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Thermodynamic Inequalities in Continuum Mechanics
Abstract
The constitutive relations for the transport of heat, stress, electric charge, etc., in a continuum must be chosen so that the second law of thermodynamics is not violated; the constraints take the form of inequalities, typically requiring the entropy generated within a material element to be non-negative. The paper is concerned with this concept—its history, the physical principles on which it depends, how to apply it when second-order or non-linear effects are important and how it is widely misused in modern continuum mechanics.
The history is reduced to the contributions of five leading thermodynamicists—Clausius, Maxwell, Gibbs, Boltzmann and Duhem. The object here was to try to discover which form of the inequality one should regard as being fundamental. One important conclusion is that entropy S must be defined simultaneously with the identification of the inequality, and that in general this cannot be done until the constitutive equations are known. The empirical element enters with the notion of irreversibility, which is given a precise meaning with the aid of the motion reversed parity η(x), a variable x having η = +1 or η = −1 if, when time and motions are reversed, x → x or x → −x. The macroscopic parity of x, η*(x), is obtained by first replacing x by the constitutive equation for x.
The entropy production rate Σ has both irreversible Σ(f) and reversible Σ(r) parts. It is shown that the reciprocal relations follow from the requirement that the macroscopic parity of Σ(i) must be +1.
Continuum thermodynamics is based on various principles extracted from theory developed for uniform systems, the example chosen to illustrate the ideas being the simple monatomic gas. Second-order constitutive relations are introduced, and the expressions for entropy and its production rate per unit volume, σ, obtained. It is shown that the stability condition σ ≥ 0 cannot, in general, be satisfied merely by imposing constraints on the constitutive relations. To second-order σ = σ1 + σ2, where σ1 is the usual bilinear form, and the terms in σ2 have an additional derivative. The second-order term σ2 can have both signs, and is not dissipative. The relation between this fact and the frame-dependence of constitutive relations is explained.
The final section illustrates the errors frequently found in the thermodynamic arguments appearing in books and papers on rational continuum mechanics. The principle of these is that σ ≥ 0 is interpreted as being a constraint on the constitutive relations alone. Another is the idea that the balance equations can be set aside as constraints by regarding them as mere definitions of a heat source and a body force, an error based partly on the misconception that constitutive relations should be frame-indifferent. Finally, an inequality due to Glansdorff & Prigogine is examined and found to be in error.
... For example, indicating a quantity to be p α does not necessarily mean that it is pressure or that it is rooted in a well-defined microscale pressure. Furthermore, these "consistent" approaches typically make use of rational thermodynamics, an approach that is not without controversy [32,43,49,50,55,56], and which has the feature that temperature is considered to be a primitive variable that actually has no correspondence with temperature as employed in classical irreversible thermodynamics. The challenges associated with averaging also confound the use of rational thermodynamics to provide consistent, testable models. ...
Porous medium researchers and practitioners usually rely on macroscale models to represent systems of concern. While macroscale models have often been formulated phenomenologically, the thermodynamically constrained averaging theory (TCAT) provides a means to rigorously derive closed macroscale models for a wide variety of systems. However, the TCAT approach can appear overwhelmingly complicated, the entry point for use unclear, and the advantages not self-evident. In response to these perceived shortcomings, we demonstrate several aspects of TCAT macroscale model formulations for single-fluid flow in a porous medium. Specifically, we illustrate an essentially exact macroscale model derived from a rigorous connection across scales, show how an entropy inequality can be used to derive an approximate macroscale form for the Stokes-flow regime, and examine models in the transition flow regime. A special emphasis is placed upon leveraging available results, and other application opportunities are summarized.
... 70,71 An additional controversy is more foundational and concerned with the so-called "axioms of continuum mechanics." [72][73][74] Interested readers are referred to the publications, and we note that even the regular (i.e., non-fluctuating) multicomponent shock mixture models are used within narrow regions of applicability for well-studied materials and their specific applications. ...
The notion of plane shock waves is a macroscopic, very fruitful idealization of near discontinuous disturbance propagating at supersonic speed. Such a picture is comparable to the picture of shorelines seen from a very high altitude. When viewed at the grain scale where the structure of solids is inherently heterogeneous and stochastic, features of shock waves are non-laminar and field variables, such as particle velocity and pressure, fluctuate. This paper reviews select aspects of such fluctuating nonequilibrium features of plane shock waves in solids with focus on grain scale phenomena and raises the need for a paradigm change to achieve a deeper understanding of plane shock waves in solids.
