What is the relation between the force and voltage in a piezoelectric sensor?

To get an output, you disturbe a lattice. This is not a continuous event but a transient one. When you do the opposite by powering a latttice, you get a non linear movement that is repeatable within limiting the movement to non-damage ranges. Many precision optical bench manipulators use this piezo electric effect to provide micron movements in a predictable and repeatable maner. This is a non linear movement vs electrical input gadget.