Theory of Elasticity (Foundations of Engineering Mechanics)

3540245561......Page 1
Theory of Elasticity......Page 3
Copyright Page......Page 4
Anatolii I. Lurie......Page 7
Foreword......Page 10
Translator's preface......Page 12
Table of Contents......Page 14
Part I
Basic concepts of
continuum mechanics......Page 29
1.1.1 Systems of coordinates in continuum mechanics......Page 30
1.1.2 External forces......Page 33
1.1.3 Internal forces in the continuum......Page 34
1.1.4 Equilibrium of an elementary tetrahedron......Page 36
1.1.5 The necessary conditions for equilibrium of a continuum......Page 39
1.2.1 Component transformation, principal stresses and principal invariants......Page 45
1.2.2 Mohr's circles of stress......Page 48
1.2.4 Examples of the states of stress......Page 51
1.3.1 Representation of the stress tensor......Page 55
1.3.3 The necessary condition for equilibrium......Page 56
1.3.5 Elementary work of external forces......Page 59
1.3.6 The energetic stress tensor......Page 62
1.4.1 Moments of a function......Page 64
1.4.3 The cases of n = 0 and n = 1......Page 65
1.4.4 The first order moments for stresses......Page 66
1.4.5 An example. A vessel under external and internal pressure......Page 67
1.4.6 An example. Principal vector and principal moment of stresses in a plane cross-section of the body......Page 68
1.4.7 An estimate of a mean value for a quadratic form of components of the stress tensor......Page 69
1.4.9 An estimate of the specific intensity of shear stresses......Page 71
1.4.10 Moments of stresses of second and higher order......Page 72
1.4.11 A lower bound for the maximum of the stress components......Page 73
1.4.12 A refined lower bound......Page 74
2.1.1 Outline of the chapter......Page 78
2.1.2 Definition of the linear strain tensor......Page 79
2.2.1 Compatibility of strains (Saint- Venant's dependences)......Page 82
2.2.2 Displacement vector. The Cesaro formula......Page 84
2.2.4 The Volterra distortion......Page 88
2.3.1 Vector basis of volumes v and V......Page 90
2.3.2 Tensorial gradients ∆R and ∆r......Page 93
2.3.3 The first measure of strain (Cauchy-Green)......Page 94
2.3.4 Geometric interpretation of the components of the first strain measure......Page 96
2.3.5 Change in the oriented surface......Page 97
2.3.6 The first tensor of finite strain......Page 98
2.3.7 The principal strains and principal axes of strain......Page 100
2.3.8 Finite rotation of the medium as a rigid body......Page 101
2.4.1 The second measure of finite strain......Page 102
2.4.3 The second tensor of finite strain (Almansi-Hamel)......Page 104
2.5.1 Strain measures and the inverse tensors......Page 105
2.5.2 Relationships between the invariants......Page 106
2.5.3 Representation of the strain measures in terms of the principal axes......Page 107
2.5.4 The invariants of the tensors of finite strain......Page 109
2.5.5 Dilatation......Page 110
2.5.6 Similarity transformation......Page 111
2.5.7 Determination of the displacement vector in terms of the strain measures......Page 112
2.6.1 Affine transformation......Page 114
2.6.2 A plane field of displacement......Page 115
2.6.3 Simple shear......Page 117
2.6.4 Torsion of a circular cylinder......Page 119
2.6.5 Cylind rical bending of a rectangular plate......Page 120
2.6.6 Radial-symmetric deformation of a hollow sphere......Page 122
2.6.7 Axisymmetric deformation of a hollow cylinder......Page 124
Part II
Governing equations of
the linear theory of
elasticity......Page 125
3.1.1 Statement of the problem of the linear theory of elasticity......Page 126
3.1.2 Elementary work......Page 128
3.1.3 Isotropic homogeneous medium of Hencky......Page 129
