Extrusion is an important working process. A simple form of extrusion used for non-ferrous metals involves a smooth square die.
We define extrusion ratio, R = ratio of areas.
for plane strain (R > 1), e.g. R = 4 = 75% reduction in area.
For a square die with sliding on the die face in plane strain, the hodograph can be constructed as follows:
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Click here for a full mathematical analysis of this hodograph.
An alternative approach to an extrusion hodograph assumes there is a 'dead metal' zone.
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Then 
After similar algebra to the previous example, we obtain
Minimising RHS, 
After more algebra, it is found that
Note that for low R (< 4) this value is less than that for sliding on the die face, even if the die face is frictionless.
For R < 4 this is a better upper bound solution for extrusion problems.
Click here for a full mathematical analysis of this hodograph.
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