Continuum Thermodynamics [1st ed.] 978-3-030-11156-4, 978-3-030-11157-1

Table of contents :
Front Matter ....Pages i-xi
Heat Conduction (Paolo Podio-Guidugli)....Pages 1-21
Thermomechanics (Paolo Podio-Guidugli)....Pages 23-31
The Principle of Virtual Powers (Paolo Podio-Guidugli)....Pages 33-40
A Virtual Power Format for Thermomechanics (Paolo Podio-Guidugli)....Pages 41-48
A Physical Interpretation of Thermal Displacement (Paolo Podio-Guidugli)....Pages 49-54
Back Matter ....Pages 55-76

Citation preview

SISSA Springer Series 1

Paolo Podio-Guidugli

Continuum Thermodynamics

SISSA Springer Series Volume 1

Editor-in-chief Gianni Dal Maso, SISSA - Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy Series editors Fabrizio Catanese, Institut für Mathematik, Universität Bayreuth, Bayreuth, Germany Boris Dubrovin, SISSA - Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy Max Gunzburger, Department of Scientific Computing, Florida State University, Tallahassee, FL, USA

The SISSA Springer Series publishes research monographs, contributed volumes, conference proceedings and lectures notes in English language resulting from workshops, conferences, courses, schools, seminars, and research activities carried out by SISSA: https://www.sissa.it/. The books in the series will discuss recent results and analyze new trends focusing on the following areas: geometry, mathematical analysis, mathematical modelling, mathematical physics, numerical analysis and scientific computing, showing a fruitful collaboration of scientists with researchers from other fields. The series is aimed at providing useful reference material to students, academic and researchers at an international level.

More information about this series at http://www.springer.com/series/16109

Paolo Podio-Guidugli

Continuum Thermodynamics

123

Paolo Podio-Guidugli Accademia Nazionale dei Lincei Rome, Italy

ISSN 2524-857X ISSN 2524-8588 (electronic) SISSA Springer Series ISBN 978-3-030-11156-4 ISBN 978-3-030-11157-1 (eBook) https://doi.org/10.1007/978-3-030-11157-1 Library of Congress Control Number: 2018968388 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Alla piccola Anna, che non era nata al tempo delle mie lezioni alla SISSA ma lo è al momento in cui licenzio queste pagine, perché le sfogli se un giorno si domanderà che cosa mai girasse nella testa di suo nonno

Preface

These notes reflect the contents of four two-hour lectures given at SISSA on May 3, 4, 7, and 8, 2018. Just like those lectures, they begin in medias res: Some basic notions in continuum mechanics, which a well-intentioned reader should but may not be familiar with, are collected in a final appendix, to be browsed also by those who need to familiarize themselves with the notations that I use. I have tried to maintain a reasonable level of mathematical rigor. In fact, the envisaged audience comprises mathematical modelers - or mathematical modelers to be - of continuous material bodies, be they mathematicians, physicists, or mathematically versed engineers. Continuum thermodynamics is an important topic in phenomenological physics, a field theory that aims to account for the phenomenology targeted by classical statistical thermodynamics and more, in that, at variance with the latter theory, it deals with processes that typically are neither spatially homogeneous nor sequences of equilibrium states. In the continuum theory of heat conduction, the constitutive laws furnish a mathematical characterization of the macroscopic manifestations of those fluctuations in position and velocity of microscopic matter constituents that statistical thermodynamics considers collectively. For us today, as in 1857 for Rudolf Clausius, heat conduction is nothing but a macroscopic evidence of “that kind of [microscopic] motion that we call heat” [10]. It is only natural that, after heat conduction per se, we devote our attention to thermomechanics, a slightly more complex paradigmatic example of a field theory where microscopic and macroscopic manifestations of motion become intertwined. In my SISSA lectures, the material in Chaps. 1, “Heat Conduction”, and 2, “Thermomechanics”, has been covered in fair detail; not so, due to the tyranny of time, that in Chap. 3, “The Principle of Virtual Powers”. This section is a convenient precursor to the material in Chap. 4, “A Virtual Power Format for Thermomechanics”, which I could not cover at all in the SISSA lectures but is included here because in my view it helps to put the contents of Chap. 2 into perspective. For the sake of completeness, I have also included the final Chap. 5, “A Physical Interpretation of Thermal Displacement”, a notion used formally in the preceding chapter. vii

viii

Preface

It is a pleasant duty for me to acknowledge SISSA’s support and hospitality, as it is to acknowledge with gratitude many valuable discussions with my lifetime friend Gianpietro Del Piero. Rome, Italy October 2018

Paolo Podio-Guidugli

Contents

1 Heat Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Energy Balance and Entropy Imbalance . . . . . . . . . . . . . . 1.2 The Clausius-Duhem Inequality . . . . . . . . . . . . . . . . . . . . 1.3 Internal Dissipation, Helmholtz Free Energy, Reduced Dissipation Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Constitutive Restrictions via the Coleman-Noll Procedure . 1.5 Specific Heat. The Gibbs Relations . . . . . . . . . . . . . . . . . 1.6 The Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 More on the Heat Equation . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 Gradient-Flow Deduction . . . . . . . . . . . . . . . . . . . 1.7.2 Time Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Nonstandard Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.1 On Proportionality of Energy and Entropy Inflows 1.8.2 Hyperbolic Heat Conduction . . . . . . . . . . . . . . . . 1.9 Order Parameters. State and Substate Variables . . . . . . . .

...... ...... ......

1 1 4

. . . . . . . . . . .

5 6 8 9 11 11 12 14 14 18 21

... ...

23 23

. . . . .

. . . . .

24 25 27 28 29

3 The Principle of Virtual Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The PVP as a Source of Balances and Representations . . . . . . . . . 3.2 The PVP as a Source of Evolution Equations . . . . . . . . . . . . . . . .

33 33 37

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

2 Thermomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Referential and Current Densities . . . . . . . . . . . . . . . . . . . . . 2.2 Energy Balance, Entropy Imbalance, and Reduced Dissipation Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Referential Versions of the Governing Laws . . . . . . . . . . . . . 2.4 The Constitutive Equations of Thermoelasticity . . . . . . . . . . . 2.5 The Constitutive Equations of Thermoviscoelasticity . . . . . . . 2.6 Internal Dissipation in Purely Mechanical Circumstances . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . .

ix

x

Contents

3.3 The PVP as a Dimension-Reduction Tool . . . . . . . . . . . . . . . . . . 3.3.1 From 3D to 1D: Beam Theory . . . . . . . . . . . . . . . . . . . . . 3.3.2 From 3D to 2D: Plate Theory . . . . . . . . . . . . . . . . . . . . . .

38 39 40

4 A Virtual Power Format for Thermomechanics . . . . . . . . . . 4.1 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Balance and Imbalance Laws of Thermal Conduction 4.2.1 Axiom of Conservation of Internal Action . . . . . . . 4.2.2 Dissipation Axiom . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

41 41 43 45 47

5 A Physical Interpretation of Thermal Displacement . . . . . . . 5.1 From Helmholtz and de Broglie to Einstein and Langevin 5.2 How to Interpret Thermal Displacement Following the Einstein Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 How to Interpret Thermal Displacement Following the Langevin Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

...... ......

49 49

......

50

......

53

Appendix: Basic Notions of Continuum Mechanics . . . . . . . . . . . . . . . . .

55

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

. . . . .

. . . . .

. . . . .

. . . . .

About the Author

Paolo Podio-Guidugli studied Nuclear Engineering at the University of Pisa. He has served as a Full Professor of Mechanics of Materials and Structures at the Universities of Ancona, Pisa, and Roma TorVergata. He is a fellow of the Accademia Nazionale dei Lincei. He has published more than 240 papers and 7 books, and co-edited 4 books. His research interests are in foundational and applied issues in rational continuum physics: elasticity (general and applied to rods, plates, shells, and nanotubes), plasticity, multiscale modeling of condensed matter, materials (oriented, with internal constraints, with elastic range), phase interface motion, phase segregation by atomic rearrangement, crack propagation, deformable ferromagnets, strain and superconductivity, and thermodynamics.

xi

Chapter 1

Heat Conduction

The central purpose of this chapter is to derive the Heat Equation, that is, the equation that describes temperature evolution in rigid conductors; this is done in Sect. 1.6, after the indispensable preliminaries are covered in Sects. 1.1–1.5. This derivation, which is of interest by itself, best exemplifies how constitutive issues are dealt with in modern continuum mechanics. A crucial assumption—that the entropy inflow in a body part is equal to the inflow of internal energy times the reciprocal of the absolute temperature—is discussed in Sect. 1.8.1. Section 1.7 contains some complementary information about the heat equation, an hyperbolic version of which is derived in Sect. 1.8.2 in a manner different from usual.

1.1 Energy Balance and Entropy Imbalance Under the assumptions that the rigid body under study is not acted upon by any force system and kept at rest, the energy balance can be stated as (internal energy)· = heating,

(1.1)

or rather, more formally, as ˙ E(P) = Q(P) for each body part P,

(1.2)



where

,

E(P) =

(1.3)

P

© Springer Nature Switzerland AG 2019 P. Podio-Guidugli, Continuum Thermodynamics, SISSA Springer Series 1, https://doi.org/10.1007/978-3-030-11157-1_1

1

2

1 Heat Conduction

with  the volume density of internal energy, and where time changes of internal energy are associated with a heat inflow (q, r ) consisting of a vectorial influx q and a scalar source r :   q · n + r. (1.4) Q(P) = − ∂P

P

Accordingly, as a part-wise formal statement of the verbal balance (1.1) we pose 



·

 =−

 ∂P

P

q·n+

r for each body part P,

(1.5)

P

whence, by localization, we arrive at the point-wise energy balance ˙ = −divq + r.

(1.6)

Remark 1.1 The use of the term ‘influx’ for q, and of the minus sign before the boundary integral, is consistent with our choice of n as the outward normal to the boundary of a body part: as a consequence, heat flows into the part in question whenever q · (−n) > 0. The use of the letter r for the volumetric heat supply alludes to the typical radiative source. A basic role in the theory of heat conduction—in fact, in any theory where an attempt is made to model mathematically dissipation and irreversiblity—is played by the part-wise entropy imbalance ˙ H(P) ≥ D(P) for each body part P,

(1.7)



where

η

H(P) =

(1.8)

P

is the entropy of the body part P and where entropy changes with time are bounded below by   h·n+ s, (1.9) D(P) = − ∂P

P

with the disorder inflow (h, s) consisting of a vectorial influx h and a scalar source s. A verbal statement of Eq. (1.7) is (entropy)· ≥disordering,

(1.10)

while Eq. (1.5) is paralleled by 

·

η ≥− P



 ∂P

h·n+

s for each body part P. P

(1.11)

1.1 Energy Balance and Entropy Imbalance

3

Inequality (1.7) is the form taken in the present context by the Second Law of thermodynamics; we postulate it alongside with the form (1.2) of the First Law. The presumption implicit in the different format of these two laws is that, at variance with internal energy, a specific external inflow does not suffice to balance entropy of a body part; to achieve this, an internal entropy production should be included, under form of an always nonnegative, spontaneous contribution to entropy growth: ˙ (P) := H(P) − D(P) ≥ 0 , (P) =

 δ.

(1.12)

P

Note that (1.12) defines the internal entropy production. As we shall see, to close the theory of heat propagation (and, in fact, to close any other classical thermodynamic theory of material behavior), we need not specify constitutively the internal entropy production. Philosophically, this is a rather subtle, but important, point: beside energy (internal and kinetic) and entropy, no further macroscopic descriptor of microscopic phenomena has to be introduced in the continuum picture. Remark 1.2 We have referred to the (influx, source) pairs associated with internal energy and entropy as to their inflows. Inflows pertain to extensive entities: a physical quantity is termed extensive if it can be regarded as the density with respect to the volume measure of a certain property of a body; it is termed intensive otherwise.1 Here, e.g., energy and entropy are extensive and temperature intensive; mass is extensive, color intensive; and so on. We read in [68], Sect. 260, Ftn. 2, that the distinction between extensive and intensive variables was first made by Maxwell in 1876, who referred to the former as ‘magnitudes’ representing “a physical quantity, the value of which, for a material system, is the sum of its values for the parts of the system”, while the latter “denote the intensity of certain physical properties of a substance”. Needless to say, set additivity does not imply absolute continuity with respect to volume measure, a stronger property that allows for an associated integral representation. Remark 1.3 To base the theory of heat conduction on balance of energy and imbalance of entropy is in agreement with two famous cosmological dictums by Rudolf Clausius: • Die Energie des Weltalls ist konstant. • Die Entropie des Weltalls strebt einem Maximum zu. In [47], reference is made to [10], Clausius’es 1864 paper in which he discusses “the different forms of the fundamental equations of the mechanical theory of heat”. Clausius, as Helmholtz and Boltzmann after him as well as many others, including Einstein [14–16], was rightly convinced that heating was nothing but a gross

1 More generally, a quantity is extensive if it has a density with respect to one or another measure of

spatial extension of the region a material body occupies, be it the volume, area, or length, measure.

4

1 Heat Conduction

phenomenon resulting from the motions of molecules, a macroscopic manifestation of “that kind of motion that we call heat” [9]. In the light of our versions (1.5) and (1.11) of energy balance and entropy imbalance, we may replace Clausius’es Weltall by a more down-to-earth isolated system, i.e., a body for which both q · n ≡ 0 on ∂ B and r ≡ 0 in B: for such a body, under the customary assumption that the inflows of energy and entropy are proportional, (1.5) and (1.11) reduce to   ·   ·  = 0 and η ≥ 0. (1.13) B

B

1.2 The Clausius-Duhem Inequality It is quite customary to deem the entropy and energy inflows—said differently, the inflows of disorder and heat—to be proportional: (h, s) = ϑ −1 (q, r ), ϑ > 0,

(1.14)

via the coldness, that is, the inverse ϑ −1 of the absolute temperature. This proportionality assumption reflects the experimental fact that an energy inflow has different entropic effects according to the temperature at which it takes place: more precisely, this assumption encodes that those effects are less and less pronounced as temperature gets bigger and bigger.2 Later on, in Sect. 1.8.1, we shall show that (1.14) must hold true when heat conduction is in a precise sense isotropic; and that it may be violated in case of less pronounced material symmetry [55]. By localization, (1.11) and (1.14) yield the entropy imbalance: η˙ ≥ −div(ϑ −1 q) + ϑ −1r,

(1.15)

an inequality which is standard to refer to as the point-wise version of the ClausiusDuhem inequality: 

η P

2 This

·

 ≥−

ϑ −1 q · n +

∂P



ϑ −1r for each body part P.

(1.16)

P

fact was evident to Gibbs: on p.7 of [61] we read the following pertinent quotation of Gibbs: “…heat received at one temperature is by no means the equivalent of the same amount of heat received at another temperature …”.