3.2.1 Internal energy of a linearly deformed body......Page 132
3.2.2 Isothermal process of deformation......Page 134
3.2.4 Specific strain energy. Hencky 's media......Page 136
3.3.1 Elasticity moduli......Page 138
3.3.2 Specific strain energy for a linear- elastic body......Page 140
3.3.3 Clapeyron's formula. Limits for the elasticity moduli.......Page 142
3.3.4 Taking account of thermal terms. Free energy......Page 144
3.3.5 The Gibbs thermodynamic potential......Page 146
3.3.6 Equation of thermal conductivity......Page 148
4.1.1 Fundamental relationships......Page 150
4.1.2 Boundary conditions......Page 151
4·1.3 Differential equations governing the linear theory of elasticity in terms of displacements......Page 152
4.1.4 Solution in the Papkovich-Neuber form......Page 154
4.1.5 The solution in terms of stresses. Beltrami 's dependences......Page 158
4.1.6 Krutkov 's transformation......Page 160
4.1.7 The Boussuiesq-Galerkin solution......Page 162
4.1.8 Curvilinear coordinates......Page 163
4.1.9 Orthogonal coordinates......Page 165
4.1.10 Axisymmetric problems. Love's solution......Page 167
4.1.11 Torsion of a body of revolution......Page 169
4.1.12 Deformation of a body of revolution......Page 170
4.1.13 The Papkovich-Neuber solution for a body of revolution......Page 173
4.1.14 Account of thermal components......Page 174
4.2.1 Stationarity of the potential energy of the system......Page 177
4.22 The principle of minimum potential energy of the system......Page 179
4.2.3 Ritz's method......Page 182
4.2.4 Galerkin's method (1915)......Page 184
4.2.5 Principle of minimum complementary work......Page 185
4.2.6 Mixed stationarity principle (E. Reissner, 1961)......Page 189
4.2.7 Variational principles accounting for the thermal terms......Page 191
4.2.8 Saint- Venant 's principle. Energetic consideration......Page 192
4.3.1 Formulation and proof of the reciprocity theorem (Betti, 1872)......Page 196
4.3.2 The influence tensor. Maxwell's theorem......Page 197
4.3.3 Application of the reciprocity theorem......Page 199
4. 3.4 The reciprocity theorem taking account of thermal terms......Page 202
4.3.5 The influence tensor of an unbounded medium......Page 203
4.3.6 The potentials of the elasticity theory......Page 207
4.3.7 Determining the displacement field for given external forces and displacement vector of the surface......Page 209
4.4.1 Kirchhoff 's theorem......Page 213
4.4.2 Integral equations of the first boundary value problem......Page 216
4.4.3 Integral equations of the second boundary value problem......Page 218
4-4.4 Comparison of the in tegral equations of the first and second boundary value problems......Page 221
4.4.5 Theorem on the existence of solutions to the second external and first internal problems......Page 222
4.4.6 The second internal boundary value problem (II(i))......Page 223
4.4.7 Elastostatic Robin's problem......Page 225
4.4.8 The first external boundary value problem (I(e))......Page 227
4.5.1 Overview of the content......Page 228
4.5.2 Determination of the state of stress in terms of the barrier constants......Page 229
4.5.3 The reciprocity theorem......Page 231
4.5.5 The case of a body of revolution......Page 233
4. 5.6 Boundary value problem for a double-conn ected body of revolution......Page 236
Part III
Special problems of the
linear theory of elasticity......Page 239
5.1.1 Singularities due to concentrated forces......Page 240
5.1.2 The system of forces distributed in a small volume. Lauricella's formula......Page 242
5.1.3 Interpretation of the second potential of elasticity theory......Page 248
5.1.4 Boussinesq's potentials......Page 249
