Determining average number of fragments created upon explosion Suppose you're analyzing some 3 dimensional object of any sort of shape, and this object explodes from some internal force (e.g, like a grenade). How would you go about determining the average number of fragments that the shape would be broken up into upon the explosion created by this internal force? 
 A: Unless you are designing it to fragment, for example with grooves*, you won't get a reliable answer. It will depend on too many unknowns, like crystal structure, manufacturing stresses, shape and velocity of the explosive etc. etc. etc. 
The empirical research could be fun though - make 500, blow them up, count the bits.


*

*this is precisely why some grenades are knobby. Others contain ball bearings.

A: It depends entirely on the structure of the device. As the outer surface deforms, the material will start to separate along lines of relative weakness and stress concentration. In the classic WWII "pineapple" grenade, grooves cut in the outer surface caused the formation of large pieces of shrapnel. In more modern versions, the explosive core is surrounded with a ball of wire, with the wire notched at regular intervals. As the wire is deformed it tends to break at the notches, creating many more (but smaller) fragments.
If the object is nominally homogenous, other effects come into play. If the explosive core has any ridges or grooves or irregularities of any sort, these will produce stress points which will provide a pattern for the resulting fragments.
Even with nominal homogeneity, the object may well have internal structure, and this will also provide starting points for fragmentation. Such internals might include impurities, voids produced by casting, and even strain zones caused by uneven cooling of the object as it was formed. 
In general, the less distinct the irregularities, the less uniform their effects, so it's almost impossible to predict what will happen in these cases.
An extreme case, though, occurs when the explosive is powerful enough. In the case of nukes, for instance, case structure just doesn't matter. The bomb is completely vaporized, producing no shrapnel at all.
