## Abstract

Onsager’s principle of maximum dissipation (PMD) has proven to be an efficient tool to derive evolution equations for the internal variables describing non-equilibrium processes. However, a rigorous treatment of PMD for several simultaneously acting dissipative processes is still open and presented in this paper. The coupling or uncoupling of the processes is demonstrated via the mathematical structure of the dissipation function. Examples are worked out for plastic deformation and heat flux.

## 1. Introduction

Onsager was the first to introduce the *Principle of maximum dissipation* (PMD) (*Rate*), later denoted as PMD, in Onsager (1931) for heat conduction and in Onsager (1945) for diffusion. This principle was rediscovered by Svoboda and Turek in the Nineties of the previous century and put on a rigorous basis by Svoboda *et al.* (2005). In the meanwhile a wide field of applications has opened, e.g. Svoboda *et al.* (2006) for multicomponent diffusion, Fischer *et al.* (2003) for grain growth and coarsening and Fratzl *et al.* (2005) for non-convex dissipation. In a detailed treatment Hackl & Fischer (2008) have studied the PMD both concerning its mathematical structure and in relation to inelastic evolution processes by dissipation potentials. The goal of this paper is to apply the PMD to problems where inelastic processes, heat and mass fluxes are present simultaneously.

For the sake of consistency the same notation is used as in Hackl & Fischer (2008) when possible. We use the following notation for matrices **A**, *A*_{ij} and **B**, *B*_{ij}, vectors **a**, *a*_{j} and **b**, *b*_{j}, **A**·**a**=*A*_{ij}*a*_{j}, **a**·**b**=*a*_{j}*b*_{j}, **A** : **B**=*A*_{ij}*B*_{ij} with the standard summation convention. We choose the specific symbol ‘:’ for the contraction of two tensors to stay in accordance with the continuum-mechanical notation, since most of our examples use tensor-valued quantities as variables. Any rate (material time derivative) of a quantity is marked by a dot, e.g. . We work in the actual configuration and use the ∇-operator as divergence operator, e.g. ∇·**a**, or as a gradient operator, e.g. ∇*a*.

We introduce as external variables the strain tensor ** ε** and the temperature

*T*. As internal variables we specify the plastic strain tensor

*ε*_{p}and the concentrations

*c*

_{i}of the

*n*individual components, with the constraint according to the molar volume

*Ω*, which is assumed to be constant,1.1assembled in a vector

**c**and some other internal variables (like the equivalent plastic strain)

*z*

_{i}, assembled in a vector

**z**.

For the sake of simplicity, we use the additive decomposition ** ε**=

*ε*_{e}+

*ε*_{p}of the strain tensor and relate the stress tensor

**to**

*σ*

*ε*_{e}by Hooke’s law.

We assign to the temperature *T* a heat flux vector **q** and introduce as conjugate quantity to *T* the entropy *s*.

For the sake of simplicity, we will assume only substitutional components (*i*=1,…,*n*) and assign to each component a chemical potential *μ*_{i} and a mass flux vector **j**_{i}. Furthermore, we neglect the role of vacancies. Then, owing to the conservation of lattice positions we have the constraint1.2If the reader is interested in the role of additional interstitial components and vacancies, details can be taken from Svoboda *et al.* (2006, §3).

Finally, we assume that the volume element d*V* is constant in time, d(d*V*)/d*t* = 0, which allows for the simplest possible derivation of the entropy production in the following context. Furthermore, we introduce mass conservation for each component as1.3for details see Fischer & Simha (2004, §3).

## 2. Problem formulation

### (a) Dissipation

We define the Helmholtz-free energy *ψ*,2.1and the internal energy *e*,2.2Since matter flows into/out of a representative volume element, the First Law of Thermodynamics follows as2.3where *h* represents a further energy source, e.g. irradiation.

The Second Law of Thermodynamics is written as2.4Both relations (2.3) and (2.4) can be taken from the open literature, e.g. from De Groot & Mazur (1969). Introduction of internal energy (2.2) in the first law of thermodynamics (2.3), combining this relation with Helmholtz free energy (2.1) and using2.5yield after some analysis (see also De Groot & Mazur (1969), chapter III, §2 and Svoboda *et al.* (2005)),2.6We use now from continuum mechanics2.7Since *ε*_{p} is a deviatoric tensor, i.e. tr *ε*_{p}=0, it holds ** σ** :

*ε*_{p}=

**s**:

*ε*_{p}, where

**s**=

**−1/3**

*σ**tr*

*σ***I**denotes the stress deviator. Furthermore, we introduce with mass conservation (1.3)2.8and find for equation (2.6)2.9The quantity has been denoted in Hackl & Fischer (2008) as ‘dissipation’

*Q*. From the physical point of view, it is better to use the term and to denominate it as ‘entropy production’, which is equivalent to the negative rate of Gibbs energy, divided by

*T*, see e.g. Svoboda

*et al.*(2006). Here, we would like to mention that the chemical part of the Gibbs energy is evaluated in linear non-equilibrium thermodynamics from local equilibrium data, and the requirement of a global equilibrium need not be met.