5.1.5 Thermoelastic displacements......Page 251
5.1.6 The state of stress due to an inclusion......Page 253
5.2.1 The problems of Boussinesq and Cerruti......Page 257
5.2.2 The particular Boussinesq problem......Page 258
5.2.3 The distributed normal load......Page 260
5.2.4 Use of the Papkooich-Neuber functions to solve the Boussinesq-Cerruti problem......Page 261
5.2.5 The influence tensor in elastic half-space......Page 264
5.2.6 Thermal stresses in the elastic half-space......Page 267
5.2.7 The case of the steady-state temperature......Page 269
5.2.8 Calculation of the simple layer potential for the plane region......Page 270
5.2.9 Dirichlet 's problem for the half-space......Page 272
5.2.10 The first boundary value problem for the half-space......Page 274
5.2.11 Mixed problems for the half-space......Page 275
5.2.12 On Saint-Venant's principle. Mises's formulation......Page 277
5.2.13 Superstatic system of forces......Page 279
5.2.14 Sternberg's theorem (1954)......Page 280
5.3.1 Statement of the problem......Page 282
5.3.2 The first boundary value problem......Page 283
5.3.3 The elastostatic Robin's problem for the sphere......Page 285
5.3.4 Thermal stresses in the sphere......Page 286
5.3.5 The second boundary value problem for the sphere......Page 289
5.3.6 Calculation of the displacement vector......Page 292
5.3.7 The state of stress at the centre of the sphere......Page 294
5.3.8 Thermal stresses......Page 295
5.3.9 The state of stress in the vicinity of a spherical cavity......Page 297
5.3.10 The state of stress in the vicinity of a small spherical cavity in a twisted cylindrical rod......Page 298
5.3.11 Action of the mass forces......Page 299
5.3.12 An attracting sphere......Page 301
5.3.13 A rotating sphere......Page 302
5.3.14 Action of concentrated forces......Page 304
5.3.15 The distributed load case......Page 307
5.4.1 Integral equation of equilibrium......Page 308
5.4.2 Tension of the hyperboloid of revolution of one nappe......Page 312
5.4.3 Torsion of the hyperboloid......Page 315
5.4 .4 Bending of the hyperboloid......Page 316
5.4.5 Rotating ellipsoid of revolution......Page 317
5.5.1 Elastostatic Robin 's problem for the three-axial ellipsoid......Page 320
5.5.2 Translatory displacement......Page 321
5.5.3 Distribution of stresses over the surface of the ellipsoid......Page 322
5.5.4 Rotational displacement......Page 325
5.5.5 Distribution of stresses over the surface of the ellipsoid......Page 327
5.5.6 An ellipsoidal cavity in the unbounded elastic medium......Page 329
5.5.7 The boundary conditions......Page 332
5.5.8 Expressing the constants in terms of three parameters......Page 334
5.5.9 A spheroidal cavity in the elastic medium......Page 336
5.5.10 A circular slot in elastic medium......Page 338
5.5.11 An elliptic slot in an elastic medium......Page 340
5.6.1 The problem of the rigid die. Boundary condition......Page 344
5.6.2 A method of solving the problem for a rigid die......Page 348
5.6.3 A plane die with an elliptic base......Page 353
5.6.4 Displacements and stresses......Page 356
5.6.5 A non-plane die......Page 358
5.6.6 Displacements and stresses......Page 361
5.6.7 Contact of two surfaces......Page 363
5.7.1 Differential equation of equilibrium of a circular cylinder......Page 370
5.7.2 Lame's problem for a hollow cylinder......Page 375
5.7.3 Distortion in the hollow cylinder......Page 376
5.7.4 Polynomial solutions to the problem of equilibrium of the cylinder......Page 379
5.7.5 Torsion of a cylinder subjected to for ces distributed over the end faces......Page 382
5.7.6 Solutions in terms of Bessel functions......Page 386
5.7.7 Filon's problem......Page 390
