# Journal Club Theme of October 2007: Irreversible thermodynamics of continuous media

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The second half of the last century saw exciting developments in formulating mathematically rigorous yet physically well founded theories for irreversible thermodynamics in a continuum mechanics framework. The development took place within communities with widely different scientific backgrounds and motivations. In general, two divergent schools existed (and possibly exit): one of them having its roots in the works of I. Prigogine and S. R. de Groot, and the other initiated by C. Truesdell, B. D. Coleman and W. Noll. The former group took as their starting point, the Gibbs relation, where they assumed that even in the out of equilibrium situation, concepts like entropy are meaningful, if only locally, and can be expressed as functions of the state variables. The entropy change for each macroscopic region is decomposed into an external and an internal part, and as the second law of thermodynamics, the positiveness of the internal part is postulated. The entropy production emerges in the form of the product of ‘fluxes’ and the corresponding ‘generalized forces’. The kinetic laws (relating fluxes and forces) are then empirically obtained such that they satisfy the restriction of the second law. The formulation of linear thermodynamics of irreversible processes is one in which a linear relationship is assumed between the force and the flux.

‘Rational thermodynamics’, as called by the second school, attempted to formulate and present the theory in an axiomatic and mathematically rigorous framework.  They diverge from the classical theory to assume temperature and entropy to be primitive variables, not dependent on something else (like mass, length etc.). This enabled them to postulate the existence of these quantities in situations far from equilibrium also. The Clausius-Duhem inequality (as the second law) is taken as the starting point and upon identifying the state variables, constitutive laws, as well as the restrictions on the kinetic laws are obtained. Therefore one major point on which this formulation differs from the one described above is that constitutive laws (like the relationship between stress and strain) are also generated as a consequence of the second law. The overall results of both the theories remaining same,  the approaches have come to differ widely. A good introduction to various points of view in continuum thermodynamics can be gained from the following two collections of papers:

• ‘Foundations of Continuum Thermodynamics’, Domingos J. J., Nina M. N. R., and Whitelaw J. H. (Eds.), John Wiley & Sons, New York – Toronto (1973). [This is a proceedings of an international symposium held at Bussaco, Portugal, in which many major researchers from both schools presented their work and had discussions at length]

• ‘New Perspectives in Thermodynamics’, Serrin J. (Ed.), Springer-Verlag (1986) [This collection has articles written by researchers from the rational thermodynamics school and gives an excellent overview on a wide variety of topics]

In selecting a list of papers for including in this discussion, I have been biased from my personal interests in the areas of spatio-temporal pattern formation and plastic evolution. Both of these subjects offer wide application of the concepts from irreversible thermodynamics of continuous media and therefore have a large literature associated with them.  I mention in particular two papers in each of these disciplines:

a) Far from equilibrium spatial pattern formation:

Prigogine I., Nobel lecture, Dec 8th 1977.

- Pattern formation outside of equilibrium, Cross, M. C. and Hohenberg P. C., Reviews of Modern Physics, 65(3), 851, 1993.

Prof. Prigogine in his well written lecture has given an elementary overview of his central ideas regarding irreversible thermodynamics. Of main interest is his thesis of  the existence of organization (or order) as a result of the non-equilibrium nature of the system. The second article is a comprehensive review (273 pages) of the literature on spatiotemporal formation in systems far from equilibrium and applications are provided for in the areas of biology, chemistry, optics, fluid dynamics and solid mechanics. Their departure point is a nonlinear partial differential equation, which represents the far from equilibrium evolution and can be identified with the kinetic laws mentioned above.

b) Plastic evolution in solids:

- The thermodynamics of plastic evolution and generalized entropy, Bridgman P. W., Reviews of Modern Physics, 22(1), 56, 1950.

, Hartley, C. R., Philosophical Magazine, 83 (31-34), 3783, 2003.

