Tensors in mechanics and elasticity
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York, NY [u.a.]
Acad. Press
1966
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| Ausgabe: | 2. print. |
| Schriftenreihe: | Engineering physics
2 |
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Datensatz im Suchindex
| _version_ | 1819761925080219648 |
|---|---|
| adam_text | Table of Conienfs
Preface to First French Edition v
Preface to English Translation vii
Translator’s Preface ix
General Works on Tensors xviili
CHAPTER I — General Remarks and First Examples 1
1 Introduction 1
2 The Role of Tensors and Their Usefulness I
3 Some Examples of Tensors 3
4 The Stress Tensor in Elasticity 7
5 How the Stress Tensor Transforms under a Change of Cartesian
Axes 10
6 Tensors of the Second Rank; Symmetry or Antisymmetry;
Degeneracy 12
7 What Is a Matrix? 15
8 Different Reference Spaces; Affine Vector Space, Metric Space 17
9 Thermodynamic Diagrams as Examples of Affine Geometry 20
CHAPTER II — Vector Geometry, Definition of Tensors 25
1 Axioms of Vector Geometry 25
2 Transformations of Rectilinear Axes 27
3 Covariance and Contra variance; Rectilinear Axes 30
4 Summations and Dummy Indices 33
5 Co variant Vector and Linear Form; Dual Space 34
6 General Definition of a Tensor 36
7 Methods of Forming Tensors; Operations of Elementary Algebra 38
8 The Contracted Product; A Criterion for Tensor Character 40
9 Examples: Force, Momentum, Velocity 42
10 Distinction between Tensors and Matrices 43
xi
TABLE OF CONTENTS
xii
11 Curvilinear Coordinates 44
12 Symmetry and Antisymmetry 46
13 Examples of Antisymmetric Tensors Formed by Means of Vector
Products Exterior Products 49
14 Remarks on the Use of Tensors of Definite Symmetry and the
Rule of Section 8 51
CHAPTER III — Pseudo-Tensors in Vector Geometry; Tensor Densities
and Capacities 53
1 Types of Pseudo-Tensors and Their Use 53
2 Reduction of Antisymmetric Tensors to Pseudo-Tensors of a
Different Rank; the Pseudo-Scalar 54
3 Another Example of a Pseudo-Scalar; Scalar Densities and
Capacities 56
4 Tensor Densities and Capacities 57
5 Examples of Pseudo-Tensors; the Volume Element Represents
the Typical Scalar Capacity 58
6 The Antisymmetric Tensor with Two Indices in Three-Dimen-
sional Space Reduces to a Pseudo-Tensor 60
7 Summary of General Results 63
8 These Pseudo-Tensors Yield Axial Vectors in a Three-Dimen-
sional Euclidean Space 65
9 General Method of Forming Pseudo-Tensors of the Two Types,
Capacities or Densities 66
10 Examples of These Transformations 69
CHAPTER IV — The Principal Differential Operators in Vector Geo-
metry 71
1 Introduction 71
2 The Ordinary Definitions of the Operators: Gradient, Curl
Divergence, Laplacian A 72
3 Nature of the Precautions to be Taken 75
4 Gradient and Curl 76
5 Geometric Meaning of the Curl in Three Dimensions 79
6 The Curl in an Arbitrary Number of Dimensions 82
7 The Divergence Operator 84
8 Algebraic Confirmation of the Role of the Divergence 86
TABLE OF CONTENTS
xiii
9 The Divergence Applied to Tensor Densities of Rank Higher
Than One 88
10 Examples of Divergences; Alternate Notation 89
CHAPTER V — Postulate of Parallel Displacement; Covariant Derivative
in Affine Geometry 91
1 Method of H Weyl 91
2 Postulate of Parallel Displacement: Affine Relationship, Geodesic
Coordinates 91
3 Covariant Derivative of a Contravariant or Covariant Vector 95
4 Covariant Derivative of an Arbitrary Tensor 98
5 Co variant Derivatives of Pseudo-Tensors 100
6 Remarks on the Divergences of Tensor Densities of Arbitrary Rank 102
7 Absolute Derivative of a Vector, Geodesic Lines 104
8 Displacement of a Quantity to a Finite Distance and Inte-
grability Conditions 105
9 Finite Displacement of a Tensor, the Integrable Case 107
10 Displacement over a Closed Path of an Arbitrary Tensor or
Pseudo-Tensor Ill
11 Geometric Meaning of These Equations: Curvature of Space 112
12 A Space of Null Curvature is Linear 115
CHAPTER VI — Metric Geometry, Riemann Space 117
1 Elementary Definitions 117
2 General Definition; Fundamental Metric Tensor 118
3 Simple Examples Curvilinear Coordinates in a Three-Dimen-
sional Euclidean Space 121
4 Interpretation of the gik Their Geometric Meaning 125
5 Displacement of Indices; Covariant or Contravariant Components
of the Same Vector or Tensor; Absolute Value; Scalar Product 128
