Rigid, perfectly plastic bending

4 St 2 M r 1 Curvature Figure 6.10 The moment curvature diagram for a rigid, perfectly plastic sheet bent without tension. t 2 t 2 S − S y e M s 1 Figure 6.11 Stress distribution for an elastic, perfectly plastic sheet bent without tension.

6.5.3 Elastic, perfectly plastic bending

For curvatures beyond the limiting elastic curvature 1ρ e and below that where the moment reaches the fully plastic moment M p , an elastic, perfectly plastic model, as in Section 6.4.1, is often used. The model is illustrated in Figure 6.6b; the flow stress is constant and for plane strain, σ 1 = 2 √ 3 σ f = S. The stress distribution is illustrated in Figure 6.11; for y y e , the material is plastic with a flow stress S. As the curvature increases, y e decreases and at any instant is given by ε b y =y e = y ρ = S E ′ i.e. y e = S E ′ 1 1ρ = m t 2 6.22 From Equation 6.19, m = 1ρe 1ρ and 1 ≥ m ≥ 0. The equilibrium equation, from Equation 6.16, is M = 2 y e E ′ y ρ y dy + t 2 y e Sy dy = St 2 12 3 − m 2 6.23 The moment, curvature characteristic is shown in Figure 6.12 and it may be seen that this is tangent to the elastic curve at the one end and to the fully plastic curve at the other. Bending of sheet 89 It may be seen that with this non-strain-hardening model, the moment still increases beyond the limiting elastic moment and reaches 1.5M e before becoming constant. For this reason, elastic plastic bending is usually a stable process in which the curvature increases uniformly in the sheet without kinking. It may be shown that for materials that do not fit this elastic, perfectly plastic model, for example aged sheet having a stress–strain curve as shown in Figure 1.4, the moment characteristic is different and kinking may occur. In real materials it is very difficult to predict precisely the moment curvature character- istic in the region covered by the bold curve in Figure 6.12 from tensile data. The moment characteristic is extremely sensitive to material properties at very small strain and these properties often are not determined accurately in a tension test. M r 1 Curvature 4 St 2 6 St 2 Figure 6.12 Moment curvature diagram for an elastic, perfectly plastic sheet bent without tension.

6.5.4 Bending of a strain-hardening sheet

If a power law strain-hardening model of the kind shown in Section 6.4.3 and Figure 6.6d is used, the stress distribution will be as shown in Figure 6.13. The whole section is assumed to be deforming plastically and the stress at some distance, y, from the middle surface is σ 1 = K ′ ε n 1 ≈ K ′ y ρ n 6.24 t 2 t 2 y M s 1 Figure 6.13 Stress distribution for a power-law-hardening sheet bent without tension. The equilibrium equation can be written as M = 2K ′ 1 ρ n t 2 y 1 +n dy = K ′ 1 ρ n t n +2 n + 2 2 n +1 6.25 90 Mechanics of Sheet Metal Forming