Continuum Mechanics Through the Eighteenth and Nineteenth Centuries: Historical Perspectives from John Bernoulli (1727) to Ernst Hellinger (1914) (Solid Mechanics and Its Applications Book 214)

Conceived as a series of more or less autonomous essays, the present book critically 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, id est, roughly during first the "Age of reason" and next the "Birth of the modern world". The emphasis is rightly placed on the original 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 breakthroughs as well as the secondary ones. It highlights the role of scientists who left essential prints in this history of scientific ideas. The book shows how the formidable developments that blossomed in the twentieth century (and perused in a previous book of the author in the same Springer Series: "Continuum Mechanics through the Twentieth Century", Springer 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 by the author of most critical texts in French or German are given to the benefit of the readers Preface 6 Contents 7 1 General Introduction: About the Contents and Form of this Book 12 Abstract 12 References 15 Web site 17 2 A Glimpse at the Eighteenth Century: From John Bernoulli to Lagrange 18 Abstract 18 2.1...John Bernoulli and the Principle of Virtual Work 18 2.2...‘‘Bernoulli’s Theorem’’ by Daniel Bernoulli 20 2.3...D’Alembert and the Metaphysical Notion of Force 21 2.4...The Notion of Internal Pressure and the Fundamental Equations of Hydrodynamics 24 2.5...Linear Momentum and Moment of Momentum: Newtonian Versus Variational Formulations 27 2.6...Calculus of Variations and Analytical Mechanics: Lagrange 28 2.7...The Age of Reason: Conclusion and Things to Come 31 A.1. Appendix A 33 A.2. Appendix B 36 A.3. Appendix C 37 References 42 3 What Happened on September 30, 1822, and What Were its Implications for the Future of Continuum Mechanics? 44 Abstract 44 3.1...Introduction 44 3.2...Preliminary Remarks 45 3.3...The Main Contents of Cauchy’s 1822/1828 Memoir 46 3.4...Cauchy’s Stress and Hyperstresses 50 3.5...Stress as a Secondary Notion 51 3.6...Stress as Part of Stress-Energy-Momentum 53 3.7...The Nec Plus Ultra: The Eshelby Stress and the Pseudo Tensor of Energy-Momentum 56 3.8...Conclusion 59 Acknowledgments 59 A.0. Appendix A 60 A.0. Appendix B 61 References 63 4 Piola and Kirchhoff: On Changes of Configurations 66 Abstract 66 4.1...Introduction 66 4.2...Piola’s Contribution 67 4.2.1 Some Words of Caution 67 4.2.2 The Strategy of Piola 68 4.2.3 Introduction of the ‘‘Piola Format’’ by Piola 68 4.2.4 Stresses as Lagrange Multipliers 72 4.3...The Role of Kirchhoff 76 4.4...Conclusion 77 References 77 5 Duhamel’s Pioneering Work in Thermo-elasticity and Its Legacy 80 Abstract 80 5.1...The Roots of Duhamel’s Contribution 80 5.2...Basic Ideas of Duhamel’s 1837 Paper 81 5.3...Duhamel’s Strategy and Equations of ‘‘Thermo-elasticity’’ 81 5.4...Comments and Further Developments 83 5.5...Conclusion 85 References 85 6 From Cauchy to Boussinesq via Barré de Saint-Venant 87 Abstract 87 6.1...Introduction 87 6.2...Cauchy and Lamé 88 6.3...Barré de Saint-Venant 89 6.3.1 On his Name 89 6.3.2 Some Biographical Elements 90 6.3.3 Saint-Venant’s Scientific Achievements 91 6.3.3.1 Elasticity 91 6.3.3.2 Anelastic Behaviour 93 6.3.3.3 Dynamics and Impact 94 6.3.3.4 Fluid Mechanics 95 6.3.3.5 Vector Calculus 96 6.3.3.6 Principles of Mechanics 97 6.3.3.7 Publication of Books 98 6.4...Boussinesq: A Disciple of Saint-Venant 100 6.4.1 Contributions to Fluid Mechanics 100 6.4.2 Contributions to the Mechanics of Deformable Solids 101 6.5...Conclusion 103 References 104 7 Helmholtz Interpreted and Applied by Duhem 109 Abstract 109 7.1...Hermann von Helmholtz and Pierre Duhem 109 7.2...Free Energy (‘‘Freie Energie’’) 111 7.3...Helmholtz-Duhem Electrodynamics 114 7.4...Stability 117 7.5...Conclusions 118 References 120 8 About the Cosserats’ Book of 1909 123 Abstract 123 8.1...Preliminaries 123 8.2...The Main Contents of the Cosserats’ Book 126 8.3...Reception and Influence of the Cosserats’ Book 130 8.4...Concluding Remarks 133 A.0. Appendix A 134 A.0.0 Theory of Deformable Bodies 137 References 143 9 