1.3 Internal Dissipation, Helmholtz Free Energy, Reduced Dissipation Inequality

5

1.3 Internal Dissipation, Helmholtz Free Energy, Reduced Dissipation Inequality Under the constitutive assumption (1.14), the internal entropy production takes the form: (1.17) δ := η˙ + div(ϑ −1 q) − ϑ −1r ≥ 0. The internal dissipation γ is defined to be γ := ϑ δ,

(1.18)

that is, temperature × entropy production. With the use of (1.17) and (1.6), definition (1.18) yields: (1.19) 0 ≤ γ = ϑ η˙ − ˙ − ϑ −1 q · grad ϑ. On introducing the Helmholtz free energy: ψ :=  − ηϑ,

(1.20)

we arrive at the following reduced dissipation inequality: ψ˙ ≤ −ηϑ˙ − ϑ −1 q · grad ϑ,

(1.21)

restricting the free-energy growth (the modifier ‘reduced’ alludes to the fact that the source term has been eliminated from the entropy inequality, thanks to the energy balance). Thermodynamic processes are strictly dissipative whenever γ > 0, so that inequality (1.21) is satisfied strictly. Remark 1.4 Strictly dissipative and irreversible are not synonyms: a palindrome is reversible, and yet reading it is a dissipative process, no matter if from left to right or viceversa. Under time reversal, time derivatives in (1.19)2 change sign whereas all the rest is presumed insensitive to that transformation; on using a self-explanatory notation, we have that − → γ −← γ− = 2(ϑ η˙ − ); ˙ (1.22) hence, internal dissipation changes under time reversal according to the sign of the constitutive construct (ϑ η˙ − ˙ ).3 As we shall see shortly, the constitutive choices leading to the heat equation imply, among other things, the Gibbs relation ˙ = ϑ η, ˙

3 An

[18].

interesting discussion of time reversal transformations is found in Sect. 10.3 of [17]; see also

6

1 Heat Conduction

with which one concludes that, in that material class, internal dissipation is insensitive to time reversal. Remark 1.5 Both the balance (1.6) and the imbalance (1.15) are insensitive to constant-in-time additive normalizations of, respectively, the energy density  and the entropy density η. Consistently, as remarked in footnote 5, p. 70 of [66], every statement involving the free energy density should be “insensitive to addition of a linear function of ϑ to ψ” (here our notations for temperature and free energy have been used). Indeed, on taking ψ0 = 0 − η0 θ , the dissipation inequality (1.21) would stay the same if written for ψ + ψ0 and η + η0 .

1.4 Constitutive Restrictions via the Coleman-Noll Procedure A message implicit in the dissipation inequality (1.21), repeated here for the reader’s convenience: ψ˙ ≤ −ηϑ˙ − ϑ −1 q · grad ϑ, is that the entities in the need of constitutive specifications are free energy, entropy, and heat flux; and that, in addition to temperature, also its space and time derivatives may count to determine whether those entities evolve consistently with (1.21) in a process. Therefore, it would seem that, in faithful application of the Principle of ˙ should be included in the list of state variables Equipresence, all of ϑ, grad ϑ, and ϑ, from which free energy, entropy, and heat flux, are supposed to depend.4 In only apparent contrast, we make the following constitutive assumptions: (ϑ, grad ϑ), η =  ψ =ψ η(ϑ, grad ϑ), q =  q(ϑ, grad ϑ),

(1.23)

with which the dissipation inequality (1.21) takes the form (∂ϑ ψ + η)ϑ˙ + ∂grad ϑ ψ · grad ϑ˙ + ϑ −1 q · grad ϑ ≤ 0.

(1.24)

We now require, in the manner introduced by B.D. Coleman and W. Noll in 1963 [12], that (1.24) be satisfied whatever the local continuation of any conceivable process, ˙ grad ϑ) ˙ at whatever state (ϑ, grad ϑ). This that is, in the present case, whatever (ϑ,

4 In

Sect. 96 of [67], Truesdell‘s Principle of Equipresence is stated and illustrated as follows: “a quantity present as an independent variable in one constitutive equation should be so present in all, unless, of course, its presence contradicts some law of physics or rule of invariance. This principle forbids us to eliminate any of the ‘causes’ present from interacting with any other as regards a particular ‘effect.’ It reflects on the scale of gross phenomena the fact that all observed effects result from a common structure such as the motions of molecules.”

1.4 Constitutive Restrictions via the Coleman-Noll Procedure

7

requirement is satisfied if and only if the constitutive mappings in (1.23) satisfy the following requirements:  is independent of grad ϑ; (i) ψ

  (ϑ); (ii)  η(ϑ, grad ϑ) = −ψ

(1.25)

(iii) the energy influx opposes the temperature gradient (and so does the entropy influx), in the sense that q(ϑ, grad ϑ) · grad ϑ ≤ 0 for all ϑ and grad ϑ. ϑ −1

(1.26)

It is important to realize on what presumptions our present application—in fact, every application—of the Coleman-Noll procedure is based: although deducing from the dissipation inequality restrictions on the constitutive mappings is a matter of algebra, the indispensable quantification involved is a matter of physics, in that it is granted by making two equally crucial and often left tacit assumptions, the one about the geometry of the state space, which should allow for “sufficiently free” local continuation of each path in it, the other about the availability of external controls allowing to realize those paths without violating the balance laws in force. In the case of rigid heat conductors, temperature is the only state variable. The only relevant balance law is the balance of internal energy; in principle, the source term r is at our disposal to satisfy (1.6), perhaps only locally and shortly, for any given temperature process.5 Moreover, what is needed to substantiate the free-localcontinuation assumption is to show that, for whatever choice of a quadruplet (ϑo > 0, ϑ˙ o , go , g˙ o ), a temperature process (x, t) → ϑ(x, t) can be found such that, at a chosen (xo , to ) pair, ˙ o , to ) = ϑ˙ o , grad ϑ(xo , to ) = go , grad ϑ(x ˙ o , to ) = g˙ o . ϑ(xo , to ) = ϑo , ϑ(x One such process is ϑ(x, t) = 0 (x) + 0 (t) + 1 (x) 1 (t), provided that

5 As

remarked by Gurtin ([27], p. 53), the Coleman-Noll procedure “…is based on the premise that the second law be satisfied in all conceivable processes, irrespective of the difficulties involved in producing such processes in the laboratory. The rational application of this procedure requires external forces and supplies that may be assigned arbitrarily to ensure satisfaction of the underlying balances in all processes. This may seem artificial, but it is no more artificial than theories based on virtual power, a paradigm that requires arbitrary variations, not guaranteed to be consistent with the resulting evolution equations, granted a constitutive description. The Coleman-Noll procedure makes explicit the external fields needed to support the ‘virtual processes’ used, and in so doing ensures that these external fields, whether virtual or not, enter the theory in a thermodynamically consistent manner.”

8

1 Heat Conduction

(i) the mappings 0 , 0 , 1 and 1 are chosen such that 0 (xo ) = ϑo , 1 (xo ) = 0, grad 0 (xo ) = go , grad 1 (xo ) = g˙ o , ˙ o (to ) = ϑ˙ o , ˙ 1 (to ) = 1 0 (to ) = 0, 1 (to ) = 0, (the second of these eight conditions is missing in Sect. 39 of [28], from where this argument is taken, with minor changes); (ii) the mappings 0 and 0 are strictly positive-valued; (iii) for p chosen such that 0 < p < ϑo , |1 (x) 1 (t)| < p in a neighborhood of (xo , to ). Remark 1.6 Had we included ϑ˙ in the list of state variables, a reasoning completely analogous to the one that excludes the dependence of ψ on grad ϑ would allow us to exclude its dependence on ϑ˙ as well. However, a dependence of entropy and heat flux on ϑ˙ would not be incompatible with (1.21) in the Coleman-Noll sense, with the rather surprising effects discussed in Sect. 1.8.2. Remark 1.7 In Sect. 1.8.2 we shall deal with a procedure, due to I. Müller and I.-S. Liu and different from Coleman-Noll’s, to achieve constitutive information consistent with the First and Second Laws.

1.5 Specific Heat. The Gibbs Relations It follows from the definition (1.20) of free energy and (1.25) that the internal energy depends only on temperature: (ϑ) + ϑ  (ϑ) − ϑ ψ   (ϑ).  = (ϑ) := ψ η(ϑ) = ψ

(1.27)

By definition, the specific heat is  λ(ϑ) :=    (ϑ). Therefore, in view of (1.27),    (ϑ), λ(ϑ) = ϑ  η  (ϑ) = −ϑ ψ

(1.28)

1.5 Specific Heat. The Gibbs Relations

9

and we see that positivity of the specific heat mapping  λ entails both that the entropy  is concave. mapping  η is monotone increasing and that the free-energy mapping ψ Furthermore, it follows from (1.25) that ˙ ψ˙ = −ηϑ.

(1.29)

Consequently, in view of (1.20), the time rates of entropy and energy are proportional via the coldness: ˙ = ϑ η, ˙ (1.30) just as their inflows, so that, as anticipated, time reversal induces no change of internal dissipation (recall (1.22)). Relations (1.29) and (1.30) are often referred to as the Gibbs Relations. They predict that, in all material bodies whose constitution is specified by (1.23), those heat-conduction processes that are isentropic involve no changes in internal energy, while those that are isothermal involve no changes in free energy (needless to say, the converse statements also hold). That is not all. As a consequence of (1.29) and of positivity of absolute temperature, the remain of the dissipation inequality (1.21) is 0 ≤ −q · grad ϑ. (1.31) Moreover, in view of (1.30), the energy balance (1.6) can be written in terms of temperature, entropy rate and internal energy inflow: ϑ η˙ = −divq + r,

(1.32)

and the internal dissipation (1.19) takes the form γ = −ϑ −1 q · grad ϑ.

(1.33)

 q(ϑ, grad ϑ) · grad ϑ ≤ 0

(1.34)

1.6 The Heat Equation It can be shown that inequality

is equivalent to the following representation for the energy influx:  q(ϑ, grad ϑ) = − C(ϑ, grad ϑ) grad ϑ,

(1.35)

10

1 Heat Conduction

where  C is a mapping delivering the conductivity tensor C in a manner consistent with (1.31), that is to say, a mapping satisfying grad ϑ ·  C(ϑ, grad ϑ) grad ϑ ≥ 0 for all ϑ and grad ϑ.

(1.36)

With (1.30) and (1.35), the point-wise energy6 balance (1.32) takes the form ϑ( η(ϑ))· = div( C(ϑ, grad ϑ) grad ϑ) + r.

(1.37)

Equation (1.37) generalizes the inhomogeneous version of the classical Heat Equation λϑ˙ = χ ϑ, (1.38) which obtains upon choosing, in addition to r ≡ 0,  η(ϑ) = λ log ϑ + a time constant, with λ > 0 the specific heat, and

(1.39)

 C(ϑ, grad ϑ) = χ I, with χ > 0 the conductivity.

Note that7 this constitutive choice for the conductivity tensor implies that there is local internal dissipation wherever the temperature gradient does not vanish. Verify that the constitutive choice (1.39) for entropy is equivalent to (ϑ) = −λ ϑ(log ϑ − 1) for free energy ψ

(1.40)

 (ϑ) = λϑ for internal energy.

(1.41)

and

6 Representation

(1.35) of the heat influx is granted by an application of a lemma, whose simple proof can be found in [6]: Lemma. Let V be a finite-dimensional vector space, let f o ∈ V , and let f be a C 1 -mapping into V from an open neighborhood O of the origin in V , such that v · ( f (v) − f o ) ≥ 0, ∀ v ∈ O. Then, f has the representation f (v) = f o + F(v)v, with

 f (o) = f o ,

F(v) = 0

(here D denotes differentiation). I denotes the identity tensor.

7 Here

1

D f (εv)dε, v · F(v)v ≥ 0, ∀ v ∈ O

1.6 The Heat Equation

11

Finally, note that positivity of specific heat, although physically plausible, is not dictated by the thermodynamic compatibility requirement embodied in the ColemanNoll procedure, whereas positivity of conductivity is. Accordingly, both the archetypal parabolic PDE u˙ = u (1.42) and its so-called backward parabolic counterpart u˙ = −u

(1.43)

are thermodynamically compatible. An initial-value problem ruled by Eq. (1.42) is equivalent to a final-value problem ruled by (1.43); for the latter, however, an initialvalue problem is hardly found well-posed, in the sense that solutions, whenever there are some, usually blow up in a finite time.

1.7 More on the Heat Equation 1.7.1 Gradient-Flow Deduction Equation (1.38) can be arrived at in a meta-thermodynamical manner, as the L 2 −gradient flow of the Dirichlet functional  F{P; ϑ} := P

1 χ |grad ϑ|2 , χ > 0, 2

(1.44)

where P is a space domain with boundary ∂ P. Note that the relevance of such a functional to heat conduction is suggested by matching the standard Fourier Law  q(ϑ, grad ϑ) = −χ grad ϑ,

(1.45)

with the sign constraint guaranteeing thermodynamic consistency of the scalar product of heat inflow and temperature gradient, so has to have that  q(ϑ, grad ϑ) · grad ϑ = −χ |grad ϑ|2 ≤ 0 and, consequently, to be driven both to individuate in spatial non-uniformity the dissipation mechanism implicit in Fourier conduction and to think of equilibrium as a situation in which spatial uniformity of temperature is realized. Note also that the variational derivative of this functional is  δ F{P; ϑ}[δϑ] = (−χ ϑ)δϑ . P

12

1 Heat Conduction

The presumption8 underlying gradient-flow derivations of evolution PDEs of parabolic type is that minimization of a given functional characterizes equilibrium states (in this case, the equilibrium temperature field of a rigid heat conductor occupying the space region P); and that any evolution process from given initial conditions can be regarded as the consequence of some perturbation of one or another equilibrium state, soon to be more and more approached as a consequence of a built-in dissipation mechanism. In fact, one finds that the Dirichlet integral plays the role of a Liapunov function, because, along whatever process that solves (1.38),  d ˙ = −χ (ϑ)ϑ˙ F{P; ϑ} = δ F{P; ϑ}[ϑ] dt P   2 −1 2 = −χ λ |ϑ| = −λ ϑ˙ 2 ≤ 0. P

P

This suggests that, as t → ∞, the temperature field tends to coincide with the minimizers of the energy functional (1.44), that is to say, with the solutions of the corresponding Euler-Lagrange equation.

1.7.2 Time Asymptotics Temperature measures are made with respect to one or another chosen zero, be it the freezing point of water or the absolute zero. And, the classical heat equation is insensitive to an additive normalization. For ϑ0 a chosen positive constant, it follows from (1.38), after multiplication of both sides by (ϑ − ϑ0 ), that λ

d  dt

 P

     1 (ϑ − ϑ0 )2 = λ (ϑ − ϑ0 )ϑ˙ = χ (ϑ − ϑ0 )ϑ 2 P P    |grad ϑ|2 + (ϑ − ϑ0 )∂n ϑ . =χ − P

8 The

(1.46)

∂P

variational derivative δ F{P; ϑ} at ϑ of functional F is defined as follows: δ F{P; ϑ}[δϑ] :=

 d F {P; ϑ}  α , α=0 dα

Fα {P; ϑ} := F{P; ϑ + α δϑ},

for all variation fields δϑ such that δϑ ≡ 0 over ∂ P. The stationarity condition on F expressed by the vanishing of its variational derivative for whatever admissible variation leads to the EulerLagrange equation δ F{P; ϑ} = 0. In the case of the Dirichlet functional (1.44), the Euler-Lagrange equation yields, after localization, the archetypal elliptic PDE: ϑ = 0 in P, associated with the names of Poisson and Laplace.

1.7 More on the Heat Equation

13

Under the assumption that (ϑ − ϑ0 )∂n ϑ ≡ 0 over ∂ P, Equation (1.46) reduces to 1d  2 dt



  (ϑ − ϑ0 )2 = −χ /λ |grad ϑ|2 . P

(1.47)

P

Provided the specific heat be positive—a requirement that, as remarked in Sect. 1.6, is not necessary to guarantee constitutive compatibility with the reduced dissipation inequality – and on recalling the Poincaré-Wirtinger inequality 

 (ϑ − ϑ0 ) ≤ C(P)

|grad ϑ|2 ,

2

P

(1.48)

P

we deduce from (1.47) and (1.48) that 1d  2 dt

 (ϑ − ϑ0 ) P

2



 λ C(P) −1  ≤− (ϑ − ϑ0 )2 , χ P

(1.49)

that is the L 2 -norm of (ϑ − ϑ0 ) decays exponentially.9 This result can be regarded as a toy instance of the heat death of the Universe (recall Remark 1.3) vaticinated in various forms by Clausius (and by Kelvin, Rankine, Helmholtz, …) way before, after Hubble’s observations, modern cosmologists gave attention to the issue. What is physically relevant in the conclusion implied by (1.49) is that the temperature field is asymptotically uniform over a no matter how large but bounded region P, whatever asymptotic value is chosen by way of an additive rescaling. In the views of the founders of continuum thermodynamics, dissipation is generally inevitable when mechanical energy is transferred from a material body to another because of their relative motion: no dissipation no motion, i.e., universal quiet! It would be wise to stay away from indulging to such extrapolations of a mathematical result that, for one, hangs on the assumption of boundedness of P and L 2 -integrability over P of the temperature field.

