Thermodynamics with Internal Variables. Part II

J. Non-Equilib. Thermodyn. Vol. 19 (1994), pages 250-289

Thermodynamics with Internal Variables Part II. Applications Gerard A. Maugin1·3, Wolfgang Muschik2, 1 Wissenschaftskolleg zu Berlin, Berlin, Germany 2 Institut für Theoretische Physik, Technische Universität Berlin, Berlin, Germany Registration Number 638

Abstract

The second part of this synthetic work presents and discusses the most spectacular and successful applications of the irreversible thermodynamics with internal variables. These include viscosity in both fluids and solids (in the former case, in complex fluids and structurally complex flows), viscoplasticity and rate-independent plasticity in small and finite strains, damage and cyclic plasticity, electric and magnetic relaxation, magnetic and electric hysteresis, normal and semi-conduction, superconductivity of deformable solids, and ferrofluids. In all cases the internal variables of interest are given some physical significance in terms of quantities defined at a sublevel of description. The relevant internal variables may be as varied as second-order tensors, real or complex valued scalars, and polar or axial vectors. Furthermore, the role played by internal variables in wave-propagation problems is emphasized through appropriate examples. The presentation ends with reaction-diffusion problems. This is illustrated by damage, plastic-strain localization and models for nerve-pulse dynamics. Globally, we have all ingredients of a truly post-Duhemian irreversible thermodynamics of complex behaviors. 9. A wealth of applications4

As there apparently are no limits to the choice of internal variables and phenomena to be represented by these, the class of possible applications is itself theoretically infinite and it is practically limited only by the imagination of scientists. However, the most 3

Permanent affiliation: Laboratoire de Modelisation en Mecanique, (URA 229 CNRS), Universite Pierre-et-Marie Curie, Paris, France. 4 The sections of this second part are numbered consecutively to those of the first part. Similarly, new bibiographical references start with item numbered 93, the first 92 references being listed at the end of Part I [93]. J. Non-Equilib. Thermodyn. Vol. 19, 1994, No. 3 © Copyright 1994 Walter de Gruyter · Berlin · New York

Thermodynamics with internal variables. Part II. Applications

251

interesting among these applications are those which exhibit a strong interplay between the evolution of internal variables and the behavior related to observable variables. The following sections have for purpose to illustrate these most fruitful developments in relation to wide classes of irreversible dynamical behaviors. Thus special attention is paid to viscosity and its various manifestations, in both fluids and solids, plasticity and plastic-like phenomena, similar processes concerning electromagnetic properties whether the mechanical behavior is that of solids or fluids, and dynamics and wave properties in so far as they are strongly influenced by the existence of a dissipative behavior described by means of internal varibles. In the remainder of this review we shall admit that the Clausius-Duhem inequality is a priori written either as

(9.1) or 0,

(9.2)

where we recall that ρ is the actual density of matter in the present configuration Kt at time r, ψ is the Helmholtz free energy per unit mass in Kt, η is the entropy density per unit mass in Kt, θ is the absolute temperature, q is the heat flux vector, S is the entropy flux vector, a superimposed dot denotes the material time derivative, and p(i) is the elementary power developed by internal forces. In pure mechanics for simple continua, the latter are the classical symmetric stresses, and p(i) reads (tr = trace)

(9.3) wherein ].

(9.4)

Here σ is the Cauchy stress tensor, í is the material velocity, and D is the tensor of strain rates. In general, S, q and θ are related by the general expression S = 0- a q + k,

(9.5)

where k is called the extra entropy flux. The latter vanishes in many applications (as a matter of fact, in the absence of diffusion of the internal variables), whence (9.2) reduces to the simplified form (9.1). For an incompressible fluid for which J7-v = trD = 0, we can rewrite (9.1) in the form - ( W+ NO) + σά:Ό + 0q· F(0" x ) ^ 0, J. Non-Equilib. Thermodyn. Vol. 19, 1994, No. 3

(9.6)

252

G. A. Maugin, W. Muschik

where σά is a deviatoric (tracefree) stress tensor and there exists a field of mechanical pressure p such that

'=ρψ, Ν = ρη.

(9.7)

For general continua in finite deformations, we have

(9.8)

where χ is the motion, F is its gradient, and p0 is the density at the reference configuration KR. As p0 does not depend on time, introducing the first Piola-Kirchhoff stress tensor Ô by

\

(9-9)

we show that (9.1) transforms to -(W+NO) + Pm + eQ-VR(e-l)>0,

(9.10)

wherein

Q = J F F- 1 -q,

FA = F r -F.

(9.11)

Transformed forms such as (9.6) and (9.10) are easily established if (9.2) holds to start with. In the following applications we shall assume that the axiom of locaraccompanying state (LAS) holds good so that entropy is defined as in equation (4.19). Furthermore, very often heat dissipation Öéê is separately considered from the so-called intrinsic dissipation # intr . 10. Viscosity in fluids 10.1. Ν on-Newtonian fluids Viscosity is the most ubiquitous dissipative mechanical behavior. Its essential feature is the manifestation of a time scale and it is, therefore, akin to relaxation. No wonder, then, that its proper framework often be that of internal-variable theory. Both fluids and solids are capable of exhibiting viscosity to a larger or lesser degree. Here we concentrate on fluids. We distinguish fluids from solids through the fact that the former, contrary to the latter, do not exhibit any privileged (reference) configuration ("they have a tendency to occupy their actual container more or less rapidly"). We may also say that their reference configuration is constantly re-actualized. J. Non-Equilib. Thermodyn. Vol. 19, 1994, No. 3

