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SOME BASIC PROBLEMS OF THE

MATHEMATICAL THEORY OF

ELASTICITY

* FUNDAMENTAL EQUATIONS

PLANE THE OEY OF ELASTICITY

TOESION AND BENDING

BY

N. I. M USK HE LISHVILI

FOURTH, CORRECTED AND AUGMENTED EDITION

TRANSLATED FR dM THE RUSSIAN •

J R M RA D OK

N OOR D HOFF IN T E R N A T ION A L PU B L ISHIN G

LEYDEN



CONTENTS

P A R T I - Fundamental equations of the mechanics of an elastic body. 1

CHAPTER 1. ANALYSIS OF STRESS.

§ 1. Body forces 5

§ 2. Stress 6

§ 3. Components of stress. Dependence of stress on the

orientation of the plane 7

§ 4. Equations, relating components of stress 10

§ 5. Transformation of coordinates. Inv arian t qua dratic

form. Stress tensor 14

§ 6. Stress surface. Principal stresses 17

§ 7. Determination of principal stresses and axes . . . . 22

§ 8. Plane stress 23

CHAPTER 2. ANALYSIS OF STRAIN.

§ 9. General remarks 28

§ 10. Affine transformation 29

§ 11. Infinitesimal affine transform ation 31

§ 12. Decomposition of infinitesimal tran sform ations into

pure deformation and rigid body motion 32§ 13. The inva riant qu ad ratic form, connected with defor-

mation. The stra in surface, p rincipal a xes. Tra ns-

formation of coordinates 38

§ 14. General deformation 41

§ 15. Determination of displacements from components of

strain. Saint-Venant's condition of compatibility . . 44

CHAPTER 3. T H E FUNDAMENTAL LAW OF THE THEORY OF ELASTICI-TY; THE BASIC EQUATIONS.

§ 16. The fundamental law of the theory of elasticity (gener-

alized Hooke's Law) 52

§ 17. Isotropic bodies 56



XXII CONTENTS

§ 18. The basic equations for the statics of an elastic iso-

tropic body 60

§ 19. The simplest cases of elastic equilibrium. The basicelastic constants 62

§ 20. The fundamental boundary value problems of static

elasticity. Uniqueness of solution 66

§ 21. Basic equations in terms of displacement components 73

§ 22. Equations in terms of stresses 74

§ 23. Remarks on the effective solution of the fundamental

problems. Saint-Venant s principle 77

§ 24. Dynamic equations. The fundamental problems of thedynamics of an elastic body 78

R T - e n e r a l f o r m u l a e of t h e p l a n e t h e o r y of e l a s t i c i t y . . . . 8 5

C H A P T E R 4. B A S I C EQUATIONS OF THE P L A N E THEORY OF E L A S -

T I C I T Y .

§ 25. P lane s t r a in 89

§ 26. D ef o r ma t i o n of a th in p late under forces act ing in its

p l an e 92

§ 27. Bas i c equ a t ions of the p l an e t h eo r y of e l as t i c i ty . . . 96

§ 28. R e d u c t i o n to the case of ab sen ce of body forces . . . 101

C H A P T E R 5. S T R E S S FUNCTION. C O M P L E X R E P R E S E N T A T I O N OF

T H E GENERAL S O L U T I O N OF THE EQUATIONS OF THE P L A N E

THEORY OF ELASTICITY.

§ 29. St ress funct ion 105

§ 30. D e t e r m i n a t i o n of d i sp lacem ent s f rom the s t ress funct ion 107§ 31. C o mp l ex r ep r e sen t a t i o n of b iharmonic func t ions . . 110

§ 32. C o mp l ex r ep r e sen t a t i o n of d i sp l acemen t s and s t resses 113

§ 33. The p h y s i ca l m ean i n g of the funct ion /. E x p r e s s io n s for

the resul tant force and m o m e n t 116

§ 34. A r b i t r a r i n es s in the defini t ion of the i n t r o d u ced

func t ions 118

§ 35. Gene ra l fo rmulae for f in ite mu l t ip ly conn ecte d regions 121

§ 36. Case of infini te regions 126§ 37. Some proper t ies fo l lowing f rom the an a l y t i c ch a r a c t e r

of the so lu t ion . On an a l y t i c co n t i n u a t i o n ac r os s a

g iven con tour 131

§ 38. T r an s f o r ma t i o n of r ec t i l i near coord ina tes 137



CONTENTS XXIII

§ 39. Polar coordin ates 140

§ 40. The fundam ental bou nda ry value problems. Uniqueness

of So lutio n 141§ 41 . Re duction of the fund am ental problem s to problems of

complex function the ory 147

§ 41a Sup plem entary rem arks 156

§ 42. Concept of the regu lar solution. Uniqueness of a regular

solution 158

§ 43. On concentrated forces, applied to the boundary . . . 162

§ 44. Dependence of the sta te of stress on the elastic con stan ts 164

CHAPTER 6. MU LTI-VALUED DISPLACEMENTS. THERMA L STRESSES.