9 The decay rate, in addition to the physical constants, depends on the region

P in a way that, roughly speaking, can be shown to be smaller and smaller as P gets larger and larger.

14

1 Heat Conduction

1.8 Nonstandard Issues 1.8.1 On Proportionality of Energy and Entropy Inflows We have seen in Sect. 1.2 that the energy inflow is customarily assumed equal to the entropy inflow multiplied by the absolute temperature. Here, after [55], on continuing to restrict attention for simplicity to heat propagation in rigid conductors, (i) an argument is given in support of taking the source terms proportional; (ii) it is proven that the same must be done for the influx terms in case conduction is, in a sense that is made precise, isotropic, but generally not in case of inferior symmetry.10 We return to relations (1.7)–(1.9) and state a notion of internal entropy production, which makes no commitment to (1.14), namely,  δ = η˙ − − divh + s ≥ 0.

(1.50)

Just as (1.17), this inequality is a confession of ignorance: we cannot exclude that entropy changes have internal causes (as instead we did for energy changes), but we confess to be unable to model them explicitly. However, we do presume that entropy is to furnish a macroscopic measure of microscopic disorder in matter, and that all evolutions of an isolated system are order → disorder transformations, along which entropy cannot decrease (and energy has to stay constant: once more, recall Remark 1.3). If we replace (1.17) by (1.50), the reduced dissipation inequality (1.21) becomes  ψ˙ ≤ −ηϑ˙ − h · grad ϑ + − div(q − ϑh) + (r − ϑs) ,

(1.51)

indicating that an additional quantity, the entropy inflow, is in the need of constitutive specification. Remark 1.8 Modifying the constitutive prescription of the energy influx in terms of temperature and entropy influx by the introduction of a gauge vector field w: q = ϑh + curl w, with w the object of a specific constitutive prescription, would neither affect (1.51) nor change anything at interior points; at the boundary, this modification would have point-wise effects, although of no global import. Remark 1.9 For

10 For

θ := ψ˙ + ηϑ˙ + h · grad ϑ,

a similar discussion with similar results within the framework of the Green-Naghdi Type III theory of heat propagation, the reader is referred to [5]; see also [4].

1.8 Nonstandard Issues

15

the part-wise version of inequality (1.51) is  θ ≤ E(P),

(1.52)

P





where E(P) := −

∂P

(q − ϑh) · n +

(r − ϑs)

(1.53)

P

is excess inflow of body part P. Note that this part-wise statement of the free-energy growth inequality (1.51) is irreducible by any reasonable requirement of invariance under observer changes.11 Note also that, if (1.52) is written for P a pillbox of thickness ρ and orientation r centered at a point x ∈ P, then, in the limit as ρ → 0, [[q − ϑh]](x) · r = 0,

(1.54)

for all x and r. In particular, when two material bodies are in thermal contact at a point of their common boundary, that is, when both q and h do not jump at that point [34, 45], then also ϑ does not. This result is in line with Caratheodory’s concept of temperature; continuity of temperature at such an ideal material wall is the content of the so-called Zero-th Law of thermodynamics (see Chaps. 2 and 15 of [47]). Remark 1.10 That the proportionality assumption expressed in (1.14) should not be regarded tenable for whatever material class has been first suggested by Müller, in 1971 [45], followed by Liu, in 1972 [35] and 1973 [36]. Müller and Liu proposed a thermodynamic format within which to test whether or not (1.14) fits a given constitutive class. The Müller-Liu format is achieved by giving the energy balance (1.6), with r = 0, the role of a constraint to be appended to the entropy imbalance (1.50), with s = 0, after multiplication by a so-called ‘Lagrange multiplier’ , a positive-valued field to be the object of a constitutive prescription [37, 38, 46]. The following inequality is arrived at:  η˙ + divh −  ˙ + divq ≥ 0,  > 0.

(1.55)

When constitutive mappings are chosen for energy, entropy, their influxes, and the Lagrange multiplier, one can try and use a procedure of Coleman-Noll type to see whether the multiplier  may be shown to be, if not equal, at least proportional to coldness [45]. Interesting results, quite in line with those to be derived here below, have been found: e.g., for isotropic solids and for fluids whose response depends on the standard list of state variables (that is to say, deformation gradient and its time rate, plus temperature and its time rate) it has been proved that h=

ϑ −1 q, 11 The

notion of observer change is shortly recapped in the Appendix, Sect. A.4.

(1.56)

16

1 Heat Conduction

with

ϑ a monotone increasing function of the absolute temperature (see [62], p.165; [34, 40], and Chap. 2 of [47]). Those who find assumption (1.14) too simplistic to be applied for all material bodies may wonder whether it could be proved to hold for some material class sufficiently large to strictly include rigid heat conductors. In my mind, a discussion of the proportionality issue should be split into one for sources and one for influxes.

1.8.1.1

Proportionality of External Sources

It seems to me that there are good reasons to take energy and entropy sources proportional for whatever material class. Here is how I argue this. When balance and imbalance laws for energy and entropy are posited, both fields r and s play the role of available controls. With a view toward making the Coleman-Noll procedure feasible, the former field is meant to be at our disposal to guarantee energy balance in every admissible constitutive process, that is, to induce in a constitutively described material element whatever local continuation of a given admissible process we please.12 Under assumption (1.14), specification of the entropy source field in a given constitutive process is automatic, and a Coleman-Noll use of inequality (1.51) (where s does not appear anymore) becomes possible. Even if one is reluctant to postulate proportionality of external influxes, I believe that energy and entropy sources should not be thought of as mutually independent control fields. If they were, given that (1.51) incorporates balance of energy whatever r , it would be possible to falsify that inequality for some choice of s. Hence, r and s must be mutually dependent: the simplest selection is r = ϑs.

1.8.1.2

Proportionality of External Influxes

Consider a class of rigid heat conductors whose free energy, entropy, and entropy inflow mappings all depend smoothly on temperature and its time and space first ˙ of the first two state variables derivatives (in the following, for short, the pair (ϑ, ϑ) is denoted by σ ). If the constitutive mappings  q and  h delivering the energy and entropy influxes are both isotropic, that is to say, if they both satisfy Q f(σ, grad ϑ) =  f(σ, Qgrad ϑ) for all (σ, grad ϑ, Q) ∈ IR+ × IR × V × Orth, (1.57)

12 For

the same reason, in the case of more encompassing theories than pure heat conduction, a control field is needed for each additional balance law.

1.8 Nonstandard Issues

17

then consistency in the Coleman-Noll sense with the free-energy growth inequality (1.51) requires that  q(σ, grad ϑ) = ϑ  h(σ, grad ϑ). (1.58) This statement is proved in [55], where it is shown that, under the isotropy assumption, both  q and  h have a representation of the form  f(σ, grad ϑ) =  f (σ, |grad ϑ|) grad ϑ,

 f (·, ·) ∈ IR;

(1.59)

and that, consequently, div(q − ϑh) = ( q − ϑ h)ϑ + grad ( q − ϑ h) · grad ϑ. But then, for inequality (1.51) to hold in the Coleman-Noll sense at a given state (σ, grad ϑ) for arbitrary choices of ϑ,  q = ϑ h. Thus, when heat conduction is isotropic in the above sense, proportionality of energy and entropy influxes holds true. Not so, as we proceed to show, in case of lesser symmetry in the conduction phenomenology. Let G be any nontrivial proper subgroup of Rot. Then, there are constitutive mappings  q and  h delivering the energy and entropy influxes, both satisfying (1.57) with Orth replaced by G, for which consistency in the Coleman-Noll sense with the free-energy growth inequality (1.51) does not imply that (1.58) holds. To prove this, choose Q ∈ G arbitrarily, and pick  q,  h such as they both admit the following representation:  f(σ, grad ϑ) =

f (σ, |grad ϑ|, |grad ϑ · w|)grad ϑ + f¯(σ, |grad ϑ|, |grad ϑ · w|)w, where the unit vector w individuates the axis of rotation Q. Then, ¯ · w. h) · grad ϑ + grad (q¯ − ϑ h) div(q − ϑh) = (

q − ϑ

h)ϑ + grad (

q − ϑ

Hence, a Coleman-Noll argument does yield

q = ϑ

h, but it does not forbid an influx discrepancy in the direction of w: ¯ .  q(σ, grad ϑ) − ϑ h(σ, grad ϑ) = (q¯ − ϑ h)w

18

1 Heat Conduction

Remark 1.11 The arguments here used13 seem robust enough to work within more general continuum mechanical contexts than the classical theory of heat propagation in rigid conductors within which we have presented them: for one, they do work unaltered whenever energy and entropy inflows are taken dependent only on the spatial and temporal evolution of the temperature field and, moreover, thermal response symmetry is assessed independently of other response symmetries (this is done, for example, in classical thermomechanics), no matter how unlikely may seem, say, that conduction isotropy coexists with mechanical anisotropy.

1.8.2 Hyperbolic Heat Conduction The parabolic nature of the diffusion Eq. (1.38) implies that thermal signals propagate at infinite speed. In the account [32, 33] of the literature up to 1990 due to Joseph and Preziosi, we read how this undesired feature can be eliminated by coupling (1.6) not with Fourier’s law q = −χ grad ϑ (1.61) but rather with one or another version of Cattaneo’s proposal [8] of 1948: τ q˙ + q = −χ grad ϑ,

(1.62)

where τ > 0 is the relaxation time for the solutions of the hyperbolic equation one arrives at doing this, namely, λτ ϑ¨ + λϑ˙ = χ ϑ.

(1.63)

In 1968, Gurtin and Pipkin [29] noticed that Cattaneo’s assumption constitutes a special dependence of the heat flux on the history of the temperature gradient, whose general integral representation of Boltzmann-Volterra type is   q(x, t) = −

t −∞

k(t − s) grad  ϑ (x, s)ds = −grad





t −∞

 k(t − s) ϑ (x, s)ds ; (1.64)

they also showed that Cattaneo’s prescription (1.62) is recovered for  k(t − s) = (κ/τ ) exp − (t − s)/τ .

13 The

corresponding excess inflow is: E(P) := −

 ∂P

 ¯ (q¯ − ϑ h)n · w.

(1.60)

1.8 Nonstandard Issues

19

Apparently, ever since the paper by Gurtin and Pipkin appeared, the continuum mechanical research on heat waves somehow concentrated on choosing one or another weighted space of temperature (or temperature gradient) histories, although proposals of other nature were advanced by Coleman, Fabrizio, and Owen [11] and by Morro and Ruggeri [44]; an analytical study of a model generalizing those assembled in [11] and [44] is found in [48]. I here propose a route to (1.63) that has nothing in common with any of those quoted above. As anticipated in Remark 1.6, we take three steps along the Coleman-Noll path, the first two being unusual. Firstly, we lay down the following provisional constitutive assumptions: ˙ ˙ (ϑ), η =  ψ =ψ η(ϑ, grad ϑ, ϑ). q = q(ϑ, grad ϑ, ϑ),

(1.65)

with a view toward filtering out their unphysical contents, if any. Note that, at variance with what is done in the classical theory of heat conduction and in hopes of generalizing the parabolic heat Eq. (1.38), we have let both entropy and heat flux depend on the temperature’s time derivative. Secondly, we split the entropy mapping  η into a ‘standard’ part and a ‘nonstandard’ part: ˙ ˙ = ηns (ϑ, grad ϑ, ϑ),  η(ϑ, grad ϑ, ϑ) ηs (ϑ, grad ϑ) +  with

(1.66)

η(ϑ, grad ϑ, 0),  ηs (ϑ, grad ϑ) :=  ˙ :=  ˙ − η(ϑ, grad ϑ, ϑ) η(ϑ, grad ϑ, 0).  ηns (ϑ, grad ϑ, ϑ)

(1.67)

,  Thirdly, we require that the dissipation inequality (1.21) be satisfied for ψ η and  q as in (1.65)−(1.67) whatever the local continuation of any admissible process, that ˙ grad ϑ) ˙ at any given state (ϑ, grad ϑ). is, in the present case, for any choice of (ϑ, We quickly find that this requirement is satisfied if and only if (i) the standard entropy is determined by the free energy: (ϑ);  ηs (ϑ, grad ϑ) = −∂ϑ ψ

(1.68)

(ii) the reduced dissipation inequality ˙ ϑ˙ + ϑ −1  ˙ · grad ϑ ≤ 0 q(ϑ, grad ϑ, ϑ)  ηns (ϑ, grad ϑ, ϑ)

(1.69)

is satisfied along all admissible processes (cf. (1.26)). Let us now make the classical constitutive choice in (1.40) for the free-energy density, whence, in view of (1.68),  ηs (ϑ, grad ϑ) = λ log ϑ

20

1 Heat Conduction

for the standard part of the entropy (cf. (1.39)) and, in view of (1.68), ˙ ˙ = λϑ + ϑ   (ϑ, grad ϑ, ϑ) ηns (ϑ, grad ϑ, ϑ)

(1.70)

for the internal energy. Moreover, let us pick ˙ η0 ≥ 0, ˙ = η0 ϑ −1 ϑ,  ηns (ϑ, grad ϑ, ϑ)

(1.71)

for the non-standard entropy, and ˙ = −q0 grad ϑ, q0 > 0,  q(ϑ, grad ϑ, ϑ)

(1.72)

for the energy influx (cf. (1.61)). Then, provided the source term r is null, the energy balance (1.6) takes the form η0 ϑ¨ + λϑ˙ = q0 ϑ,

(1.73)

which coincides with (1.63) provided one sets η0 = λτ, q0 = χ . The standard heat Eq. (1.38) is recovered by setting η0 = 0, that is, for τ = 0. Under the constitutive circumstances stipulated just above, the dissipation inequality (1.69) takes the form (1.74) η0 ϑ˙ 2 − q0 |grad ϑ|2 ≤ 0, a restriction on the type of processes that the present theory admits, heat waves included (in particular, temperature must be constant in time where and when it is constant in space). Moreover, definitions (1.17) and (1.18) yield the following expression for the internal dissipation: γ = −η0 ϑ˙ 2 + q0 |grad ϑ|2 ; thus, the internal dissipation associated with the standard theory of heat conduction (namely, γ s = q0 |grad ϑ|2 ) is in the present theory mitigated because of the introduction of a non-standard part of the entropy. Finally, it follows from (1.73) that d 1 dt 2 whence, provided

    η0 ϑ˙ 2 + q0 |grad ϑ|2 + λϑ˙ 2 = −

 P

P

ϑ˙ q · n ≡ 0 over the boundary of P,

∂P

ϑ˙ q · n ,

(1.75)

1.8 Nonstandard Issues

21

we deduce that the theory’s Liapunov function is: t → P (t) :=

1 2



 η0 ϑ˙ 2 (x, t) + χ |grad ϑ|2 (x, t) d x . P

Note that condition (1.75) is fulfilled both if the body is immersed in a heat bath, that is, a large energy reservoir that keeps its boundary temperature constant in time (so that ϑ˙ ≡ 0) and if the body is thermally isolated (so that q · n ≡ 0).

