In my case it would depend on the opponent.
I always keep track of my play style and how it's perceived in the table and keep as much information about my opponents as possible.
Don't enable reads on you
When you push with your AA do you do the same when you have AK,AJ,KQ etc...?
If you don't, than your move could be a read, your opponent could guess your range like AA, KK, QQ.
The same question can be asked when you call. If you call only with monsters a 40BB allin and a player behind to act, than it will be easy to read your hand.
Is he a fish, fun player, gambler or good player?
In most cases knowing your opponent makes it easier to decide to go all in or call. Many fish, or people who want to have fun call the 40BB in some cases.
If you just call there's a risk.
Is you opponent likely to play any garbage hands? Can you extract more money from him by not shoving and let him lead the bet on flop etc...
I think it's very unlikely that he would bet after calling 40BB. Most of the time, if he wanted to gamble by paying 40 BB preflop he will miss the flop and won't continue with any further bets.
First to act
Sadly you're first to act and your opponent doesn't seem like he's gonna call even the 40BB bet since he just called BB. The best move is probably just go allin because you're first to act.
You add additional risks if you just call 40BB because he will probably check back if you check the flop. And if you have to bet on flop, your bet should be probably 40BB at least. Which again he will not call unless he has some kind of draws or hit the flop.
Reraising 80BB is silly
It's very unlikely he will call 80BB to see a flop and than fold or keep calling turn, river. By doing this again you show how strong you are, because in basically any situations you would never reraise someone to the point where he commits almost 50 % of his stack and expect him to fold later or call until river. In most cases what would happen is he shoves behind the rest of his money or folds.