Continuum Mechanics Through the Eighteenth and Nineteenth Centuries: Historical Perspectives from John Bernoulli (1727) to Ernst Hellinger (1914)

275 Pages·2014·2.707 MB·English

by  Gérard A. Maugin

Solid Mechanics and Its Applications
Gérard A. Maugin
Continuum
Mechanics Through
the Eighteenth
and Nineteenth
Centuries
Historical Perspectives from John
Bernoulli (1727) to Ernst Hellinger
(1914)
Solid Mechanics and Its Applications
Volume 214
Series editor
G. M. L. Gladwell, Waterloo, Canada
For furthervolumes:
http://www.springer.com/series/6557
Aims and Scope
Thefundamentalquestionsarisinginmechanicsare:Why?How?andHowmuch?
The aim of this series is to provide lucid accounts written by authoritative
researchersgivingvisionandinsightinansweringthesequestionsonthesubjectof
mechanics as it relates to solids.
The scope of the series covers the entire spectrum of solid mechanics. Thus it
includes the foundation of mechanics; variational formulations; computational
mechanics;statics,kinematicsanddynamicsofrigidandelasticbodies:vibrations
of solids and structures; dynamical systems and chaos; the theories of elasticity,
plasticity and viscoelasticity; composite materials; rods, beams, shells and mem-
branes; structural control and stability; soils, rocks and geomechanics; fracture;
tribology; experimental mechanics; biomechanics and machine design.
The median level of presentation is the first year graduate student. Some texts
aremonographsdefiningthecurrentstateofthefield;othersareaccessibletofinal
year undergraduates; but essentially the emphasis is on readability and clarity.
Gérard A. Maugin
Continuum Mechanics
Through the Eighteenth
and Nineteenth Centuries
Historical Perspectives from John Bernoulli
(1727) to Ernst Hellinger (1914)
123
GérardA.Maugin
InstitutJean LeRond d’Alembert
Université Pierre etMarieCurie
Paris
France
ISSN 0925-0042 ISSN 2214-7764 (electronic)
ISBN 978-3-319-05373-8 ISBN 978-3-319-05374-5 (eBook)
DOI 10.1007/978-3-319-05374-5
Springer ChamHeidelberg New YorkDordrecht London
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Preface
Itisnotbyonewayalonethatwecanarriveatsosublimeamystery
(attributedtoQuintusAureliusSymmachus,384)
Conceived as a series of more or less autonomous essays, this book exposes the
initial developments of continuum thermo-mechanics in a post-Newtonian period
extending from the creative works of the Bernoullis to the First World War, i.e.,
roughly during first the ‘‘Age of reason’’ and next the ‘‘Birth of the modern
world.’’ The emphasis is rightly placed on contributions from the ‘‘Continental’’
scientists (the Bernoulli family, Euler, d’Alembert, Lagrange, Cauchy, Piola,
Duhamel, Neumann, Clebsch, Kirchhoff, Helmholtz, Saint-Venant, Boussinesq,
the Cosserat brothers, Caratheodory) in competition with their British peers
(Green, Kelvin, Stokes, Maxwell, Rayleigh, Love,..). It underlines the main
breakthroughsaswellasthesecondaryones.Ithighlightstheroleofscientistswho
left essential prints inthis historyofscientific ideas. It showshowthe formidable
developments that blossomed in the twentieth century (and perused in a previous
book of the author: ‘‘Continuum Mechanics Through the Twentieth Century,’’
Springer SMIA series, Dordrecht, 2013) found rich compost in the constructive
foundational achievements of the eighteenth and nineteenth centuries. The pre-
WWI situation is well summarized by a thorough analysis of treatises (Appell,
Hellinger) published at that time. English translations of most critical texts in
French or German are given for the benefit of the readers.
