Translated by A. Belyaev

A.I. Lurie

Theory of Elasticity

Translated by A. Belyaev

With 49 Figures

^y Springer

Contents

Anatolii I. Lurie 3

Foreword 7

Translator's preface 9

I Basic concepts of continuum mechanics 27

1 Stress tensor 291.1 Field of stresses in a continuum 29

1.1.1 Systems of coordinates in continuum mechanics . . . 291.1.2 External forces 321.1.3 Internal forces in the continuum 331.1.4 Equilibrium of an elementary tetrahedron 351.1.5 The necessary conditions for equilibrium of a contin-

uum 381.1.6 Tensor of stress functions 42

1.2 The properties of the stress tensor 441.2.1 Component transformation, principal stresses and prin-

cipal invariants 441.2.2 Mohr's circles of stress 471.2.3 Separating the stress tensor into a spherical tensor

and a deviator 50

12 Contents

1.2.4 Examples of the states of stress 501.3 Material coordinates 54

1.3.1 Representation of the stress tensor 541.3.2 Cauchy's dependences 551.3.3 The necessary condition for equilibrium 551.3.4 Another definition of the stress tensor 571.3.5 Elementary work of external forces 581.3.6 The energetic stress tensor 611.3.7 Invariants of the stress tensor 63

1.4 Integral estimates for the state of stress 631.4.1 Moments of a function 631.4.2 Moments of components of the stress tensor 641.4.3 The cases of n = 0 and n = 1 641.4.4 The first order moments for stresses 651.4.5 An example. A vessel under external and internal

pressure 661.4.6 An example. Principal vector and principal moment

of stresses in a plane cross-section of the body . . . 671.4.7 An estimate of a mean value for a quadratic form of

components of the stress tensor 681.4.8 An estimate for the specific potential energy of the

deformed linear-elastic body 701.4.9 An estimate of the specific intensity of shear stresses 701.4.10 Moments of stresses of second and higher order . . . 711.4.11 A lower bound for the maximum of the stress com-

ponents 721.4.12 A refined lower bound 73

2 Deformation of a continuum 772.1 Linear strain tensor 77

2.1.1 Outline of the chapter 772.1.2 Definition of the linear strain tensor 78

2.2 Determination of the displacement in terms of the linearstrain tensor 812.2.1 Compatibility of strains (Saint-Venant's dependences) 812.2.2 Displacement vector. The Cesaro formula 832.2.3 An example. The temperature field 852.2.4 The Volterra distortion 87

2.3 The first measure and the first tensor of finite strain . . . . 892.3.1 Vector basis of volumes v and V 892.3.2 Tensorial gradients VR and Vr 922.3.3 The first measure of strain (Cauchy-Green) 932.3.4 Geometric interpretation of the components of the

first strain measure 952.3.5 Change in the oriented surface 96

Contents 13

2.3.6 The first tensor of finite strain 972.3.7 The principal strains and principal axes of strain . . 992.3.8 Finite rotation of the medium as a rigid body . . . . 1002.3.9 Expression for the tensor of finite strain in terms of

the linear strain tensor and the linear vector of rotationlOl2.4 The second measure and the second tensor of finite strain . 101

2.4.1 The second measure of finite strain 1012.4.2 The geometric meaning of the component of the sec-

ond measure of strain 1032.4.3 The second tensor of finite strain (Almansi-Hamel) . 103

2.5 Relation between the strain measures 1042.5.1 Strain measures and the inverse tensors 1042.5.2 Relationships between the invariants 1052.5.3 Representation of the strain measures in terms of the

principal axes 1062.5.4 The invariants of the tensors of finite strain 1082.5.5 Dilatation 1092.5.6 Similarity transformation 1102.5.7 Determination of the displacement vector in terms of

the strain measures I l l2.6 Examples of deformations 113

2.6.1 Affine transformation 1132.6.2 A plane field of displacement 1142.6.3 Simple shear 1162.6.4 Torsion of a circular cylinder 1182.6.5 Cylindrical bending of a rectangular plate 1192.6.6 Radial-symmetric deformation of a hollow sphere . . 1212.6.7 Axisymmetric deformation of a hollow cylinder . . . 123

