Small strips of various shapes are drawn, however, here we shall take the case of drawing of rectangular strip in plane strain condition. This condition is realized when the ratio width/thickness >>1. Analysis of drawing of non-circular shapes in general by slab method is given here. For analysis the slab method is adapted. Figure 9.21 shows the scheme of the process, in which a slab of width dx and at a distance x from the end of conical portion of die is acted upon by the following stresses which are similar to that in wire drawing.
(i) Axial stress σx acting on smaller flat face and σx + dσx on the bigger flat face of the slab.
(ii) The pressure px acting normally on the contacting surface of the slab with the die.
(iii) The frictional stress τx acting tangentially on the contacting surface in a direction opposite to that of metal flow.
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The equilibrium of these forces in the axial direction gives the following equation.
In the above analysis the stress due to shear deformation at the entry and exit is not included. It may be calculated as given below. Consider a small element dy.dx at a distance y from the central line of strip (Fig. 9.22). This suffers a shear deformation θ. When die angle is small we may take θ ≈ tan θ = y/A. (Fig. 9.22)
The optimum die angle may be calculated from Eqn. (9.50) for particular reduction and friction conditions. At the optimum die angle the drawing stress is minimum. Therefore, it may be obtained by differentiation of σd with respect to α and equating it to zero or by numerical method.
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Example:
A 10 mm wide and 4 mm thick strip is drawn through straight taper dies to 3 mm thickness in plane strain condition. Each die is inclined at an angle of 10 degrees to the central plane of the strip. Determine the drawing pull at the end of inclined portion of die if the average yield stress of strip metal 180 N/mm2 and coefficient of friction is equal to 0.06.
Solution:
