## Abstract

We study the evolution of systems described by internal variables. After the introduction of thermodynamic forces and fluxes, both the dissipation and dissipation potential are defined. Then, the principle of maximum dissipation (PMD) and a minimum principle for the dissipation potential are developed in a variational formulation. Both principles are related to each other. Several cases are shown where both principles lead to the same evolution equations for the internal variables. However, also counterexamples are reported where such an equivalence is not valid. In this case, an extended PMD can be formulated.

## 1. Introduction

We consider a mechanical or physical system, which is described by a set of external variables ** X** and internal variables

**, both assembled in vectors**

*x***={**

*X**X*

_{1},

*X*

_{2}, …,

*X*

_{m}} and

**={**

*x**x*

_{1},

*x*

_{2}, …,

*x*

_{n}}. The material time derivatives d

**/d**

*X**t*and d

**/d**

*x**t*, with

*t*being the time, are denoted as and , respectively. Furthermore, we introduce a specific Helmholtz free energy . Then, according to the elementary thermodynamics, the dissipation

*Q*is given by(1.1)Here, and later on, we use the following notation for a matrix

**,**

*A**A*

_{ij}and vectors

**,**

*a**a*

_{j}and

**,**

*b**b*

_{j}, , , with the standard summation convention. We choose the specific symbol for the contraction of two vectors to stay in accordance with continuum-mechanical notation, since most of our examples use tensor-valued quantities as variables. The vector

**is denoted as the vector of the thermodynamic forces, , therefore**

*q***is also denoted as the vector of thermodynamic fluxes. The second law enforces that must hold.**

*v*We consider two well-known examples for *Q*.

The first one is classical rate-independent plasticity in the form of perfect plasticity with ** x** consisting of , the plastic strain within a small strain setting. The thermodynamic forces are the components of the stress deviator

**. The dissipation**

*s**Q*

_{p}is as follows:(1.2)where , with being the yield stress. The term is often denominated as the equivalent plastic strain rate . Hence, we have . Note that

*Q*

_{p}is homogeneous of order 1 in

**(see Carstensen**

*v**et al*. (2002) or Mielke (2003) for details).

It should also be mentioned that a viscoelastic eigenstrain, in the simplest case represented by a relation , with being a given matrix, yields a dissipation being homogeneous of the order of 2 as(1.3)A second example is the diffusion process of *n* components in a bulk material. The thermodynamic velocities are the fluxes and the corresponding forces are , with being the chemical potential of constituent *i*. If we use, for the sake of simplicity, a one-dimensional setting, then we can arrange the fluxes and forces as and , respectively, with *z* being the length coordinate. The dissipation *Q*_{d} is as follows:(1.4)(For details, see Svoboda *et al*. (2002, 2006).) The matrix ** R** is positive definite with the coefficients

*R*

_{ij}. The dissipation

*Q*

_{d}is now homogeneous of the order of 2 in

**and, therefore, a quadratic form. An orthogonal point transformation to a basis formed by the eigenvectors of**

*v***allows us to write (1.4) in a diagonal form. We assume that this has already been carried out. The coefficients**

*R**R*

_{ii}can be taken from Svoboda

*et al*. (2002, §7).

Onsager was now obviously the first one who introduced a variational formulation for *Q*, denoted here as the *principle of maximum dissipation* (*rate*; PMD), in Onsager (1931) for heat conduction and Onsager (1945) for diffusion. Later, Ziegler (1963) showed that this principle can also be applied to nonlinear problems. Svoboda & Turek (1991) ‘rediscovered’ this principle and successfully exploited it in a series of papers (Svoboda *et al*. 2002, 2006) to find the evolution equations for the fluxes as(1.5)The matrix ** L** is the so-called ‘Onsager matrix’, which is symmetric due to the positive-definite matrix

**and reflects the well-known ‘Onsager reciprocal relations’. For details, the reader is referred to a paper by Svoboda**

*R**et al*. (2005) with respect to a general presentation of the principle and its application for heat flux in combination with diffusional fluxes.

