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I would like to create a computer simulation which would fold two plane-ish surfaces into two crumpled-ish surfaces. This in order to simulate the earths crust movement and collisions resulting in mountain chains, vallies and hills etc.

Is there any existing calculations or models for this? Are these models already put into computer simulations or are they just available on paper?

What is the math behind this?

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I am afraid that you aren't being specific enough to really answer your question: What about mountain chians valleys and hills are you interested in? There are simple and complex models dating back from the 1960s to present. Are you interested in learning about the material properties of the crust, or just topography in general? What about weathering effects? What about the flow laws that govern the lower and middle crust (depending on who you talk to).

I would look into the field of dynamic topography and how that is calculated.

Two books that I think might be useful to read are Fowler and Tourcotte and Schubert, they will help give you an overview of geodynamics as a field, and what engineering solutions we apply to Earth. In the 1-D and 2-D flow sections there is much about the crust.

A good starting paper might be Bird 1991 (sorry, its probably behind a pay wall, but I couldn't find an open pdf on it).

As far as codes (and consquently mathematics) The two sections of CIG I would look at are S-T Dynamics and Mantle Convection.

I am sorry I couldn't provide a comprehensive answer, but I think the sources I've linked may give you a good start and thinking about this problem.

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    $\begingroup$ Actually the appropriate section in CIG is Long-term tectonics (e.g., codes such as GALE). Short term tectonics addresses deformation during earthquake cycle (0-100s of years). Mantle convection models can simulate longer term deformation but over larger spatial scales and therefore cannot resolve small scale (in space) processes such as mountain/plateau building. $\endgroup$
    – stali
    May 28, 2015 at 15:59
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    $\begingroup$ Perhaps. But I cannot in good conscious recommend anyone use gale: perhaps one of the biggest failures of the geodynamics community. $\endgroup$
    – Neo
    May 28, 2015 at 16:02
  • $\begingroup$ Why is that? Any specific reasons? $\endgroup$
    – stali
    May 29, 2015 at 2:56
  • $\begingroup$ Gale was the communities attempt to make Underworld a community driven code but as the few developers of underworld developed for underworld and its best features never made it to gale, forking the two softwares. Gale, like underworld, is extremely hard to navigate: the learning curve is very very steep. Though it has powerful techniques, there are very few users of these two codes and as such it is not recommended to use that software. $\endgroup$
    – Neo
    May 29, 2015 at 4:42
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Over longer time scales (hundred thousands to millions of years) deformation of crust can be simulated as a viscous fluid. Basically this amounts to simulating stokes flow (i.e., the math behind the problem). Thin viscous sheet models are also fairly common.

Over shorter time periods (months to thousands of years) viscoelastic models are more appropriate. For even shorter time periods (seconds to days) elastic/poroelastic rheologies are appropriate. That is why ground motion following slip on faults can be simulated simply using elastodynamics.

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There are several codes which can probably model what you seem to describe. A few that come to mind (in no particular order) are Underworld, Elefant (C Thieulot) and Elvis (T Gerya). The choice really depends on the exact model you would like to run.

For a discussion paper describing Elefant: http://www.solid-earth-discuss.net/6/1949/2014/sed-6-1949-2014.html

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