A friend and I were discussing flop and turn play when 3-betting AK pre-flop.
Assume a typical 1/2 live game, with large pre-flop bet sizing, and 100bb stacks. Pre-flop raises are often in the ball park of $8-$15, 3 bet sizing will fall between $25-$50, and often gravitate towards the middle of the two values. A typical c-bet will be 1/2 to 3/4 pot, which means an additional 1/2 to 3/4 pot will be left behind on the turn.

As a default if hero decides to bet the flop (meaning about half of his stack is now invested), should hero be prepared to shove most turn cards, or go into check/fold mode on the turn if he does not improve?

Against, tighter opponents and stronger 3-bet calling ranges, Hero is not likely betting the flop unless he hits, because he is often behind.

Against looser opponents (both good and bad), Hero is c-betting a larger number of flops that miss, but are not unfavorable. Once Villains 3-bet calling range widens, AK is still ahead on a good number of flops. When hero bets these flops and gets called, should he be more inclined to shove most turn cards? (Especially if hero has position and is checked to on the turn).

This question can't be answered in a vacuum. Position and the board matter. PF action matters. Post a sample hand or two and you'll get much more useful advice.

You already answered your own question ...

1) What is Hero's 3-bet range?
2) What is V calling range?
3) Did the board hit either of these ranges?
4) Can Hero 'sell' an overpair/set/TP from his image?
5) Stacks?
6) Position?
7) History?

There is really not much that can go wrong with c-betting (except the occasional raise), but you may have to be willing to double barrel on a lot of boards ... which brings stacks into play. GL

It depends.