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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.

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  • $\begingroup$ A keyword for internet search is probably "nonlinear elasticity" $\endgroup$
    – Trimok
    Jun 13, 2013 at 19:12
  • $\begingroup$ @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. $\endgroup$
    – Paul
    Jun 13, 2013 at 21:09

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... under what circumstances one would use a continuous function to model spatially varying elastic moduli in a solid material.

Seismology. Seismic wave velocity depends upon the elastic moduli of the earth's interior. Although you'll find plenty of examples of layer models of the earth, you'll also find examples where regions of the earth's interior is represented by a continuous function.

Look for references on Theoretical Seismology.

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  • $\begingroup$ Cool! Out of curiosity, how deep might those layers be? Could we find them fairly close to the surface of the earth? What manner of "layers" might I want to look up with these properties? $\endgroup$
    – Paul
    Jun 13, 2013 at 21:12
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    $\begingroup$ Seismic wave velocity is proportional to the square root of modulus divided by density. Rock density generally increases with depth. If the elastic moduli were constant velocity would decrease with depth. Generally, the velocity of seismic waves are observed to increase with depth - because the moduli increase even faster. $\endgroup$
    – Mark Rovetta
    Jun 14, 2013 at 5:23

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