1 / 13

Equations of state R. Wentzcovitch U. of Minnesota VLab tutorial

Equations of state R. Wentzcovitch U. of Minnesota VLab tutorial. EoS relates P,V,T in materials EoS of minerals are necessary to build Earth models In this lecture: isothermal EoS only (Eos parameters are functions of T).

murray
Download Presentation

Equations of state R. Wentzcovitch U. of Minnesota VLab tutorial

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Equations of stateR. WentzcovitchU. of MinnesotaVLab tutorial EoS relates P,V,T in materials EoS of minerals are necessary to build Earth models In this lecture: isothermal EoS only (Eos parameters are functions of T) Poirier’s “Introduction to the Physics of the Earth Interior”, Cambridge Press

  2. The definition of the bulk modulus offers an EoS (with K=cte=K0) -This is only a naive example of how to generate EoS. -K is not cte. It varies with P, except for really ifinitesimal volume changes.

  3. Murnaghan EoS • It can be similarly derive assuming is cte

  4. Strains • Eulerian strain (f>0 for compression) • Lagrangian strain (ε<0 for compression) OK for ε→0 • Hencky strain (logarithmic strain) For hydrostatic compression

  5. For f → 0 • One more relation to be used: Therefore Bulk modulus with Now we will expand the free energy in term of (eulerian) strains and derive relationships P(V), K(V), K’(V)… F=af2+bf3+cf+…

  6. Birch Murnaghan EoS (2nd order) • 2nd order expansion of the free energy F=af2 • Recall that • Therefore with with

  7. Therefore for f→0 LM assemblage Murnaghan EoS overestimate P for non-infinitesimal strain

  8. Birch-Murnaghan 3rd order Take into account: with Then one gets: At P=0 (f=0), K=K0 K’=K0’ with → 2 eq.s for 2 unknowns, a and b If K0’=4 we recover the 2nd order BM

  9. One needs measurements in a larger pressure range to fit a 3rd order EoS • There are trade offs between Ko and Ko’ • If the pressure range is small Ko’ is usually constrained to 4.

  10. Vinet EoS • This EoS is based on a different expansion of F l is a scaling length • Defining and changing variables (r→V) in F • Replace a, l, K0, and K0’ from relations above and get

  11. Logarithmic EoS • Expand F in powers of Hencky strains εH like Birch-Murnaghan • To 2nd order in εH one gets • And to 3rd order one gets

  12. Comparison between parameters offered by various EoS

  13. Summary • EoSs based on expansions of F in terms of strain (finite strain EoSs) give order dependent parameters (trade offs). • High order EoS require data in larger pressure ranges. • The Vinet EoS is good for any pressure range.

More Related