In the notation of this paper, the PMD can be formulated as follows.

*From all admissible fluxes* *v**(possibly constrained by conservation and boundary conditions), these fluxes are taken which maximize* *subjected to the equivalence relation*

Here the maximization is taken with respect to fixed ** X** and

**. Possible constraints can be taken into account by either directly taking the admissible**

*x***from a suitably restricted set or using Lagrange multipliers. Let us mention two examples. In the case of diffusion of several components the fluxes of the substitutional components are constrained by the lattice postitions (Svoboda**

*v**et al*. 2002, §6), or the fluxes are zero at the surface of a closed system (Svoboda

*et al*. 2006, §3).

A second way to exploit a given dissipation is a variational formulation introducing a dissipation potential , which enters a minimum principle named the *minimum principle for the dissipation potential* (MPDP). In this case one looks from all admissible fluxes ** v**, maybe constrained by some conservation and boundary conditions, for those ones, which minimize with , being the rate of the Helmholtz free energy

*ψ*. This principle appears in various forms already in works by Martin & Ponter (1966), Maier (1969), Halphen & Nguyen (1975) and Maugin (1992), and is widely used in the literature. More recent treatments, especially in connection with the description of microstructure, can be found in Hackl (1997), Ortiz & Repetto (1999) and Carstensen

*et al*. (2002). Obviously, equivalent formulations are still being developed (e.g. the recent work by Yang

*et al*. (2006

*a*,

*b*) and Fischer & Svoboda (2007)). Independently, Petryk (2003) has introduced an energy rate to be minimized, with

*Q*being a homogeneous function of the order of 1. Then, he extended the energy minimization to the second-order rate and could show that some symmetry conditions must be required for

*Q*(in his notation

*D*).

In the notation of this paper, the MPDP can be formulated as follows.

*From all admissible fluxes* *v**(possibly constrained by conservation and boundary conditions), these fluxes are taken which minimize*

Here minimization is taken with respect to fixed ** X**,

**and**

*V***. Concerning possible constraints, the same remark holds true as in the case of PMD. Obviously, it holds(1.6)The attractiveness of MPDP lies in its property of being a pure minimization principle. This turns out to be advantageous for describing the onset of an evolution of microstructures in materials (see §5).**

*x*The goal of this paper is now to check under which circumstances both principles, the PMD and MPDP, are equivalent yielding finally the same evolution equations for the variable ** x**.

## 2. Problem formulation

### (a) Principle of maximum dissipation

The PMD can be formulated as follows:(2.1)Introducing a Lagrange multiplier *λ*, a Lagrangian *L*_{v} can be written as(2.2)The derivative yields(2.3)Multiplication with ** v** and using (1.1) yields(2.4)

Since one needs the evolution equation for the internal variables , equation (2.4) must be inverted yielding finally as . This can, for example, be done very easily in the case of diffusion, see (1.4), yielding , with *Q* being homogeneous of the order of 2.

In the case of a homogeneity of the order of 1 as in rate-independent plasticity, see (1.2), the dissipation *Q* is non-differentiable at the origin, and the evolution equation (2.4) becomes a differential inclusion (see Carstensen *et al*. 2002; Mielke 2003),(2.5)Here, is called the subdifferential of *Q* with respect to ** v** (see Moreau 1963; Moreau 1968; Rockafellar 1970). From relation (2.5), it can be seen that we always have and that

**can be different from**

*v***0**only if . In this case, we have , with . This means that in the isotropic case considered here the vector

**(e.g. the components of the plastic strain tensor) is parallel to the vector**

*v***(e.g. the components of the stress deviator). However, the consistency parameter**

*q**λ*must be derived by a further observation (see appendix C).