### Remark 2.1.

The expression for remains the same, if , where denotes the material velocity. In this case the material time derivative of a quantity *a* must be replaced by according to the Transport Theorem, see e.g. Fischer & Simha (2004) and applied for e.g. *c*_{i},*e*,*s*.

### (b) Principle of maximum dissipation for coupled processes

Before introducing the PMD, we inspect the thermodynamic kinetic variables, which are and the components of , , and the fluxes **j**_{2},…,**j**_{n}. We have discarded and the flux **j**_{1} of component 1, addressed as lattice-forming element, by applying equations (1.1) and (1.2). According to this fact, we introduce modified chemical potentials as for *i*=2,…,*n* and replace in equation (2.9) after reformulation of 2.10Moreover, let us collect the thermodynamic kinetic variables formally in a vector and the thermodynamic state variables formally in a vector **x**={** ε**,

*T*,

*ε*_{p},

*c*

_{2},…,

*c*

_{n},

**z**}. Then the dissipation

*Q*, which reflects the internal dissipative processes during evolution of the microstructure, can be written in the form

*Q*(

**x**;

**v**). The reader is referred to Svoboda

*et al.*(2002) for the specific contributions to

*Q*with respect to the mass fluxes

**j**

_{2}to

**j**

_{n}of the substitutional elements.

We formulate the PMD as2.11where .

Introducing a Lagrange multiplier *λ*, a Lagrangian *L*_{v} can be written as2.12The stationarity conditions ∂*L*_{v}/∂**v**=0 immediately yield with respect to , and the so-called Coleman–Noll conditions2.13and with respect to (for *i*=2,…,*n*):

*variation with respect to* **q**:2.14*variation with respect to* :2.15*variation with respect to* :2.16*variation with respect to* **j**_{i}:2.17Note that we have , where *V* denotes the region occupied by the material body considered. This means that the variation of this term contributes to the formulations of boundary conditions only. Since we are mainly interested in the derivation of field equations this issue will not be further discussed here.

We express now the second terms in equations (2.14)–(2.17) by the partial derivatives of Q, e.g. asand find with (the side condition in equation (2.11))2.18We can express the thermodynamic forces, i.e. the quantities work-conjugate in expression (2.9) to the thermodynamic kinetic variables collected in **v**, as2.19
2.20
2.21and
2.22The set of equations (2.19)–(2.22) corresponds to equation (2.4) in Hackl & Fischer (2008). Since one needs evolution equations for **v**, the equations (2.19)–(2.22) must be inverted with respect to **v**. Details will be discussed in the next section.

### Remark 2.2.

It should be mentioned that Ziegler derived an equation similar to equation (2.18) in his book (Ziegler 1977, §15.1). However, he investigated the case of maximum dissipation by looking for orthogonality conditions with respect to a dissipation surface *Q*=*Q*_{0}, saying that the thermodynamic forces are parallel to the gradient of *Q*_{0}. Kestin & Rice (1970) criticized this assumption, already with respect to an earlier version of it, unless *Q* happens to be a homogeneous function in the rates of the internal variables, or more general, a function of a homogeneous function of them. The variational concept followed in this paper, however, is not restricted to some selected types of functions for *Q*.

### (c) Principle of maximum dissipation for non-coupled processes

One may argue, however, that the entropy production is related to dissipation in a separate way for the various physical processes present. To be specific let us introduce contributions to generated by and , respectively, as2.23and2.24Let us assume now that the dissipation *Q* can be summed up by the contributions in equation (2.23) as2.25We formulate a modified PMD aswhereIntroducing Lagrange multipliers once again, we obtain a Lagrangian2.26The further procedure goes along the same lines as in §2*b*, the main difference being that we obtain separate expressions for the Lagrange multipliers2.27with2.28The corresponding evolution equations are then only weakly coupled via the constitutive functions and given by2.29
2.30
2.31and
2.32

## 3. Interpretation of evolution equations

### (a) Quadratic forms for Q

The simplest exploitation of the set of evolution equations (2.19)–(2.22) occurs, if *Q* is a quadratic form in **v**. In this case represents a viscous and not a rate-independent plasticity term. If we use Euler’s theorem for homogeneous functions, we have3.1In this case, we can easily invert the relations (2.15)–(2.18) and find *Q* as the sum of quadratic forms with the positive definite matrices **R**_{T},**R**_{p},**R**_{z} and **R**_{c} as3.2and3.3With respect to **j**_{i} we refer to Svoboda *et al.* (2002, 2006), since one has to insert the last term of for , yielding the so-called symmetric Onsager matrix for diffusion, **L**, which is outlined in both references above and in detail in Svoboda *et al.* (2005, §3), and concentrate on **q**, and as thermodynamic parameters.

One observes classical evolution rules, as Fourier’s law, linear viscoplasticity, etc. All the internal processes only interact *in a weak sense* via the dependence of **R**_{T},**R**_{p},**R**_{z},**R**_{c} on **x**.