6 Geometric Meaning of These Operations; Components of a Vector
along the Coordinate Lines and Perpendicular Projection 131
7 The Tangent Euclidean Space; Reduction of the Table gik to
Diagonal Form 133
8 How Does the Determinant g Transform under Changes of Axes ? 136
9 The Expressions g 112 and |g|-1/2 as Types of Scalar Densities
and Capacities 137
XIV
TABLE OF CONTENTS
CHAPTER VII — Differential Operators and the Covariant Derivative in
Metric Geometry 141
1 Extension of the Formulas of Chapter IV 141
2 The Laplacian A 142
3 Some Applications 144
4 Comparison of Ordinary Vector Notations and Tensor Notation;
Essential Differences in the Definitions 145
5 The Problem of Displacement of Measuring Units; Gage
Invariance 148
6 Covariant Derivative in Metric Geometry in Riemann Space
Christoff el Symbols 150
7 The Covariant Derivatives of the gik and of the Determinant g
Vanish; All Scalar Densities or Capacities Have Vanishing
Co variant Derivatives 153
8 Geometric Results and Meaning of the Rules of Parallel Displace-
ment; Geodesic Coordinates 156
9 Properties of Geodesics; Minimum Length 157
10 Examples 161
11 Curvature of a Riemann Space The Riemann-Christoffel Tensor 166
12 The Contracted Tensor of Ricci and Einstein and the Displace-
ment of Pseudo-Tensors around a Closed Path 169
13 Bianchi’s Identities 171
14 Normal Coordinates of Riemann; A Space of Null Curvature Is
Euclidean 173
15 Riemann Curvature; Mean Curvature of Ricci 175
CHAPTER VIII — The Application of Riemann Geometries to Analytical
Mechanics 177
1 Usefulness of Riemann Geometry in Classical Mechanics 177
2 d’Alembert’s Principle 178
3 Lagrange’s Equations 181
4 The Case of Time-Independent Constraints 183
5 Geometric Interpretation 184
6 Time-Independent Constraints with Potential Energy; How to
Reduce a Mechanical Problem to a Search for a Geodesic 188
7 Geodesics in “Space-Time”; Allusion to Relativistic Mechanics 191
8 Time-Dependent Holonomic Constraints or Moving Reference
Systems 194
TABLE OF CONTENTS xv
9 Systems with Variable Constraints; Geometric Interpretation
in Space-Time 197
10 Discussion and Examples 200
11 Lagrange’s Principle of Least Action 206
12 Conservative Systems with Time-Independent Holonomic
Constraints 211
13 Analytical Mechanics Compared with Geometrical Optics
Maupertuis’ Principle and Fermat’s Principle 215
14 Hamilton’s Equations, General Form 217
15 Hamilton’s Equations; Conservative Systems 222
16 Discussion and Examples 227
17 Some Consequences and Extensions of the Principle of Least
Action 231
18 A Thermodynamic Analogy; Distinction between Heat and
Mechanical Work 235
19 A Very Slow Transformation Leads to Boltzmann’s Formula 238
20 Adiabatic Transformation; Adiabatic Invariant of Ehrenfest 240
21 Radiation Pressure 243
CHAPTER IX — The Transition to Wave Mechanics 247
1 The Introduction of Quanta into Physics 247
2 Energy, Frequency and Mass 248
3 Frequency Assigned to Hamilton’s Waves; Wavelength and
Momentum 250
4 Physical Optics and Geometrical Optics; Fermat’s Principle 253
5 Formation of a Wave Equation for Mechanics 258
6 General Method of Formation of the Wave Equation in
Schrödinger’s Mechanics 261
7 Commutation Rules 264
8 Propagation of a Group of Waves; Phase Velocity and Group
Velocity 265
9 Groups of Waves in Space 270
10 Wave Groups in Wave Mechanics 274
CHAPTER X — Elasticity 277
1 Tensors in Elasticity 277
2 Review of Some Definitions 278
xvi TABLE OF CONTENTS
3 Elastic Stresses 281
4 Resultant Force on a Volume Element 284
5 Study of Deformations in Cartesian Coordinates 287
6 Strains General Definition 292
7 Isotropic Body Invariants of the Strain 296
8 Definition of a Potential Energy Density for a Strained Solid 299
9 The Elastic Coefficients; Voigt’s Coefficients; Lame’s Notation
The Cauchy Relations 304
10 Relation between the Stresses and Strains 310
11 The Solid Medium in Motion; Euler’s Coordinates 315
,12 Example of the Study of a Body under an Initial Strain Role
of an Initial Internal or External Pressure on the Elastic
Properties 317
13 Equations of Motion in Lagrange’s Coordinates; Boussinesq’s
Formulas 323
14 A General Minimum Principle 327
CHAPTER XI — Elastic Waves in Solids
1 Propagation of Elastic Waves in a Crystal 329
2 Proper Vibrations of a Rectangular Parallelepiped; Cyclic
Conditions 335
3 Reflection of an Elastic Wave at a Plane Surface 342