§ 45. M ulti-valued displacem ents. Dislocations 167

§ 46. Th erm al stresses 170

CHAPTE R 7. TRANSFORM ATION OF THE BASIC FORMULAE FOR CON-

FORMAL MAPPING.

§ 47. Conformal transfo rm ation 176

§ 48. Simple examples of conformal mapping.

1°. Bilinear function 1802°. Pa sca l's limacon 185

3°. Ep itrocho ids 186

4°. H ypotro choid s 187

5°. Ell iptic rings 188

§ 49. Curvilinear coordina tes, conne cted w ith conformal

transfo rm ations into circular regions 190

§ 50. Transfo rm ation of th e formulae of the plane theo ry of

elasticity 192§ 51 . Boun dary conditions in the image regions 194

PART I I I - Solution of sever l problems of the plane theory of el sticity

by means of power series 197

CHAPTER 8. O N FO UR IER SERIES.

§ 52. On Fo urie r series in com plex form 199

§ 53. On th e convergence of Fo urie r series 202

CHAPTER 9. SOLUTION FOR REGION S, BOUND ED BY A CIRCLE.

§ 54. Solution of the first fund am ental problem for the

circle 204



XXIV CONTENTS

§ 55. Solution of the second fundamental problem for the

circle 207

§ 56. Solution of the first fundamental problem for theinfinite plane w ith a circular hole 208

§ 56a. Exam ples.

1°. Uni-directional tension of a plate, weakened by a

circular hole 211

2°. Bi-ax ial tension 214

3°. Uniform normal pressure applied to the edge of a

circular hole 214

4°. A concentrated force, applied at a point of theinfinite plane 215

5°. Co ncentrated couple 216

§ 57. On the general problem of concentrated forces . . . 217

§ 58. Some cases of equilibrium of infinite pla tes , con tainin g

circu lar discs of different m ater ial 221

1°. Infinite plate with a circular hole into which an

elastic circular disc with an originally larger radiushas been inserted 222

2°. Stretching of plates with inserted or attached rigid

discs 224

3°. Stretching of plates with inserted or a ttac he d

elastic discs 226

CHAPTER 10. T H E CIRCULAR RING.

§ 59. Solution of the first fundamental problem for the

circula r ring 230

§ 59a. Exam ples.

1°. Tub e subject to uniform extern al and internal

pressures 235

2°. Stress distribution in a ring, rotating about itscentre 236

§ 60. M ulti-valued displace ments in the case of a circular ring 237

§ 61 . Sup plem ent. Bending of a curved beam 242

§ 62. Thermal s tresses in a hollow circular cylinder . . . . 246



CONTENTS XXV

CHAPTER 11. APPLICATION OF CONFORMAL MAPPING.

§ 63. Case of simply connected regions 250§ 64. Example of application of mapping on to a circular ring.

Solution of the fundamental problems for a continuous

ellipse 257

PART IV - On Cauchy integrals 265

CHAPTER 12. FUNDAMENTAL PROPERTIES OF CAUCHY INTEGRALS.

§ 65. Notation and terminology 267

§ 66. Cauchy integrals 270§ 67. Values of Cauchy integrals on the path of integration.

Principal value 271

§ 68. Boundary values of Cauchy integrals. The Plemelj

formulae 276

§ 69. The derivatives of Cauchy integrals 279

§ 70. Some elementary formulae, facilitating the calculation

of Cauchy integrals 281

§ 71. On Cauchy integrals, taken along infinite straight lines 286§ 72. On Cauchy integrals, taken along infinite straight lines

(continued) 296

CHAPTER 13. BOUNDARY VALUES OF HOLOMORPHIC FUNCTIONS.

§ 73. Some general propositions 298

§ 74. Generalization 301

§ 75. Harnack s theorem 301

§ 76. Some special formulae for the circle and the half-plane 303§ 77. Simple applications: solutions of the fundamental

problems of potential theory for a circle and half-plane 308

PART V - Application of Cauchy integrals to the solution of boundary

problems of plane elasticity 315

CHAPTER 14. GENERAL SOLUTION OF THE FUNDAMENTAL PRO-

BLEMS FOR REGIONS BOUNDED BY ONE CONTOUR.

