The error you get is because the system is not restraint properly, you have to fix at least one point in the first picture where you have only sliding and in the second picture there are too many fixtures. I would fix one face where you apply the load and apply the load on the other end of the rod. The other faces you may or may not apply sliding but this depends on your application, it does not matter at this point.
I'm not sure if that's the main cause but please correct your stress-strain curve...
The first point should be always your elastic limit which is 60MPa base on your example.
I don't think so... if that is the first point then I wouldn't have any elastic behavior in the part.