Table Of ContentProgress in Mathematical Physics
Volume 37
Editors-in-Chief
Anne Boutet de Monvel, Université Paris VII Denis Diderot
Gerald Kaiser, The Virginia Center for Signals and Waves
Editorial Board
D. Bao, University of Houston
C. Berenstein, University of Maryland, College Park
P. Blanchard, Universität Bielefeld
A.S. Fokas, Imperial College of Science, Technology and Medicine
C. Tracy, University of California, Davis
H. van den Berg, Wageningen University
Alfredo Bermúdez de Castro
Continuum
Thermomechanics
Birkhäuser Verlag
Basel · Boston · Berlin
Author:
Alfredo Bermúdez de Castro
Facultad de Matemáticas
Universidade de Santiago de Compostela
Campus Universitario Sur
15782 Santiago de Compostela
Spain
e-mail : [email protected]
2000 Mathematics Subject Classification 74A, 74J, 76A, 76N, 80A
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C.,
USA
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graphic data is available in the Internet at <http://dnb.ddb.de>.
ISBN 3-7643-7265-6 Birkhäuser Verlag, Basel – Boston – Berlin
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To my wife, Ana
Contents
Preface xi
1 General Definitions, Conservation Laws 1
1.1 Motion of a Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Conservation of Mass. . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Balance of Linear and Angular Momentum . . . . . . . . . . . . . 5
1.4 Balance of Energy. First Principle of Thermodynamics . . . . . . . 7
1.5 Second Principle of Thermodynamics. The Clausius-Duhem
Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Lagrangian Coordinates 13
2.1 The Piola-KirchhoffStress Tensors . . . . . . . . . . . . . . . . . . 13
2.2 The Conservation Equations in LagrangianCoordinates . . . . . . 14
3 Constitutive Laws 17
3.1 Thermodynamic Process. Material Body . . . . . . . . . . . . . . . 17
3.2 Coleman-Noll Materials . . . . . . . . . . . . . . . . . . . . . . . . 18
4 The Principle of Material Frame-Indifference 27
4.1 Change in the Observer.The Indifference Principle . . . . . . . . . 27
4.2 Consequences for Coleman-Noll Materials . . . . . . . . . . . . . . 28
5 Replacing Entropy with Temperature 33
5.1 The Conservation Equations in Terms of Temperature . . . . . . . 33
6 Isotropy 37
6.1 The Extended Symmetry Group . . . . . . . . . . . . . . . . . . . 37
6.2 Isotropic Bodies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
7 Equations in Lagrangian Coordinates 43
viii Contents
8 Linearized Models 47
8.1 Linear Approximation of the Motion Equation . . . . . . . . . . . 47
8.2 Linear Approximation of the Energy Equation . . . . . . . . . . . 52
8.3 Isotropic Linear Thermoviscoelasticity . . . . . . . . . . . . . . . . 54
9 Quasi-static Thermoelasticity 57
9.1 Statement of the Equations . . . . . . . . . . . . . . . . . . . . . . 57
9.2 Time Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 57
9.3 A Particular Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
10 Fluids 61
10.1 The Concept of Fluid, First Properties . . . . . . . . . . . . . . . . 61
10.2 Motion Equation. Thermodynamic Pressure . . . . . . . . . . . . . 63
10.3 Energy Equation, Enthalpy . . . . . . . . . . . . . . . . . . . . . . 64
10.4 Thermodynamic Coefficients and Equalities . . . . . . . . . . . . . 66
10.5 Gibbs Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
10.6 Statics of Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
10.7 The Boussinesq Approximation, Natural Convection . . . . . . . . 78
11 Linearized Models for Fluids, Acoustics 81
11.1 General Equations, Dissipative Acoustics . . . . . . . . . . . . . . 81
11.2 The Isentropic Case, Non-Dissipative Acoustics . . . . . . . . . . . 85
11.3 Linearized Models under Gravity . . . . . . . . . . . . . . . . . . . 87
12 Perfect Gases 93
12.1 Definition, General Properties . . . . . . . . . . . . . . . . . . . . . 93
12.2 Entropy and Free Energy . . . . . . . . . . . . . . . . . . . . . . . 94
12.3 The Compressible Navier-Stokes Equations . . . . . . . . . . . . . 97
12.4 The Compressible Euler Equations . . . . . . . . . . . . . . . . . . 98
13 Incompressible Fluids 101
13.1 Isochoric Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
13.2 Newtonian Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
13.3 Ideal Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
14 Turbulent Flow of Incompressible Newtonian Fluids 105
14.1 Turbulence Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
14.2 The k−(cid:1) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
15 Mixtures of Coleman-Noll Fluids 109
15.1 General Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
15.2 Mixture of Perfect Gases . . . . . . . . . . . . . . . . . . . . . . . . 113
Contents ix
16 Chemical Reactions in a Stirred Tank 119
16.1 Chemical Kinetics. The Mass Action Law . . . . . . . . . . . . . . 119
16.2 Conservation of Chemical Elements . . . . . . . . . . . . . . . . . . 122
16.3 Reacting Mixture of Perfect Gases . . . . . . . . . . . . . . . . . . 123
17 Chemical Equilibrium of a Reacting Mixture of Perfect Gases
in a Stirred Tank 125
17.1 The Least Action Principle for the Gibbs Free Energy . . . . . . . 125
17.2 Equilibrium for a Set of Reversible Reactions,
Equilibrium Constants . . . . . . . . . . . . . . . . . . . . . . . . . 126
17.3 The Stoichiometric Method . . . . . . . . . . . . . . . . . . . . . . 131
18 Flow of a Mixture of Reacting Perfect Gases 135
18.1 Mass Conservation Equations . . . . . . . . . . . . . . . . . . . . . 135
18.2 Motion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
18.3 Energy Conservation Equation . . . . . . . . . . . . . . . . . . . . 137