§ 78. Reduction of the fundamental problems to functional

equations 317

§ 79. Reduction to Fredholm equations. Existence theorems 323

§ 79a. On some other applications of the preceding integral

equations 333



XXVI CONTENTS

CHAPTER 15. SO LUTION OF THE FUNDAMENTAL PROBLEMS FOR

REGIONS MAPPED ON TO A CIRCLE BY RATIONAL FUNCTIONS.

EXTENSION TO APPROXIMATE SOLUTION FOR REGIONS OFGENERAL SHAPE.

§ 80. Solu tion of the first fundam enta l problem for th e circle 334

§ 80a. Exa m ples.

1°. Circular disc under concentrated forces, applied to

its bou nda ry 338

2° . Disc under concentrated forces and couples acting

at internal points 342

3°. Ro tating disc with attac hed discrete masses . . . 345

§ 81. Solution of the second fundamental problem for the

circle 346

§ 82. Solution of the first fundamental problem for the

infinite plane w ith an elliptic hole 347

§ 82a. Exa m ples.

1°. Stretching of a plate with an elliptic hole . . . 351

2°. Ellip tic hole the edge of wh ich is sub ject to uniformpressure 353

3°. Ellip tic hole th e edge of which is subject t o uniform

tangential stress T 354

4° . Elliptic hole (or straight cut) part of the edge of

which is subject to uniform pressure 354

5°. Approximate solution of the problem of bending

of a str ip (beam) w ith an elliptic hole 358

§ 83. Solution of the second fundamental problem for theinfinite plane w ith an elliptic hole 361

§ 83a. Exa m ples.

1°. Uni-directional tension of an infinite plate with a

rigid elliptic cen tre 363

2° . Case when th e elliptic cen tre is not allowed to ro tat e 365

3° . Case when a couple with given moment acts on the

elliptic kernel 366

4° . Case when a force acts on the centre of the elliptickernel 366

§ 84. General solution of the fundamental problems for

regions, mapped on to the circle by the help of

polynomials 366



CONTENTS XXVII

§ 85 . Ge nera l i za t ion to the case of t r ans fo rm at ion s b y mea ns

of ra t ion al funct ions 374

§ 85 . So lu t ion of t he second fund am en ta l p rob lem . On thes olu tio n of t h e m i xe d f u nd a m e n ta l p ro b le m . . . . 3 79

§ 87 . Oth er m eth od s of so lu t ion of t he fund am enta l p rob lem s 379

§ 87a . E xa m ple . So lu t ion of t he fi rs t funda m enta l p rob lem

for an infini te pla ne w ith a circ ula r hole 380

§ 8 8 . Fu r t h e r ex amp l es . A p p l i ca t io n t o so me o t h e r b o u n d a r y

prob lems 384

§ 89 . Ap p l i ca t ion to the ap pro x im ate so lu t ion of t he genera l

case 385

C H A P T E R 16. S O L U T I O N O F T H E F U N D A M E N T A L PR O B L E M S E O R T H E

H A L F - P L A N E A N D F O R S E M I - I N F I N I T E R E G I O N S .

§ 90 . Ge neral formu lae an d prop osi t ion s for the hal f -p lane 391

§ 9 1 . T h e g e n e ra l f or m u la e fo r s em i -i nf in it e r eg io n s . . . . 3 97

§ 92 . Basic form ulae, con nec ted wi th conforma l t ransfor-

m at io n on to the ha l f -p lane 399

§ 93 . Solu t ion of the f i r s t fundamental problem for the

hal f -p lane 402

§ 93a . E xa m ple 406

§ 9 4. S o lu tio n of t h e s ec on d f u n d a m e n t al p r ob le m . . . . 4 09

§ 95 . Solu t ion of the fundamental problems for regions ,

mapped on to the ha l f -p l ane by means o f r a t iona l

funct ion s . Case of a para bol ic co nto ur 411

C H A P T E R 17. S O M E G E N E R A L M E T H O D S O F S O L U T I O N O F B O U N D A R Y

V A L U E P R O B L E M S . G E N E R A L I Z A T I O N S .

