There is a difference between a proper "armour piercing" bullet and a bullet that will pierce armour. It also depends on what kind of armour we are talking about.
Typically bullets designed to pierce armour use a hardened steel penetrator.
Piercing armour can be accomplished with a conventional lead-copper bullet if it has the correct combination of velocity and shape.
The bullet that most typifies that is the CF 9mm ammo. It was originally loaded very hot for use in SMG's. The bullet is just pointed enough and it has just enough velocity to allow it to penetrate a level 2A kevlar vest.
Some of the very high velocity sintered copper frangible bullets also possess enough velocity to penetrate kevlar armour.
At one time there was a machined copper bullet known as a THV that was specifically designed to penetrate armour by way of a very high velocity. The weird looking bullet fourth from the left is a THV. The second and third bullets from the left are also solid copper armour piercing designs.