II Governing equations of the linear theory of elas-ticity 125

3 The constitutive law in the linear theory of elasticity 1273.1 Isotropic medium 127

3.1.1 Statement of the problem of the linear theory of elas-ticity 127

3.1.2 Elementary work 1293.1.3 Isotropic homogeneous medium of Hencky 130

3.2 Strain energy 1333.2.1 Internal energy of a linearly deformed body 1333.2.2 Isothermal process of deformation 1353.2.3 Adiabatic process 1353.2.4 Specific strain energy. Hencky's media 137

3.3 Generalised Hooke's law 139

14 Contents

3.3.1 Elasticity moduli 1393.3.2 Specific strain energy for a linear-elastic body . . . . 1413.3.3 Clapeyron's formula. Limits for the elasticity moduli. 1433.3.4 Taking account of thermal terms. Free energy . . . . 1453.3.5 The Gibbs thermodynamic potential 1473.3.6 Equat ion of thermal conductivity 149

4 Governing relationships in the linear theory of elasticity 1514.1 Differential equations governing the linear theory of elasticity 151

4.1.1 Fundamental relationships 1514.1.2 Boundary conditions 1524.1.3 Differential equations governing the linear theory of

elasticity in terms of displacements 1534.1.4 Solution in the Papkovich-Neuber form 1554.1.5 The solution in terms of stresses. Beltrami's depen-

dences 1594.1.6 Krutkov's transformation 1614.1.7 The Boussinesq-Galerkin solution 1634.1.8 Curvilinear coordinates 1644.1.9 Orthogonal coordinates 1664.1.10 Axisymmetric problems. Love's solution 1684.1.11 Torsion of a body of revolution 1704.1.12 Deformation of a body of revolution 1714.1.13 The Papkovich-Neuber solution for a body of revolution 1744.1.14 Account of thermal components 175

4.2 Variational principles of statics for a linear elastic body . . 1784.2.1 Stationarity of the potential energy of the system . . 1784.2.2 The principle of minimum potential energy of the

system 1804.2.3 Ritz's method 1834.2.4 Galerkin's method (1915) 1854.2.5 Principle of minimum complementary work 1864.2.6 Mixed stationarity principle (E. Reissner, 1961) . . . 1904.2.7 Variational principles accounting for the thermal termsl924.2.8 Saint-Venant's principle. Energetic consideration . . 193

4.3 Reciprocity theorem. The potentials of elasticity theory . . 1974.3.1 Formulation and proof of the reciprocity theorem

(Betti, 1872) 1974.3.2 The influence tensor. Maxwell's theorem 1984.3.3 Application of the reciprocity theorem 2004.3.4 The reciprocity theorem taking account of thermal

terms 2034.3.5 The influence tensor of an unbounded medium . . . 2044.3.6 The potentials of the elasticity theory 208

Contents 15

4.3.7 Determining the displacement field for given externalforces and displacement vector of the surface . . . . 210

4.3.8 On behaviour of the potential of the elasticity theoryat infinity 212

4.4 Theorems on uniqueness and existence of solutions 2144.4.1 Kirchhoff's theorem 2144.4.2 Integral equations of the first boundary value problem 2174.4.3 Integral equations of the second boundary value prob-

lem 2194.4.4 Comparison of the integral equations of the first and

second boundary value problems 2224.4.5 Theorem on the existence of solutions to the second

external and first internal problems 2234.4.6 The second internal boundary value problem (IlM) . 2244.4.7 Elastostatic Robin's problem 2264.4.8 The first external boundary value problem (I*-6-*) . . 228

