Most people - I'd sat actually all - use chest (rib cage) and diaphragm at different times, and both together when breathing deeply, even if they do not realise it.
Just put a hand over your bottom rib on the front of your chest, breathe out and then take as deep a breath as you can.
You will feel the rib move upward slightly. That's "chest breathing", as you call it. The angle of your ribs changes slightly, from downward to nearer straight out, making your chest deeper and expanding your lungs.
You _cannot_ use the full capacity of your lungs unless you use both chest and diaphragm together.