18.4 Conservation of Elements . . . . . . . . . . . . . . . . . . . . . . . 140
18.5 Equilibrium Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . 141
18.6 The Case of Low Mach Number . . . . . . . . . . . . . . . . . . . . 142
19 The Method of Mixture Fractions 145
19.1 General Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
19.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
19.3 The Adiabatic Case . . . . . . . . . . . . . . . . . . . . . . . . . . 149
19.4 The Case of Equilibrium Chemistry . . . . . . . . . . . . . . . . . 149
20 Turbulent Flow of Reacting Mixtures of Perfect Gases,
The PDF Method 153
20.1 Elements of Probability . . . . . . . . . . . . . . . . . . . . . . . . 153
20.2 The Mixture Fraction/PDF Method . . . . . . . . . . . . . . . . . 155
A Vector and Tensor Algebra 161
A.1 Vector Space. Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
A.2 Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
A.3 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
A.4 The Affine Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
B Vector and Tensor Analysis 173
B.1 Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . 173
B.2 Curves and Curvilinear Integrals . . . . . . . . . . . . . . . . . . . 175
B.3 Gauss’ and Green’s Formulas. Stokes’ Theorem . . . . . . . . . . . 177
B.4 Change of Variable in Integrals . . . . . . . . . . . . . . . . . . . . 178
B.5 Transport Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 178
B.6 Localization Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 179
x Contents
B.7 Differential Operators in Coordinates . . . . . . . . . . . . . . . . . 179
B.7.1 Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . 179
B.7.2 Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . 182
B.7.3 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . 184
C Some Equations of Continuum Mechanics in Curvilinear Coordinates 189
C.1 Mass Conservation Equation . . . . . . . . . . . . . . . . . . . . . 189
C.2 Motion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
C.3 Constitutive Law for Newtonian Viscous Fluids in Cooordinates . 191
D Arbitrary Lagrangian-Eulerian (ALE) Formulations of the Conservation
Equations 195
D.1 ALE Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
D.2 Conservative ALE Form of Conservation Equations . . . . . . . . . 197
D.2.1 Mixed Conservative ALE Form of the Conservation
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
D.3 Mixed Nonconservative Form of ALE Conservation Equations . . . 200
Bibliography 203
Index 205
Preface
This book is intended to be an extensionof Gurtin’s book on continuum mechan-
ics [5] by including the laws of thermodynamics and thus making it possible to
studythe mechanicalbehaviourofmaterialbodies,the responseofwhichinvolves
variablessuch as entropy or temperature. In order to do that our departure point
is Coleman and Noll’s article [3] on the thermodynamics of elastic materials with
heat conduction and viscosity which has been extended for the purpose at hand
to the case of nonhomogeneous materials.
The present book has been used for many years as a textbook for gradu-
ate and undergraduate mathematics students at the University of Santiago de
Compostela.
The first Chapter revisits the conservation principles of continuum thermo-
mechanics,thatis,theconservationofmass,linearandangularmomentumbalance
and the first two principles of thermodynamics: namely, energy conservation and
entropy inequality. All principles are introduced in integral form and in Eulerian
coordinates. Local forms consisting of partial differential equations are then ob-
tained. Writing these local equations in Lagrangian coordinates is the subject of
Chapter 2.
Chapter 3 deals with the constitutive laws of continuum thermomechanics.
Firstthe notionofamaterialbody characterisedbyits constitutive classis given.
Then we introduce a general material body defined by Coleman and Noll in the
above referenced article. By imposing the second principle of thermodynamics,
we prove some relations to be satisfied by the response functions of such a mate-
rial.Then,in Chapter4,the principle ofmaterialframe-indifferenceis introduced
andits consequencesforthe responsefunctions ofthe Coleman-Nollmaterialsare
established. In Chapter 5, the partial differential equations governing a thermo-
dynamic process are written replacing entropy with temperature.
Chapter 6 is devoted to isotropy. By using the representation theorems for
isotropic tensor and vector-valued functions, we obtain simple forms for the re-
sponse functions of Coleman-Noll materials. In Chapter 7, the equations satisfied
by each thermodynamic process of these materials are written in Lagrangian co-
ordinates. We also show that inviscid Coleman-Noll materials are hyperelastic.
The linear approximations of these equations about a static reference state
arededucedinChapter8,assumingthatthegradientofthe displacementandthe
difference of temperature with respect to a reference state are both small. This
is rigorously done through careful computation of the derivatives of the response
functions.Isotropicmaterialsarespecificallyconsidered.Thus,weobtainthe par-
tial differential system for linear thermoviscoelasticity; its numerical solution by
incremental methods, in the inviscid quasi-static case, is addressed in Chapter 9.
Fluids are the subject of Chapter 10 where they are introduced as partic-
ular Coleman-Noll materials when the extended symmetry group is the unimod-
ular group. We define the classical thermodynamic variables like specific heat,
soundspeed, volumetric thermal expansion,and write the conservationequations
