Unless I'm misunderstanding what you mean by 'internal tension', this depends on factors such as the stiffness and thickness of the rubber, since the edges will have to be beveled in order to get the same curvature inside the pieces as at the boundaries. Otherwise you'll have a polyhedron which is topologically the same as a hemisphere; but a hemisphere is topologically the same as a flat disk with no curvature.

Intuition inclines me to agree with zeocin -- well, almost. Just three triangles meeting in a point will do, no rounding-off is required; moreover, I'd venture that the triangles ought to be equilateral. Proving that this arrangement has the minimal internal tension would require more information, however.

I'm kinda confused by the question. What sort of pieces do we have, and how are we allowed to attach them? Are we working with support struts here? (Since otherwise, three pieces of rubber will just flop around)

Okay, going along the same lines, what about cutting the hemisphere in a spiral shape (plus a small disk at the apex) and then unwinding it? By making the spiral infinitesimally thin, the curvature of the strip should go to zero. Then just snip it at two arbitrary points to make it into three pieces as requested.

Re:

i am going to venture to say, if you have to use three, and not two, that something along the lines of a baseball, but you would have to figure it out a little more than that

a baseball is made out of two pieces that are ALL curves, and that is what makes it more stable than a round ball, which usually is done with between three and six lozenges with chopped off tips sewn/glued to two round things at top and bottom. With three pieces, I think it gets more complicated. The easiest way to get a sphere is to do four triangles, with curved sides, but the points where they meet will be funky. So i am going to guess something like a baseball, two "Y" shaped pieces, and one round piece. Or maybe three "Y" shaped pieces, but i am not sure how to do it exactly. The other thing to try would be to make something like this (__) and two small spheres for top and bottom, glueing together the ends of the parentheses, and glueing the underscore to the spheres.