# lecture3.ppt - What is VLab

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```					               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)

Poirier’s “Introduction to the Physics of the Earth Interior”, Cambridge Press
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.
Murnaghan EoS
• It can be similarly derive assuming

is cte
Strains
• Eulerian strain (f>0 for compression)

• Lagrangian strain (ε<0 for compression)
OK for ε→0

• Hencky strain (logarithmic strain)

For hydrostatic compression
• One more relation to be used:
For f → 0

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+…
Birch Murnaghan EoS (2nd order)
• 2nd order expansion of the free energy F=af2
with

• Recall that

• Therefore

with
Therefore for f→0

LM assemblage

Murnaghan EoS
overestimate P for
non-infinitesimal
strain
Birch-Murnaghan 3rd order

Take into account:                           with

with

Then one gets:

At P=0 (f=0), K=K0
→ 2 eq.s for 2 unknowns, a and b
K’=K0’

If K0’=4 we recover the 2nd order BM
• 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.
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
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
Comparison between parameters offered by various EoS
Summary

• EoSs based on expansions of F in terms of
strain (finite strain EoSs) give order dependent

• High order EoS require data in larger pressure
ranges.

• The Vinet EoS is good for any pressure range.

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