... We apply the method of Coleman and Noll [3] to derive constitutive relationships from the second law of thermodynamics. This method has been criticized [21] and shown to be more restrictive than Mueller's method [19]. However, the criticisms are not shared by all experts in the field [19,20], and for our purposes the use of either method is satisfactory. ...
A thermo-chemo-electro-mechanical formulation of quasi-static finite deformation of swelling incompressible porous media is derived from a mixture theory including the volume fraction concept. The model consists of an electrically charged porous solid saturated with an ionic solution. Incompressible deformation is assumed. The mixture as a whole is assumed locally electroneutral. Different constituents following different kinematic paths are defined: solid, fluid, anions, cations and neutral solutes. Balance laws are derived for each constituent and for the mixture as a whole. A Lagrangian form of the second law of thermodynamics for incompressible porous media is used to derive the constitutive restrictions of the medium. The material properties are shown to be contained in one strain energy function and a matrix of frictional tensors. A principle of reversibility results from the constitutive restrictions. Existing theories of swelling media should be evaluated with respect to this principle.
... where S denotes entropy of a thermodynamic state. Although the interpretation of entropy as a state variable has been a matter of controversy (Woods, 1981(Woods, , 1982, we shall assume that entropy can be accommodated through the concept of an internal variable. The first law of thermodynamics prescribes the local equation of energy balance, which can be adapted in the incremental form _ W ðN; I 1 ; I 2 ; SÞ ¼ T ij _ D ij À r Áh þ _ WðN; I 1 ; I 2 Þ; ðA:2Þ ...
Important effects resulting from chemical exposure of a polymeric material are the alterations in its stress–strain response, its sensitivity to strain-rate effects and the development of irreversible strains. The paper presents the results of experiments conducted on membranes subjected to chemical exposure and discusses the development of a single constitutive model that is applicable to both untreated and chemically-treated materials. The constitutive modelling of both categories of polymeric materials takes into consideration the influence of large strains, strain-rate effects and irreversible effects.
A macroscopic continuum theory of two-phase saturated porous media is derived by a purely variational deduction based on the least Action principle. The proposed theory proceeds from the consideration of a minimal set of kinematic descriptors and
keeps a specific focus on the derivation of most general medium-independent governing equations, which have a form independent from the particular constitutive relations and thermodynamic constraints characterizing a specific medium.
The kinematics of the microstructured continuum theory herein presented employs an intrinsic/extrinsic split of volumetric strains and adopts, as an additional descriptor, the intrinsic scalar volumetric strain which corresponds to the ratio between solid true densities before and after deformation.
The present theory integrates the framework of the Variational Macroscopic Theory of Porous Media (VMTPM) which, in previous works, was limited to the variational treatment of the momentum balances of the solid phase alone. Herein, the derivation of the complete set momentum balances inclusive of the momentum balance of the fluid phase is attained on a purely variational basis.
Attention is also focused on showing that the singular conditions, in which either the solid or the fluid phase are vanishing, are consistenly addressed by the present theory, included conditions over free solid-fluid surfaces.
In this work, we perform a thermodynamical analysis of an isothermal relaxation process inside the context of elasto-viscoplasticity under the assumption of small deformations. Departing from the one-dimensional Bingham rheological model and assuming that some key decompositions on stress, free energy, and dissipation are possible we demonstrate that, among all equilibrium states satisfying the yield criterion, the actual equilibrium state, given a prescribed total deformation, is the one which minimizes the viscous energy dissipated during the relaxation process. From the one-dimensional motivation we extend the formulation to the multidimensional hardening case. Therefore, based on concepts of thermodynamics with internal variables (TIV) and some results of convex analysis, we arrive at a constrained minimization problem whose solution are state equations for the “over quantities”, or for the equilibrium state, which are identical to those obtained from the minimum complementary free energy principle. This result brings some physical aspects which can aid the understanding of relaxation processes in elasto-viscoplastic media, providing physical foundations which can help the understanding of well established viscoplastic models, as well as the proposing of constitutive formulations and solution algorithms for this class of materials.
