To understand what happens with an evolution stone after it’s used, it might be worth it to start with how they’re used in the first place. You see, evolution stones are actually charged with elemental energies, much in the same way pokémon themselves are. These energies are actually present within our planet and are often thought to be either byproducts, strains, or close cousins to aura, or the basic essence of every living being, depending on whom you ask.
When an evolution stone is used on a compatible pokémon, the pokémon draws that energy within themselves to unlock the part of its genetic code that enables evolution (in much the same way other pokémon might do it when they gain enough battle experience, when they’re happy enough, and so on and so forth). Thus, when a stone is used, it enters an inert state and must be reintroduced to a known spring of compatible energy (that is, veins of its same type of evolution stone) in order to recharge. Of course, recharging is a very long process; if done naturally, it could take thousands of years for a single stone to regain enough energy to be usable again.
For this reason, stones are normally thought to be, well … less impressive or valuable upon use. Most trainers will simply throw them away (or toss them into the wild), but some stone collectors will keep used specimens. Additionally, some people fashion them into jewelry, as they’re essentially safe to have around like pokémon (meaning, pokémon won’t attempt to use them as evolution stones) at that point, and they’re not often seen as a valuable enough stone to steal. However, it’s worth it to note that used evolution stones are duller than their charged counterpart, so their value compared to charged stones is a matter of hot debate among jewelers and jewelry collectors.
In short, they don’t break. They just become inert, and what one does with them afterwards is really a matter of personal taste.
The Pokédex According to Bill 