Thermodynamics with internal variables. Part II. Applications

253

The classical incompressible viscous behavior is that of Newtonian fluids for which classical T.I.P is sufficient [13], Generally, on account of the axiom of local state, T.LP based on (9.6) yields the celebrated Newtonian-Stokesian constitutive equation σά = 2ηνΌ, trD = 0, ^0,

(10.1)

where ηυ is the (possibly temperature dependent) viscosity (shear) coefficient. Any fluid whose constitutive equation deviates from this is called a non-Newtonian fluid. The modern science of rheology (e.g., [94]) is essentially concerned with the construction of models, and the subsequent study of the flows, of these fluids. The Deborah number, already introduced in Part I, is the characteristic parameter of rheology. In the spirit of internal-variable theory, for non-Newtonian fluids (10.1)! is modified to read (10.2) where the internal variable á is governed by an evolution equation (10.3) which, together with the traceless extra stress σρ, is subjected to satisfy the statement of the Clausius-Duhem inequality. The formulation (10.2)-(10.3) fits in the general conceptual frame sketched in Part I [93]. It is conceived as an alternative to the time functional formulation (10.4) which has come to represent the general constitutive equation of simple fluids following the works of Coleman, Noll and others in the 1960s-1970s [95]. With appropriate continuity assumptions various approximations of (10.4) provide hereditary constitutive equations in the manner of Volterra. Among these we single out the BernsteinKearsley-Zapas (BKZ) [96] model for which an explicit form of (10.4) is s

o

(10.5)

wherein B:= FFT, φί = dir/dli9

i = 1,2.

(10.6)

Here / f are the first two invariants of the Finger tensor B and if is a kind of potential. A special case of Hf is said to be separable [97] whenever we can write ir(s9Il9I2) = m(at)V(Il9I2)9

(10.7)

where m can be represented by a linear combination of memory functions of the exponentially decreasing type. A remarkable property of the model (I0.5)-(l0.7) is that J. Non-Equilib. Thermodyn. Vol. 19, 1994, No. 3

254

G- A. Ma gin, W. Muschik

B satisfies an "objective" (i.e., invariant by time-dependent rotations at time f) differential equation: (10.8) where fc is linear affine in its two variables. It follows from the separability condition that the stress σά itself satisfies a series of rheological evolution equations (for short DRM = differential rheological models). The above model refers to an elastic fluid (since  is a measure of elastic strain). No microstructure, no image relating to any physical vision, intervene in this formal construction. We keep in mind from this model that  satisfies an evolution equation, (10.8). The situation is quite different with models based on internal variables which directly relate to a microstructure. In this case the tensorial order of á is a determining factor in the precise formulation of the coupled equations (10.2) and (10.3). This is illustrated by so-called complex fluids. 10.2. Complex fluids

Complex fluids are those fluids for which we are aware of the multicomponent composition and/or the existence of a microstructure. The latter gives rise to a microscopically sensible alteration in viscosity properties. These fluids include dilute and semi-dilute polymeric solutions, polyelectrolyte solutions, liquid crystals, suspensions of rigid or deformable particles, emulsions, etc. They are excellent candidates for a thermodynamical description by means of internal variables as the latter then can be more or less readily identified, usually at some mesoscopic scale of observation. Furthermore, a relationship with a statistical theory can often be established, at least formally; but the imposition of restrictions by the second law usually brings more severe requirements on the resulting constitutive-evolution equations than does the averaging based on the statistical approach. Several choices are possible for the internal variable and we shall classify the models according to the tensorial order of the selected internal variable. A previous review on the subject is that of Maugin and Drouot [98] to whom the reader is referred for greater detail for works up to 1987. A. The internal variable is an anelastic strain Several authors [99-101] must be credited for the original idea of introducing a tensor as an internal variable in viscoelastic materials and we owe to Green and Naghdi [102] and Perzyna [103] the introduction of an intermediate configuration KL and the idea to relate the tensorial internal variable to one of the elements of the kinematical decomposition thus introduced (at least in viscoplasticity). In the case of interest here (viscoelasticity of complex fluids), Leonov [104] has exploited the above ideas by considering the multiplicative decomposition of E. H. Lee: F = Fe¥a, where none of the elements Fe (for elastic) and Fa (for anelastic) are true gradients. From the definition of  - equation (10.6) - it is shown that the following kinematical identity holds true (compare (10.8)) B:=D,B-(BD + BB) = 0,

(10.9) J. Non-Equilib. Thermodyn. Vol. 19, 1994, No. 3

Thermodynamics with internal variables. Part II. Applications

255

where Dj denotes the Zaremba-Jaumann co-rotational derivative defined by £>, =  - Ù Â + ÂÙ, Ù:=( Fv)fl = £ [ Fv -( Fv)7']-

(10.10)

Let >.

(10.11)

Then one shows that Bc satisfies the following differential equation: (10.12) One can view equations (10.9) and (10.12) as evolution equations. In particular, following Leonov [104], if we consider Be as a tensorial internal variable, and note that for a fluid behavior W cannot depend on B (see [95]), we can only take W as a function of Be and Θ, and the procedure sketched out in Part I leads to an intrinsic-dissipation inequality ^0,

(10.13)

where (d = deviatoric part)

dW

V

A

dW

«