5.7.8 Homogeneous solutions......Page 392
5.7.9 Boundary conditions on the end faces......Page 395
5.7.10 Generalised orthogonality......Page 399
6.1.1 Statement of Saint- Venant's problem......Page 405
6.1.2 Integral equations of equilibrium......Page 406
6.1.3 Main assumptions......Page 407
6.1.4 Normal stress σz in Saint- Venant's problem......Page 408
6.1.5 Shear stresses Txz and Tyz......Page 409
6.2.1 Introducing the stress function......Page 411
6.2.2 Displacements in Saint- Venant's problem......Page 414
6.2.3 Elastic line......Page 417
6.2.4 Classification of Saint- Venant's problems......Page 419
6.2.5 Determination of parameter α......Page 421
6.2.6 Centre of rigidity......Page 424
6.2.7 Elementary solutions......Page 426
6.3.1 Statement of the problem......Page 429
6.3.2 Displacements......Page 431
6.3.3 Theorem on the circulation of shear stresses......Page 433
6.3.4 Torsional rigidity......Page 435
6.3.5 The membrane analogy of Prandtl (1904)......Page 437
6.3.6 Torsion of a rod with elliptic cross-section......Page 439
6.3.7 Inequalities for the torsional rigidity......Page 441
6.3.8 Torsion of a rod having a rectangular cross-section......Page 443
6.3.9 Closed-form solutions......Page 445
6.3.10 Double connected region......Page 447
6.3.11 Elliptic ring......Page 449
6.3.12 Eccentric ring......Page 451
6.3.13 Variational determination of the stress function......Page 454
6.3.14 Approximate solution to the problem of torsion......Page 458
6.3.15 Oblong profiles......Page 463
6.3.16 Torsion of a thin-walled tube......Page 467
6.3.17 Multiple-connected regions......Page 470
6.4.1 Stresses......Page 474
6.4.2 Bending of a rod with elliptic cross-section......Page 476
6.4.4 Rectangular cross-section......Page 478
6.4·5 Variational statement of the problem of bending......Page 482
6.4.6 The centre of rigidity......Page 484
6.4.7 Approximate solutions......Page 486
6.4.8 Aeroioil profile......Page 488
6.5.1 Statement of the problem......Page 490
6.5.2 Distribution of normal stresses......Page 493
6.5.3 Tension of the rod......Page 494
6.5.5 Stresses σx , σy , T xy......Page 499
6.5.6 Determining σ0z......Page 502
6.5.7 Bending of a heavy rod......Page 503
6.5.8 Mean values of stresses......Page 505
6.5.9 On Almansi 's problem......Page 507
7.1.1 Plane strain......Page 508
7.1.2 Airy' stress junction......Page 511
7.1.3 Differential equation for the stress junction......Page 512
7.1.4 Plane stress......Page 513
7.1.5 The generalised plane stress......Page 516
7.1.6 The plane problem......Page 517
7.1.7 Displacements in the plane problem......Page 518
7.1.8 The principal vector and the principal moment......Page 520
7.1.9 Orthogonal curvilinear coordinates......Page 521
7.1.11 Representing the biharmonic function......Page 522
7.1.12 Introducing a complex variable......Page 524
7.1.13 Transforming the formulae of the plane problem......Page 525
7.1.14 Goursat's formula......Page 527
7.1.15 Translation of the coordinate origin......Page 529
7.2.1 Statement of the plan e problem for beam and bar......Page 530
7.2.2 Plane Saint- Venant's problem......Page 532
7.2.3 Operator representation of solutions......Page 534
7.2.4 Stress junction for the strip problem......Page 536
7.2.5 The elementary theory of beams......Page 539
7.2.6 Polynomial load (Mesnager, 1901)......Page 540
7.2.7 Sinusoidal load, solutions of Ribiere (1898) and Filon (1903)......Page 542
7.2.8 Concentrated force (Karman and Seewald, 1927)......Page 546
7.2.9 Bar with a circular axis loaded on the end faces (Golovin, 1881)......Page 551
7.2.10 Loading the circular bar on the surface......Page 555
7.2.11 Cosinusoidal load......Page 558
