Dissemination of IT for the Promotion of Materials Science (DoITPoMS)

# • 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. 