Manufacturing Processes  MFRG 315  3.2 Metal Forming 

3.2.5 Redundant Work
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 90^{o}. 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
nondeforming triangular prisms or wedges, see diagram.

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: v_{BC} = v; v_{BA} = 1.4142 v; v_{CA} = v_{DC} = 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: 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

References:
1. 'Introduction to Manufacturing Processes', J A Schey, McGrawHill International, 1987.
David J Grieve, 24th October 2008.