1.9 Order Parameters. State and Substate Variables There are continuum mechanical theories where the state variables have psychologically different statuses: although they all belong to one and the same list, some are meant to carry gross information about a large collection of ‘substates’. For example, in classical thermoelasticity, temperature has a psychological status different from displacement gradient: the latter measures macroscopic local changes in shape, volume, or orientation, the former is meant to account for infinitely many, grossly equivalent substates of microscopic agitation. Moreover, there is often a need (or a convenience) to include a list of order parameters in the notion of state. Order parameters have a role not different from temperature, in that they also account for certain substate changes that we do envisage as physically relevant and yet we can not, and at times we choose not to describe in greater detail. However, as the modifier ‘order’ suggests, the relative substate changes are rather in microscopic organization than in microscopic agitation: think, for example, of the order parameter in theories of phase segregation, like AllenCahn’s [51]. In my opinion, it would help clarity and definiteness to introduce the habit of distinguishing between state and substate variables (such as, respectively, displacement gradient and temperature in thermoelasticity), while referring collectively to all fields of constitutive relevance for a given material class as to the constitutive variables. An alternative nomenclature that was quite popular in the good old days, intermediate variables, has a motivation not so easy to figure out: it alludes to the fact that those variables mediate the space-time dependence of the fields they determine through constitutive prescriptions (for example, in thermoelasticity the stress field depends on space x and time t via the values taken at (x, t) by displacement gradient, temperature, and temperature gradient). Thanks to Walter Noll, who introduced it in his thesis, the more appropriate term ‘constitutive’ has supplanted ‘intermediate’.

Chapter 2

Thermomechanics

In this chapter, after we put together the balance and imbalance laws of thermomechanics in Sects. 2.2 and 2.3, we further exemplify how a systematic use of the Coleman-Noll sieve leads to thermodynamically consistent constitutive prescriptions of increasing complexity, elastic in Sect. 2.4 and viscoelastic in Sect. 2.5.

2.1 Referential and Current Densities When dealing with rigid heat conductors, we did not need differentiate between referential and current descriptions. Here we do. Whenever a deformation of gradient F occurs, any of the state functions—the internal energy, say—has three types of densities: per unit current volume (), per unit referential volume (), and per unit mass (˜ ). The assumption that mass is conserved can be expressed as follows in terms of the measures of mass, referential volume, and current volume: dm = ρ dv = ρ dv,

(2.1)

where ρ and ρ denote mass densities per unit volume, referential and current, and ρ = (det F)ρ.

(2.2)

The volume densities  and  relate to the mass density ˜ as follows:  = ρ ˜ ,  = ρ ˜ ,

(2.3)

 = (det F).

(2.4)

whence, with the use of (2.2),

© Springer Nature Switzerland AG 2019 P. Podio-Guidugli, Continuum Thermodynamics, SISSA Springer Series 1, https://doi.org/10.1007/978-3-030-11157-1_2

23

24

2 Thermomechanics



Moreover,



·

=



Pt

·





·

˙ .

=

P



and



(2.5)

P



ρ ˙˜ .

=

Pt

(2.6)

Pt

2.2 Energy Balance, Entropy Imbalance, and Reduced Dissipation Inequality In classical thermomechanics, the continuum theory which deals with deformable heat conductors, the energy balance is written as (E(P) + K(P))· = Q(P) + ni (P), for each body part P,

(2.7)

where one chooses 

 ρ ˜ and K(P) =

E(P) = Pt

Pt

1 ρv·v 2

for internal energy and kinetic energy, while heating and noninertial power are presumed to have the expressions (1.4) and  ni (P; c, dni )[v] := Given that



∂ Pt

c·v+

dni · v. Pt

˙ K(P) = −in (P)

(recall (A.10)), and that, in view of (A.22), (A.28) and (A.29), ni (P) + in (P) = s (P), (2.7) can be given the following alternative form: ˙ E(P) = Q(P) + s (P), for each body part P,

(2.8)

where, on recalling the symmetry of the Cauchy stress and the definition of the stretching tensor,  s T · D. (2.9)  (P) = Pt

2.2 Energy Balance, Entropy Imbalance, and Reduced Dissipation Inequality

25

This part-wise balance localizes as follows: ρ ˙˜ = −divq + r + T · D at each point of Bt .

(2.10)

Moreover, the entropy imbalance is given the same form as for rigid heat conductors, and proportionality of energy and entropy inflows is accepted. Consequently, the point-wise version of the Clausius-Duhem inequality is supposed to hold at each body point, and the definitions of entropy production and internal dissipation do not change. What changes, as a direct consequence of the new form (2.10) of the point-wise energy balance, is the reduced dissipation inequality, which becomes: ˙ ≤ −ρ η˜ ϑ˙ − ϑ −1 q · grad ϑ + T · D.  ρψ

(2.11)

Just as (1.21), inequality (2.11) is seen as a constitutive restriction, whose role is to filter out inadmissible classes of material response.

2.3 Referential Versions of the Governing Laws At each point of Bt , in addition to the energy balance (2.10) and the dissipation inequality (2.11), the momentum balance holds: divT + dni = ρ v˙ .

(2.12)

When coupled with a set of constitutive equations for stress, internal energy, and heat flux, consistent with the dissipation inequality, the balance equations of energy and momentum govern the evolution of whatever material body in the chosen constitutive class, that is, determine what motion and temperature fields evolve from a given set of initial conditions of position, velocity, and temperature, subject to a given set of traction and/or confinement boundary conditions. We now derive the referential versions of the three basic laws, the balances of momentum and energy and the dissipation inequality, in this order. It is convenient to start from the part-wise version of (2.12), 

 ∂ Pt



Tn +

ρ v˙ ,

dni = Pt

(2.13)

Pt

and introduce a new stress measure, Piola’s. The Piola stress P restitutes the contact force field per unit referential area: 

 ∂P

Pn :=

∂ Pt

Tn , whatever the body part P.

(2.14)

26

2 Thermomechanics

It follows from this definition that P = TF∗ , where

(2.15)

F∗ := (det F)F−T

denotes the cofactor of F.1 It also follows from this definition that the stress power expended on a body part can be given the referential forms 



s (P)[v] =

˙ P · F.

P · ∇v = P

(2.16)

P

Then, on setting2 ni

d = (det F)dni , the balance (2.13) can be given its referential version 

 ∂P



ni

Pn +

ρ v˙ ,

d = P

(2.17)

P

which localizes as follows at each point of B: ni

˙ p := ρv; Div P + d = p,

(2.18)

here Div denotes the referential divergence operator, and p denotes the so-called linear (or translational) momentum. Quite similarly, consider the part-wise form of the energy balance 

ρ ˙˜ = − Pt



 ∂ Pt



q·n+

r+

T·D

Pt

Pt

1 The

cofactor mapping ‘pushes forward’ oriented area elements in the reference configuration into their current counterparts: n d a¯ → n da = F∗ n d a¯ . In particular, da = j d a, ¯ where j := |F∗ n| is the area jacobian. is how to transform  s (P)[v] = T·D

2 Here

Pt

into (2.16): ˙ (det F)T · D = (det F)T · Ł = (det F)T · FF

−1

= TF∗ · F˙ = P · ∇v.

2.3 Referential Versions of the Governing Laws

27

and make use of the transformation law for oriented area elements given in footnote 2 to get    ∂ Pt

q·n =

and of (2.16), to arrive at 

∂P

˙¯ = −

P

F∗ T q · n =

∂P

q · n, q := F∗ T q,



 ∂P



q·n+

r+ P

P · ∇v

(2.19)

P

for every body part, and to ˙¯ = −Div q + r + P · ∇v

(2.20)

at each point of B. Finally, as is easy to check, the referential dissipation inequality reads: ˙ ψ˙ ≤ −η¯ ϑ˙ − ϑ −1 q · ∇ϑ + P · F. (2.21)

2.4 The Constitutive Equations of Thermoelasticity Guided by the developments of Sect. 1.4, after a glance to the dissipation inequality (2.21), we take the free energy and the entropy per unit referential volume, the referential heat flux, and the Piola stress, to depend on temperature, referential temperature gradient, and deformation gradient: (F, ϑ, ∇ϑ), η¯ =  ψ =ψ η(F, ϑ, ∇ϑ), q =  q(F, ϑ, ∇ϑ), P =  P(F, ϑ, ∇ϑ). (2.22) It follows that the dissipation inequality (2.21) takes the form ¯ ϑ˙ + ∂∇ϑ ψ · ∇ ϑ˙ + ϑ −1 q · ∇ϑ ≤ 0. (∂F ψ − P) · F˙ + (∂ϑ ψ + η)

(2.23)

The Coleman-Noll procedure yields that  is independent of ∇ϑ,  (F, ϑ), and  (F, ϑ); ψ η(F, ϑ) = −∂ϑ ψ P(F, ϑ) = ∂F ψ (2.24) moreover, once again the energy and entropy influxes oppose the temperature gradient:  q(F, ϑ, ∇ϑ) · ∇ϑ ≤ 0 for all ϑ and grad ϑ, (2.25) as a consequence of the fact that, for all materials whose response is delimited by (2.24), the internal entropy production is q(F, ϑ, ∇ϑ) · ∇ϑ ≥ 0. − ϑ −2

(2.26)

28

2 Thermomechanics

Just as for rigid conductors, we see that the only source of internal dissipation is the flow of heat against the temperature gradient, a flow that is now influenced by the deformation gradient. Remark 2.1 On recalling the representation of the heat influx given in Sect. 1.6, it would not be difficult to show that the heat flux is null as long as the temperature gradient is—and hence, in view of (2.26), as long the internal production of entropy is null – whatever the values of the state variables F and ϑ: no piezocaloric effect occurs in thermoelasticity. Sadi Carnot would have said that the heat flux is null as long as the transfer of caloric fluid is null. As is well-known, the caloric fluid was a form of pseudo-matter imagined by Antoine Lavoisier to explain heat transfer, a weightless ‘subtle gas’ capable of passing through ordinary matter, whatever the aggregation state of the latter. This is why these days the adjective ‘piezocaloric’ is used in the place of ‘piezothermal’, less exoteric but unduly oblivious of the early history of thermodynamics. On accepting a quantum-mechanical perspective, one might adjourn and extend Lavoiser’s picture by introducing some sort of discreteness via a specific waveparticle duality. For example, when studying crystals in the framework of modern solid-state physics, continuum-mechanical thermal energy is associated with lattice vibrations, which are quantized and given a particle representation as phonons. The absence of a piezocaloric effect prompts the conclusion that lattice deformations do not affect lattice vibrations, at least for as much as lattice deformations are measured by the deformation gradient within the macroscopic framework of thermoelasticity. This ‘fact’ would seem to support an Andersen-Parrinello-Rahman approach to simulate the behavior of particle systems. Remark 2.2 In the present thermoelastic context, the Gibbs relation can be adjourned as follows: ψ˙ = −η¯ ϑ˙ + P · F˙ (2.27) and

˙ ˙ = ϑ η˙¯ + P · F.

(2.28)

We shall return to these relations in Sect. 2.6.

2.5 The Constitutive Equations of Thermoviscoelasticity Let us now assume that the fields in the need of a constitutive prescription depend also on the time rate of the deformation gradient: ˙ ϑ, ∇ϑ), . . . , P =  ˙ ϑ, ∇ϑ). (F, F, ψ =ψ P(F, F,

(2.29)

2.5 The Constitutive Equations of Thermoviscoelasticity

29

It follows that the dissipation inequality (2.21) takes the form ¯ ϑ˙ + ∂∇ϑ ψ · ∇ ϑ˙ − ϑ −1 q · ∇ϑ ≤ 0. (∂F ψ − P) · F˙ + ∂F ψ · F¨ + (∂ϑ ψ + η)

(2.30)

Once again we apply the Coleman-Noll procedure. Just as in the thermoelastic case, we find that (F, ϑ). (F, ϑ) and η¯ =  (2.31) ψ =ψ η(F, ϑ) = −∂ϑ ψ But, the residual dissipation inequality, (∂F ψ − P) · F˙ − ϑ −1 q · ∇ϑ ≤ 0,

(2.32)

does not allow us to conclude that the third relation in (2.24) holds, because both ˙  depends on F. factors in the first product vary in a process continuation, when P However, we can still extract from (2.32) further constitutive information. It is enough to split the Piola stress into a nondissipative part  P(F, 0, ϑ, ∇ϑ) Pnd (F, ϑ, ∇ϑ) :=  and a dissipative part ˙ ∇ϑ) :=  ˙ ϑ, ∇ϑ) −   P(F, F, Pnd (F, ϑ, ∇ϑ Pd (F, Fϑ, to deduce from inequality (2.32) that  (F, ϑ), Pnd (F, ϑ) = ∂F ψ and to further reduce that inequality to one, − ϑ −1 q · ∇ϑ + Pd · F˙ ≥ 0,

(2.33)

featuring two sources of internal dissipation, the one thermal the other mechanical.

2.6 Internal Dissipation in Purely Mechanical Circumstances The presumption implicit in the treatment of entropic issues that led us to the proportionality assumption (1.14) is that temperature-scaled energy inflows are the only foreseen external fount of disorder, or rather, to put it in less radical a form, that all other founts are regarded as negligible with respect to the one accounted for. Thinking of various examples of stress-induced phase tranformations, one may find this presumption risky, if not wrong. It seems to me that an advantage of dealing

30

2 Thermomechanics

with rigid heat conductors first—as we did here—is that who proceeds the other way around might overlook this discussable feature of classical thermomechanics, and take the elimination of the stress power density from the dissipation inequality as an obvious and innocuous consequence of the involved reduction in generality. Now, if attention is restricted to isothermal processes, that is, if ϑ(y, t) ≡ ϑ0 , then the point-wise dissipation inequality (2.21) reduces to ˙ ≤ T · D.  ρψ

(2.34)

This ‘purely mechanical’ dissipation inequality has been postulated to regulate constitutive choices in contexts where thermal effects are ignored, such as phase transformations by atomic rearrangement. Here is a way to argue (2.34) (see Appendix A-1 of [A2]) or, for what it matters, its equivalent referential version ˙ ψ˙ ≤ P · F.

(2.35)

For hyperelastic solids, σ (F, x) P = ∂F σ, σ =  where the constitutive mapping  σ delivers the energy stored per unit referential volume about point x when the deformation gradient field takes the value F at that point. Consequently, the stress power equals the time rate of the stored energy: (s) (P) =



P · F˙ = ( P



σ )· , P

 so that P σ is conserved in a deformation cycle for whatever body part P. For Newtonian fluids, a class of incompressible materials, the stress response is specified as follows: T = −π 1 + μD, divv = 0, μ > 0, with π the pressure field arising as a reaction to the incompressibility constraint, and μ the viscosity; no energy is stored in any process, and the stress power equals the mechanical dissipation: (s)





 (P) =

T·D= Pt

μ D · D ≥ 0. Pt

Guided by these3 two well-known examples, for purely mechanical theories of Cauchy continua one is driven to define internal dissipation in a body part in motion

3 Recall

˙ −1 = divv. that I · D = I · FF

2.6 Internal Dissipation in Purely Mechanical Circumstances

31

as the difference, assumed to be non negative, of the stress power and the rate of change of an internal potential  φ

(P) = Pt

to be constitutively specified: ˙ ≥ 0. s (P) − (P) In other words, one is driven to postulate the following dissipation inequality: 

φ

·

Pt

or rather, after localization,

 ≤

T · D, for each body part P,

(2.36)

Pt

˙ ≤ T · D;  ρφ

(2.37)

and to require that the constitutive choices for the internal potential and the Cauchy stress T be consistent with (2.37) in the sense of [12]; alternatively, one can require that the internal potential and the Piola stress be chosen consistent with the referential version of (2.37), namely, ˙ φ˙ ≤ P · F. (2.38) For (2.38)) to agree with (2.21), an interpretation of φ must be provided in the light of a parent theory accounting for thermal phenomena. When thermoelasticity is the parent theory chosen for hyperelasticity, relations (2.28) and (2.27) respectively imply that success is guaranteed if φ is interpreted either as the restriction of the specific internal energy to the isentropic process class or as the restriction of the specific free energy to the isothermal process class; by definition, the internal entropy production is null only in the second case. Now, typical real-life processes are isentropic when they take place in such a short time span that heat conduction can be neglected; real-life isothermal processes are instead usually slow. In both cases, we just showed how hyperelasticity can be given a position with respect to thermoelasticity. It is important to realize that there is no need to do so to validate the former theory: Karl Popper has taught us that an internally coherent scientific theory should be validated by falsifying some of its predictions [59]. But, it is good to know in what practical circumstances it makes sense for us to have recourse to hyperelasticity theory for successful predictions.

