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Gibbs free energy and constitutive equations

I don't know if my understanding of the following is correct, so I would appreciate your comments. I have an expression for Gibbs free energy of a material G=G(σ,T,...). Is it correct to derive the equation for strain as ε=∂G/∂σ ?


Zhigang Suo's picture

Yes.  A typical derivation does as follows.  Start with internal energy as a function of strain and entropy

(1)     U=U(ε,S),

where ε is the strain and S is the entropy.  The differential form of the function is

(2)     dU=σdε+TdS.

Define the Gibbs free energy by

(3)     G=U-σε-TS

A combination of (2) and (3) gives

(4)    dG=-εdσ-SdTd.

Thus, we can regard the Gibbs free energy as a function G=G(σ,T), and obtain equations of state as

(5)    ε=-∂G(σ,T)/∂σ

(6)    S=-∂G(σ,T)/∂T

This change of variable is given a physical interpretation in my notes on pressure.


  I was wondering about step three (G = U - se - TS).  Does this also "work" in the case where the stress is not a monotonically increasing function of strain?





Prof. Dr. Sanjay Govindjee
University of California, Berkeley

Zhigang Suo's picture

Dear Sanjay, I believe so.  An example is the van der Waals fluid.  Here pressure is not a monotonic function of volume.  When you evaluate the Gibbs free energy, you get a multi-valued function. This function is plotted in textbooks of thermodynamics; for example, on p. 119 of Pippard's Elements of Classical Thermodynamics. 

Dear Zhigang and others.

I am dealing with constitutive models at various levels in my PhD-study on shape memory alloys. In this respect thermodynamics of materials is of great importance. However, I have looked through a lot of books on mechanics, and never found any books that cover thermodynamics of materials truly in-depth. Do you have any suggestions to books that might give a thorough description of the subject?



Jim S. Olsen

Zhigang Suo's picture

Here are several good textbooks on thermodynamics:

The list easily goes on.

However, no book comes to mind with a good coverage of mechanics and thermodynamics together.  I'd be curious if others know any.

Amit Acharya's picture

I think this is a superb introduction to thermodynamics of solids:



Ericksen, Jerald L.

Introduction to the Thermodynamics of Solids



Verlag : 

Springer New York



Let me suggest the following books

1)  The Mechanics and Thermodynamics of Continuous Media (Theoretical and Mathematical Physics) by Miroslav Silhavy


2) In the Handbuch der Physik series, there are various useful bits in the Truesdell, Noll, Toupin, etc. articles.

3) At a more elementary level you may wish to consult the book of G.A. Maugin:

Thermomechanics of Nonlinear Irreversible Behaviours: An Introduction
(World Scientific Series on Nonlinear Science, Series a, Vol 27)


  The Thermomechanics of Plasticity and Fracture (Cambridge Texts in Applied Mathematics)



Prof. Dr. Sanjay Govindjee
University of California, Berkeley

Thank you both for your suggestions. I will definitely look into this material, hopefully it will help enhance my understanding of the thermodynamic part of material modelling.




Jim S. Olsen

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