4.5 State of stress in a double-connected volume 2294.5.1 Overview of the content 2294.5.2 Determination of the state of stress in terms of the

barrier constants 2304.5.3 The reciprocity theorem 2324.5.4 The strain energy of distortion 2344.5.5 The case of a body of revolution 2344.5.6 Boundary value problem for a double-connected body

of revolution 237

III Special problems of the linear theory of elastic-ity 241

5 Three-dimensional problems in the theory of elasticity 2435.1 Unbounded elastic medium 243

5.1.1 Singularities due to concentrated forces 2435.1.2 The system of forces distributed in a small volume.

Lauricella's formula 2455.1.3 Interpretation of the second potential of elasticity

theory 2515.1.4 Boussinesq's potentials 2525.1.5 Thermoelastic displacements 2545.1.6 The state of stress due to an inclusion 256

5.2 Elastic half-space 2605.2.1 The problems of Boussinesq and Cerruti 2605.2.2 The particular Boussinesq problem 2615.2.3 The distributed normal load 263

16 Contents

5.2.4 Use of the Papkovich-Neuber functions to solve theBoussinesq-Cerruti problem 264

5.2.5 The influence tensor in elastic half-space 2675.2.6 Thermal stresses in the elastic half-space 2705.2.7 The case of the steady-state temperature 2725.2.8 Calculation of the simple layer potential for the plane

region 2735.2.9 Dirichlet's problem for the half-space 2755.2.10 The first boundary value problem for the half-space 2775.2.11. Mixed problems for the half-space 2785.2.12 On Saint-Venant's principle. Mises's formulation . . 2805.2.13 Superstatic system of forces 2825.2.14 Sternberg's theorem (1954) 283

5.3 Equilibrium of the elastic sphere 2855.3.1 Statement of the problem 2855.3.2 The first boundary value problem 2865.3.3 The elastostatic Robin's problem for the sphere . . . 2885.3.4 Thermal stresses in the sphere 2895.3.5 The second boundary value problem for the sphere . 2925.3.6 Calculation of the displacement vector 2955.3.7 The state of stress at the centre of the sphere . . . . 2975.3.8 Thermal stresses 2985.3.9 The state of stress in the vicinity of a spherical cavity 3005.3.10 The state of stress in the vicinity of a small spherical

cavity in a twisted cylindrical rod 3015.3.11 Action of the mass forces 3025.3.12 An attracting sphere 3045.3.13 A rotating sphere 3055.3.14 Action of concentrated forces 3075.3.15 The distributed load case 310

5.4 Bodies of revolution 3115.4.1 Integral equation of equilibrium 3115.4.2 Tension of the hyperboloid of revolution of one nappe 3155.4.3 Torsion of the hyperboloid 3185.4.4 Bending of the hyperboloid 3195.4.5 Rotating ellipsoid of revolution 320

5.5 Ellipsoid 3235.5.1 Elastostatic Robin's problem for the three-axial el-

lipsoid 3235.5.2 Translatory displacement 3245.5.3 Distribution of stresses over the surface of the ellipsoid3255.5.4 Rotational displacement 3285.5.5 Distribution of stresses over the surface of the ellipsoid3305.5.6 An ellipsoidal cavity in the unbounded elastic medium3325.5.7 The boundary conditions 335

Contents 17

5.5.8 Expressing the constants in terms of three parameters 3375.5.9 A spheroidal cavity in the elastic medium 3395.5.10 A circular slot in elastic medium 3415.5.11 An elliptic slot in an elastic medium 343

5.6 Contact problems 3475.6.1 The problem of the rigid die. Boundary condition . . 3475.6.2 A method of solving the problem for a rigid die . . . 3515.6.3 A plane die with an elliptic base 3565.6.4 Displacements and stresses 3595.6.5 A non-plane die 3615.6.6 Displacements and stresses 3645.6.7 Contact of two surfaces 3665.6.8 Hertz's problem in the compression of elastic bodies 371

5.7 Equilibrium of an elastic circular cylinder 3735.7.1 Differential equation of equilibrium of a circular cylin-

der 3735.7.2 Lame's problem for a hollow cylinder 3785.7.3 Distortion in the hollow cylinder 3795.7.4 Polynomial solutions to the problem of equilibrium

of the cylinder 3825.7.5 Torsion of a cylinder subjected to forces distributed

over the end faces 3855.7.6 Solutions in terms of Bessel functions 3895.7.7 Filon's problem 3935.7.8 Homogeneous solutions 3955.7.9 Boundary conditions on the end faces 3985.7.10 Generalised orthogonality 402

