I just had my iPhone 6s refusing to respond (black screen, no reaction on any of the individual buttons). Slight stress peak, as a working phone is pretty important to me 😉
The solution was to hold the home button (the round button on the front bottom) and the power button (right upper side button) at the same time for about 10 seconds (just wait until the phone shows the Apple logo).
The iPhone then reboots, asks for your passphrase and runs as usual. Interestingly, it apparently keeps the SIM card powered, as I did not need to enter the PIN for the SIM card.
Reference: Apple’s article on this.