The relation ensures that for fixed *A*, ** q**, and therefore

**, is normal to the surface in**

*v***-space. This is in accordance with the so-called normality rule in plasticity. It is interesting to note that here the normality rule is a strict outcome of maximization of the dissipation. This is not the case if one starts (e.g. as in Lubliner (1984)) with the rather classical formulation for any admissible**

*q*

*q*^{*}. In Lubliner (1984), the construction of a local kinematic stability postulate is necessary, which then allows us to introduce a normality rule.

Introducing *A* to be independent of ** X**, the construction above corresponds to an associative elastic–plastic material law, with yield function given by . Then, from (2.5), it follows that(2.6)together with the Kuhn–Tucker conditions,(2.7)It is possible to include non-associative plasticity into the formulation as well by allowing a dissipation of the form . This can be done, for instance, by choosing in the previous example. This leaves equation (2.5) unchanged. Hence, taking , we have(2.8)On the other hand, it now holds that . Let us now solve for . Substitution gives , and we have(2.9)where .

This principle can also be formulated with respect to the thermodynamic forces, ** q**, by defining the dissipation in the form and requiring(2.10)This leads to equivalent evolution equations in the form(2.11)if

*R*is given via the Legendre transform of

*Q*as(2.12)(For details, see Svoboda

*et al*. (2005) and Fischer & Svoboda (2007).)

The PMD is equivalent to minimizing the rate of free energy with respect to the same constraints, hence we have(2.13)Introducing a Lagrange multiplier *λ*, a Lagrangian *L*_{v} can be written with (1.6)(2.14)Since , the extremum assumed is actually a minimum here. The derivative yields(2.15)which obviously is equivalent to equation (2.3).

Minimization of free energy as a universal principle governing irreversible thermodynamics is strongly advocated in Müller & Weiss (2005, ch. 7).

### (b) Minimum principle for the dissipation potential

The MPDP can be formulated as follows:(2.16)Introducing a Lagrangian gives(2.17)The derivative yields, with , the relation(2.18)An inverse relation can be found by installing a Legendre-conjugate dissipation potential(2.19)leading to the relation(2.20)

## 3. Interrelation of the principles

The most prominent feature of MPDP is that ** q** is obviously a gradient of a dissipation potential with respect to

**(see (2.18)). In general, however, this need not be the case for the PMD. Now, the question is whether a potential can be assigned to any given and vice versa, so that the relations (2.4) and (2.18) are equivalent yielding(3.1)At this point, we should mention that all results we are going to derive in this and the following section concerning the potentials**

*v**Q*and

*Δ*hold identically for

*R*and

*J*defined in (2.12) and (2.19), respectively.

### (a) Given dissipation Q, check for the existence of dissipation potential *Δ*

A necessary and sufficient condition for to be a gradient (3.1) is given by the integrability or exactness equations(3.2)Inserting *q*_{i} from (3.1) as after some differential operations and comparison of both sides of (3.2) leads to the condition(3.3)If, and only if, relation (3.3) is satisfied with respect to Q, then the dissipation potential *Δ* can be constructed by integrating with respect to a variable and using (1.1) as(3.4)This agrees with Petryk's minimization of (Petryk 2003) since, there *Q* is a homogeneous function of the order of 1 in yielding . The fact that for the viscous case (1.4) and for rate-independent plasticity was already stated by Moreau (1970).

Let us, for reference, summarize the result obtained stating the following.

*For a given* *, there exists a* *such that equation* *(3.1)* *holds if, and only if, equation* *(3.3)* *is satisfied. In this case*, *is given via equation* *(3.4)*.

If *Q* can be represented as a differentiable function *f* of a homogeneous function of the order of in , i.e. it holds that and , then it can be shown that relation (3.3) is satisfied (see appendix A), and *Δ* can be constructed along (3.4) as . In the case of rate-independent plasticity (see (1.2)), *Δ* follows as *Q*_{p}. In the case of a diffusional or viscoplastic process (see (1.4)) *Δ* follows as .