Saint-Venant's problem 4096.1 The state of stress 409

6.1.1 Statement of Saint-Venant's problem 4096.1.2 Integral equations of equilibrium 4106.1.3 Main assumptions 4116.1.4 Normal stress az in Saint-Venant's problem 4126.1.5 Shear stresses TXZ and ryz 413

6.2 Reduction to the Laplace and Poisson equations 4156.2.1 Introducing the stress function 4156.2.2 Displacements in Saint-Venant's problem 4186.2.3 Elastic line 4216.2.4 Classification of Saint-Venant's problems 4236.2.5 Determination of parameter a 4256.2.6 Centre of rigidity 4286.2.7 Elementary solutions 430

6.3 The problem of torsion 4336.3.1 Statement of the problem 433

18 Contents

6.3.2 Displacements 4356.3.3 Theorem on the circulation of shear stresses 4376.3.4 Torsional rigidity 4396.3.5 The membrane analogy of Prandtl (1904) 4416.3.6 Torsion of a rod with elliptic cross-section 4436.3.7 Inequalities for the torsional rigidity 4456.3.8 Torsion of a rod having a rectangular cross-section . 4476.3.9 Closed-form solutions 4496.3.10j Double connected region 4516.3.11 Elliptic ring 4536.3.12 Eccentric ring 4556.3.13 Variational determination of the stress function . . . 4586.3.14 Approximate solution to the problem of torsion . . . 4626.3.15 Oblong profiles 4676.3.16 Torsion of a thin-walled tube 4716.3.17 Multiple-connected regions 474

6.4 Bending by force 4786.4.1 Stresses 4786.4.2 Bending of a rod with elliptic cross-section 4806.4.3 The stress function of S.P. Timoshenko 4826.4.4 Rectangular cross-section 4826.4.5 Variational statement of the problem of bending . . 4866.4.6 The centre of rigidity 4886.4.7 Approximate solutions 4906.4.8 Aerofoil profile 492

6.5 Michell's problem 4946.5.1 Statement of the problem 4946.5.2 Distribution of normal stresses 4976.5.3 Tension of the rod 4986.5.4 Shear stresses T\X,T\/Z 5036.5.5 Stresses ax,ay,Txy 5036.5.6 Determining <J°Z 5066.5.7 Bending of a heavy rod 5076.5.8 Mean values of stresses 5096.5.9 On Almansi's problem 511

7 The plane problem of the theory of elasticity 5137.1 Statement of the plane problems of theory of elasticity . . . 513

7.1.1 Plane strain 5137.1.2 Airy' stress function 5167.1.3 Differential equation for the stress function 5177.1.4 Plane stress 5187.1.5 The generalised plane stress 5217.1.6 The plane problem 5227.1.7 Displacements in the plane problem 523

Contents 19

7.1.8 The principal vector and the principal moment . . . 5257.1.9 Orthogonal curvilinear coordinates 5267.1.10 Polar coordinates in the plane 5277.1.11 Representing the biharmonic function 5277.1.12 Introducing a complex variable 5297.1.13 Transforming the formulae of the plane problem . . 5307.1.14 Goursat's formula 5327.1.15 Translation of the coordinate origin 534

7.2 Beam and bar with a circular axis 5357.2.1 Statement of the plane problem for beam and bar . 5357.2.2 Plane Saint-Venant's problem 5377.2.3 Operator representation of solutions 5397.2.4 Stress function for the strip problem 5417.2.5 The elementary theory of beams 5447.2.6 Polynomial load (Mesnager, 1901) 5457.2.7 Sinusoidal load, solutions of Ribiere (1898) and Filon

(1903) 5477.2.8 Concentrated force (Karman and Seewald, 1927) . . 5517.2.9 Bar with a circular axis loaded on the end faces (Golovin,

