Effect of strain-hardening Drawing the flange

There are thus two opposing factors determining the drawing stress, one increasing the stress due to hardening of the material and the other reducing the stress as R becomes smaller. Usually the drawing stress will increase initially, reach a maximum and then fall away as shown in Figure 8.6. h s r i Figure 8.6 Typical characteristic of drawing stress versus punch travel for a strain-hardening material.

8.2.2 Effect of friction on drawing stress

There are two separate ways in which friction will affect the drawing stress. One is at the die radius. Section 4.2.5 gives an analysis of an element sliding around a radius. If the changes in thickness are neglected, Equation 4.13a can be written in terms of the stress; i.e. dσ φ σ φ = μ dφ 8.5 At the die radius, as shown in Figure 8.7, we obtain by integration σ φ = σ r i exp μ π 2 8.6 Friction between the blankholder and the flange will also increase the drawing stress. It is a reasonable approximation to assume that the blankholder force B will be distributed around the edge of the flange as a line force of magnitude B2π R o per unit length, as shown in Figure 8.8. The friction force on the flange, per unit length around the edge, is thus 2μB2π R o . This can be expressed as a stress acting on the edge of the flange, i.e. σ r r =R o = μB π R o t 8.7 where t is the blank thickness. σ r i s f r d Figure 8.7 Sliding of the flange over the die radius. 120 Mechanics of Sheet Metal Forming t p R o t m B 2p R o 2p R o 2pR o B B m B s = Figure 8.8 Friction arising from the blankholder force, assumed to act at the outer edge. Both of these factors will increase the stress required to draw the flange. This stress can be determined by various numerical techniques or by approximate models that simplify the effect of strain-hardening and thickness change in the flange Integrating Equation 8.4 for the new boundary condition, given by Equation 8.7, we obtain σ r i = σ f av. ln R o r i + μB π R o t 8.8 and applying Equation 8.6, the stress in the wall is σ φ = 1 η σ f av. ln R r i + μB π Rt exp μπ 2 8.9 Equation 8.9 is an approximate one that neglects, among other things, the energy dissipated in bending and unbending over the die radius. For this reason, an efficiency factor η has been added; this will have a value less than unity.

8.2.3 The Limiting Drawing Ratio and anisotropy

To determine the Limiting Drawing Ratio some method of determining the average flow stress σ f av. , the current thickness and the maximum permissible value of wall stress in Equation 8.9 is necessary. This is beyond the scope of this work, but some comments can be made about the influence of different properties on the Limiting Drawing Ratio. If, in the first instance, we neglect strain-hardening, then the maximum drawing stress will be at the start of drawing; if the flow stress is σ f = Y = constant, the wall stress to initiate drawing is, from Equation 8.9, σ φ = 1 η Y ln R o r i + μB π R o t exp μπ 2 8.10 As indicated, this must not exceed the load-carrying capacity of the wall. If the wall deforms, the punch will prevent circumferential straining, so the wall must deform in plane strain. The stress at which it would deform depends on the yield criterion. Figure 8.9 illustrates yielding in plane strain for various criteria. The line OA indicates the loading path in the cup wall. For a Tresca yield condition, Figure 8.9a, the stress in the wall, σ φ , will have the value Y and substitution in Equation 8.10 gives the condition for the maximum blank size, i.e. ln R o r i + μB π R o t Y exp μπ 2 = η Cylindrical deep drawing 121