As I understand it, compliance is the derivative of position with quasistatic force, and in a linear system this is a constant. Nonlinear systems have compliance that varies with position; think of a probe touching skin with subcutaneous fat over muscle and bone. Compliance is high at first because of the fat, but with larger displacements the fat is squashed out enough that now you’re measuring the compliance of the muscle, which increases rapidly with further displacement. I suspect that this kind of force-displacement curve is important to the tactile sensations of softness, hardness, etc.
Generally I think that, if you want to plot the curve, you want to plot the derivative of force against displacement, which is to say, the compliance as a function of displacement.
Along a second dimension, we have the frequency at which the force is being applied; for some systems, resonant modes will give very large “compliances” at some frequencies and smaller ones at other frequencies, tending to the quasistatic limit as the frequency approaches zero. The degree to which the system exhibits these resonances gives you some information about lossiness in its elasticity. A steel spring might have the same compliance as a cotton pillow, and you can taper the spring to make its compliance vary with displacement (as mattress makers do routinely), but the cotton pillow will lack any sharp resonant peaks because it’s very lossy, so you can easily distinguish them. This is noticeable if you just tap both systems with a coin. Other systems have other characteristic resonances. Nonlinearity will produce vibrations at other frequencies, including harmonics and subharmonics, vaguely like Raman emission but in the acoustic domain, potentially adding a third dimension.
This sort of “viscoelasticity spectroscopy” or “compliance spectroscopy” is potentially useful for a number of different purposes:
Teledildonics. Nonlinear compliance is crucial to reproducing the tactile sensation of touching a human body.
Somatic and haptic interfaces; by distinguishing a finger pressing a button from a palm or a floor pressing it, software can distinguish between different actions to take. Even pressing a button with the same finger at different angles can be detected. Tiny total internal reflection compound parabolic reflectors could be integrated into a button to provide a simultaneous optical interface for coupling LED illumination in and out of a finger without the discomfort occasioned by the LED bumps traditional in pulse oximeters. This, too, has obvious masturbatory uses: a vibrator can be programmed to respond to not only the pressure it’s put under but also the compliance curve of the tissues it’s stimulating.
Material identification. This is somewhat trickier, because the compliance spectrum of an object depends on many things other than the material it’s made of; for example, what shape it is, how firmly it’s being held, and what it’s being held in. But with the whole 2-D spectrum, it might be feasible to tease out at least a guess.
Material characterization. If you make a standard-geometry coupon of the material being tested and hold it in a standard way, the above variations go away, and you can compute quantitative viscoelastic properties of the material. Alternatively, at high enough frequencies, it should be possible to characterize just the material in the region around the probe.
Object identification. You can use compliance spectroscopy to distinguish multiple objects, even if they are made of the same material.