1881) 5567.2.10 Loading the circular bar on the surface 5607.2.11 Cosinusoidal load 5637.2.12 Homogeneous solutions 565

7.3 Elastic plane and half-plane 5677.3.1 Concentrated force and concentrated moment in elas-

tic plane 5677.3.2 Flamant's problem (1892) 5707.3.3 General case of normal loading 5737.3.4 Loading by a force directed along the boundary . . . 5757.3.5 The plane contact problem 5777.3.6 Constructing potential to 5797.3.7 A plane die 5827.3.8 Die with a parabolic profile 5837.3.9 Concentrated force in the elastic half-plane 583

7.4 Elastic wedge 5867.4.1 Concentrated force in the vertex of the wedge . . . . 5867.4.2 Mellin's integral transform in the problem of a wedge 5887.4.3 Concentrated moment at the vertex of the wedge . . 5927.4.4 Loading the side faces 595

7.5 Boundary-value problems of the plane theory of elasticity . 5997.5.1 Classification of regions 5997.5.2 Boundary-value problems for the simply-connected

finite region 5997.5.3 Definiteness of Muskhelishvili's functions 6027.5.4 Infinite region with an opening 603

20 Contents

7.5.5 Double-connected region. Distortion 6077.5.6 Representing the stress function in the double-connected

region (Michell) 6087.5.7 Thermal stresses. Plane strain 6107.5.8 Plane stress 6127.5.9 Stationary temperature distribution 6157.5.10 Cauchy's theorem and Cauchy's integral 6187.5.11 Integrals of Cauchy's type. The Sokhotsky-Plemelj

, formula 6207.6 Regioas with a circular boundary 622

7.6.1 Round disc loaded by concentrated forces 6227.6.2 The general case of loading round disc 6257.6.3 The method of Cauchy's integrals 6277.6.4 Normal stress OQ on the circle 6297.6.5 Stresses at the centre of the disc 6317.6.6 A statically unbalanced rotating disc 6327.6.7 The first boundary-value problem for circle 6357.6.8 The state of stress 6397.6.9 Thermal stresses in the disc placed in a rigid casing 6417.6.10 Round opening in an infinite plane 6437.6.11 A uniform loading on the edge of the opening . . . . 6467.6.12 Tension of the plane weakened by a round opening . 6467.6.13 Continuation of $ (z) 6487.6.14 Solving the boundary-value problems of Subsections

7.6.2 and 7.6.10 by way of the continuation 6507.7 Round ring 653

7.7.1 The stresses due to distortion 6537.7.2 The second boundary-value problem for a ring . . . 6547.7.3 Determining functions $ ( ( ) , * (C) 6557.7.4 Tube under uniform internal and external pressure

(Lame's problem) 6577.7.5 Thermal stresses in the ring 6577.7.6 Tension of the ring by concentrated forces 6597.7.7 The way of continuation 660

7.8 Applying the conformal transformation 6657.8.1 Infinite plane with an opening 6657.8.2 The method of Cauchy's integrals 6677.8.3 Elliptic opening 6707.8.4 Hypotrochoidal opening 6727.8.5 Simply connected finite region 6747.8.6 An example 6787.8.7 The first boundary-value problem 6797.8.8 Elliptic opening 6837.8.9 Double-connected region 6857.8.10 The non-concentric ring 687

Contents 21

IV Basic relationships in the nonlinear theory ofelasticity 691

8 Constitutive laws for nonlinear elastic bodies 6938.1 The strain energy 693

8.1.1 Ideally elastic body 6938.1.2 The strain potentials 6948.1.3 Homogeneous isotropic ideally elastic body 697

8.2 The constitutive law for the isotropic ideally-elastic body . 6988.2.1 General form for the constitutive law 6988.2.2 The initial and the natural states 7008.2.3 Relation between the generalised moduli under the

different initial states 7008.2.4 Representation of the stress tensor 7028.2.5 Expressing the constitutive law in terms of the strain

tensors 7048.2.6 The principal stresses 7068.2.7 The stress tensor 7088.2.8 The stress tensor of Piola (1836) and Kirchhoff (1850) 7108.2.9 Prescribing the specific strain energy 711