7.2.12 Homogeneous solutions......Page 560
7.3.1 Concentrated force and concentrated moment in elastic plane......Page 562
7.3.2 Flamant's problem (1892)......Page 565
7.3.3 General case of normal loading......Page 568
7.3.4 Loading by a force directed along the boundary......Page 570
7.3.5 The plane contact problem......Page 572
7.3.6 Constructing potential w......Page 574
7.3.7 A plane die......Page 577
7.3.9 Concentrated force in the elastic half-plane......Page 578
7.4.1 Concentrated force in the vertex of the wedge......Page 581
7.4.2 Mellin's integral transform in the problem of a wedge......Page 583
7.4.3 Concentrated moment at the vertex of the wedge......Page 587
7.4.4 Loading the side faces......Page 590
7.5.2 Boundary-value problems for the simply-connected finite region......Page 594
1.5.3 Definiteness of Muskhelishvili's functions......Page 597
7.5.4 Infinite region with an opening......Page 598
7.5.5 Double-connected region. Distortion......Page 602
7.5.6 Representing the stress junction in the double-connected region (Michell)......Page 603
7.5.7 Thermal stresses. Plane strain......Page 605
7.5.8 Plane stress......Page 607
7.5.9 Stationary temperature distribution......Page 610
7.5.10 Cauchy's theorem and Cauchy's integral......Page 613
7.5.11 Integrals of Cauchy's type. The Sokhotsky-Plemelj formula......Page 615
7.6.1 Round disc loaded by concentrated forces......Page 617
7.6.2 The general case of loading round disc......Page 620
7.6.3 The method of Cauchy's integrals......Page 622
7.6.4 Normal stress σ0 on the circle......Page 624
7.6.5 Stresses at the centre of the disc......Page 626
7.6.6 A statically unbalanced rotating disc......Page 627
7.6.7 The first boundary-value problem for circle......Page 630
7.6.8 The state of stress......Page 634
7.6.9 Thermal stresses in the disc placed in a rigid casing......Page 636
7.6.10 Round opening in an infinite plane......Page 638
7.6.12 Tension of the plane weakened by a round opening......Page 641
7.6.13 Continuation of Φ (z)......Page 643
7.6.14 Solving the boundary-value problems of Subsections 7.6.2 and 7.6.10 by way of the continuation......Page 645
7.7.1 The stresses due to distortion......Page 648
7.7.2 The second boundary-value problem for a ring......Page 649
7.7.3 Determining functions Φ (S), ψ (S)......Page 650
7.7.5 Thermal stresses in the ring......Page 652
7.7.6 Tension of the ring by concentrated forces......Page 654
7.7.7 The way of continuation......Page 655
7.8.1 Infinite plane with an opening......Page 660
7.8.2 The method of Cauchy's integrals......Page 662
7.8.3 Elliptic opening......Page 665
7.8.4 Hypotrochoidal opening......Page 667
7.8.5 Simply connected finite region......Page 669
7.8.6 An example......Page 673
7.8.7 The first boundary-value problem......Page 674
7.8.8 Elliptic opening......Page 678
7.8.9 Double-connected region......Page 680
7.8.10 The non-concentric ring......Page 682
Part IV
Basic relationships in the
nonlinear theory of
elasticity......Page 685
8.1.1 Ideally elastic body......Page 686
8.1.2 The strain potentials......Page 687
8.1.3 Homogeneous isotropic ideally elastic body......Page 690
8.2.1 General form for the constitutive law......Page 691
8.2.3 Relation between the generalised moduli under the different initial states......Page 693
8.2.4 Representation of the stress tensor......Page 695
8.2.5 Expressing the constitutive law in terms of the strain tensors......Page 697
8.2.6 The principal stresses......Page 699
8.2.7 The stress tensor......Page 701
8.2.8 The stress tensor of Piola (1836) and Kirchhoff (1850)......Page 703
8.2.9 Prescribing the specific strain energy......Page 704
