Problem

A direct extrusion process starts with a $6061-0$ Aluminum billet with diameter $4$ in $(10\mathrm{.}16cm)$ and length of $50$ cm and ends with a rectangular C $-$Channel as shown in Figure $1.$ Use the Johnson formula with $a=0\mathrm{.}7$ and $b=1\mathrm{.}2$ to estimate the extrusion strain; the strength coefficient of $6061-0$ Aluminum is $205$ MPa and the strain$-$hardening exponent is $0\mathrm{.}20.$ In this process, $5$ cm will be left as a butt out of the whole length of the billet.

a$)$ Calculate the extrusion ratio.

b$)$ Calculate the extrusion force at the beginning of the extrusion process and at the end of the extrusion process.

c$)$ Calculate the power needed to execute this extrusion process knowing that the billet will be extruded in $30$ s$.$

Figure $1$ The base of the C$-$Channel is $10$ cm $,$ its height is $5$ cm and the wall thickness is $1\mathrm{.}25$ cm $($all around$).$