8.3 Representing the constitutive law by a quadratic trinomial . 7148.3.1 Quadratic dependence between two coaxial tensors . 7148.3.2 Representation of the energetic stress tensor 7148.3.3 Representation of the stress tensor 7158.3.4 Splitting the stress tensor into the spherical tensor

and the deviator 7178.3.5 Logarithmic strain measure 721

8.4 Approximations of the constitutive law 7248.4.1 Signorini's quadratic constitutive law 7248.4.2 Dependence of the coefficients of the quadratic law

on the initial state 7278.4.3 The sign of the strain energy 7298.4.4 Application to problems of uniaxial tension 7318.4.5 Simple shear 7328.4.6 Murnaghan's constitutive law 7338.4.7 Behaviour of the material under ultrahigh pressures 7348.4.8 Uniaxial tension 7368.4.9 Incompressible material 7378.4.10 Materials with a zero angle of similarity of the deviators739

8.5 Variational theorems of statics of the nonlinear-elastic body 7418.5.1 Principle of virtual displacements 7418.5.2 Stationarity of the potential energy of the system . . 7438.5.3 Complementary work of deformation 7478.5.4 Stationarity of the complementary work 749

22 Contents

8.5.5 Specific complementary work of strains for the semi-linear material 750

9 Problems and methods of the nonlinear theory of elasticity7559.1 The state of stress under affine transformation 755

9.1.1 The stress tensor under affine transformation . . . . 7559.1.2 Uniform compression 7579.1.3 Uniaxial tension 7589.1.4 Simple shear 759

9.2 Elastic layer 7619.2.1 Cylindrical bending of the rectangular plate 7619.2.2 Compression and tension of the elastic strip 7659.2.3 Equations of statics 7679.2.4 Compression of the layer 7699.2.5 Tension of the layer 770

9.3 Elastic cylinder and elastic sphere 7719.3.1 Cylindrical tube under pressure (Lame's problem for

the nonlinear elastic incompressible material) . . . . 7719.3.2 Stresses 7729.3.3 Determination of the constants 7749.3.4 Mooney's material 7769.3.5 Cylinder "turned inside out" 7779.3.6 Torsion of a circular cylinder 7799.3.7 Stresses, torque and axial force 7829.3.8 Symmetric deformation of the hollow sphere (Lame's

problem for a sphere) 7859.3.9 Incompressible material 7879.3.10 Applying the principle of stationarity of strain energy 789

9.4 Small deformation in the case of the initial loading 7919.4.1 Small deformation of the deformed volume 7919.4.2 Stress tensor 7949.4.3 Necessary conditions of equilibrium 7969.4.4 Representation of tensor <~> - 7989.4.5 Triaxial state of stress 80L9.4.6 Hydrostatic state of stress 803!9.4.7 Uniaxial tension 806.9.4.8 Torsional deformation of the compressed rod . . . . 807.

9.5 Second order effects 810-9.5.1 Extracting linear terms in the constitutive law . . . 8109.5.2 Equilibrium equations 8149.5.3 Effects of second order 8159.5.4 Choice of the first approximation 8219.5.5 Effects of second order in the problem of rod torsion 8239.5.6 Incompressible media 8269.5.7 Equilibrium equations 827

Contents 23

9.6 Plane problem 8299.6.1 Geometric relationships 8299.6.2 Constitutive equation 8319.6.3 Equations of statics 8329.6.4 Stress function 8339.6.5 Plane stress 8359.6.6 Equilibrium equations 8389.6.7 Constitutive equation 8399.6.8 System of equations in the problem of plane stress . 8419.6.9 Using the logarithmic measure in the problem of plane

strain 8429.6.10 Plane strain of incompressible material with zero an-

gle of similarity of deviators 8449.6.11 Example of radially symmetric deformation 847

