Strain of an element

3 1 Principal directions s 1 , e 1 s 2 = s 1 2 e 2 = 0 T 1 T 2 Figure 4.4 An element of the sheet sliding on the face of the punch.

4.2.2 Thickness of the element

The current thickness can be expressed in terms of the strain, ε 1 , i.e. t = t expε 3 = t exp [ −1 + βε 1 ] = t exp −ε 1 4.3

4.2.3 Stress on the element

The state of stress on the element is σ 1 ; σ 2 = ασ 1 ; σ 3 = 0 4.4 and as β = 0, from Equation 2.14, α = 12. An effective stress–strain law must be chosen. In this case we shall choose one dis- playing a definite initial yield stress: σ = Kε + ε n 3.7 Combining this with Equations 4.2, we obtain σ = K ε + 2 √ 3 ε 1 n 4.5 From this, the major stress σ 1 is obtained using Equation 2.18b, i.e. σ 1 = σ 1 − α + α 2 = 2σ √ 3 4.6

4.2.4 Tension or traction force at a point

As shown in Figure 3.7, for a given material and initial sheet thickness, the tension, or force per unit width at a point can be expressed as a function of the strain at that point. The major principal tension, T 1 , in the sectioning plane is, from the above equations, T 1 = σ 1 t = K ε + 43 1 + β + β 2 ε 1 n √ 1 − α + α 2 t exp −ε 1 4.7 48 Mechanics of Sheet Metal Forming which, for the plane strain case in which β = 0 and α = 12, gives T 1 = 2Kt √ 3 ε + 2 √ 3 ε 1 n exp −ε 1 4.8 and T 2 = T 1 2 4.9 Differentiating Equation 4.8, indicates that the tension reaches a maximum at a strain of ε ∗ 1 = n − √ 3 2 ε 4.10 where the ‘star’ denotes a limit strain for the process. Clearly when the tension reaches a maximum at a point, the sheet will continue to deform at that point under a falling tension. Other regions of the sheet will unload elastically and the sheet will fail at the point where the tension maximum occurred.

4.2.5 Equilibrium of the element sliding on a curved surface

We now consider a larger element of arc length ds as shown in Figure 4.5. If the tool surface is curved, there will be a contact pressure p and if the sheet is sliding along the surface, there will be a frictional shear stress μp, where μ is the coefficient of friction. Both the tension and the thickness will change because of the frictional force. dq dq d s Sliding T 1 + d T 1 T 1 + d T 1 T 1 dq t + dt p p R m p pR dq m p R dq a b c d t T 1 T 1 Figure 4.5 a An element sliding on a tool face. b Thickness of the element. c Forces on the