In the seminal work of Kestin & Rice (1970, §8.5), it was concluded that the plastic strain rate (in our case ** v**) is normal to a surface , if the dissipation is given as , where is a homogeneous function of the order of in

**. In our notation, this can be seen in the following way: here**

*q***is given by equation (2.11). Owing to the homogeneity condition above, equation (3.3) now holds for**

*v**R*as a function of and we can use formula (3.4) to construct a potential , such that holds. Hence, is given as a gradient and we can take

*K*=

*J*.

If *Q* can be represented as a differentiable function *f* of an expression of the form , where denotes a linear operator, precisely, if it holds , then it can be shown that relation (3.3) is satisfied (see appendix A), and *Δ* can be constructed along (3.4) as(3.5)where(3.6)

(A counterexample.) If *Q* is given as the sum of two homogeneous functions of a different order as , then it can immediately be shown by using (A 3) that (3.3) is satisfied only ifThis is generally not the case! If we consider a rate-independent plastic process (*F*_{1}) coupled with a creep process (*F*_{2}), (1.2) and (1.3), then we haveBoth expressions need not to be identical.

### (b) Given dissipation potential *Δ*, check the validity of PMD

For a given dissipation potential, *Q* follows from (1.1) and (3.1) as(3.7)In order for the PMD to give the same ** q** as the MPDP, the identity (3.1) has to hold, which we reformulate as(3.8)We replace

*Q*in equation (3.8) using (3.7) and its derivativeand obtain, by comparison of both sides, the condition(3.9)Note that this identity is different from (3.3). Once again, we summarize the result obtained as follows.

*For given* *and* *defined by equation* *(3.7)*, *the thermodynamic forces* *are identical, i.e*. *(3.1)* *holds if, and only if, relation* *(3.9)* *is satisfied*.

If *Δ* can be represented as a differentiable function *g* of a homogeneous function of the order of in , i.e. it holds and , then it can be shown that relation (3.9) is satisfied (see appendix B).

Let us assume that *Δ* can be represented as a differentiable function *g* of a term of the form , where denotes a linear operator. To be more precise, let ; then, it can be shown that relation (3.9) is satisfied (see appendix B), and *Q* can be constructed along (3.7) as(3.10)where(3.11)

(A counterexample.) However, if *Δ* is given as the sum of two homogeneous functions of a different order as , then it can also immediately be shown along the concept of §3*a* that (3.9) is satisfied if, and only if,And this is generally not the case (see also example 3.3).

## 4. An extended problem solution

### (a) Reconstruction of a dissipation potential *Δ*

One may ask the question, if there exists a function , in general different from *Q*, which ensures that ** q** is a gradient of a yet unknown potential

*Δ*with respect to

*v*, when maximized subject to the relation , (see (1.1)). We denote this problem as the

*extended principle of maximum dissipation*(

*rate*; EPMD) formulated as(4.1)with the Lagrangian (4.2)The derivative yields(4.3)Multiplication with and using (1.1) yield(4.4)(4.5)and finally for a component

*q*

_{i},(4.6)One can immediately see from (4.6) that is a gradient with respect to if the multiplier of is a quantity independent of , say . Then, is related to

*Q*as(4.7)If a function satisfying (4.7) exists at all, then it can immediately be constructed by integration with respect to a variable using the relationand as(4.8)A simple inspection then shows that defined by (4.8) indeed satisfies (4.7). The potential

*Δ*follows now directly from (4.6) asand, consequently,(4.9)Any integration constant independent of can be added.

Insertion of (4.8) yields the following relation for *Δ* in an analogous way to *Δ* (see (3.4)) as:(4.10)Note that *Δ* is now independent of .

It is important to remark that is in general not uniquely defined. There are more possibilities to render the r.h.s. of equation (4.6) a gradient than the one proposed in equation (4.7).