### (b) General forms for Q

The situation is qualitatively different, if *Q* is not a homogeneous function of order *m*, but in the simplest case, a sum of homogeneous functions of a different order. A prominent example is rate-independent plasticity with dissipation , where and *σ*_{0} is the yield stress, see e.g. Hackl & Fischer (2008, §§1 and 2) together with heat flux represented by .

We have then3.4and3.5Then (1+*λ*)/*λ* follows as3.6The quantity *f* is not a constant but varies with the process. We have 1≤*f*≤2. All the internal processes interact in a strong sense.

Then, we find for the heat flux from equation (2.19)3.7and the plastic strain rate3.8Generally a solution can only be found in an incremental procedure. It should be mentioned that in the last case discussed, a different established variational principle, based on the dissipation potential, see e.g. Hackl & Fischer (2008, §2*b*), is *not* equivalent to the PMD.

### Remark 3.1.

Observe that in general we have . This means that the Onsager symmetry relations do not hold in this case. However, they do hold for the limit cases or , since then *f* assumes a constant value. Of course, the Onsager symmetry relations hold in the weakly coupled case, too.

### (c) An example

We study the coupling of heat flux and rate-independent plasticity, as outlined in §3*b* above. Let us for simplicity consider the one-dimensional case leading to a formulation with scalar variables only such as , the heat flux *q*,∇*T*, and the stress deviator component *s*=2*σ*/3 with being the plastic strain rate and *σ* the stress in the longitudinal direction. For we have , for **R**_{T} we write *R*. Combining equation (3.6) with equation (3.7) yields3.9which is a cubic equation with respect to *q* and linear with respect to and can be solved analytically. We can distinguish two limiting cases3.10and obtain with equation (3.8)3.11As specific data we select a problem, as it may be realistic for steel with *σ*_{0}=3×10^{7} N m^{−2}, *T*=300 K, ∇*T*=100 K m^{−1} and *R*=1.5×10^{−4} s N^{−1} which corresponds to *λ*=45 N K s^{−1}, if we equalize *q*_{ref}=−*λ*∇*T*.

As one can see from figure 1 the ratios *q*/*q*_{ref} and *σ*/*σ*_{0} vary between the values 1/2 and 1. The following observation is of interest: If obtains a realistic value, say , then *σ*/*σ*_{0} is nearly unaffected by the coupling and obtains a value 1. However, the absolute value of *q* is decreased nearly by a factor 1/2 in relation to the heat flux vector for no plasticity present, i.e. *q*_{ref}. This points, at least qualitatively, to the fact that the microstructure, which has changed owing to plastic deformation, retards the heat flux, which is in accordance with the physics of phonons as carriers of the heat.

### Remark 3.2.

It should be mentioned that already Ziegler (1977) investigated in §15.3 of his book the coupling of heat flow and a deformation process dealing with the gradients to a surface *Q*=*Q*_{0}. Compared with his concept the variational principle at hand presents a very simple answer with respect to coupling by inspecting equation (2.18), or in other words, if the ratio *Q*/*N* is a constant or not.

### Remark 3.3.

It seems to be interesting to note that coupled processes may exist although for each individual process a contribution to the total dissipation is addressed depending only on the individual thermodynamic kinetic variables characterizing only this process. As outlined in the open literature, e.g. by Haase (1969, §4*b*–*d*) many coupled processes are dealt within the framework of thermodynamics of irreversible processes. However, the coupling is usually described for *Q* being a quadratic form in all the thermodynamic kinetic variables. An uncoupling in the space of thermodynamic kinetic variables is possible, if the positive definite matrix, on which *Q* is based, is transformed to its eigendirections. That means a linear transformation may lead to an uncoupling. However, a coupling via equation (2.18) with *Q*/*N* varying during a process cannot be decoupled.

## 4. Conclusion

The PMD is established in order to derive evolution equations for heat flux, plastification, fluxes of matter and further internal processes, represented by a further set of internal variables. It is shown that the evolution equations for each individual process obtain interaction terms with respect to all other processes. Only if the dissipation is a homogeneous function of the same order for all thermodynamic kinetic variables (generalized fluxes), a decoupling of the processes takes place. Specifically the coupling of heat flux and plasticity is investigated.

## Acknowledgements

F.D.F. and J.S. acknowledge gratefully the financial support by the Austrian Federal Government (in particular from the Bundesministerium für Verkehr, Innovation und Technologie and the Bundesministerium für Wirtschaft und Arbeit) and the Styrian Provincial Government, represented by Österreischische Forschungsförderungsgesellschaft mbH and by Steirische Wirtschaftsförderungsgesellschaft mbH, within the research activities of the K2 Competence Centre on ‘Integrated Research in Materials, Processing and Product Engineering’, operated by the Materials Center Leoben Forschung GmbH in the framework of the Austrian COMET Competence Centre Programme as well as funding through DFG project AB.314/1 (FOR 741). J.S. acknowledges gratefully the financial support by the Research Plan of Institute of Physics of Materials (project CEZ:AV0Z20410507). The cooperation of authors (F.D.F. and J.S.) is also supported in the frame of COST action P19.

- Received April 1, 2010.
- Accepted June 29, 2010.

- © 2010 The Royal Society