9.7 Semi-linear material 8499.7.1 Equilibrium equations for the semi-linear material . 8499.7.2 Conserving the principal directions 8499.7.3 Examples: cylinder and sphere 8509.7.4 Plane strain 8529.7.5 State of stress under a plane affine transformation . 8569.7.6 Bending a strip into a cylindrical panel 8579.7.7 Superimposing a small deformation 8609.7.8 The case of conserved principal directions 8659.7.9 Southwell's equations of neutral equilibrium (1913) . 8669.7.10 Solution of Southwell's equations 8689.7.11 Bifurcation of equilibrium of a compressed rod . . . 8719.7.12 Rod of circular cross-section 8739.7.13 Bifurcation of equilibrium of the hollow sphere com-

pressed by uniformly distributed pressure 874

V Appendices 879

A Basics tensor algebra 881A.I Scalars and vectors 881A.2 The Levi-Civita symbols 883A.3 Tensor of second rank 885A.4 Basic tensor operations 888A.5 Vector dyadic and dyadic representation of tensors of second

rank 891A.6 Tensors of higher ranks, contraction of indices 893A.7 Inverse tensor 896A.8 Rotation tensor 898A.9 Principal axes and principal values of symmetric tensors . . 900A. 10 Tensor invariants, the Cayley-Hamilton theorem 903

24 Contents

A. 10.1 Principal axes and principal directions of non-symmetrictensors 908

A.11 Splitting the symmetric tensor of second rank in deviatoricand spherical tensors 912A.11.1 Invariants of deviator 913

A.12 Functions of tensors 914A.12.1 Scalar 914A.12.2 Tensor function of tensor Q 915A.12.3 Gradient of scalar with respect to a tensor 916A. 12.4 Derivatives of the principal invariants of a tensor with

respect to the tensor 916A.12.5 Gradient of an invariant scalar 917

A. 13 Extracting spherical and deviatoric parts 918A.14 Linear relationship between tensors 923

B Main operations of tensor analysis 925B.I Nabla-operator 925B.2 Differential operations on a vector field 926B.3 Differential operations on tensors 928B.4 Double differentiation 930B.5 Transformation of a volume integral into a surface integral . 932B.6 Stokes's transformation 934

C Orthogonal curvilinear coordinates 937C.I Definitions 937C.2 Square of a linear element 938C.3 Orthogonal curvilinear coordinate system, base vectors . . . 939C.4 Differentiation of base vectors 941C.5 Differential operations in orthogonal curvilinear coordinates 943C.6 Lame's dependences 946C.7 Cylindrical coordinates 947C.8 Spherical coordinates 948C.9 Bodies of revolution 949CIO Degenerated elliptic coordinates 951C.ll Elliptic coordinates (general case) 953

D Tensor algebra in curvilinear basis 959D.I Main basis and cobasis 959D.2 Vectors in an oblique basis 960D.3 Metric tensor 961D.4 The Levi-Civita tensor 963D.5 Tensors in an oblique basis 963D.6 Transformation of basis 964D.7 Principal axes and principal invariants of symmetric tensor 965

Contents 25

E Operations of tensor analysis in curvilinear coordinates 969E.I Introducing the basis 969E.2 Derivatives of base vectors 970E.3 Covariant differentiation 972E.4 Differential operations in curvilinear coordinates 975E.5 Transition to orthogonal curvilinear coordinates 977E.6 The Riemann-Christoffel tensor 978E.7 Tensor inc P 982E.8 Transformation of the surface integral into a volume one . . 983

F Some information on spherical and ellipsoidal functions 985F.I Separating variables in Laplace's equation 985F.2 Laplace's spherical functions 987F.3 Solutions Qn (fi) and qn (s) 990F.4 Solution of the external and internal problems for a sphere . 993F.5 External and internal Dirichlet's problems for an oblate el-

lipsoid 995F.6 Representation of harmonic polynomials by means of Lame's

products 996F.7 Functions Sf) (p) 998

F.8 Simple layer potentials on an ellipsoid 999

Bibliographic References 1005

Index 1039