The car will not even crank or turn over?

We have had the engine starter replaced before at the Chrysler dealer but the start problems persists. It always happens after driving awhile and then you shut the engine down. If you try to start it up minutes or even hours later, the car will not even crank or turn over and gvies the same symptom as a dead batter. But the battery is very strong. The radio, lights, etc come on. If you wait long enough, like maybe 5 hours or overnight it will start the next day. This problem is a random occurrence.