You do know that this is an unusual problem, right?
How old is the tub valve? is it possible that you've got disintegrating rubber parts in the tub? It sounds like old neoprene rubber. It's also possible that there's an old bit of rubber that fell off a valve somewhere in the house. I'd take the tub valve apart (after turning off the water) and try turning the water back on long enough to flush the valve out. With the rubber parts removed from the valves, if you still get flakes, it might be in the pipes.
There have been problems with rubber-lined water heater flex connectors.