Aluminium bar cold forming calculation

[glasvegas]

I'm working on a problem that involves calculating force required to form a component. The problem is as follows (diagram attached):

A round aluminium bar, with the initial diameter d0 = 36 mm, has to be cold formed into the component shown in using a combined forward/backward extrusion process. Calculate the force required to form this component. Assume that friction accounts for 55% of the total force and that equation of the work hardening curve for aluminium is σ = 519(0.001 + ε)0.27 MPa.

d1=40 mm
h1 = 90
d2 = 26
h2 = 40
d3 = 31
h3 = 40
α = 45°

I know how to work out ε and then realize that I should integrate σ to get unit plastic work but can't perform this integral. Am a missing some trick or easy way of doing this?

Any help appreciated.

 

[Navin A S]

The solution to this problem involves a few steps. First, you need to calculate the strain ε for the entire workpiece (d1-d3). This can be done by comparing the original diameter d0 with the final diameter of the component. Then, you need to calculate the unit plastic work using the equation of the work hardening curve. This is done by integrating the strain-stress relationship (σ = 519(0.001 + ε)0.27 MPa). Finally, you need to calculate the force required to form the component. This can be done by multiplying the unit plastic work with the total area of the component and then multiplying it by the coefficient of friction which is 0.55 in this case.