Chapter 3

The Principle of Virtual Powers

Hodiernal continuum mechanics is multiscale and multiphysics, in that it deals with situations where interdependent phenomena take place at different scales in material bodies that often must be regarded as an interactive composition of more than one of the basic structures of phenomenological physics, that is to say, the mechanical, thermal, electrical, and magnetic structures. Plurality of scales and physical structures calls for various adjustments of the standard modeling format. Problem by problem, there is a need to lay down, in a manner as systematic as possible, the relevant evolution, balance and imbalance laws, making sure that no structure interaction is overlooked. This can be done by accepting a format modeled after a suitable generalization of the standard, purely mechanical virtual-power format. We are going to exemplify that format for the mechanical and thermal structures, separate in this chapter and combined in Chap. 4, after [54]. The related literature is so copious that an attempt at completeness is doomed to fail; the reader is referred to [13], where reasoned reference to papers [2, 3, 21, 22, 43, 65] is made.

3.1 The PVP as a Source of Balances and Representations We describe the basic mechanical structure pertaining to the class of the so-called simple (or Cauchy’s) continua, under form of a part-wise Principle of Virtual Powers. Our present developments are taken freely from [58], where a few more references are found. We begin by introducing formally contact forces as dual of velocities, and stresses as dual of velocity gradients; such dualities are specified by two linear, continuous and bounded functionals of power expenditure, the one external the other internal to any chosen body part. Precisely, the internal and external expenditures of virtual power have the following forms: © Springer Nature Switzerland AG 2019 P. Podio-Guidugli, Continuum Thermodynamics, SISSA Springer Series 1, https://doi.org/10.1007/978-3-030-11157-1_3

33

34

3 The Principle of Virtual Powers

 δi (P)[δu] :=

S · grad δu, 

δe (P)[δu] :=

Pt

∂ Pt

(3.1) t · δu ,

for all subbodies P and for all test(≡ virtual) velocity fields δu over Bt . It is important to understand that, in a virtual-power approach, the only primitive object is the space of test velocities; both stresses and contact forces are secondary notions. The collection of test velocities Virt is supposed to include all realizable velocities (that is, all velocity fields v = ∂t f , for f a deformation), and to be closed under the operation of observer change, in the sense that, if δu ∈ V ir t, then w + Q δu = δu+ ∈ V ir t

(3.2)

for all translations w and all rotations Q. In both relations (3.1), time plays the role of a parameter; in the second one, actions at a distance are ignored, because they are inessential to the purpose of the present discussion; with this proviso, when written for realizable velocities, the right sides of relations (3.1) have the same form as those defining, respectively, the stress power and the external force power. The internal field S is meant to measure the mechanical interaction of a material element of subbody P with its immediate adjacencies (this is why the first spatial gradient of the velocity field that scatters those interactions appears in (3.1)1 ); the external field t should account for the mechanical action exerted on P, at each point of its boundary, by its complement with respect to the material universe the body B belongs to. Accordingly, given that we are here dealing with the current expressions of virtual-power expenditures, S and t can be identified with, respectively, the Cauchystress field T and the contact-force field c considered in Sect. 2.2; from now on in this section, we write T for S, and c for t. Both the stress T and the contact force c are required to be indifferent to observer changes: (3.3) T+ = QTQT and c+ = Qc for all rotations Q; both power expenditures are required to be invariant under observer changes. Given (3.2) and the second of (3.3), invariance of the external power expenditure over an arbitrary subbody implies that the contact-force field be balanced, i.e., that  ∂ Pt

c = 0 for all subbodies P.

Moreover, given that (gradδu)+ = Q(gradδu)QT + W for all skew-symmetric tensors W,

(3.4)

3.1 The PVP as a Source of Balances and Representations

35

invariance of the internal power expenditure over an arbitrary subbody implies that the Cauchy-stress field be symmetric-valued: T = TT at all points of B.

(3.5)

The mutual consistency of the stress and contact-force fields is the consequence of postulating the following Principle of Virtual Powers: δi (P)[δu] = δe (P)[δu], for all P ⊂ B and for all δu ∈ Virt.

(3.6)

Needless to say, the PVP is an invariant statement; the twofold quantification it features deserves some comments. Asking that (3.6) holds for all body parts is much stronger a requirement than demanding, as is usually done, that it holds for the whole body: in fact, this enhanced quantification implies that the values taken by T and c are interdependent at all points of B, and not only at its boundary points. With the use of a standard integration-byparts lemma, we find that (3.6) can be written as follows: 

 (−divT) · δu +

∂ Pt

Pt

 Tn · δu =

∂ Pt

c · δu,

(3.7)

c · δu,

(3.8)

for all P ⊂ B and for all δu ∈ Virt; in particular, 

 (−divT) · δu + Bt

∂ Bt

 Tn · δu =

∂ Bt

for all δu ∈ Virt. Now, while both (3.7) and (3.8) imply the classical point-wise balances − divT = 0 at all points of Bt and Tn = c at all points of ∂ Bt ,

(3.9)

it is (3.7), which has to hold for all body parts, that implies that Tn = c at all points of Bt and for all unit vectors n,

(3.10)

and not only at the points of ∂ Bt and for n the outward unit normal field, as specified by (3.9)2 and implied by the weaker statement (3.8). A first direct consequence of (3.10) is that c = cˆ (y, n), namely, that the contact-force field at any point y ∈ Bt should be thought of as depending on the orientation of the plane chosen through that point to detect contact interactions. Secondly, it follows from (3.10) that the fields T and c carry the same

36

3 The Principle of Virtual Powers

information, so that (3.10) can be regarded as a basic consistency condition for the pair of dynamic entities (c, T) introduced in (3.1) as dual to the kinetic entities (δu, gradδu). Indeed, given the stress mapping  T, (3.10) yields cˆ (y, n) =  T(y)n;

(3.11)

conversely, given the contact-force mapping cˆ , the stress field can be constructed by means of a formula that we repeat here for the reader’s convenience:  T(y) =

3 

cˆ (y, ni ) ⊗ ni

(3.12)

i=1

(recall that the triplet of mutually orthogonal unit vectors ni is chosable arbitrarily). In summary, the double quantification in our formulation of the Principle of Virtual Powers yields two types of consequences: not only balances, at interior and boundary points, but also representations, for contact force in terms of stress and conversely. To guarantee such a twofold crop is, I believe, the main virtue of a virtual-power approach to the derivation of the basic laws of whatever branch of continuum physics. Remark 3.1 Alternatively, the PVP can be written in the following referential form: 





P · ∇δu = P

d · δu + P

∂P

c · δu,

(3.13)

for all body parts and all test velocity fields δu over B. Here P is the Piola stress and c = j c, with j the area jacobian; note that we have reinstalled the virtual power expenditure of the distance force d; the dependence on time is parametric and not displayed; the fields P and c carry the same information, in the sense that 3 

 c(x, ni ) ⊗ ni .

(3.14)

δi (B)[δu] = δe (B)[δu], for all δu ∈ Virt

(3.15)

 c(x, n) =  P(x)n,  P(x) =

i=1

Remark 3.2 In its weaker body-wise form

(compare with (3.6)), the PVP is viewed as the mechanical forerunner of a mathematically precise statement of force and moment balances; as such, trivially but importantly, it is more general than the Euler-Lagrange equation for the stationary points of an energy functional, because it neither obscures nor precludes the contribution of dissipative forces to those balances.

3.2 The PVP as a Source of Evolution Equations

37

Fig. 3.1 The space-time cylinder B × (0, T ) and a typical part of it

3.2 The PVP as a Source of Evolution Equations The PVP can be given a dynamical version from which evolution equations for the relevant kinematical fields can be extracted, together with appropriate initial conditions; all is needed to do this is to single out and characterize constitutively inertia forces. With reference to Fig. 3.1, let P be a typical part of a body point-wise identified with the referential space region B, and let I = (ti , t f ) be a subinterval, whose boundary is ∂ I = {ti , t f }, of the time interval (0, T ); the boundary of P × I consists of the union of P × ∂ I and ∂ P × I . The external virtual power-expenditure is defined to be 

˙ + (d · δu + p · δu) ni

δe (P)[δu] := P×I



 c · δu +

∂ P×I

p · δu ,

(3.16)

P×∂ I

ni

where d denotes the non-inertial distance force, and p the translational momentum, per unit referential volume, and where 



  p f (x) · δu(x, t f ) + pi (x) · δu(x, ti ) .

p · δu := P×∂ I

(3.17)

P

The internal virtual power-expenditure is  δi (P) =

P · ∇δu . P×I

(3.18)

38

3 The Principle of Virtual Powers

The dynamic version of the Principle of Virtual Powers is δi (P)[δu] = δe (P)[δu],

(3.19)

for all test functions such that δu|t=t f ≡ 0 in P and for all subbodies P. By repeated localization, we deduce that p˙ = Div P + d

ni

in P × I,

and that, in addition to the boundary condition Pn = c in ∂ P × I , the following initial condition prevails: p(x, ti ) = pi (x) x ∈ P.

3.3 The PVP as a Dimension-Reduction Tool We now want to show how the balance equations of both beam and plate theories can be derived from the standard weak version of the PVP, provided the specialty in shape of the body classes in question is exploited (see pp. 214–215 of [53]). In an attempt to avoid unnecessary technical complications so as to let the essential conceptual features emerge, we consider the cylindrical body shown in Fig. 3.2, for which, on taking all contact loads null for simplicity, the standard PVP reads

Fig. 3.2 The right cylinder S × (0, l). From [57], reproduced with permission

3.3 The PVP as a Dimension-Reduction Tool

39



 P · ∇δu = S×(0,l)

d · δu, for all test fields δu

(3.20)

S×(0,l)

(cf. (3.13)).

3.3.1 From 3D to 1D: Beam Theory Pick the test fields δu B = v(z)e2 + (w(z) + y ϕ(z))e3 , and restrict attention to load fields of the form d = p(x, ¯ y, z)e2 + q(x, ¯ y, z)e3 . Then, (3.20) reduces to a not completely standard one-dimensional PVP: for all scalar test fields v, w, and ϕ, 

l



 ϕ + N w + Mϕ  = Tv + T 

0



l

( pv + qw + cϕ),

0

where 

 := P23 , T

T :=



 P32 ,

S



N :=

S

P33 , and M := S

y P33 , S

, the normal force N , and the bending are called, respectively, the shear forces T and T moment M, and where    p, ¯ q := q, ¯ and y q, ¯ p := S

S

S

are the transverse load, the axial load, and the bending couple, per unit length. Test invariance of this statement yields both the balance equations prevailing at each point z ∈ (0, l), namely,  = c, −T  = p , −N  = q , and − M  + T and the boundary conditions at z = 0 and z = l, which consist of assignments of ⎧ ⎨

⎫ ⎧ ⎫ shear force T ⎬ ⎨ vertical displacement ⎬ normal force N axial displacement either or . ⎩ ⎭ ⎩ ⎭ bending moment M rotation about e1

40

3 The Principle of Virtual Powers

Remark 3.3 Our one-dimensional PVP is not powerful enough to imply that the  are equal, just as the standard three-dimensional PVP does not shear forces T and T imply that the stress field is symmetric.

3.3.2 From 3D to 2D: Plate Theory This time, all one has to do is (i) to pick the test fields δu P = w(x, y)e3 + z ϕ(x, y), ϕ ⊥ e3 , and the load fields: d = q(x, ¯ y, z)e3 ; (ii) to insert these fields into (3.20); (iii) to exploit the inherent test-field quantification. Remark 3.4 Needless to say, the plate test-field δu P are ‘Reissner-Mindlin inspired’, just as the beam test-field δu B in Sect. 3.3.1 are ‘Timoshenko inspired’ (see, e.g., [52]).

Chapter 4

A Virtual Power Format for Thermomechanics

When regarded as the composition of two material body structures, the one mechanical the other thermal, thermomechanics epitomizes multiphysics theories. The kinetic variables are the mechanical displacement u and the thermal displacement α, two smooth fields over the closure with respect to the product topology of the space-time cylinder B × (0, T ) shown in Fig. 3.1. The time derivatives of these displacement fields are, respectively, the velocity v := u˙ and the temperature ϑ := α; ˙ note that, at the present level of generality, we do not require that the mapping t → α(·, t) be strictly monotonic increasing, i.e., that temperature be positive. In that thermal displacement and temperature gather information about a lower scale than mechanical displacement and velocity, thermomechanics is also a multiscale theory. We postpone our account of the attempts to give thermal displacement a physical interpretation until next chapter.

4.1 The General Case The format we propose for thermomechanics, which is taken from [54], hinges on three axioms: the first yields the mechanical and thermal balance laws of the theory, as well as the accompanying initial and boundary conditions; the second yields a generalized version of the balance of energy; the third and last, a generalized version of the imbalance of entropy. In this section, we formulate the first axiom; the other two will be formulated in the next subsection, in the simpler version leading to a generalized set of basic laws for thermal conduction, which, as we show in closing, reduce to the standard laws via a number of convenient identifications and ad hoc assumptions. © Springer Nature Switzerland AG 2019 P. Podio-Guidugli, Continuum Thermodynamics, SISSA Springer Series 1, https://doi.org/10.1007/978-3-030-11157-1_4

41

42

4 A Virtual Power Format for Thermomechanics

Axiom of Virtual Powers. The kinetic processes of a thermomechanical body are pairs of twice continuously differentiable mappings:   (x, t) → u(x, t), α(x, t) ,

(4.1)

the former vector-valued, the latter scalar-valued; by time differentiation, kinetic processes generate pairs of realizable velocities:   (x, t) → v(x, t), ϑ(x, t) .

(4.2)

The dynamics of a thermomechanical body is specified by a pair of linear, bounded and continuous functionals, the internal and external expenditures of virtual power, defined over a given collection of continuously differentiable virtual velocities   (x, t) → δu(x, t), δα(x, t) .

(4.3)

With slight abuse of notation, we continue to denote by Virt this collection, which includes all realizable velocities and is closed under the operation of observer change, in the sense that, (δu)+ = w + q(δu), (δα)+ = δα, for all translations w and all rotations q. Note that a change of thermomechanical observer does not affect the macroscopic perception of the thermal displacement. Again with reference to Fig. 3.1 in Sect. 3.2, the internal virtual power-expenditure is defined to be the following functional over Virt:  δi (P)[(δu, δα)] :=

(P · ∇δu + h δα + g · ∇δα) ;

(4.4)

P×I

the external virtual power-expenditure is  δ (P)[(δu, δα)] :=  + e

ni ˙ + d δα + π δα) ˙ (d · δu + p · δu P×I    c · δu + c δα + (p · δu + π δα) ,

∂ P×I

where

(4.5)

P×∂ I

 (p · δu + π δα) : = p f (x) · δu(x, t f ) + π f (x) δα(x, t f )

∂I

(4.6)

+ pi (x) · δu(x, ti ) + πi (x) δα(x, ti ), for all x ∈ B. The Axiom of Virtual Powers is the requirement that δi (P)[(δu, δα)] = δe (P)[(δu, δα)]

(4.7)

4.1 The General Case

43

for each part P × I of B × (0, T ) and for each virtual velocity pair defined over the closure of P × I and such as to vanish at the end of I itself. Clearly, on restricting attention to the following class of virtual velocities:   (x, t) → δu(x, t), 0 ,

(4.8)

Equation (4.7) reduces to (3.19). On the other hand, as we are going to see in the next subsection, perusal of the virtual-velocity class complementing (4.8):   (x, t) → 0, δα(x, t)

(4.9)

yields the balance and initial/boundary conditions pertaining to the thermal structure per se. Remark 4.1 When given a PVP format, classical thermomechanics is an example of multivelocity theory that turns out to be especially simple, because the two types of virtual velocities, mechanical and thermal, are not interdependent This would not be the case if they were bound to obey a mutual scale-bridging constraint: φ(δu, δα) = 0, such as, say, temperature-dependent compressibility. There are also multivelocity theories—not to be dealt with here—in which, in addition to the standard constitutive coupling exemplified in Sects. 2.4 and 2.5 in the case of thermomechanics, balance coupling occurs—that is, coupling between coexistent physical structures that manifests itself when the balance laws are formulated.

