May I suggest you precut the pieces of wood to almost the breaking point. Then the force of the axe should create a good break.
Pre-split the wood and lightly re-attach with a bit of putty or other adhesive. It will easily 'split' when chopped on stage. If you want sound to accompany the split you could record, but if you use a large log base to split the wood on and dull axe, you can get your sound when the axe hits the base. If your setting is so intimate you do not want to go with an authentic axe, but want to use one made of hard wood, that sound might be enough for you as well.