Prof. Bridgman’s classic paper raised many interesting and practical issues in formulating a thermodynamic theory for plastic evolution. He severely criticized the existing methods of classical thermodynamics in their inability to model processes which are ‘completely surrounded by irreversibility’. He proposed to use a new thermodynamic parameter of state, which can be measured but not ‘controlled’ (for e.g. dislocations). The evolution of such a parameter is purely irreversible. Researchers familiar with the current trends in dislocation based plastic modeling will easily recognize this as of the central importance in these models.  In the second paper, a recent model of dislocation based continuum plasticity has been proposed. After carefully formulating the associated kinematics, thermodynamic considerations are outlined, based on which empirical kinetic laws are proposed.

## Comments

### Is "rational thermodynamics" rational?

Dear Anurag,

Thanks for the topic! It will be nice to learn about the "rational thermodynamics" and its implications. I agree that the "rational thermodynamics" enjoys a somewhat elegant mathematical structure. Does it have any connection to the physical reality? What are the 'primitive' temperature and entropy in physical terms? What experiments favor the theoretical concepts?

Regards,

Kosta

### Re: Rational thermodynamics

Dear Kosta,

Thank you for your question. The results obtained via 'rational thermodynamics', in most cases, match with the ones obtained through classical thermodynamics (Prigogine etc.) and which are indeed verified by experiments. So, going by the consequences of a theory, no one can put a doubt at the validity of such a formulation. The task that 'rational thermodynamics' assumes, is to provide a logical (or axiomatic) structure to the theory and thus to (attempt to) clarify various obscurities which have always surrounded the foundations of thermodynamics.

As explained by Truesdell, temperature and entropy can be understood in physical terms as the measure of 'hotness' and the amount of heat exchanged at constant temperature respectively.

I hope that my remarks are helpful,

Anurag

### dislocation patterning

Anurag,

This a a very interesting post.....I am not too familiar with research on plasticity other than what I gather from talks here and there however I have recently developed an active research interest in pattern formation. Certainly the paper by Cross and Hohenberg is required reading for anyone looking into this topic. In a recent work (# 34), my collaborators and I applied some methods from pattern formation to self-assembly of nanostructure. The nonlinear PDE that emerges for such systems is a Cahn-Hilliard type equation derived sometime back by Zhigang Suo and Wei Lu (# 109). We performed nonlinear stability analysis on that equation and (I think) obtained some interesting insights. Currently, we are looking into developing strategies to force such a non-linear evolving system to exhibit "perfect patterns". I will report on the latter shortly....meanwhile, after reading your post, it struck me that several tools of pattern formation should also be applicable to dislocation evolution and patterning. Do you know if there is any work on that?

### reaction diffusion and dislocation patterning

Pradeep,

My two cents: look at the long line fo works initiated by Aifantis. The main issue, in my opinion, is that dislocation theory and plasticity are well established theories - making the physical connections between anything one suggests and at least dislocation theory is very important. This, if you think about it, is not easy at all.

Finally, energy-driven pattern formation is one source of microstrcuture development. We should not forget the paradigmatic case of turbulence where it is the interaction of strong nonlinear transport with diffusion. Unfortunately, the mathematics of dealing with nonlinear wave-phenomena in systems is orders of magnitude more difficult than reaction diffusion systems with lots of interesting stuff like solutions of determinsitic systems only being able to be interpreted in statistical terms etc. - but (un)fortunately gravitation, tubulence, is all about nonlinear wave-interactions and we have to deal with this.

It is my opinion that the final theory of plasticity that will lead to predictions of patterning will be of this sort - very transport dominated.

### Rational thermodynamics

Anurag,

"...no one can put a doubt at the validity of such a formulation". I can and do . I have read papers where authors started promising thermodynamics-based results and ended with physically suspicious conclusions.

The Truesdell explanation of the temperature and entropy is not clear, especially, concerning entropy as an amount of the heat exchange. Does that mean that heat is an amount of the entropy exchange?

To be more precise, I mean the statistical foundations of the 'rational thermodynamics'. For example, the statistical foundations of the classical thermodynamics, which can be found in the 5th volume of Landau and Lifshitz for example, are more or less comprehensible...

