Summary 4f handbook jackhumechanicsofsheetmetalformingsecond

is always valid, but the equation for forces tangential to the sheet, dT φ dr − T θ − T φ r = 0 7.4 neglects the effect of friction between the sheet and the tooling. In integrating this equation, we have assumed that the effective tension T is constant. This introduces further error, although as mentioned, it is usually a better assumption than assuming that the material is rigid, perfectly plastic. If the material obeys a stress strain law σ = Kε + ε n , an appropriate value for T could be T = Kε n t 7.14 where t is the initial thickness. If ε is very small, Equation 7.14 would probably under- estimate the effective tension and some discretion should be exercised.

7.7 Exercises

Ex. 7.1 In a hole expansion process, the inner edge is unloaded and the meridional tension at the outer radius r = r is T φ = 2T 3. If the sheet is fully plastic, what is the current ratio r r ? [Ans : 3 ] Ex. 7.2 For the nosing operation shown in Figure 7.9 a, given the boundary condition T φ = 0 at r i , show that the meridional tension is distributed as follows: T φ = −T 1 − r i r and T θ = −T . Ex. 7.3 In a flaring operation, what is the range of r for which the equation T φ = T ln r r is valid? [Ans : r i ≤ r ≤ er i ] 116 Mechanics of Sheet Metal Forming 8 Cylindrical deep drawing

8.1 Introduction

In Chapter 7, a simple approach to the analysis of circular shells was given. Here we examine in greater detail the deep drawing of circular cups as shown in Figure 8.1. This can be viewed as two processes; one is stretching sheet over a circular punch, and the other is drawing an annulus inwards. The two operations are connected at the cylindrical cup wall, which is not deforming, but transmits the force between both regions. The simple analysis in Section 7.5.2 gave a limit to the size of disc that could be drawn as e = 2.72 times the punch diameter. This over-estimates the Limiting Drawing Ratio and in this chapter we investigate various factors that influence the maximum blank size. In the single-stage process shown in Figure 8.1, the greatest ratio of height to diameter in a cup is usually less than unity; this is determined by the Limiting Drawing Ratio. Deeper cups may be made by redrawing or by thinning the cup wall by ironing and these processes are studied.

8.2 Drawing the flange

The flange of the shell can be considered as an annulus as shown in Figure 8.2; the stresses on an element at radius r are shown in Figure 8.3. The equilibrium equation for this element is, in the absence of friction, σ r + dσ r t + dt r + dr dθ = σ r tr dθ + σ θ t dr dθ Die T f Blank Blank- holder Punch a b Figure 8.1 a Drawing a cylindrical cup from a circular disc. b Transmission of the stretching and drawing forces by the tensions in the cup wall. 117