Kluitenberg–Verhás Rheology of Solids in the GENERIC Framework

1 Introduction

The internal variable approach of non-equilibrium thermodynamics, with a symmetric tensorial internal variable, provides a distinguished model family – the Kluitenberg–Verhás model family [1] (covering the Hooke, Kelvin–Voigt, Maxwell, Poynting–Thomson, and Jeffrey models as special cases) – for the rheology of solids. This family is significant not only from a theoretical perspective but also for experimental applications [2], [3], [4], [5]. GENERIC (General Equation for Non-Equilibrium Reversible–Irreversible Coupling) is an attractive general framework for non-equilibrium thermodynamical models (see, e. g., [6], [7], [8], [9]). Whenever a new non-equilibrium thermodynamical model emerges, it is advantageous and recommended to check how it suits the frame of GENERIC. Here, we investigate how the internal variable formulation leading to the Kluitenberg–Verhás model family can be represented in GENERIC.

For the main part of the paper, specific entropy is treated as one of the state variables – a choice natural from principal aspects. Later, in an alternative version, temperature is used, instead – which may be more convenient for certain engineering applications.

We believe that the relationship between the internal variable framework and GENERIC may be fruitful for both approaches, providing

1. insight concerning the theoretical side,

2. wider applicability,

3. suggestions for novel numerical methods (see, e. g., [10], [11] for such a promising direction), and

4. a beneficial connection of such numerical approaches with analytical results (e. g., [12]).

Notably, certain aspects of our treatment could be presented at some more general, systematic, and methodological level (for example, performing the deviatoric–spherical separation of tensors in a multiplicative split form [13]). Here, we follow a simple direct approach.

2 Necessary elements I: The internal variable formulation of rheology of solids

We start with a summary and generalization of the internal variable approach to the Kluitenberg–Verhás rheological model family of solids [1]. The discussion is generalized in that the derivation in [1] neglected thermal expansion and started from Hookean elasticity, while the version here is free of those restrictions, only isotropy of the material being assumed.

The small-strain regime is considered, where strain ε is small (|ε|≪1), there is no need to distinguish spacetime and material manifold variables and vectors/tensors – accordingly, aspects of objectivity and spacetime compatibility [14], [15] are not addressed here – mass density ϱ is constant, and one can relate the time derivative (partial coinciding with substantial) of strain with the symmetrized gradient^[1] of the velocity field v, as follows:

Due to the isotropy of the material, the deviatoric–spherical decomposition of symmetric tensors plays here an important role (the spherical part of, e. g., strain is proportional to the identity tensor, εs=13trε1, while εd=ε−εs is its deviatoric part). With s0 denoting mass-specific entropy, our variables will be

The balance of linear momentum is

with the divergence of stress σ0 on the rhs, where σ0 is related to a (mass-)specific internal energy eint,0εd,εs,s0 as shown by the partial derivatives^[2]

Rheology is a behavior that mostly manifests in the mechanical aspect so, for an internal variable description of it, in conformity with the notion that strain and stress are second-order symmetric tensors, we introduce a symmetric tensorial internal variable ξ. Mechanical effects of rheology are to be embodied by a ξ-dependent extension of stress, i. e.,

Correspondingly, specific internal energy expressed in terms of the extended collection of variables,

is of the form

The balance of internal energy is

where je denotes heat current density, and the only source term considered is related to mechanical power. Substituting (4), (5), and (7) into (10), on the one hand we obtain

and on the other hand

The rhs of (11) is to be equal to the rhs of (12), which leads to

Since the balance of the extended entropy is to be of the form

with entropy current density js chosen to be the usual js=1Tje, and taking entropy production πs into account, in the light of (13), we can write