-Kosta

### Re: Rational thermodynamics

Kosta,

As far as I understand, any physical formulation which requires phenomenological input, has a capability to produce "physically suspicious conclusions". For example, if we try to model metal plasticity using the framework of classical thermodynamics (with volume rather than strain as a state variable), without properly understanding the nature of the phenomenon, then as a result we obtain meaningless conclusions. These meaningless conclusions can not be in turn (wholly) blamed on the formalism used, but should (also) be seen as the result of misuse of the formalism, without properly taking into consideration, the additional paprameters which would be needed for a meaningful conclusion. Therefore the fault, in many cases, lies not in the approach of the formalism used, but rather on the application of this approach to meet certain ends.

Again, as I understand, Truesdell means that a change in 'entropy' is a measure of heat exchanged at constant temperature.

On the issue of the statistical foundations of rational thermodynamics, I fear, I am not suffciently informed to make a definite comment.

Anurag

### Anurag,We get to an

Anurag,

We get to an interesting point. The formalism cannot be misused if it really exists. There are many "thermodynamic" formalisms and this fact questions the soundness of the 'rational thermodynamics'. The variety of the formulations of the 'rational thermodynamics' is a consequence of its vague physical grounds. Is it?

Kosta

### Formulations and their misuse

Kosta,

It is very apt to quote Prof. P W Bridgman here, who remarked that "there are almost as many formulations of the second law as there have been discussions of it". Does it mean that the second law stands on 'vague physical grounds'?

I agree, that a multitude of formulations can indeed be a reason for vagueness, but it might also be an indicator of the richness 'the aim' of all such formulations might posess.

Anurag

### Re

Well, I like your optimistic attitude...

Generally, I think that mathematics should please physics and not vice versa as it often happens in Solid Mechanics. It is interesting that there is a lot of abstract theories in Solid Mechanics contrary to Fluid Mechanics. Is that due to the Truesdell influence?

### multitude of models

Kosta,

I would tend to agree with Anurag that a mutitude of models often indicates a certain 'richness' and depth of the underlying problem. The ultimate theory is simply too hard to lay down. This probably happens most often when homogenization and multiple scale behavior are involved within nonlinear dynamical phenomena. Seems to me thermodynamics fits this bill perfectly, as does most questions in mesoscopic and macroscopic solid mechanics - fracture, fatigue, plasticity, you name it.

There are many models in these fields and Clifford Truesdell actually had nothing to do with any of these fields (OK, may be just a very little bit with hypoelasticity as a possible model of plasticity).

You draw attention to Fluid Mechanics. In my opinion, fluid mechanics is actually simpler than solid mechanics (I know now I have said something controversial!) because the standard version (Newtonian viscous fluid) is only geometrically nonlinear, whereas solid mechanics is geometrically as well as materially nonlinear. As soon as it gets to the situation where you need homogenization, i.e. turbulence, there are just as many theories/models as there are in solid mechanics. The same could be said of rheology (non-Newtonial viscous fluids) where athere are as many power laws as in solid mechanics.

And finally, I don't know about you but I have spent a fair amount of time with the book "Rational Thermodynamics" edited by Truesdell and found it quite profitable. I think there are very deep essays there by Feinberg, Owen, Coleman which are substantial and precise (no vagueness here) works on questions of existence and (non)uniqueness of entropy out of equilibrium starting from a more primitive point than the version of the second law generally used in RT. There are also nice articles by Truesdell comparing predictions of the kinetic theory and those from plain phenomenological continuum mechanics like the Navier Stokes equations ( of the things I can remember).

I work in plasticity and find the subject tremendously difficult - it helps to keep an open mind and learn from wherever I can, be it Hill or Truesdell or whoever (OK, with Truesdell you do have to get past some of the annoying polemics sometimes), as long as there is some hope that what they have to say will help solve my problem.

Finally, from what I remember, Truesdell follows Eckart in defining rate of change of entropy from the notion of an absolute limit to the ratio of the rate at which heat can be supplied to a body to the temperature at which this heat is supplied. I am doing this from memory and there is a possibility I don't have it precisely right, but I am no expert in the area and I am sure there are much better people than me to set this record straight. (Anurag, please help out).

- Amit

### Multitude as uncertainty

Amit,

I was talking about one physical phenomenon with a multitude of 'thermodynamics'. This multitude is an indicator of uncertainty rather than 'richness'.

