If I wanted to explore a discrete mathematics approach to continuum mechanics, what textbooks should I look into?
I suppose a ready answer to the question might be: "computational continuum mechanics", but usually textbooks that discuss such a subject are usually focused upon applying numerical analysis to continuous theories (i.e. the base is continuous), whereas I would like to know if there is a treatment of the subject that builds up from a base that is discrete.
I think you might be interested in peridynamics. It is an approach to continuum mechanics that is based on an integral formulation rather than a differential one where a point in the material is governed by interactions with neighboring points within a given radius. The approach itself is not discrete but the implementation is and most of the works about implementation focus on how to model the material (the interaction horizon, the forcing functions, etc) rather than the numerical methods because the numerical methods are very simple.
The field is relatively new (since 2000) so there are few books, but there are chapters in books such as Advances in Applied Mechanics. Also search for papers by Silling, one of the original papers can be found here. There are countless other papers expanding the theory and applications and it's currently a very active research area.
It seems like you want Discrete Exterior Calculus. This formalizes discrete 'approximations' to continuous ideas like differential forms and such, but without depending on an underlying continuous theory: it's developed through chains and cochains on a simplicial complex from the bottom up. Here's the main original thesis, and Here's a good overview of the field with some interesting algorithms. It's a self-contained analgoue to the associated continuous ideas, rather than being an approximation. It's actually discrete, rather than just discretized, so it seems to be what you're looking for.