Positive semidefiniteness of entropy production can be ensured for the first term via je=λ1T⊗∇←, λ≥0 (Fourier heat conduction, a vectorial part that cannot isotropically couple to the remaining, tensorial, terms; hence, for simplicity, heat conduction is neglected in what follows), and via Onsagerian equations concerning the further terms, with independent deviatoric and spherical parts because of isotropy,

with lSd=12l12d+l21d and lSs=12l12s+l21s. Hence, the four-by-four coefficient matrix in the middle is required to be positive semidefinite, which necessitates for the coefficients, using Sylvester’s criteria,

It is to be noted that, in general, the coefficient matrices ld, ls need not be symmetric nor antisymmetric, corresponding to the notion that the concrete physical interpretation of ξ may not be available and the behavior of ξ under time reflection might not be purely sign preserving/flipping.^[5]

To see that this model family covers classic rheological models like Kelvin–Voigt and Poynting–Thomson, one can start with the special case of Hooke elasticity, and eliminating the internal variable leads, in the isothermal approximation (constant l11d,s, lAd,s, ϱTlSd,s, ϱTl22d,s), to the Kluitenberg–Verhás model family [1],

3 Necessary elements II: Summary of the GENERIC framework

In GENERIC [6], [7], time evolution of the collection of state variables (fields, in case of continuum models like ours here), x, is formulated as

where the operator matrix L acts on the column vector that is the functional derivative of the energy functional E of x, and the operator matrix M acts on the column vector that is the functional derivative of the entropy functional S of x; L is required to be antisymmetric,

(T denoting transpose which, for operators, means not merely matrix index transposition but includes operator adjoint). Thanks to this and the degeneracy condition

energy is conserved, dEdt=0. In parallel, the other degeneracy requirement

ensures that the first term on the rhs of (25) does not increase entropy, and M is demanded to be positive semidefinite to lead to dSdt≥0 eventually. That LδEδx is related to reversible dynamics is manifested further by also prescribing the Jacobi identity

(A, B, C being arbitrary functionals) for the bilinear generalized Poisson bracket

Consequently, the first term on the rhs of (25) can be interpreted as a reversible – generalized Hamiltonian – time evolution contribution (vector field) while the second term (another vector field) embodies the irreversible time evolution contribution to dynamics.

Analogously to (26) and (30), imposing symmetricity for M,

induces that the bilinear product

is positive semidefinite, A,A≥0. The latter bracket (32) completes the former one (30) in the sense that time evolution for any functional A can be expressed as

A constructive and productive way to generate the irreversible contribution to dynamics is to derive it from a dissipation potential [7], [17], [18]. Assuming a dissipation potential is, on the other side, not necessary and reduces the level of generality of the GENERIC framework [19].

4 Internal variable rheology of solids realized in the GENERIC formulation

Section 2 has actually been given in a form to provide preparations for the present one, where we establish the GENERIC form for the ξ-described rheology of solids. The set of variables x is (8), the energy functional consists of the internal energy contribution (9) supplemented with the kinetic energy-related one, and the entropy functional is straightforward, i. e.,

The non-trivial task is to identify L and M. Concerning the time evolution of the state variables, we know (6), (1), (13), and (16)–(17) so we conjecture the decomposition to reversible and irreversible parts as^[6]

The governing principle for this decision for decomposition is that, since dissipation and irreversibility are related to entropy production and to the internal variable, the reversible vector field should not contain them but only pure fluid mechanics.

Then L can directly be read off from the first term on the rhs of (37), i. e.,

with ∙ denoting the “slot” where the operator acts. This L apparently fulfills the degeneracy condition LδSδx=0.

To prove antisymmetry of L, let us take the corresponding bracket (30), i. e.,

where A and B are arbitrary functionals of the state variables, and abbreviations of the kind

have been introduced. Using indices (with Einstein convention and the Kronecker symbol δij), we have

Via integration by parts and omitting surface terms,

Now, taking into consideration that the functional derivative of a scalar functional with respect to a symmetric tensor is symmetric, with respect to a deviatoric tensor is deviatoric, and with respect to a spherical tensor is spherical, the first term in the integrand can be reformulated as

and the second term can be treated analogously. Next, it is easy to show that the third term is Bv·Aεd·∇←. Further, in the terms that contain δij, the j index can be changed to i, hence, in these terms we find the gradient of the trace of a tensor. Therefore, the fourth term contains trace of a deviatoric tensor – which is traceless – so this term gives zero contribution. Finally, the fifth term contains a spherical tensor and thus can be rewritten as 13∂kAεsiiBvk=∂kAεsikBvi. To sum up, we find

and antisymmetry is revealed.