I do not know what you mean saying that "fluid mechanics is actually simpler than solid mechanics" but I feel that fluid mechanics is richer in physical phenomena than solid mechanics because particles comprising fluids are much more mobile than particles comprising solids.

Unfortunately I did not learn much from the school of RT except the desire to impose restrictions on the constitutive models of materials. The restrictions are of low value for me because their physical grounds are too uncertain.

The main premise of the 'rational thermodynamics' is that an extension of the classical thermodynamics exists to all physical processes. Why should it exist? The physical theories are restrictive. The latter is normal and we can live with that. My impression is that RT was created by pure mathematicians who regarded mechanics as a mathematical structure in the sense of Bourbaki. In the latter case the creation of RT is an achievement. Does mechanics need bourbakization?

-Kosta

### transport in fluid and solid mechanics

Kosta,

It is precisely when strong wave phenomena gets involved in both solid and fluid mechanics that the theories become really difficult - solids become diffcult due to material nonlinearity even in the static case. It is the nonlinearity from the advection term in the balance of linear momentum, whether in solids or fluids, that I refer to as the geometric nonlinearity. For solids, even in the constitutively simplest case  of nonlinear elasticity, the material law is an added source of nonlinearity and this does create serious additional problems - balance of linear momentum, even in the Lagrangian setting, becomes a quasilinear second order system.

One concrete way of seeing this is that while existence of weak solutions for Navier Stokes(for special boundary conditions) was proven long ago by Jean Leray, a corresponding global existence theorem for nonlinear elastodynamics does not exist today (for elastostatics there are results due to John Ball with suitable restrictions on the strain energy function).

The situation improves when one considers some dissipation in solids. For example, once you consider dislocations as defects, even when one is doing statics, the evolution equation for the Nye tensor looks very much like the equation fo transport of vorticity except, dislocation density moves with a velocity which is the sum of the material velocity and a stress and Nye tensor dependent velocity relative to the material where as fluid vortices move only with the material velocity. Depending upon who you talk to, one could rig things up so a small diffusion term shows up, much like in Navier-Stokes (not rigged up of course), but any way it would be a small term and the beautiful thing is that even in the static setting, one ends up having to deal with exactly a similar thing as in turbulence and expecting microstructure to appear wuld not be so far-fetched, just as it is not in turbulence. Essentially there is nothing better than nonlinear transport equations to produce fields with high gradients (if not outright shocks) from initially smooth fields, and the mathematical signature of microstructure is highly oscillatory, bounded spatial fields.

If now to this picture of dislocation transport you add material inertia (as in high strain rate deformation, e.g.) you see that the material deformation also satisfies a transport law and this also feeds into the dislocation transport....so this is why I made the comment that fluid dyanlics is actually simpler than solid dynamics.

- Amit

### Agreed

Amit,

I agree that plasticity problems are difficult. None of the existing large-strain theories makes me happy. I learned, however, to be happy without the proofs of the existence theorems: a couple of the finite element meshes giving similar solutions is enough for me .

Best,

Kosta

### Morton Gurtin's Book "Topics in Finite Elasticity"

Morton Gurtin's book "Topics in Finite Elasiticty" has about half a page comparing Incompressible Navier Stokes and the equations of solid mechanics, and he says that the equations of solid mechanics are harder. His point seems to be that the non-linearity in NS is defined by the equations, the non-linearity in solid mechanics is defined by the choice of the constitutive relation, which in general is unknown. He also, mentions that the deformations in solid mechanics do not live in a linear space, whereas the velocity (NS) does live in a linear space.

Perhaps a fairer comparison would be non-linear solid mechanics v.s. non-Newtonian fluid mech.

-Nachiket

PS: I wasn't able to verify the linear space property for N.S.  in a straight forward manner, perhaps I'm missing something, or I'm just getting rusty.

### What is solid, what is fluid?

As we all know, the Alps is flowing. So, I have to ask "what is solid, what is fluid---Isn't solid one kind of fluid?", there is more constraints (or interaction) between one part and another within solid, thus the theory on the solids should consider more than that on fluid. More consideration leads to more complication, isn't it?

wu cw

Thank you,