4 Enumeration of the Proper Vibrations of a Finite Solid 347
5 Direct Study of Interference and Standing Waves from Reflec-
tion at Plane Mirrors 352
6 Proper Vibrations of a Solid Enclosed in a Rectangular Parallel-
epiped with Smooth Rigid Walls 358
7 Influence of Higher Order Terms and Perturbations in Wave
Propagation 364
8 Examples of These Secondary Effects in Traveling Waves 367
9 The Case of Standing Waves 370
10 Radiation Pressure of Standing Waves 372
11 The Calculation of Radiation Pressure by the Boltzmann-
Ehrenfest Formula 378
12 The Radiation Stress Tensor of a Traveling Wave 380
13 Examples of Applications of These Tensors 384
TABLE OF CONTENTS xvii
14 Mean Values for Completely Diffuse Elastic Waves 388
15 Elastic Waves in a Fluid; Liquids as a Special Case of Solids 393
16 Conditions for Measuring the Radiation Pressure at a Submerged
Disk 398
17 What Can Be Said about the Elastic Coefficients A, B, C in
the Second Approximation ? 401
CHAPTER XII — Quantum Theory of the Solid State
1 Introduction; Analysis of Thermal Motion of Solids 405
2 Interpretation of the Thermal Expansion of Solids 407
3 Vibrations of a String with a Discontinuous Structure; Cutoff
Frequency 410
4 Various Possible Extensions Case of Vibrations of a Line of Atoms 414
5 Enumeration of Proper Vibrations Transition to the Two-
Dimensional Lattice 419
6 Three-Dimensional Crystal Lattice; Its Proper Vibrations 422
7 Simplifying Hypotheses for an Isotropic Solid; Debye’s Method 427
8 The Quantum Theory of Thermal Motion in a Solid 431
9 Discussion; Comparison with Experiment 438
10 Radiation Pressure and Thermal Expansion of Solids 443
11 Comparisons with Experiment, Thermal Expansion 445
12 Thermodynamics of the Ideal Solid 448
13 Discussion and Review of Essential Points 452
14 Attempt to Extend the Theory to Liquids 456
Index
|
| any_adam_object | 1 |
| author | Brillouin, Léon |
| author_facet | Brillouin, Léon |
| author_role | aut |
| author_sort | Brillouin, Léon |
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| building | Verbundindex |
| bvnumber | BV026441862 |
| classification_rvk | UF 1000 |
| ctrlnum | (OCoLC)918500078 (DE-599)BVBBV026441862 |
| discipline | Physik |
| edition | 2. print. |
| format | Book |
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| id | DE-604.BV026441862 |
| illustrated | Illustrated |
| indexdate | 2024-12-24T00:47:09Z |
| institution | BVB |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-022012802 |
| oclc_num | 918500078 |
| open_access_boolean | |
| owner | DE-188 |
| owner_facet | DE-188 |
| physical | XVIII, 478 S. graph. Darst. |
| publishDate | 1966 |
| publishDateSearch | 1966 |
| publishDateSort | 1966 |
| publisher | Acad. Press |
| record_format | marc |
| series | Engineering physics |
| series2 | Engineering physics |
| spellingShingle | Brillouin, Léon Tensors in mechanics and elasticity Engineering physics Elastizitätstheorie (DE-588)4123124-7 gnd Kontinuumsmechanik (DE-588)4032296-8 gnd Tensor (DE-588)4184723-4 gnd Elastizität (DE-588)4014159-7 gnd Mechanik (DE-588)4038168-7 gnd Tensorrechnung (DE-588)4192487-3 gnd |
| subject_GND | (DE-588)4123124-7 (DE-588)4032296-8 (DE-588)4184723-4 (DE-588)4014159-7 (DE-588)4038168-7 (DE-588)4192487-3 |
| title | Tensors in mechanics and elasticity |
| title_alt | Les tenseurs en mécanique et en elsticité |
| title_auth | Tensors in mechanics and elasticity |
| title_exact_search | Tensors in mechanics and elasticity |
| title_full | Tensors in mechanics and elasticity Leon Brillouin |
| title_fullStr | Tensors in mechanics and elasticity Leon Brillouin |
| title_full_unstemmed | Tensors in mechanics and elasticity Leon Brillouin |
| title_short | Tensors in mechanics and elasticity |
| title_sort | tensors in mechanics and elasticity |
| topic | Elastizitätstheorie (DE-588)4123124-7 gnd Kontinuumsmechanik (DE-588)4032296-8 gnd Tensor (DE-588)4184723-4 gnd Elastizität (DE-588)4014159-7 gnd Mechanik (DE-588)4038168-7 gnd Tensorrechnung (DE-588)4192487-3 gnd |
| topic_facet | Elastizitätstheorie Kontinuumsmechanik Tensor Elastizität Mechanik Tensorrechnung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022012802&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV026472753 |
| work_keys_str_mv | AT brillouinleon lestenseursenmecaniqueetenelsticite AT brillouinleon tensorsinmechanicsandelasticity |