4.2 The Balance and Imbalance Laws of Thermal Conduction On choosing the purely thermal virtual velocities (4.9), the PVP (4.7) takes the form: δi (P)[(0, δα)] = δe (P)[(0, δα)] , i.e., 



˙ + (d δα + π δα)

(h δα + g · ∇δα) = P×I

P×I



(4.10)

 c δα +

∂ P×I

π δα , (4.11) P×∂ I

where  ∂I

π δα := π f (x) δα(x, t f ) + πi (x) δα(x, ti ), for all x ∈ B;

(4.12)

44

4 A Virtual Power Format for Thermomechanics

Equations (4.11)–(4.12) are required to hold for each part P × I of B × (0, T ) and for each virtual velocity field δα defined over the closure of P × I and vanishing at the end of I . In relation (4.11), the internal fields h, g measure the thermal interactions (of the zeroth and first order, respectively) of a material element of the subbody P with its immediate adjacencies. The external field d accounts for the thermal distance interactions with P of its complement with respect to the material universe the body B belongs to; similarly, the external field c accounts for the thermal contact interactions between P and its complement through their common boundary ∂ P. In addition, the mappings π f , πi specify the external actions on the cylinder P × I at its time boundaries P × {t f } and P × {ti }. The external field π will be interpreted shortly as entropy per unit referential volume; π has a specific source d at an interior point of a subbody, and a specific flux g · n at a boundary point. The interpretation of the scalar field h will be given later on. Just as for its purely mechanical counterpart, the present axiom of virtual mechanical powers has two types of consequences: balance laws, prevailing at interior and boundary points of B × (0, T ); and reversible representations for the first-order internal interaction g in terms of the corresponding contact interaction c. To arrive at these results, it is useful to note that   (h − Div g)δα + (g · n)δα , δi (P)[(0, δα)] = ∂ P×I

P×I



and that



δ (P)[(0, δα)] = e

(d − π˙ ) δα + c δα ∂ P×I  P×I   + − π(x, ti ) + πi (x) δα(x, ti ) . P

Exploiting the quantification over Virt first, for every P × I ⊂ B × (0, T ) we have the thermal balance: π˙ = Div g − h + d in P × I , (4.13) together with the boundary condition g · n = c on ∂ P × I.

(4.14)

π(x, ti ) = πi (x) for x ∈ P .

(4.15)

and the initial condition

Remark 4.2 On setting:

4.2 The Balance and Imbalance Laws of Thermal Conduction

45

π = η, the entropy density, −h = δ, the internal entropy production, d = s, the external entropy source,

(4.16)

−g = h, the entropy influx, we recover from (4.13) the entropy balance η˙ = −Div h + s + δ.

(4.17)

Remark 4.3 Just as α is called ‘thermal displacement’ to allude at a role analogy with the mechanical displacement u, one might call π (≡ η) the thermal momentum, by analogy with the mechanical momentum p; just as the latter may be thought of as measuring the macroscopic matter’s reluctance to quiet, entropy may be thought of as measuring microscopic reluctance to order. Remark 4.4 Given any part P × I of B × (0, T ), (4.14) may be regarded, with the help of a pill-box argument, as a continuity condition: c = (−g) · (−n) , establishing the vector field −g as a measure of the entropy influx at a point of an oriented surface of normal n. In particular, this interpretation holds true at points of ∂c B, the portion of the body’s boundary where the external contact-interaction field c0 is assigned and where (4.14) specialize as follows: g · n = c0 on ∂c B × (0, T ) .

(4.18)

4.2.1 Axiom of Conservation of Internal Action Call cycle a cyclic thermodynamic process, that is to say, a closed smooth path in the state space, parameterized by time and trodden from some initial  t time ti to a final time t f > ti ; moreover, denote time integration over a cycle by ti f . Let a scalar-valued function P → C(P) be given, defined over the collection of body parts and with C(P) representable as an integral over P, and suppose that 

tf

C(P) = 0

ti

for a given cycle. Then, the function P → G(P) defined by 

t

G(P)(t) := −

C(P) ti

(4.19)

46

4 A Virtual Power Format for Thermomechanics

is such that

˙ G(P) = C(P);

(4.20)

we say that, in the given cycle, G(P) is conserved. Now, let   dϑ+ cϑ H (P) :=

(4.21)

∂P

P

be the heating supplied to or extracted from P by agencies external to it, during a thermal process. Consider the following consequence of balances (4.13), (4.14) and definition (4.21):    d  ˙ ; (4.22) H (P) = π ϑ + F(P), F(P) := (h ϑ + g · ∇ϑ − π ϑ) dt P P note that the quantity F(P) is conserved in a cycle if and only if the heating supplied to or extracted from P in that cycle is zero: 

tf

 H (P) = 0

⇔

ti

tf

F(P) = 0 .

(4.23)

ti

Next, given a cycle, imitate (4.19) and set: 

t

(P)(t) := −

 F(P) , where (P) =

ti

φ; P

call (P) the internal action in P in that cycle,1 and note that the specific internal action φ is such that ˙ φ˙ = −(h ϑ + g · ∇ϑ − π ϑ). (4.24) The Axiom of Conservation of Internal Action is the requirement that 

(P) = 0

(4.25)

hold true for all subbodies and for all cycles. On introducing the energy-like entity τ := φ + π ϑ

1 As

(4.26)

is standard with state functions in thermodynamics, the internal action is defined to within a time constant, which is irrelevant when it comes to formulate any physically significant balance or imbalance statement involving that function.

4.2 The Balance and Imbalance Laws of Thermal Conduction

and on letting

47

 τ,

T (P) :=

(4.27)

P

it follows from the Axiom of Conservation of Internal Action, with the help of (4.22) and (4.24), that T˙ (P) = H (P) . (4.28) We note here for later use the following local consequence of (4.28): τ˙ = Div (ϑg) + d ϑ .

(4.29)

4.2.2 Dissipation Axiom The Dissipation Axiom is the requirement that, whatever the kinetic process (x, t) → α(x, t), h α˙ ≤ 0 (4.30) over the space-time cylinder B × (0, T ). On recalling the second of (4.16), h, the ‘force’ conjugate to the ‘velocity’ α, ˙ can be interpreted as the negative of the internal entropy production; consistently, (−h)α˙ is interpreted as the internal dissipation. Combination of (4.30) with (4.24) yields the dissipation inequality φ˙ ≤ −π ϑ˙ + g · ∇ϑ ; (4.31) moreover, combining this inequality with (4.26) and (4.29), and provided that ϑ > 0 (an assumption we need not make to arrive at (4.31)), we find: π˙ ≥ Div g + d.

(4.32)

Relations (4.29), (4.31) and (4.32) are general counterparts of, respectively, the energy balance (1.6), the reduced dissipation inequality (1.21), and the ClausiusDuhem inequality (1.15). To recover those standard relations, it is sufficient to identify π with η and d with s, as was done in (4.16)1,3 , and, in addition, (i) to set the inflow (g, d) proportional to the heat inflow (−q, r ) through the temperature: (4.33) g = −ϑ −1 q, d = ϑ −1r , so that (4.21) takes the standard form 



H (P) :=

r+ P

∂P

q·n;

48

4 A Virtual Power Format for Thermomechanics

(ii) to identify τ and φ with, respectively, the internal energy and the Helmholtz free energy ψ per unit referential volume, whence ψ = − η ϑ.

(4.34)

In particular, modulo some of the above identifications, (4.31) takes the form ψ˙ ≤ −η ϑ˙ − ϑ −1 q · ∇ϑ,

(4.35)

the inequality which opens the way to the use of the Coleman-Noll procedure to lessen caprice in the choice of constitutive relations. Needless to say, the same constitutive information would follow from a use of (4.31) à la Coleman-Noll.

Chapter 5

A Physical Interpretation of Thermal Displacement

Until recently, the use of thermal displacement—and of its gradient—in modern narrations of continuum thermodynamics (see e.g. [4, 5, 23–25, 54]) has been interesting but at bottom formal: a physical interpretation was wanted, a discrete, and hence microscopic, scale-bridging interpretation being both unequivocal and explicit, and not, as the one put forward in [23], just allusively inferred from the well-known statistical interpretation of temperature. Such an interpretation may be found in [56], a note whose contents are faithfully reproduced in this chapter.

5.1 From Helmholtz and de Broglie to Einstein and Langevin In continuum physics, we recall from the preceding chapter, the thermal displacement is a smooth scalar field α over the closure with respect to the product topology of the space-time cylinder B × (0, T )—the Cartesian product of a referential region for the body of interest and the observation time interval—a field whose time derivative is by definition the absolute temperature T : α˙ := T .

(5.1)

This notion was introduced by Helmholtz, who pursued a purely mechanical interpretation of the objects of thermodynamics, in 1884 [30, 31], when a consistent deduction of the basic laws of that discipline from statistical mechanics was yet to come. To Helmholtz, who worked within the framework of analytical mechanics, the thermal displacement was a coordinate entering the Lagrangian of a mechanical system only through its time derivative, whose momentum had to be a motion constant: a ‘monocyclic’ variable for him, a ‘fast’ variable for de Broglie [7], in that © Springer Nature Switzerland AG 2019 P. Podio-Guidugli, Continuum Thermodynamics, SISSA Springer Series 1, https://doi.org/10.1007/978-3-030-11157-1_5

49

50

5 A Physical Interpretation of Thermal Displacement

only its time derivative is observable at the macroscopic time scale and is therefore includible among a system’s Lagrangian coordinates. In fact, both thermal displacement and temperature are macroscopic variables which gather information about microscopic events that, with Einstein [14], can be referred to collectively as ‘thermal motion’. For a particle system at statistical equilibrium, temperature is shown proportional to the kinetic-energy’s expectation, a conglomerate information about the particles’ velocity fluctuations. In [56], a statistical interpretation for the thermal displacement in a fluid is proposed in terms of the mean value of the squared diffusive displacement from their common injection point of a system of particles in Brownian motion suspended in that fluid. To my surprise, this interpretation was at the time when [56] was written new, although it is arrived at following the same path of reasoning employed by Einstein more than a century before to link Brownian motion to Fickian diffusion.1 Einstein’s path will be followed closely in Sect. 5.2 here below; in Sect. 5.3, again taking freely from [56], it will be shown how to arrive at a completely analogous interpretation following the different path proposed by Langevin in 1908, three years after Einstein’s first paper.

5.2 How to Interpret Thermal Displacement Following the Einstein Path “Einstein’s explanation of the nature of Brownian motion must be regarded as the beginning of stochastic modelling of natural phenomena”, we read on p. 3 of [20]. The explanation is achieved regarding Brownian motion as “required by the molecularkinetic theory of heat” [14] or, in the words of Clausius [9], as a manifestation of “that kind of motion that we call heat”. In fact, in the introduction of [14] Einstein goes so far as to declare that, were the predictions of his model shown incorrect, this fact would provide “a weighty argument” against that theory.2 Einstein works in a one-dimensional space setting, with no conceptual loss. In Sects. 1–3 of [14], the diffusion of spherical particles suspended in a viscous fluid at rest is studied, under the assumption that suspended particles, not differently from a solute’s molecules, induce an osmotic pressure on a confining wall permeable to the fluid but not to them; the main result is the so-called Einstein-Smoluchowski formula for the diffusion coefficient: (5.2) do = μ k B T,

1 See [14, 15], both edited with notes by R. Fürth and translated by A.D. Cowper in [16], where three other related papers by Einstein are found. 2 The role that Einstein’s work on Brownian motion had in making the so-called ‘atomistic revolution’ successful is scholarly discussed in [60].

5.2 How to Interpret Thermal Displacement Following the Einstein Path

51

where the mobility μ = (6π ka)−1 , a function of the particles’ radius a and the fluid’s viscosity k, is interpreted as a (drift velocity)/(driving force field) ratio, k B is Boltzmann’s constant, and T is the absolute temperature.3 In Sect. 4 of [14], the concentration c =  c(x, t) of suspended particles, whose “irregular movements …arise from [the fluid’s] thermal molecular movements”, is shown to satisfy the classic diffusion equation ∂t c = do c .

(5.3)

Equation (5.3) is arrived at under the assumptions that  c(x, ·) is of class C 1 and that 2  c(·, t) is of class C , by postulating the following stochastic balance:   c(x, t + τ ) =

+∞

−∞

 c(x + σ, t)φ(σ )dσ.

(5.4)

Here, (i) τ is the length of a time interval “very small compared to the observable time intervals, but nevertheless so large that in two successive time intervals of duration τ the motions executed by a particle can be thought of as events which are independent of each other”; (ii) σ denotes a displacement experienced by a suspended particle during the time lapse τ , due to the fluid’s thermal motion; (iii) the smooth function φ is assumed to be even and normalized:  φ(σ ) = φ(−σ ),

+∞ −∞

φ(σ )dσ = 1,

(5.5)

and to have a finite second moment,4 so that the following representation of the diffusion coefficient makes sense:  +∞ 1 σ 2 φ(σ ) dσ . (5.6) do = 2τ −∞ According to Eq. (5.4), for any chosen pair (x, t), the number of particles in a unit space cell centered at x (that is, the concentration at x) equals at time (t + τ ) the expected value, with respect to the displacement probability φ, of the number of particles found in the same space cell at the preceding instant t. We now show that a different statistical-mechanics interpretation of do can be achieved. To do so, we employ a somewhat cured version of an argument given in Sect. 4.4.1 of [19]. 3 Smoluchowski

arrived at (5.2) independently in [63], a paper which appeared shortly after Einstein’s; see also [64]. 4 That the displacement probability φ have a finite second moment, an assumption that Einstein did not bother to make explicit, is crucial to deduce from (5.4) the standard Fickian model for diffusion. ‘Anomalous’ diffusion [1]—or rather, in the terminology currently used, ‘fractional’ or ‘nonlocal’ diffusion—is arrived at with different choices of φ.

52

5 A Physical Interpretation of Thermal Displacement

We presume, as Einstein did, that all particles are injected at the initial time t = 0 at point o of En , an n-dimensional Euclidean point space occupied by the previously considered viscous fluid at rest; to formalize this presumption, we supply (5.3) with an initial condition in terms of Dirac’s delta, namely,  c(x, 0) = δ(x).