Gérard A. Maugin
v
Contents
1 General Introduction: About the Contents and Form
of this Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 A Glimpse at the Eighteenth Century: From John Bernoulli
to Lagrange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 John Bernoulli and the Principle of Virtual Work. . . . . . . . . . 7
2.2 ‘‘Bernoulli’s Theorem’’ by Daniel Bernoulli . . . . . . . . . . . . . 9
2.3 D’Alembert and the Metaphysical Notion of Force. . . . . . . . . 10
2.4 The Notion of Internal Pressure and the Fundamental
Equations of Hydrodynamics. . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Linear Momentum and Moment of Momentum: Newtonian
Versus Variational Formulations. . . . . . . . . . . . . . . . . . . . . . 16
2.6 Calculus of Variations and Analytical Mechanics:
Lagrange. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.7 The Age of Reason: Conclusion and Things to Come. . . . . . . 20
Appendix A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Appendix B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Appendix C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 What Happened on September 30, 1822, and What Were
its Implications for the Future of Continuum Mechanics? . . . . . . 33
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Preliminary Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 The Main Contents of Cauchy’s 1822/1828 Memoir. . . . . . . . 35
3.4 Cauchy’s Stress and Hyperstresses . . . . . . . . . . . . . . . . . . . . 39
3.5 Stress as a Secondary Notion. . . . . . . . . . . . . . . . . . . . . . . . 40
3.6 Stress as Part of Stress-Energy-Momentum . . . . . . . . . . . . . . 42
3.7 The Nec Plus Ultra: The Eshelby Stress and the Pseudo
Tensor of Energy-Momentum . . . . . . . . . . . . . . . . . . . . . . . 45
vii
viii Contents
3.8 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Appendix A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Appendix B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 Piola and Kirchhoff: On Changes of Configurations. . . . . . . . . . . 55
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Piola’s Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2.1 Some Words of Caution . . . . . . . . . . . . . . . . . . . . . 56
4.2.2 The Strategy of Piola . . . . . . . . . . . . . . . . . . . . . . . 57
4.2.3 Introduction of the ‘‘Piola Format’’ by Piola . . . . . . . 57
4.2.4 Stresses as Lagrange Multipliers. . . . . . . . . . . . . . . . 61
4.3 The Role of Kirchhoff. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5 Duhamel’s Pioneering Work in Thermo-elasticity
and Its Legacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.1 The Roots of Duhamel’s Contribution. . . . . . . . . . . . . . . . . . 69
5.2 Basic Ideas of Duhamel’s 1837 Paper. . . . . . . . . . . . . . . . . . 70
5.3 Duhamel’s Strategy and Equations of ‘‘Thermo-elasticity’’ . . . 70
5.4 Comments and Further Developments. . . . . . . . . . . . . . . . . . 72
5.5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6 From Cauchy to Boussinesq via Barré de Saint-Venant. . . . . . . . 77
6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2 Cauchy and Lamé. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.3 Barré de Saint-Venant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.3.1 On his Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.3.2 Some Biographical Elements. . . . . . . . . . . . . . . . . . 80
6.3.3 Saint-Venant’s Scientific Achievements . . . . . . . . . . 81
6.4 Boussinesq: A Disciple of Saint-Venant . . . . . . . . . . . . . . . . 90
6.4.1 Contributions to Fluid Mechanics. . . . . . . . . . . . . . . 90
6.4.2 Contributions to the Mechanics of Deformable
Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7 Helmholtz Interpreted and Applied by Duhem . . . . . . . . . . . . . . 99
7.1 Hermann von Helmholtz and Pierre Duhem. . . . . . . . . . . . . . 99
7.2 Free Energy (‘‘Freie Energie’’). . . . . . . . . . . . . . . . . . . . . . . 101
Contents ix
7.3 Helmholtz-Duhem Electrodynamics . . . . . . . . . . . . . . . . . . . 104
7.4 Stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.5 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
8 About the Cosserats’ Book of 1909 . . . . . . . . . . . . . . . . . . . . . . . 113
8.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
8.2 The Main Contents of the Cosserats’ Book . . . . . . . . . . . . . . 116
8.3 Reception and Influence of the Cosserats’ Book. . . . . . . . . . . 120
8.4 Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Appendix A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
