What are the differences in overtones for hard vs soft materials? I've been researching Physical Modeling lately, and so far pretty much everything makes sense from a physical/logical perspective.

The one thing that I haven't yet found a proper explanation on is the difference in overtones between soft and hard materials.  I've seen it mentioned in a few places that this stiffness is the parameter which defines the difference between metallic and nylon string on a guitar, for instance.  I've also caught that one of the two directions (hard vs soft) "spreads the harmonics."  But that's all I've got.

So, what are these differences in overtones for rigid materials?  They seem to be more spread out, I've come to recognize metallic sounds as being rather comb like.  But I'm trying to eventually write code to synthesize these sounds, and need to know some details as to the way they are spread.

Also, here's a great video in which most other aspects of physical modeling are pretty well explained, as well as the mention of more spread out overtones for harder materials (though I don't remember at what time): https://www.youtube.com/watch?v=dUcNzPhZdwk
I am not sure if this is exactly answering your question but it may help you in your modeling efforts.  The metallic strings are going to be fairly simple to model in that the differential equations you use will be only dependent on the shape of string at any given time and the tensile strength of the string.

The nylon strings exhibit non-linear viscoelastic behavior which has a time dependent component to it.  That means that is has a memory and the local velocity of the string will be important.  By memory I mean there is residual strain built in once plucked and the time for recovery will be much greater than the frequency of motion.  I believe this is what gives it a “softer” sound.  Classically viscoelastic materials are modeled with a spring and damper in parallel.  

Descretizing the string over a number of elements and using a divided differences approach to approximate the differential equations you should be able to get a theoretical sense of how these differences affect the observed signal.

Hope that is helpful for you in thinking about your problem.