Elastic, perfectly plastic model

is less than the plane strain yield stress, S σ 1 = E ′ ε 1 6.9 where the modulus of elasticity in plane strain is slightly different from the uniaxial Young’s modulus, E; i.e. E ′ = E 1 − ν 2 6.10 where ν is Poisson’s ratio. For strains greater than the yield strain, σ 1 = S 6.11 where S is constant. In isotropic materials, S is related to the uniaxial flow stress by Equation 6.6 for the von Mises yield condition.

6.4.2 Rigid, perfectly plastic model

For smaller radius bends, and where we are not concerned with elastic springback, it may be sufficient to neglect both elastic strains and strain-hardening. A rigid, perfectly plastic model is shown in Figure 6.6c, where σ 1 = S 6.12 and S is a value averaged over the strain range as indicated in Section 3.5.4.

6.4.3 Strain-hardening model

Where the strains are large, the elastic strains may be neglected and the power law strain- hardening model used, where σ 1 = K ′ ε n 1 6.13 For a material having a known effective stress–strain curve of the form σ = σ f = Kε n 6.14 the strength coefficient K ′ can be calculated using Equations 6.6. This model is illustrated in Figure 6.6d.

6.5 Bending without tension

Where sheet is bent by a pure moment without any tension being applied, the neutral axis will be at the mid-thickness. This kind of bending is examined here for several types of material behaviour. In these cases, a linear strain distribution as illustrated in Figure 6.3 is assumed and the equilibrium equations, Equations 6.7 and 6.8, will apply. 86 Mechanics of Sheet Metal Forming e 1 s 1 a b c S S 1 Figure 6.7 Linear elastic bending of sheet showing the material model a, the strain distribution b, and the stress distribution c.

6.5.1 Elastic bending

The material model is illustrated in Figure 6.7a where the yield stress is S. The stress–strain relation is given by Equation 6.9 and for the strain distribution shown in Figure 6.7b, the stress distribution in Figure 6.7c will be obtained. The stress at a distance y from the neutral axis, from Equations 6.4 and 6.13, is σ 1 = E ′ ε b = E ′ y ρ 6.15 The moment at the section, from Equation 6.8, is M = t 2 −t2 E ′ y ρ y dy = 2 E ′ ρ t 2 y 2 dy = E ′ ρ t 3 12 6.16 From Equation 6.15, we have E ′ ρ = σ 1 y and Equation 6.16 can be written in the familiar form for elastic bending: M I = σ 1 y = E ′ 1 ρ 6.17 6 St 2 E ′t 2 S M r 1 Moment Curvature Figure 6.8 Moment curvature diagram for elastic bending. Bending of sheet 87