8.3.2 Representation of the energetic stress tensor......Page 707
8.3.3 Representation of the stress tensor......Page 708
8.3.4 Splitting the stress tensor into the spherical tensor and the deviator......Page 710
8.3.5 Logarithmic strain measure......Page 714
8.4.1 Signorini's quadratic constitutive law......Page 717
8.4.2 Dependence of the coefficients of the quadratic law on the initial state......Page 720
8.4.3 The sign of the strain energy......Page 722
8.4.4 Application to problems of uniaxial tension......Page 724
8.4.5 Simple shear......Page 725
8.4.6 Murnaghan's constitutive law......Page 726
8.4.7 Behaviour of the material under ultrahigh pressures......Page 727
8.4.8 Uniaxial tension......Page 729
8.4.9 Incompressible material......Page 730
8.4.10 Materials with a zero angle of similarity of the deviators......Page 732
8.5.1 Principle of virtual displacements......Page 734
8.5.2 Stationarity of the potential energy of the system......Page 736
8.5.3 Complementary work of deformation......Page 740
8.5.4 Stationarity of the complementary work......Page 742
8.5.5 Specific complementary work of strains for the semi-linear material......Page 743
9.1.1 The stress tensor under affine transformation......Page 747
9.1.2 Uniform compression......Page 749
9.1.3 Uniaxial tension......Page 750
9.1.4 Simple shear......Page 751
9.2.1 Cylindrical bending of the rectangular plate......Page 753
9.2.2 Compression and tension of the elastic strip......Page 757
9.2.3 Equations of statics......Page 759
9.2.4 Compression of the layer......Page 761
9.3.1 Cylindrical tube under pressure (Lame's problem for the nonlinear elastic incompressible material)......Page 763
9.3.2 Stresses......Page 764
9.3.3 Determination of the constants......Page 766
9.3.4 Mooney's material......Page 768
9.3.6 Torsion of a circular cylinder......Page 771
9.3.7 Stresses, torque and axial force......Page 774
9.3.8 Symmetric deformation of the hollow sphere (Lame's problem for a sphere)......Page 777
9.3.9 Incompressible material......Page 779
9.3.10 Applying the principle of stationarity of strain energy......Page 781
9.4.1 Small deformation of the deformed volume......Page 783
9.4.2 Stress tensor......Page 786
9.4.3 Necessary conditions of equilibrium......Page 788
9.4.4 Representation of tensor O......Page 790
9·4.5 Triaxial state of stress......Page 793
9.4.6 Hydrostatic state of stress......Page 795
9.4.7 Uniaxial tension......Page 798
9.4.8 Torsional deformation of the compressed rod......Page 799
9.5.1 Extracting linear terms in the constitutive law......Page 802
9.5.2 Equilibrium equations......Page 806
9.5.3 Effects of second order......Page 807
9.5.4 Choice of the first approximation......Page 813
9.5.5 Effects of second order in the problem of rod torsion......Page 815
9.5.6 Incompressible media......Page 818
9.5.7 Equilibrium equations......Page 819
9.6.1 Geometric relationships......Page 821
9.6.2 Constitutive equation......Page 823
9.6.4 Stress function......Page 825
9.6.5 Plane stress......Page 827
9.6.6 Equilibrium equations......Page 830
9.6.7 Constitutive equation......Page 831
9.6.8 System of equations in the problem of plane stress......Page 833
9.6.9 Using the logarithmic measure in the problem of plane strain......Page 834
9.6.10 Plan e strain of incompressible material with zero angle of similarity of deviators......Page 836
9.6.11 Example of radially symmetric deformation......Page 839
9.7.2 Conserving the principal directions......Page 841
9.7.3 Examples: cylinder and sphere......Page 842
9.7.4 Plane strain......Page 844
9.7.5 State of stress under a plane affine transformation......Page 848
9.7.6 Bending a strip into a cylindrical panel......Page 849