Since L is independent of the state variables and is antisymmetric, the generalized Poisson bracket also satisfies the Jacobi identity [6].

Now let us turn towards the irreversible side. The operator matrix M can be constructed from the second term of the time evolution equation (37) and the degeneracy condition (27); we find

with^[7]

One can notice that this M is not symmetric – see the elements that contain l12d vs. l21d, as well as the ones with l12s vs. l21s. As mentioned in Section 2, when we have no additional microscopic or experimental information about ξ and about the corresponding coefficients ld, ls, then we cannot exclude that antisymmetric parts lAd, lAs appear in the dynamics.

On the other side, positive semidefiniteness can be shown by reformulating the integrand of the irreversible bracket A,A to a quadratic expression. More closely, we can form a matrix that contains non-negative elements and the Onsagerian coefficients, and (19)–(20) just prove to be the conditions that ensure positive semidefiniteness. The calculation is straightforward but lengthy.

Actually, the whole realization of ξ-based rheology provided above is straightforward (if lengthy), and is expected to work for non-constant coefficient matrices ld, ls as well. However, specifically for constant coefficients, an alternative version is also possible, i. e., implementing the antisymmetric part of the coefficient matrices, that is, the constants lAd, lAs, in the reversible part of the time evolution.

For this case, let us use the prepared (21)–(22) form of the Onsagerian equations. Rearranging the time evolution equation is straightforward, and we find for the alternative reversible operator matrix L′

while the elements of the alternative irreversible operator matrix M′ are very similar to (48): we just have to change all l12d and l21d to lSd and, similarly, l12s and l21s to lSs.

The degeneracy criteria, antisymmetry of L′, the Jacobi identity for the generalized Poisson brackets, and positive semidefiniteness for M′ prove to be satisfied. Moreover, in this case the symmetric property of M′ also holds.

We repeat that this alternative realization is valid only for constant Onsagerian coefficients as otherwise the Jacobi identity would be violated.

This latter variant appears rather counter-intuitive from a principal point of view but may be beneficial for numerical solutions, e. g., to have as much symplectic part in the numerical scheme as possible – see [10], [11] about the importance of this.

5 Temperature as state variable

For mechanical engineering applications and evaluations of experiments (see, e. g., [2]), it can be beneficial to use temperature, instead of entropy, as state variable. Then the collection of state variables is

To keep the discussion as concrete and basic as possible, let us choose the simplest constitutive equation for the internal energy, linear in temperature with constant specific heat c, containing elastic energy related to Hooke’s law,

and neglecting thermal expansion – which is manifested in the separation of strain dependence and temperature dependence, i. e.,

Temperature has the same relationship to specific entropy as seen in Section 4, now utilized in the opposite direction (i. e., what is a variable and what is a function): The thermodynamical consistency condition ds˜0dT=1Tde˜thdT that follows from the Gibbs relation (and which is the manifestation of the third equation of (4)) leads to

with auxiliary constants s˜aux, Taux, and the extension (7) induces

or, expressing temperature,

Now the energy and entropy functionals are

We perform a transformation of variables x→x˜, by which the transformation (operator) matrix Q=δx˜δx is accompanied. This Q can be used to establish the relationship between the original and transformed reversible and irreversible operator matrices [6], i. e.,

In the present current special case, we change only the fourth state variable (from s to T), so only the fourth row of Q contains non-trivial elements. Furthermore, since (55) does not contain non-local (gradient) terms, we can realize the transformation directly in the form