# Modeling elastic moduli as a continuous function in space for a single solid material

I've read a number of solid mechanics papers where a single material is modeled with constant elastic moduli (lame parameters $\lambda$, $\mu$). I've also seen composite materials modeled with distinct elastic moduli, separated by a sharp, discontinuous interface. I haven't really seen any papers where a single material is modeled by elastic moduli that vary continuously in space. I imagine the constant parameters are used for simplicity and are valid under a great number of situations. I'm curious if there are any research papers where the elastic moduli for a single material are modeled as a continuous function of space.

My primary goal for this question is to understand under what circumstances one would use a continuous function to model spatially varying elastic moduli in a solid material. I realize that this may be a bit of an open question, as there may be many different circumstances where it is done. Any guidelines, examples, heuristics, anecdotal evidence and/or references are certainly welcome.

• A keyword for internet search is probably "nonlinear elasticity" – Trimok Jun 13 '13 at 19:12
• @Trimok: I wouldn't necessarily characterize it as non-linear. At least, not from a mathematical point of view because I'm not assuming that the material property is a function of the state variables, just the spatial variable. – Paul Jun 13 '13 at 21:09