Toughness is the ability to resist fracturing under impact, including repeated impacts (and including bending back and forth repeatedly without any impact at all, but let's not complicate matters).
Striking a blade with a nail is not different from striking the nail with the blade. If you shot a nail at a blade with some sort of gun and caused the blade to fracture that would be a measure of toughness, and it would indicate whether the blade would fracture if you hit a nail while chopping.
Toughness can determine how many times you can roll an edge and steel it back to shape before it fractures. (The edge is also being abraded while you use it, so that might force you to regrind the edge before it reaches its toughness limit.)
Whether an edge rolls to one side or is indented straight back (ie equally to both sides) depends on a number of factors. Edge geometry is the easiest to change -- if your edge rolls resharpen to a slightly less acute angle, repeat until it stops rolling. Another factor is how much side force the edge is subjected to -- if you chop very accurately, not much, but if you're clumsy -- and we all get clumsier when we're tired or in a hurry -- then the edge will be subjected to greater side forces.
Bending to one side subjects the inner side of the bend to stress (compression) and the outside of the bend to strain (tension, stretching). Materials vary in compression strength and tensile strength independently (for instance, steel has great tensile strength but lesser compression strength; concrete has great compression strength but poor tensile strength (that's why the two materials are often used in combination)).
So rolling an edge involves tension as well as compression ... the side of the edge that's not being compressed contributes to resisting rolling by resisting tension. A blade with greater tensile strength would resist edge rolling better. Tensile strength also contributes to indentation straight back and to the Rockwell test, though -- the surface of the metal is being stretched, and it resists that.
Looking back over my reasoning so far ... it looks like I've established that tensile strength is a factor in rolling an edge, in indenting an edge straight back, and in the Rockwell hardness test -- but does that factor have the same importance in resisting all three forms of plastic deformation? I doubt it's exactly the same. Is it significantly different, though? I don't seem to be able to answer that question from here in my chair ... it's time for some experimentation.
Note that elasticity has the same effect in the Rockwell hardness test and in chopping -- you measure the depth of the indentation in the Rockwell test after you withdraw the diamond point and the metal has a chance to rebound; you observe the indentation in the edge after hitting a knot after the metal has a chance to rebound, too.
-Cougar :{)
[This message has been edited by Cougar Allen (edited 06-25-2000).]