A thermodynamic analysis of tokamak plasmas is developed, starting from a single-fluid model. Some general propertiesof entropy production in the plasma are discussed. An approach is employed that is more general than the one based on Onsager symmetry relationships. This choice appears to be justified by experiments. The concept of ‘thermodynamic stability’ is introduced, and a general criterion for the stability of stationary states far from thermodynamic equilibrium is assessed. An upper limit on the particle density is found to exist as a necessary condition in order to prevent disruptions. Analysis of the entropy balance shows that this limit is modified when neutral-beam heating is applied. Radial temperature profiles are shown to be stable against external heating under specific experimental conditions, inparticular when ion and electron temperatures are comparable. An offset scaling law for the energy confinement time TE = TE(Pinp) (with Pinp total input power) is obtained for stationary states when stability against additional heating is imposed.
The second-order transport terms in a monatomic gas, originally derived by Burnett from Boltzmann's equation via the Chapman–Enskog iteration, have the curious property of being dependent on the observer's reference frame. However, frame-indifference has long been accepted as an essential property of constitutive equations in continuum mechanics. Various attempts have been made to find errors in the kinetic theory, but these have been countered by physical accounts of the framedependent terms that are independent of the Chapman–Enskog theory.
It is shown in this paper that both the Chapman–Enskog theory and the physical models are based on an inappropriate definition of the peculiar velocity. The Chapman–Enskog theory is easily corrected, and the result is a kinetic theory in harmony with the principle of material frame-indifference. A brief survey of the debate on this topic is presented, and corrected expressions are given for the Burnett terms.
A new theory is developed for the transport of energy across a strong magnetic field. The mechanism is micro-convection by sheared fluid elements, and it proves to be much more effective in heat transport than classical conduction. The equivalent thermal conductivity is calculated for the electron gas in a cylindrical plasma carrying an axial current. When the plasma pressure can be neglected, this conductivity is about 400 times greater than the classical value under certain typical conditions.
A cellular timing mechanism is proposed based on the spiralling motion of membranous blocks of molecules as they pass sequentially through dielectric zones within a cell. The zones can change with altered membrane surface conditions causing altered sequences and timing in metabolic processing and can relate to oncogenesis.
A general model of thermomechanical behaviour is introduced, based on Bridgman’s perception of the role of thermodynamic forces,
Eckart’s finite anelastic deformation concept, Ziegler’s dissipation potential and a greatly simplifying assumption of ‘small
elastic deformation’. An apparently new non-linear extension to the ‘thermorheologically simple’ viscoelastic model is obtained
and a fairly comprehensive elastoviscoplastic model presented.
The problem of determining the states of contacting, rubbing and wearing bodies is resolved in the present work using the constitutive equations for materials. Measures of deformation are formulated for a body and a two-dimensional micropolar interfacial layer. Definitions for the thermodynamical processes in the body and the layer are introduced. Constitutive equations and linearized theories for the thermoelastic and hyperelastic bodies are derived. Constitutive equations and linearized theories for the layer are formulated for the micropolar thermoelastic material and the micropolar fluid. The thermo viscous fluid layer is given as an example of non-polar layer theory.
Examples of coupled irreversible processes like the thermoelectric phenomena, the transference phenomena in electrolytes and heat conduction in an anisotropic medium are considered. For certain cases of such interaction reciprocal relations have been deduced by earlier writers, e.g., Thomson's theory of thermoelectric phenomena and Helmholtz' theory for the e.m.f. of electrolytic cells with liquid junction. These earlier derivations may be classed as quasi-thermodynamic; in fact, Thomson himself pointed out that his argument was incomplete, and that his relation ought to be established on an experimental basis. A general class of such relations will be derived by a new theoretical treatment from the principle of microscopic reversibility. (§§1-2.) The analogy with a chemical monomolecular triangle reaction is discussed; in this case a a simple kinetic consideration assuming microscopic reversibility yields a reciprocal relation that is not necessary for fulfilling the requirements of thermodynamics (§3). Reciprocal relations for heat conduction in an anisotropic medium are derived from the assumption of microscopic reversibility, applied to fluctuations. (§4.) The reciprocal relations can be expressed in terms of a potential, the dissipation-function. Lord Rayleigh's "principle of the least dissipation of energy" is generalized to include the case of anisotropic heat conduction. A further generalization is announced. (§5.) The conditions for stationary flow are formulated; the connection with earlier quasi-thermodynamic theories is discussed. (§6.) The principle of dynamical reversibility does not apply when (external) magnetic fields or Coriolis forces are present, and the reciprocal relations break down. (§7.)