9.7.7 Superimposing a small deformation......Page 852
9.7.8 The case of conserved principal directions......Page 857
9.7.9 Southwell's equations of neutral equilibrium (1913)......Page 858
9.7.10 Solution of Southwell's equations......Page 860
9.7.11 Bifurcation of equilibrium of a compressed rod......Page 863
9.7.12 Rod of circular cross-section......Page 865
9.7.13 Bifurcation of equilibrium of the hollow sphere compressed by uniformly distributed pressure......Page 866
Part V Appendices......Page 871
A.1 Scalars and vectors......Page 872
A.2 The Levi-Civita symbols......Page 874
A.3 Tensor of second rank......Page 876
A.4 Basic tensor operations......Page 879
A.5 Vector dyadic and dyadic representation of tensors of second rank......Page 882
A.6 Tensors of higher ranks, contraction of indices......Page 884
A.7 Inverse tensor......Page 887
A.8 Rotation tensor......Page 889
A.9 Principal axes and principal values of symmetric tensors......Page 891
A.10 Tensor invariants, the Cayley-Hamilton theorem......Page 894
A.11 Splitting the symmetric tensor of second rank in deviatoric and spherical tensors......Page 903
A.11.1 Invariants of deviator......Page 904
A.12.1 Scalar......Page 905
A.12.2 Tensor junction of tensor Q......Page 906
A.12.4 Derivatives of the principal invariants of a tensor with respect to the tensor......Page 907
A.12.5 Gradient of an invariant scalar......Page 908
A.13 Extracting spherical and deviatoric parts......Page 909
A.14 Linear relationship between tensors......Page 914
B.1 Nabla-operator......Page 915
B.2 Differential operations on a vector field......Page 916
B.3 Differential operations on tensors......Page 918
B.4 Double differentiation......Page 920
B.5 Transformation of a volume integral into a surface integral......Page 922
B.6 Stokes's transformation......Page 924
C.1 Definitions......Page 927
C.2 Square of a linear element......Page 928
C.3 Orthogonal curvilinear coordinate system, base vectors......Page 929
C.4 Differentiation of base vectors......Page 931
C.5 Differential operations in orthogonal curvilinear coordinates......Page 933
C.6 Lame's dependences......Page 936
C.7 Cylindrical coordinates......Page 937
C.8 Spherical coordinates......Page 938
C.9 Bodies of revolution......Page 939
C.10 Degenerated elliptic coordinates......Page 941
C.11 Elliptic coordinates (general case)......Page 943
D.1 Main basis and cobasis......Page 949
D.2 Vectors in an oblique basis......Page 950
D.3 Metric tensor......Page 951
D.5 Tensors in an oblique basis......Page 953
D.6 Transformation of basis......Page 954
D.7 Principal axes and principal invariant s of symmetric tensor......Page 955
E.1 Introducing the basis......Page 958
E.2 Derivatives of base vectors......Page 959
E.3 Covariant differentiation......Page 961
E.4 Differential operations in curvilinear coordinates......Page 964
E.5 Transition to orthogonal curvilinear coordinates......Page 966
E.6 The Riemann-Christoffel tensor......Page 967
E.7 Tensor inc P......Page 971
E.8 Transformation of the surface integral into a volume one......Page 972
F.1 Separating variables in Laplace's equation......Page 973
F.2 Laplace's spherical functions......Page 975
F.3 Solution Qn (µ) and qn (s)......Page 978
F.4 Solution of the external and internal problems for a sphere......Page 981
F.5 External and internal Dirichlet 's problems for an oblate ellipsoid......Page 983
F.6 Representation of harmonic polynomials by means of Lame's products......Page 984
F.7 Functions s(k)i (p)......Page 986
F.8 Simple layer potentials on an ellipsoid......Page 987
Bibliographic References......Page 992
Index......Page 1025