# Manufacturing Processes - MFRG 315 - 3.2 Metal Forming

3.2.5 Redundant Work
Even after determining tool stresses and forces accounting for the effects of friction, in some processes the the calculations may give values that are too low. In some processes, drawing and forging and particularly piercing and extrusion, external constraint requires significant internal distortion of the workpiece beyond that strictly necessary for the shape change. This effect can be seen in the adjacent diagram where a narrow, infinitely wide punch is indenting an infinite plane. (nonhomogeneous) deformation occurs.

For plane strain deformation a method known as 'slip line field analysis' can be used which gives good agreement with experimental results. A slip line field is a two dimensional vector diagram which shows the directions of maximum shear stress, identifies with the directions of slip, at any point. There are always two such directions as shear is accompanied by a complementary shear at 90o. The yield flow shear stress 'k' is assumed to act along these slip lines.

The deformation occuring during the indentation can be represented in simplified form by slip lines due to the movement of non-deforming triangular prisms or wedges, see diagram.

The wedge under the punch moves down with velocity v and displaces the two adjacent wedges horizontally with velocities vbc, see velocity diagram. As there is no sliding of the workpiece on the tool the effect of the effect of friction is zero in this case. These push the outer two wedges upward so that they form bulges at the sides of the indentation.

Work is done by shearing along boundaries AB, BC, AC, CD, BG, BF, FG and EF.
The shearing force along the wedge boundaries per unit width, w, is equal to the yield flow shear stress, k, times the length of the boundary. Fro the velocity diagram the velocities along these boundaries are given by:

vBC = v; vBA = 1.4142 v; vCA = vDC = v/1.4142.

The total power to be provided by the pressure p, along AG is the sum of all the shearing powers:

Lpv = 2k(Lv + 1.4142 Lv/1.4142 + Lv/2 + Lv/2)

The terms in the brackets correspond to the boundaries BC, AB Ac and AD respectively and each has to be counted twice. This simplifies to:

p = 6 k
or using the von Mises criterion: p = 6 Y / 1.732 = 3.46 Y

Due to the simplification, this result is slightly higher than the exact solution which is given as p = 5.14 k = 2.97 Y

3.2.6 Relationship Between Hardness and UTS
The analysis in the previous section indicates that there may be a relationship between the hardness and the UTS of metals. The relationship, which is reasonably good for steels, is that when converted into the SAME units: BHN = 2.84 UTS.
Or UTS (in MPa) = 3.45 BHN.
This is not quite linear as it depends upon the degree of strain hardening, which depends upon the load used.

References:
1. 'Introduction to Manufacturing Processes', J A Schey, McGraw-Hill International, 1987.

David J Grieve, 24th October 2008.