9 Caratheodory: Thermodynamics and Topology . . . . . . . . . . . . . . 137
9.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
9.2 On Caratheodory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
9.3 The Standard Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . 139
9.4 Caratheodory’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
9.5 Reception of Caratheodory’s Axiomatics. . . . . . . . . . . . . . . . 144
9.6 Toward Irreversible Thermodynamics. . . . . . . . . . . . . . . . . . 145
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
10 On Duhem’s Energetics or General Thermodynamics . . . . . . . . . 149
10.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
10.2 On Duhem’s Style and General View of Science . . . . . . . . . . 151
10.3 Some of Duhem’s Creative Works in Thermo-mechanics . . . . 153
10.3.1 Physical Chemistry. . . . . . . . . . . . . . . . . . . . . . . . . 153
10.3.2 Fluid Mechanics, Viscous Fluids . . . . . . . . . . . . . . . 154
10.3.3 Deformable Solids . . . . . . . . . . . . . . . . . . . . . . . . . 155
10.3.4 General ‘‘Thermo-mechanics’’ and Thermodynamic
Potentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
10.3.5 Contributions to Mathematics. . . . . . . . . . . . . . . . . . 157
10.4 Short Analysis of Some Exemplary Writings. . . . . . . . . . . . . 158
10.4.1 On the General Treatise on Energetics . . . . . . . . . . . 158
10.4.2 On the Evolution of Mechanics . . . . . . . . . . . . . . . . 159
10.4.3 On Permanent Alterations and Hysteresis . . . . . . . . . 162
10.5 Influence on Contemporaries and Later on . . . . . . . . . . . . . . 168
10.6 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
Appendix A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
Appendix B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
x Contents
11 A Course of Continuum Mechanics at the Dawn
of the Twentieth Century (Volume III of Appell’s Treatise
on Rational Mechanics) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
11.1 Prolegomena: On Paul Appell . . . . . . . . . . . . . . . . . . . . . . . 187
11.2 Setting the Stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
11.3 The Contents of Appell’s Volume on the Mechanics
of Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
11.3.1 Some Words of Introduction . . . . . . . . . . . . . . . . . . 190
11.3.2 Vector Analysis and Potential Theory. . . . . . . . . . . . 191
11.3.3 Equilibrium and Internal Motion of a Continuous
Mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
11.3.4 Hydrostatics and Stability . . . . . . . . . . . . . . . . . . . . 192
11.3.5 Deformation of Continua. . . . . . . . . . . . . . . . . . . . . 192
11.3.6 Kinematics of Continua. . . . . . . . . . . . . . . . . . . . . . 193
11.3.7 General Theorems of the Dynamics of Perfect
Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
11.3.8 Theory of Vortices. . . . . . . . . . . . . . . . . . . . . . . . . 194
11.3.9 Parallel Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
11.3.10 Barocline Fluids. . . . . . . . . . . . . . . . . . . . . . . . . . . 196
11.3.11 Elements of Elasticity Theory . . . . . . . . . . . . . . . . . 197
11.3.12 On Viscous Fluids . . . . . . . . . . . . . . . . . . . . . . . . . 198
11.4 The Cosserats’ Theory of Euclidean Action. . . . . . . . . . . . . . 199
11.5 Concluding Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Appendix: Biography of Paul Appell . . . . . . . . . . . . . . . . . . . . . . . 201
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
12 A Successful Attempt at a Synthetic View of Continuum
Mechanics on the Eve of WWI: Hellinger’s Article
in the German Encyclopaedia of Mathematics. . . . . . . . . . . . . . . 205
12.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
12.2 The Scientific Environment. . . . . . . . . . . . . . . . . . . . . . . . . 207
12.3 The Contents of Hellinger’s Article . . . . . . . . . . . . . . . . . . . 208
12.3.1 Introductory Remark. . . . . . . . . . . . . . . . . . . . . . . . 208
12.3.2 The Layout and Articulation of the Contribution . . . . 208
12.4 The Identified Fields of Marked Interest of Hellinger. . . . . . . 209
12.4.1 On General Principles of Mechanics and General
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
12.4.2 On Variational Formulations . . . . . . . . . . . . . . . . . . 210
12.4.3 On Finite Strains and Elasticity . . . . . . . . . . . . . . . . 212
12.4.4 On Oriented Media. . . . . . . . . . . . . . . . . . . . . . . . . 213
12.4.5 On One-Dimensional and Two-Dimensional
Bodies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213