# • Hodographs II

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.

\(R = \) \(\frac{{{A_0}}}{{{A_1}}}\) = \(\frac{H}{h}\) 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:

Click here for a full mathematical analysis of this hodograph.

An alternative approach to an extrusion hodograph assumes there is a 'dead metal' zone.

Then \(p\frac{H}{2}\) = \(k\left\{ {PQ{v_{PQ}} + DQ{v_{dq}} + QR{v_{qr}}} \right\}\)

After similar algebra to the previous example, we obtain

\[\frac{p}{{2k}} = \frac{1}{{2\left( {\sin \varphi - \cos \varphi } \right)}}\left\{ {\frac{{R + 1}}{{\sin \varphi }} - 2\left( {R - 1)\cos \varphi } \right)} \right\}\]

Minimising RHS, \(\cot \varphi = 1 - \) \(\frac{2}{{\sqrt {R + 1} }}\)

After more algebra, it is found that

\[\frac{{{p_{\min }}}}{{2k}} = 2\left( {\sqrt {R + 1} - 1} \right)\]

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.

\( \Rightarrow \) For R < 4 this is a better upper bound solution for extrusion problems.

Click here for a full mathematical analysis of this hodograph.