Thickness of the element Stress on the element Tension or traction force at a point

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 element. d Resultant of tension forces acting radially inwards. Simplified stamping analysis 49 The length of the element can be expressed in terms of the tool radius and the angle subtended, i.e. ds = R dθ and the surface area for a unit width of sheet is R dθ 1 The force acting on the element radially outward is pR dθ The force tangential to the sheet due to friction is μpR dθ The tension forces are T 1 and T 1 + dT 1 . As the direction of the tension forces differs by an angle, dθ , there is a radially inward component of force, T 1 dθ , as shown in Figure 4.5d. The equilibrium equation for forces in the radial direction is T 1 dθ = pR dθ or p = T 1 R 4.11 It is useful at this stage to re-arrange Equation 4.11. Recalling that T 1 = σ 1 t , the contact pressure is p = σ 1 Rt 4.12 The contact pressure as shown in Equation 4.12 is inversely proportional to the bend ratio Rt . The radius of curvature of the punch face is likely to be several orders of magnitude greater than the thickness and even at most corner radii, it will be 5 to 10 times the thickness. The principal stress σ 1 is at most only 15 greater that the flow stress σ f , and so the contact pressure will be a small fraction of the flow stress, justifying the assumption of plane stress, except at very small radii in the tooling. The equilibrium condition for forces along the sheet is, from Figure 4.5, T 1 + dT 1 − T 1 = μp1R dθ or, combining the above equations, dT 1 T 1 = μ dθ 4.13a From Equation 4.12, the contact pressure depends on the radius ratio, but the change in tension as given in Equation 4.13a is independent of curvature and is a function of the coefficient of friction and the angle turned through, sometimes called the angle of wrap. If the tension at one point, j , in the section is known, then the tension at some other point, k, can be found by integrating Equation 4.13a, i.e. T 1k T 1j dT 1 T 1 = θ j k μ dθ 50 Mechanics of Sheet Metal Forming