Caratheodory: Thermodynamics and Topology 147 Abstract 147 9.1...Introduction 147 9.2...On Caratheodory 149 9.3...The Standard Formulation 149 9.4...Caratheodory’s Work 151 9.5...Reception of Caratheodory’s Axiomatics 154 9.6...Toward Irreversible Thermodynamics 155 References 156 10 On Duhem’s Energetics or General Thermodynamics 158 Abstract 158 10.1...Introduction 158 10.2...On Duhem’s Style and General View of Science 160 10.3...Some of Duhem’s Creative Works in Thermo-mechanics 162 10.3.1 Physical Chemistry 162 10.3.2 Fluid Mechanics, Viscous Fluids 163 10.3.3 Deformable Solids 164 10.3.4 General ‘‘Thermo-mechanics’’ and Thermodynamic Potentials 165 10.3.5 Contributions to Mathematics 166 10.4...Short Analysis of Some Exemplary Writings 167 10.4.1 On the General Treatise on Energetics 167 10.4.2 On the Evolution of Mechanics 168 10.4.3 On Permanent Alterations and Hysteresis 171 10.5...Influence on Contemporaries and Later on 177 10.6...Conclusion 181 A.1. Appendix A 181 A.2. Appendix B 185 References 192 11 A Course of Continuum Mechanics at the Dawn of the Twentieth Century (Volume III of Appell’s Treatise on Rational Mechanics) 195 Abstract 195 11.1...Prolegomena: On Paul Appell 195 11.2...Setting the Stage 196 11.3...The Contents of Appell’s Volume on the Mechanics of Continua 198 11.3.1 Some Words of Introduction 198 11.3.2 Vector Analysis and Potential Theory 199 11.3.3 Equilibrium and Internal Motion of a Continuous Mass 199 11.3.4 Hydrostatics and Stability 200 11.3.5 Deformation of Continua 200 11.3.6 Kinematics of Continua 201 11.3.7 General Theorems of the Dynamics of Perfect Fluids 202 11.3.8 Theory of Vortices 202 11.3.9 Parallel Flows 203 11.3.10 Barocline Fluids 204 11.3.11 Elements of Elasticity Theory 205 11.3.12 On Viscous Fluids 206 11.4...The Cosserats’ Theory of Euclidean Action 207 11.5...Concluding Comments 207 A.1. Appendix: Biography of Paul Appell 209 References 210 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 213 Abstract 213 12.1...Introduction 213 12.2...The Scientific Environment 215 12.3...The Contents of Hellinger’s Article 216 12.3.1 Introductory Remark 216 12.3.2 The Layout and Articulation of the Contribution 216 12.4...The Identified Fields of Marked Interest of Hellinger 217 12.4.1 On General Principles of Mechanics and General Equations 217 12.4.2 On Variational Formulations 218 12.4.3 On Finite Strains and Elasticity 220 12.4.4 On Oriented Media 221 12.4.5 On One-Dimensional and Two-Dimensional Bodies 221 12.4.6 On Thermodynamics and Dissipative Behaviours 222 12.4.7 On Newly Studied Behaviours 223 12.4.8 On Relativistic Continuum Mechanics 224 12.5...Conclusion 225 A.1. Appendix A 226 A.2. Bibliography 228 A.3. 12.8...The Basic Laws of Statics 235 A.4. 12.9...The Basic Laws of Kinetics [Dynamics] 247 References 253 13 Epilogue 256 Abstract 256 13.1...On the Method 256 13.2...The Main General Themes and Breakthroughs 258 13.3...The Breakthroughs of Second Rank 260 13.4...The Timid Steps in Coupled Fields 260 References 261 Retrospective A Gallery of Portraits of the Main Actors 263 Front Matter....Pages i-xi General Introduction: About the Contents and Form of this Book....Pages 1-6 A Glimpse at the Eighteenth Century: From John Bernoulli to Lagrange....Pages 7-32 What Happened on September 30, 1822, and What Were its Implications for the Future of Continuum Mechanics?....Pages 33-54 Piola and Kirchhoff: On Changes of Configurations....Pages 55-68 Duhamel’s Pioneering Work in Thermo-elasticity and Its Legacy....Pages 69-75 From Cauchy to Boussinesq via Barré de Saint-Venant....Pages 77-98 Helmholtz Interpreted and Applied by Duhem....Pages 99-112 About the Cosserats’ Book of 1909....Pages 113-136 Caratheodory: Thermodynamics and Topology....Pages 137-147 On Duhem’s Energetics or General Thermodynamics....Pages 149-185 A Course of Continuum Mechanics at the Dawn of the Twentieth Century (Volume III of Appell’s Treatise on Rational Mechanics)....Pages 187-204 A Successful Attempt at a Synthetic View of Continuum Mechanics on the Eve of WWI: Hellinger’s Article in the German Encyclopaedia of Mathematics....Pages 205-247 Epilogue....Pages 249-255 Back Matter....Pages 257-269