.6 assumes the von Mises yield condi- tion. If a Tresca yield criterion is assumed, σ 1 = σ f = S. The stresses on a section along the bend axis are illustrated in Figure 6.4. Clearly, at the edge of the sheet, the stress along the bend axis will be zero at the free surface and plane strain will not exist. It is usually observed that the edge of the sheet will curl as illustrated. This happens because the stress state is approximately uniaxial tension near the edges of the sheet; the minor strain will be negative near the outer surface and positive near the inner surface giving rise to the anticlastic curvature as shown. Within the bulk of the sheet, however, plane strain deformation is assumed with the minor strain along the axis of the bend equal to zero. s 1 s 2 , e 2 = 0 Free edge Figure 6.4 Stress state on a section through the sheet in plane strain bending.

6.3 Equilibrium conditions

We consider a general stress distribution on a normal section through a unit width of sheet in bending, as shown in Figure 6.5. The force acting on a strip of thickness dy across the unit section is σ 1 × dy × 1. The tension T on the section is in equilibrium with the integral of this force element, i.e. T = t 2 −t2 σ 1 dy 6.7 84 Mechanics of Sheet Metal Forming R r y t 2 t 2 s 1 × d y × 1 M T dy a b Figure 6.5 Equilibrium diagram a for a section through a unit width of sheet and b a typical stress distribution. Integrating the moment of the force element, we obtain M = t 2 −t2 σ 1 dy1y = t 2 −t2 σ 1 y dy 6.8 We note too that there is a third equilibrium equation for forces in the radial direction arising from the tension T . This is given in Section 4.2.5 by Equation 4.11.

6.4 Choice of material model

For the strain distribution given by Equation 6.3, the stress distribution on a section can be determined if a stress strain law is available. In general, the material will have an elastic, plastic strain-hardening behaviour as shown in Figure 6.6a. In many cases, it is useful to approximate this by a simple law and several examples will be given. The choice of material model will depend on the magnitude of the strain in the process. The strain will depend mainly on the bend ratio, which is defined as the ratio of the radius of curvature to sheet thickness, ρt.

6.4.1 Elastic, perfectly plastic model

If the bend ratio is not less than about 50, strain-hardening may not be so important and the material model can be that shown in Figure 6.6b. This has two parts, i.e. if the stress s 1 e 1 E ′ S S s 1 = K′e n 1 a s 1 e 1 b s 1 e 1 c s 1 e 1 d Figure 6.6 Material models for bending. a An actual stress–strain curve. b An elastic, perfectly plastic model. c A rigid, perfectly plastic model. d A strain-hardening plastic model. Bending of sheet 85