(5.7)

The consequent isotropic evolution of particle concentration is described by the following solution of (5.3):  |x|2  , x := x − o, C0 = (4π do )−n/2 ;  c(x, t) = C0 t −n/2 exp − 4do t

(5.8)

here we have disposed of constant C0 by requiring that, at each time t, the nonnegative-valued field c(·, ˆ t) is normalized:  En

c(x, ˆ t) = 1,

(5.9)

so that we can regard it as a probability measure over En ; note that the stochastic balance (5.4) is consistent with both normalization conditions (5.5)2 and (5.9). Our crucial move is to imitate Einstein in introducing the integral  En

c(x, ˆ t)|x|2 ,

that is, the squared distance covered by suspended particles at time t, weighted over their concentration in the spot they have reached at that time.5 We next multiply Eq. (5.3) by |x|2 , and integrate over En . We find: ∂t

 En

c(x, ˆ t)|x|

2



 = do

En

 = do

 |x| Div (∇c) = do 2

En

En

  Div (|x|2 ∇c) − 2x · ∇c

   Div (|x|2 ∇c) − 2 ∇(cx) − c n = 2n do ;

here we have made use of the normalization condition (5.9). We conclude that  1 ∂t |x|2 , |x|2  := do = c(x, ˆ t)|x|2 . (5.10) n 2n E

5 Precisely,

Einstein introduces “die Würzel aus dem arithmetischen Mittel der Quadrate der Verrückungen in Richtung der X -Achse” (“the square root of the arithmetic mean of the squares of displacements in the direction of the X -axis” [16], p. 17).

5.2 How to Interpret Thermal Displacement Following the Einstein Path

53

This formula provides an interpretation of the macroscopic diffusion coefficient do in terms of the time rate of a statistical-mechanics observable that can be computed by post-processing microscopic information gathered in a molecular-dynamics simulation. Now, on combining (5.10) with (5.2), we find: ∂t |x|2  = 2n μ k B T,

(5.11)

whence, with the use of definition (5.1) of thermal displacement, we deduce that α ∝ |x|2 .

(5.12)

Relations (5.11) and (5.12) together make evident how the diffusing particles collectively serve as sort of a Brownian thermometer gauging the thermal motion of the fluid they are suspended in.

5.3 How to Interpret Thermal Displacement Following the Langevin Path On reading Einstein’s and Smoluchowski papers [14, 63], Langevin [41] conceived an original way of modelling Brownian motion: in his words, “J’ai pu constater tout d’abord qu’une application correcte de la méthode de M. Smoluchowski conduit à retrouver la formule de M. Einstein exactement et, de plus, qu’il est facile de donner, par une méthode toute différente, une démonstration infiniment plus simple.”6 If Einstein’s 1905 paper is the beginning landmark of stochastic modelling of natural phenomena, Langevin’s paper of 1908 can rightly be regarded as the beginning landmark of stochastic calculus. Just like Einstein, Langevin works in a onedimensional space setting; at sharp variance with him, he assumes that the motion of a Brownian particle can be studied by applying the momentum balance law of macroscopic Newtonian mechanics.7 His key idea is to imagine that a Brownian particle suspended in a viscous fluid at rest is acted upon by a fluctuating force F (his force complémentaire), which adds to the standard Stokesian viscous drag and gives

6 In

[42], this statement is translated as follows: “I have been able to determine, first of all, that a correct application of the method of M. Smoluchowski leads one to recover the formula of M. Einstein precisely, and, furthermore, that it is easy to give a demonstration that is infinitely more simple by means of a method that is entirely different.” 7 This remark is not meant as a criticism: the diameter of a Brownian particle is of the order of, say, 1 µm, way above atomic size.

54

5 A Physical Interpretation of Thermal Displacement

the motion a stochastic character.8 Accordingly, the Newtonian balance of inertial and noninertial forces on a spherical particle of radius a and mass m reads:

whence

m x¨ = −μ−1 x˙ + F,

(5.13)

d 1 1 d2 m 2 (x · x) + μ−1 (x · x) = m x˙ · x˙ + F · x. 2 dt 2 dt

(5.14)

Langevin takes the arithmetic mean of this equation over a large collection of identical particles. Just as Einstein and Smoluchowski did, he appeals to the kinetic theory of gases to evaluate the mean of the kinetic energy term on the right side; moreover, he characterizes the fluctuating force by assuming that the mean of the ‘virial’ term F · x be null. Then, on denoting by |x|2 the arithmetic mean of (x · x) and by z the time derivative of the latter: d z(t) := |x|2 (t), dt he arrives at:

1 1 m z˙ + μ−1 z = k B T. 2 2

(5.15)

It follows from this equation that, for (μm)−1 t large, d 2 |x|  2μ(k B T ) dt

(5.16)

(compare with (5.11)). In view of definition (5.1), we arrive at: α ∝ |x|2 ,

(5.17)

a relation that permits an interpretation for the thermal displacement completely analogous, but not identical, to the one implied by (5.12).

8 “Sur

la force complémentaire X nous savons qu’elle est indifféremment positive et négative, et sa grandeur est telle qu’elle maintient l’agitation de la particule que, sans elle, la résistance visqueuse finirait par arrêter.” [41] (“About the complementary force X, we know that it is indifferently positive and negative and that its magnitude is such that it maintains the agitation of the particle, which the viscous resistance would stop without it.” [42], p. 1081).

Appendix

Basic Notions of Continuum Mechanics

A.1 Deformation, Motion To study deformation, we make use of an observational background consisting of two copies of one and the same typically three-dimensional Euclidean point space, the one referential Er e f , the other current Ecur . We denote by Pt the region of Ecur , with boundary ∂ Pt , occupied at the current time t by a part P of a referential space region B ⊂ Er e f ; a part is a regularly open subset of B, that is, a set P ⊂ B that coincides with the interior of its closure. With a conscious abuse of notation, we denote by B and P also a material body and its typical part (or subbody). We regard the place x a body point occupies in B as an indelible label of that point. We stipulate that Pt = f t (P), where f t , the deformation of B at time t, is an invertible C1 -mapping of B into Bt , which delivers the current position y of the typical point x ∈ B: y = f t (x). We choose a time interval T (a subset of the oriented real line) and introduce the motion mapping (x, t) → f (x, t) ∀ (x, t) ∈ B × T , (A.1) which delivers the trajectory of all points of body B; for each fixed time t, the motion mapping yields the deformation at that time: f t (·) = (·, t). We require that, for all x ∈ B, f (x, ·) is of class C2 , so that it makes sense to talk about velocity v(x, t) := ∂t f (x, t) and acceleration a(x, t) := ∂t2 f (x, t). Given the invertibility of f (·, t), the mapping x = f t−1 (y) is well defined for all (y, t) ∈ Bt × T , and hence it also makes sense to regard velocity, acceleration, and other objects of interest as fields over Bt , the space region the body occupies at the current time.

© Springer Nature Switzerland AG 2019 P. Podio-Guidugli, Continuum Thermodynamics, SISSA Springer Series 1, https://doi.org/10.1007/978-3-030-11157-1

55

56

Appendix: Basic Notions of Continuum Mechanics

A.2 Evolution/Balance Laws. Inertial Objects The fundamental evolution laws of continuum mechanics are those of momenta, translational and rotational, and of energy. Here are provisional semi-verbal versions of these laws: - (momenta evolution)

- (energy evolution)

(translational momentum)· = f or ce,

(A.2)

(r otational momentum)· = torque;

(A.3)

(energy)· = heating + wor king.

(A.4)

Oftentimes, these statements are referred to as the balances of force, of torque, and of heat and work. Indeed, (A.2) can be arrived at by postulating in the first place the force balance: (A.5) f ni + f in = 0 and then by choosing f in to be a sort of d’Alembertian acceleration force. We infer from (A.2) and (A.5) that the standard additive splitting of the total force into a noninertial part f ni  force and an inertial part f in  −(transl. mom.)· has been accepted. A similar splitting of the total torque precedes logically (A.3); the other standard additive splitting, of the total energy into internal and kinetic parts, is not displayed in (A.4). The inertial objects of standard continuum mechanics are in fact momentum, translational and rotational, and kinetic energy. In my view, both concepts–and not the momentum concept alone–have been developed to account for the interactions of bodies in our world W with the totality of bodies in Wext , the rest of the universe, that are too far and too many for their individual interactions to be singled out. To obviate the impossible task of determining those interactions, one assumes that (the totality of bodies in) Wext contributes to the force balance of each part of a body in W through inertial forces, and to the heating-and-working balance through kinetic energy. Precisely, one accepts an augmented version of Noll’s postulat d’inertie [A1], and stipulates that, for an observer of part Pt of a body Bt in translational motion with respect to a chosen privileged observer (oftentimes called the “fixed-stars observer”), the current action of Wext on W is manifested • in the translational-momentum balance, by a force term equal to minus the rate of change of momentum:  (A.6) − ( v dm)· , P ⊂ B , t

Pt

where dm is the mass measure on Bt ;

t

Appendix: Basic Notions of Continuum Mechanics

57

• in the energy balance, by a power term equal to minus the rate of change of kinetic energy:  1 (A.7) −( v · v dm)· , Pt ⊂ Bt . 2 Pt As a discussion in terms of interactions suggests, this inertia postulate has an inherent constitutive nature1 ; it does yield an interpretation of the force term in the right side of (A.2) as the mechanical action collectively exerted on a given body by all other bodies in W. However, the remarkable symmetry in the roles of inertial objects in the basic balance laws is incomplete because, as the creators of thermodynamics realized, while the momentum rate is enough to balance noninertial forces, there is also a need for a concept of internal energy to balance heating and working. Indeed, relation (A.4) may be legitimately interpreted in two ways, depending on the significations attached to the terms ‘energy’ and ‘working’: • if ‘energy’ is total (internal plus kinetic) energy and ‘working’ is power of noninertial mechanical agencies, then (A.4) can be read as (total energy)· = heating + noniner tial power ;

(A.8)

• if ‘energy’ is internal energy and ‘working’ is total (inertial plus noninertial) power, then (internal energy)· = heating + total power. (A.9) These two interpretations are consistent if (kinetic energy)· + iner tial power = 0.

(A.10)

I believe that the success of standard continuum thermomechanics as a predictive science depends crucially on the mutual consistency of the classical expressions of momentum and kinetic energy in the sense of (A.10).2

A.3 A Few Bits of Linear Algebra Symmetric and Skew- Symmetric Tensors. Tensors are linear transformations of a vector space V1 into another vector space V2 . Mostly, we shall be concerned with second-order tensors transforming a three-dimensional inner-product vector space 1 “La

loi d’inertie est regardée comme un postulat constitutif” [A1].

2 It is possible to find a rationale for such consistency that allows to deduce one expression from the

other, to identify the internal part of the energy, and ultimately to build up consistent, more general theories. The implications of postulating a relation of type (A.10) as an additional balance law in a theory otherwise based only on force and torque balances are briefly discussed in [A2]; the case of continua whose kinematic structure is more complex than the standard structure considered here is dealt with in [A3].

58

Appendix: Basic Notions of Continuum Mechanics

V into itself: V  v → Av ∈ V; we shall denote by Lin their collection, which has itself a natural structure of vector space endowed with an inner product. We denote the subcollections of symmetric and skew elements of Lin by Sym and Skw, respectively, so that Lin = Sym ⊕ Skw, (A.11) i.e., for3 each A ∈ Lin, A = sym A + skw A,   1 1 sym A := A + AT ∈ Sym, skw A := A − AT ∈ Skw; 2 2

(A.12)

here AT is the transpose of A, defined by a · Ab = AT a · b, a, b ∈ V.

(A.13)

For each w ∈ V there is one W ∈ Skw such that Wv = w × v, v ∈ V;

(A.14)

conversely, relation (A.14) associates a unique axial vector w with each skew tensor W. It follows that the dimension of the subspace Skw of Lin, a 9-dimensional vector space, is 3, the same as V; in view of (A.11), the dimension of Sym is 6; Skw and Sym are orthogonal subspaces of Lin: A · B = 0, ∀ A ∈ Skw, B ∈ Sym. Cofactor. Given A ∈ Lin, the cofactor A∗ of A is the unique element of Lin such that, whenever w ∈ V and W ∈ Skw obey (A.14), A∗ w and AWAT obey it as well:   AWAT v = (A∗ w) × v, v ∈ V. From this definition, it follows that A∗ (a × b) = Aa × Ab, a, b ∈ V;

(A.15)

moreover, if detA = 0 (that is, if A is invertible), then A∗ = (det A)A−T ,

 −1  −1 T A−T = AT = A ,

where A−1 is the inverse of A.

3 The

symbol ⊕ denotes the operation of direct sum of spaces.

(A.16)

Appendix: Basic Notions of Continuum Mechanics

59

Dyadic Product. The dyadic product of vectors a, b ∈ V is the second-order tensor defined as follows: (a ⊗ b)[c] := (b · c)a for all c ∈ V. Note that a ⊗ b maps the 3-dimensional space V into the 1-dimensional space span(a), the collection of all scalar multiples of the first factor of the dyadic product; thus, a ⊗ b is not an invertible linear transformation of V into iself. Orientation- Preserving Tensors. Let a, b, c ∈ V be such that a × b · c = 0; moreover, let L ∈Lin be invertible. Then, it is not difficult to show, with the use of (A.15), that La × Lb · Lc = (det L) a × b · c. The triplets a, b, c and La, Lb, Lc are said to have the same orientation if the triple vector products a × b · c and La × Lb · Lc have the same sign. The linear transformation L is said orientation-preserving iff det L > 0. We denote by Lin+ := {L ∈ Lin | det L > 0}

(A.17)

the collection of all (invertible and) orientation-preserving elements of Lin. Note that, in view of definition (A.17), Lin+ has the group structure but it is not a vector space. Orthogonal Tensors. Rotations.   Orth := Q ∈ Lin | QT Q = QQT = I is called the orthogonal group of Lin. According to this definition, Orth is the collection of all elements Q of Lin whose transpose QT and inverse Q−1 are equal; alternatively, Orth can be defined as the collection of all second-order tensors that preserve the inner product of vectors: Q ∈ Orth For

⇔

Qa · Qb = a · b, a, b ∈ V.

(A.18)

  Rot := R ∈ Lin+ | RT R = RRT = I ,

Orth can also be represented as the direct product of Rot and the two-element group {I, −I} consisting of the identity and the central reflection −I. The elements of Rot are called rotations because a rigid rotation about a point o is a deformation r of the form y = r (x) := o + R(x − o).

60

Appendix: Basic Notions of Continuum Mechanics

A.4 Observer Changes Translational observer changes leaving the body fixed at time t¯ – in short, t-changes in observer – have the form y + = y + (t − t¯)w, so that, at time t¯ and in fact for all times, v+ = v + w, a+ = a.

(A.19)

Similarly, for a r-change in observer, that is, a rotational observer change leaving the body fixed at time t¯, one finds that y + = o + Q(t)(y − o), with t → Q(t) the unique solution of the following initial-value problem: T ˙ (t) =  ∈ Skw, Q(t¯) = i Q(t)Q

(here i is the identity tensor), whence v+ = v + y, a+ = a − 2 y + 2v,

(A.20)

at time t¯. It follows from the first relations in (A.19) and (A.20) that a general rototranslational observer change leaving the body fixed at a chosen time t¯ induces at that time a velocity field v+ that can be visualized as the superposition of a rigid velocity field of parameters w and  on the velocity field v: v+ = v + w + y.