Let us again consider a rate-independent plastic process coupled with a viscoplastic process (1.2) and (1.4); then, *Q* is as follows:Insertion of these expressions in (4.8) and (4.10) yields

### (b) Reconstruction of a dissipation *Q*

One may also ask the question whether a dissipation *Q* exists obeying the PMD for a given set of thermodynamic forces . Then, we have to recall the definition of *Q* from (1.1) as , from which we find(4.11)Here, we have adopted the convention that contraction from the left, , is given in components as , whereas contraction from the right, , has the meaning . Using (4.11) and relation (2.4), after some manipulation, we find(4.12)

Relation (4.12) is satisfied if the components of are homogeneous functions of the order of , i.e. if it holds that , yielding .

(A counterexample.) If can be split into subvectors, with components being homogeneous functions of a different order of in , then it can be easily seen using that the identity (4.12) is in general not satisfied.

### (c) Reconstruction of dissipation potential *Δ* and extended problem solution

As already discussed in §3*a*, the existence of a dissipation potential *Δ* is related to the integrability conditions (3.2), hence a necessary and sufficient condition is given by(4.13)In this case, we obtain *Δ* according to (3.4) as(4.14)Then, following the EPMD, a function can also be constructed applying (4.9) as(4.15)

## 5. Time-incremental variational principle and prediction of microstructures

Time-incremental variational principles have proved to be an important tool for finding proper solutions in an implicit incremental procedure (see Nguyen 1977). In a time-incremental setting, the MPDP can be combined in an intriguing way with the principle of minimum potential energy. For this purpose, let the energy of the material body be given as , where *ϕ* denotes deformation and is once again a set of internal variables. Let us now consider a time-increment and let be the known values of our state variables at the beginning , and be the unknown values at the end of the increment. Within the increment, let us use the approximation , then by integrating the Lagrangian given in (2.17), we obtain(5.1)where is the volume occupied by the body. Ignoring the constant term and adding the potential of external forces , we are able to construct a combined Lagrangian of the form(5.2)Now, minimization of with respect to clearly gives the principle of minimum potential energy, while minimization with respect to gives a discretized version of equation (2.18) in the form(5.3)where we introduced the notation . This means that we obtain via the combined minimization principle(5.4)The functional depends on in a purely algebraic way. Hence, we can formally eliminate by preminimization leading to the condensed energy(5.5)Thus, we end up with a minimization problem for *ϕ*, which is equivalent to an elastic one and contains only as a parameter,(5.6)An important property of is given by the fact that, in general, it may not satisfy the condition of quasiconvexity, even if does. This means that there may exist a local fluctuation field that lowers the energy averaged over a representative volume element when compared with a homogeneous deformation. In this case, it holds that(5.7)The first example of this behaviour appeared in Carstensen *et al*. (2002) for a model of crystal plasticity. Determination of the energy-minimizing fluctuation field then leads to the so-called quasiconvex envelope(5.8)The minimizer can now be interpreted as a microstructure exhibited by the material. Note that this procedure depends on (5.4) being a pure minimization principle. The modelling of microstructures based on the minimization of energy was first proposed by Ball & James (1987). The calculation of envelopes via variational calculus is described in detail by Dacorogna (1989). Details of this procedure, especially concerning the application of numerical methods, can be found in Bartels *et al*. (2004, 2006). Related approaches and results are given in Ortiz & Repetto (1999), Miehe & Lambrecht (2003), Miehe *et al*. (2004) and Mielke (2003). One might imagine the derivation of similar principles based on the PMD, but they would not be suitable for the description of the formation of microstructures.

## 6. Conclusion

Two *a priori* formulated principles, namely the ‘principle of maximum dissipation’ (PMD) and the ‘minimum principle for the dissipation potential’ (MPDP), are investigated with respect to their correspondence. Several identities and relations are derived, which allow a direct transfer between the two. However, counterexamples are also found, where the application of the principles would lead to different results.

## Acknowledgements

F.D.F. appreciates the very valuable discussions with Prof. H. Petryk IPPT, Warsaw.

## Footnotes

- Received June 18, 2007.
- Accepted September 20, 2007.

- © 2007 The Royal Society