It seems that Rockwell hardness is definitely "nonlinear", but there are lots of types of non-linearity, of which exponential, or logarithmic are some examples. But there are others, such as power laws, etc. I don't know enough engineering/material-science to know which non-linearity best describes Rockwell hardness.
Some thoughts:
(1) HRC is basically how deep an indentation is made by an indenter which is pressed into the metal by a specific force. (See the links for details:
http://www.gordonengland.co.uk/hardness/rockwell.htm and
http://www.nist.gov/manuscript-publication-search.cfm?pub_id=853006 )
The indenter is sphero-conical in shape; it is a 120 degree (inclusive) cone with a spherical tip. The spherical tip has a radius of 0.2mm. Here is a diagram from the NIST document:
The indenter is pressed into the metal with a force of 150 kilograms (kgf).
Each point of the HRC scale represents the indenter sinking in an additional 0.002mm (2 microns).
Because they wanted bigger numbers to represent harder materials, they arbitrarily subtract from 100:
HRC = 100 - h/(0.002mm)
where h is the depth of the (permanent) indentation. Here, h is measured in millimeters. (Okay, this is an over-simplification because I'm ignoring the preliminary minor load. See
http://www.gordonengland.co.uk/hardness/rockwell.htm for details.)
So an HRC of 100 means that there was no permanent indentation made (ie: h=0mm). (In practice, HRC is only used up to about 80, because beyond that it's hard to measure.)
(2) Eventhough HRC is linear in the depth of the indentation, it is not linear in terms of the volume of metal displaced. This is partly because volume goes as length cubed, and partly because of the shape, namely the spherical tip and the conical base behind it. It's my understanding that for normal ranges of HRC (ie: 20-80 HRC), the indentation has gone deeper than the spherical tip. (Someone correct me if this isn't true.)
So the displaced volume of metal is basically the cube of the depth (ie: basically proportional to h^3 if you ignore the spherical tip).
(3) One might be tempted to say, then, that HRC has a non-linearity of h^3. But that's ignoring other stuff like how far the volume of metal gets displaced, and hundreds of other effects. It also does not account for h=0 means that the diamond indenter did nothing permanent to the surface!
In practice, hardness is sufficiently difficult to define, that it is defined in terms of test procedures (ie: Rockwell Hardness) rather than fundamental physics.
(4) I'm over-simplifying some things to make them easier to explain. Go see the links above for details.
(5) I'm neither an engineer nor a material-scientist. So I don't know what all this means! But I think it's fascinating.
(6) Many thanks to Ankerson for doing these knife tests, and for including the Rockwell Hardness in the results!
Sincerely,
--Lagrangian