(A.21)

A.5 Invariance of External Power. Translational Balance of Forces A reductio ad unum of relations (1.1) and (1.2) was carried out by Noll in 1959 [A1], where he showed that they follow from an “... axiome d’objectivité qui demande l’invariance du travail par changement de repère arbitraire.” Similar results were obtained in 1964 by Green and Rivlin [A4], who postulated the invariance under superposed rigid motions of a version of (1.3), where energy had been additively split and the mass density of kinetic energy given the standard quadratic form in

Appendix: Basic Notions of Continuum Mechanics

61

the velocity; besides force and torque balances, they derived the equation of mass conservation, an axiom in Noll’s approach.4 A declaration of invariance formalizes the prejudice that the minimal self-consistency a physical theory should have is consistency with the geometry of the underlying space-time structure: Noll’s invariance of external power (to Noll’s ‘travail’ I prefer the term ‘power’) under the group action of space-time isometries (≡ observer changes) exploits the Cartesian structure of the nonrelativistic event space used as observational background in classical mechanics. The external power expended at time t on a body part P during a motion f t by a force system (c, d) for B is a linear form in the motion velocity v:  e (P; c, d)[v] :=

 ∂ Pt

c·v +

d · v,

(A.22)

Pt

where c, the contact force, is the surface density of all contact forces exerted on ∂ Pt by its exterior), and d, the distance force, is the volume density of all distance forces acting on Pt . Remark A.1 A definition of expended power is a keystone in the construction of a mechanical theory, because it makes explicit our prejudices about the basic mechanical duality between kinematical and dynamical variables: with Newtonian terminology, between fluxes and forces, such as, respectively, v and (c, d) in (A.22). The basic force balances follow, as we now show, from requiring invariance of external power, that is, of the power expended on body parts by all mechanical agencies, whatever their nature, external to the body part under observation. A force system is t-invariant if (c, d)+ = (c, d) for all t-changes in observer. With this definition, we are in a position to state a result of central importance: Translational-Balance Theorem. Consider a body B in motion under the action of a t-invariant force system (c, d). Then, the expended power is t-invariant, that is, e (P; c+ , d+ )[v+ ] = e (P; c, d)[v], for all P ⊂ B and for all t-changes in observer, if and only if the force system is balanced:   c + d = 0, for all P ⊂ B. (A.23) ∂ Pt

4 Mass

Pt

conservation, in a general tensorial form, is a byproduct of the invariance approach to total energy splitting adopted in [A2].

62

Appendix: Basic Notions of Continuum Mechanics

The easy proof of this theorem is almost read off from the form of (e )+ := e (P; c+ , d+ )[v+ ]. We find that +

+

+



∂ Pt

+

c ·v

+



d+ · v +    e =  (P; c, d)[v] + w · c + d ;

 (P; c , d )[v ] = e

+

Pt

∂ Pt

Pt

hence, (e )+ = e for all choices of w if and only if (A.23) holds. Remark A.2 Needless to say, the argument in the above proof holds no matter if v+ − v = w is interpreted as the relative velocity of two observers in a t-change or as a translation velocity field superimposed on the field v. However, while Green & Rivlin’s superimpositions of rigid motions to the current kinematic fields yields the same results as Noll’s observer changes in this case and others, it seems to me that Green & Rivlin’s method lacks the conceptual depth of Noll’s, or at least that it somehow conceals its own conceptual depth. Remark A.3 Under the assumptions of the theorem, d+ and d are understood as the distance-force fields at time t¯ as they are perceived by two observers in a translational relative motion leaving the body fixed at time t¯. This minimal invariance requirement does not conflict with the fact that distance forces include inertial contributions: for example, if the body under study conserves its mass when in motion,5 and if the standard recipe is accepted for the inertia force experienced by the typical body part P, than such force may be written as  v˙ dm

d (P) = − in

Pt

and is therefore invariant under t-changes in observer, because the acceleration field a = v˙ is. Remark A.4 Velocity-dependent contact forces are t-invariant if and only if they depend on the relative velocity of the surfaces in contact (relative velocities are t-invariant by definition).

5 The

assumption that mass is conserved implies that    d  ρψ dv = ρ ψ˙ dv. dt Pt Pt

Appendix: Basic Notions of Continuum Mechanics

63

A.6 Cauchy Stress Cauchy’s hypothesis that the contact force density at a point of an oriented surface S in Bt depend on S only through the normal n to S at that point implies that there is a tensor field, the Cauchy stress T, such that, at each point of Bt , c = Tn, d = − divT,

(A.24)

for each balanced force system (c, d) (for a proof of this basic result, see [A5], Section 14). Cauchy devised an effective procedure to construct stress out of contact forces: given three mutually orthogonal planes of normal ni (i = 1, 2, 3) through any chosen point y of Bt , let ci = cˆ (y, ni ) be the relative contact forces: then, T(y) =

3

c(y, ni ) ⊗ ni

(A.25)

i=1

(it is not difficult to show that whatever choice of three mutually orthogonal planes yields one and the same stress field T). Note that, in view of (A.24)1 and (A.25), the contact force field and the stress field carry the same mechanical information; and that the balanced force system corresponding to a given stress field is given by relations (A.24). Remark A.5 When, as we are going to do later on, the basic balance laws are encoded into a postulated Virtual-Power Principle, where the dynamic ingredients are (force system, stress field) pairs, then relations (A.24)1 and (A.25) play the role of consistency requirements singling out the admissible (c, T) pairs. In a balanced force system, the distance force field d is thought of as defined over the interior of Bt , the contact force field c over the Cartesian product of the closure of Bt and the sphere U of vectors of unit length. For c0 (y) the contact force exerted on body B by its environment at a point y ∈ ∂ Bt , and for n(y) the outer unit normal at that same point, c and c0 must satisfy the following mutual-consistency condition: c(y, n(y)) = c0 (y);

(A.26)

with (A.24)1 , this implies the traction boundary condition: T(y)n(y) = c0 (y),

(A.27)

at each point y ∈ ∂ Bt where c0 is assigned. With reference to Fig. A.1, a simple limit argument helps interpreting (A.27) as a consequence of force balance: construct part Pt as the intersection with Bt of a right cylinder of height ε, whose axis is parallel to n(y) and whose top end is a flat open neighborhood of point y ∈ ∂ Bt ; then, on denoting by Bot the flat bottom part of ∂ Pt and by Top its curved top part,

64

Appendix: Basic Notions of Continuum Mechanics

Fig. A.1 Traction boundary condition as a consequence of part-wise force balance

and on taking both (A.24)1 and (A.26) into account, the balance equation (A.23) for such a pillbox-like part reads:  Bot

  T (−n(y)) +

Top

c0 + O(ε) = 0,

whence (A.27).

A.7 Stress Power. Rotational Balance of Forces With (A.24), an integration by parts allows us to give the right side of (A.22) the following form:    c·v + d·v = T · grad v; (A.28) ∂ Pt

Pt

we6 call

Pt

 s (P; T)[v] :=

T · grad v

(A.29)

Pt

the stress power of the body part P. The notion of stress power opens the way to establishing an invariance result that complements the Translational-Balance Theorem.

6 Recall

that



 ∂ Pt



Tn · v +

(−divT) · v = Pt

T · grad v. Pt

Appendix: Basic Notions of Continuum Mechanics

65

Rotational-Balance Theorem. Consider a Cauchy body B in motion under the action of a t-invariant, balanced force system (c, d). Assume that the contact force field c is r-invariant. Then, the following statements are equivalent: (i) the stress power is r-invariant: s (P; T+ )[v+ ] = s (P; T)[v], for all P ⊂ B and for all r-changes in observer; (ii) the Cauchy stress field is symmetric-valued: T ∈ Sym;

(A.30)

(iii) the force system is rotationally balanced: 

 ∂ Pt

(y − o) × c +

(y − o) × d = 0, for all P ⊂ B.

(A.31)

Pt

Here is a proof that (i) ⇒ (ii) ⇒ (iii). As a premiss, note that an r-change in observer leaves n as is; and that, since by hypothesis c is invariant under such changes, the same is true for T. Now, the invariance condition on the stress power reads: 

+

+



T · (grad v) = Pt

T · grad v, Pt

for (grad v)+ = grad v +  and for all  ∈ Skw, or rather:  T = 0 for all  ∈ Skw and for all P ⊂ B,

· Pt

which is tantamount to have that the stress field is symmetric-valued. But then, when written for v a rotation about a point o, (A.28) reads as follows: 

 c · (y − o) +

∂ Pt

d · (y − o) = 0 for all  ∈ Skw and for all P ⊂ B, Pt

or rather, equivalently, 

 ·

∂ Pt

(y − o) ⊗ c +

whence (A.31).

(y − o) ⊗ d = 0 for all  ∈ Skw and for all P ⊂ B, Pt

(A.32)

66

Appendix: Basic Notions of Continuum Mechanics

To see that (iii) ⇒ (ii), consider that, with the use of (A.24)1 ,      (y − o) ⊗ c = (y − o) ⊗ (Tn) = div (y − o) ⊗ T ∂ Pt ∂ Pt P   t TT + (y − o) ⊗ divT. = Pt

Pt

With this, and skipping quantifications for brevity, (A.32) yields: ·





 (y − o) ⊗ (divT + d) = 0,

T + T

Pt

Pt

whence, with (A.24)2 , ·



 TT = 0 for all  ∈ Skw, Pt

etc.; the remain of the proof can be safely omitted. Equations (A.23) & (A.31) and Eqs. (A.24) & (A.30) are, respectively, part-wise and point-wise statements of the force balance laws common to all Cauchy continua. These laws are the first products of the mechanical axiomatic approach we have chosen.

A.8 Velocity. Referential and Current Fields. Velocity Gradient. Stretching and Spin Given a motion (x, t) → y = f (x, t) = f t (x), F := ∂x f is the deformation gradient, a field over Er e f × T . Let ϕ = ϕ(x, t) denote a generic referential field; the standard notation for the result of the operation of space differentiation is ∇ϕ, and is ϕ˙ (or ϕ · ), for the result of the operation of time differentiation. The superposed-dot notation (that can be traced back to Newton) is useful when the referential field in question is given a current counterpart: ϕ= ϕ (y, t) := ϕ( f t−1 (y), t). In such a case, it is important to distinguish ∂t ϕ from ϕ˙ = (∂ y ϕ)∂t y + ∂t ϕ = (∂ y ϕ)v + ∂t ϕ.

(A.33)

The standard notation for ∂ y ϕ is grad ϕ; if, in particular, the field ϕ is scalar-valued, then grad ϕ is a vector field, and (A.33) can be written as follows: ϕ˙ = grad ϕ · v + ∂t ϕ.

(A.34)

Appendix: Basic Notions of Continuum Mechanics

67

For any scalar field ϕ that is given both a referential and a current description, the relation (A.35) ∇ϕ = FT (grad ϕ) holds true. Taking ϕ = a · b, with b any constant vector, (A.35) yields: ∇a = (grad a)F.

(A.36)

The grad operator is often employed to represent the velocity gradient L, writing L = grad v for (A.37) L(y, t) := ∂ y v( f t−1 (y), t), and noting that, with the use of (A.36), ∇v = ∂x (∂t f ) = ∂t (∂x f ) = F˙ = (grad v)F

⇒

˙ −1 . grad v = FF

(A.38)

The symmetric and skew-symmetric parts of L, D=

1 1 (L + LT ) and W = (L − LT ), 2 2

(A.39)

are called the stretching tensor and the spin tensor; the first measures the strain rate, the second the motion vorticity; in fluid mechanics, they are ubiquitous. Remark A.6 It is the matter of a straightforward algebraic calculation to show that the vector uniquely associated to the skew tensor 2W is curl v, the vorticity of the velocity field; thus, in particular, a motion is locally irrotational whenever W=0

⇔

curl v = 0.

A.9 Rigid Motions A motion of a body B is rigid if it has the following form: y = frig (x, t) := r (t) + R(t)(x − o), with r (t) ∈ Ecur and R(t) ∈ Rot, (A.40) for all x ∈ B; since r (t) is the current position of point o ∈ Er e f , relation (A.40) can be written as follows: y = Rx, (A.41) with y := y − r and x := x − o the current and referential position vectors of a typical body point. The referential velocity and acceleration fields are, respectively, ˙ − o) and y¨ = r¨ + R(x ¨ − o); y˙ = r˙ + R(x

68

Appendix: Basic Notions of Continuum Mechanics

the current velocity field is: vrig (y, t) = w(t) + (t)(y − r (t)) = w(t) + ω(t) × (y − r (t)),

(A.42)

with ω(t) the current angular velocity, and is parameterized by T ˙ (t), (or ω(t) ↔ (t)); w(t) := r˙ (t) and (t) := R(t)R

finally, the current acceleration field is: ˙ + 2 )(y − r (t)). arig (y, t) = r¨ (t) + (

(A.43)

We7 also have that ∇ frig (x, t) = R(t), for all x ∈ B, and that grad vrig (y, t) = (t), for all y ∈ Bt = frig (B, t) = R(t)B. Remark A.7 For a typical material fiber (x, e), whose image in a body motion is (y, f) = ( f (x, t), F(x, t)e), we have that ˙ L(x, t)f = (F(x, t)e)· = f, or rather,

1 f˙ = Df + curl v × f, 2

a representation of the time rate of f that splits it additively into fiber stretching and fiber spinning. For one example of use of this formula, consider the following state of uniaxial stretching: F(x, t) = λ(x, t)e ⊗ e + (1 − e ⊗ e), λ(x, t) > 0, in which

˙ −1 = (log λ)· e ⊗ e ∈ Sym FF

7 Since

RR T = 1 we have that

⇒

(RR T )· = 0

⇒

˙ T = −RR ˙T  := RR

¨ T = ˙ + 2 . RR

⇒

˙ = R, R

Appendix: Basic Notions of Continuum Mechanics

69

and hence no fibers are spun, in contrast with what happens in a rigid motion, when no fibers are stretched and all are equally spun, because ˙ T ∈ Skw. grad vrig = RR For another example, consider the following state of plane shearing: F = 1 + τ (x, t)e1 ⊗ e2 . Here

F−1 = 1 − τ e1 ⊗ e2 , F˙ = τ˙ (x, t)e1 ⊗ e2 = L,

and hence, for f1 = Fe1 = e1 and f2 = Fe2 = e2 + τ e1 , one finds that f˙1 = 0, f˙2 = τ˙ (x, t)e1 .

Bibliography [A1] W. Noll, La mécanique classique, basée sur un axiome de objectivité, pp. 47-56 of “La Méthode Axiomatique dans les Mécaniques Classiques et Nouvelles” (Colloque International, Paris, 1959). Gauthier-Villars, Paris, 1963 (reprinted in Noll’s Selecta, 1974). [A2] P. Podio-Guidugli, Inertia and invariance. Annali Mat. Pura Appl. CLXXII (IV) (1997), 103-124. [A3] P. Podio-Guidugli, La scelta dei termini inerziali per i continui con microstruttura. Rendiconti Lincei, Ser. IX, XIV (4) (2003), 319-326. [A4] A.E. Green and R.S. Rivlin, On Cauchy’s equations of motion. ZAMP 15, 290-292 (1964). [A5] M.E. Gurtin, An Introduction to Continuum Mechanics. Academic Ptess (1981).

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Index

A Area jacobian, 26 Axial vector, 58

H Helmholtz free energy, 5 Hyperelastic solids, 30

C Caloric fluid, 28 Clausius-Duhem inequality, 4 Cofactor, 26 Coldness, 4

I Inflow disorder i., 2 heat i., 2 Influx i. of a physical entity, 2 Intensive i. physical quantity, 3 Internal dissipation, 5, 30 Internal potential, 31 Isolated system, 4

D Dissipation inequality purely mechanical d. i., 30 reduced d. i., 5 E Energy free e., see Helmholtz f.e., 5 Entropy e. production, 3, 27 Extensive e. physical quantity, 3 F First Law, 3 Force contact f., 61 distance f., 61 Force system, 61 t-invariant f. s., 61 G Gibbs relations, 9

M Motion rigid, 67

N Newtonian fluids, 30

O Observer change r-change in observer, 60 t-change in observer, 60

P Phase transformations stress-induced, 29 Piezocaloric effect, 28

© Springer Nature Switzerland AG 2019 P. Podio-Guidugli, Continuum Thermodynamics, SISSA Springer Series 1, https://doi.org/10.1007/978-3-030-11157-1

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76 Principle of Equipresence, 6 Virtual Powers, for Cauchy continua, 35 Process isentropic p., 31 isothermal p., 31 R Reflection central r., 59 S Second Law, 3 Source radiative s., 2 s. of a physical entity, 2 State s. space, 7

Index Stress Cauchy stress, 63 Piola stress, 25 stress power, 64

T Temperature, 4 Tensor cofactor of a t., 58 skew t., 58 symmetric t., 58 transpose of a t., 58 Thermal displacement, 41

V Velocity realizable v., 34