The simplest way is to inflate a truck inner tube inside the laminated shell and pressure plates. It will try to ooze out the end faces, so you would need substantial bulkheads to trap it at each end of the assembly.
The mechanical expansion mechanism shown in your first image is probably a little safer. The radial links between the core 'star' and the shoes appear to be toggle links. I.e., they are just a little too long to fit nicely in the space, and the screw bears against something to move the 'star' axially and push the links into more nearly radial positions.
Google 'toggle linkage' for help with that.
Note: I'm not completely sure that the image actually does depict a quad toggle mechanism; that's just my best guess from what I can see.
Rather than trying to analyze the linkage, which is possible if your geometry is well known and consistent, it might be advisable to make the toggle links of adjustable length.
The quick and dirty way is a bucket of vise-grips, which also contain a toggle lock mechanism.
Mike Halloran
Pembroke Pines, FL, USA