§ 96 . On th e in teg ral eq ua t ion s of S . G. M ikhlin 414

§ 97 . On a general method of so lu t ion of problems for

m ul t ip ly conne c ted reg ions 416

§ 9 8 . T h e i n te g r a l e q u a t i o n s , p r op o s ed b y t h e A u t h o r . . . 4 17

§ 99 . Ap p l i ca t ion to con tours w i th co rner s 427

§ 100. On the num erica l so lu t ion of th e in te gra l eq ua t ion s of

th e p lan e the ory of e las t ic i ty 427

§ 101 . Th e in t eg ra l eq ua t io ns of D . I . She rm an-G . Laur i ce l l a 427

§ 102. So lu t ion of th e f i rs t a nd second fun da m en tal prob lem s

b y the m etho d of D . I . Sh erm an 431



XXVIII CONTENTS

§ 103. On the solution of the mixed fundamental problem and

of certain other boundary problems by means of D. I.

Sherman s method 440§ 104. Generalization to anisotropic bodies 441

§ 105. On other applications of the general representation

of solutions 441

PART VI - Solution of the boundary of the plane theory of elasticity

by reduction to the problem of linear relationship 445

CHAPTER 18. THE PROBLEM OF LINEAR RELATIONSHIP.

§ 106. Sectionally holomorphic functions 447

§ 107. The problem of linear relationship (the Hilbert problem) 448

§ 108. Determination of a sectionally holomorphic function

for a given discontinuity 449

§ 109. Application 452

§ 109a. Example 455

§110. Solution of the problem: F

= gF~ f 456

§ 1 1 1 . Case of discontinuous coefficients 468

CHAPTER 19. SOLUTION OF THE FUNDAMENTAL PROBLEMS FOR THE

HALF-PLANE AND FOR THE PLANE WITH STRAIGHT CUTS.

§ 112. Transformation of the general formulae for the half-

plane 471

§ 113. Solution of the first and second fundamental problems

for the half-plane 476§ 114. Solution of the mixed fundamental problem . . . . 478

§ 114a. Examples.

1°. Stamp with straight horizontal base 486

2°. Stamp with straight inclined base 488

3°. Effect of asymmetrically distributed forces . . . 492

§ 115. The problem of pressure of rigid stamps in the absence

of friction 492

§116. Application 496§ 116a. Examples.

1°. Stamp with straight horizontal base 501

2°. Stamp with straight inclined base 501

3°. Stamp with curved base 502



CONTENTS XXIX

§ 117. Eq uilibrium of a rigid stam p on the bou nd ary of an

elastic half-plane in th e presence of friction 504

§ 117a. Examples.

1°. Sta m p with straight horizontal base 508

2°. Sta m p with straig ht inclined base 509

§ 118. An alternativ e m ethod for the solution of the bo und ary

prob lem s for th e half-plane 510

§ 119. Pro blem of con tac t of two elastic bodies (generalized

plane problem of Hertz) 510

§ 120. Bo und ary problems for the plane with straight cuts 515

CHA PTER 20. SOLUTION OF BOUNDARY PROBLEMS FOR REGION S,

BOUNDED BY CIRCLES, AND FOR THE INFINITE PLANE, CUT

ALONG CIRCULAR ARCS.

§ 121. Transform ation of the general formulae for regions,

bou nded by a circle 525

§ 122. Solu tion of the first and second fundam enta l problem s

for th e region, bo unded by a circle 529

§ 123. The mixed fundam ental problem for a region, bou nded

by a circle 531§ 123a. Ex am ple 536

§ 124. Bo un dary problems for the plane, cut along circular arcs 538

§ 124a. Example. Extension of the plane, cu t along a circular arc 542

CHAPTER 21. SOLUTION OF THE BOUNDARY PROBLEMS FOR REGIONS,

MAPPED ON TO THE CIRCLE BY RATIONAL FUN CTIONS.

§ 125. Transform ation of th e general formulae 546§ 126. Solution of th e first and second fun dam enta l problems 552

§ 127. Solution of the mixed fundamental problem . . . . 554

§ 127a. Exa m ple. Solution of the mixed funda m ental problem

for the plane w ith an elliptic hole 558

§ 128. The problem of con tact with a rigid stam p 560

§ 128a. Examples.

1°. Circular disc 568

2°. Infinite plane w ith a circular hole 5713°. Infinite plane w ith an elliptic hole 574

P A R T VII - Extension torsion and bending of homogeneous and

compound bars 579



XXX CONTENTS

CHAPTER 22. TOR SION AND BENDING OF HOMOGENEOUS BARS

(PROBLEM OF SAINT-VENANT).

§ 129. Sta tem ent of the problem 583§ 130. Ce rtain formulae 586

§ 131. General solution of the torsion problem 587

§ 132. Complex torsion function. Stress functions 594

§ 133. On the solution of the torsion problem for certain pa rti-

cula r cases 597

§ 134. Ap plication of conformal m app ing 599

§ 134a. Examples.

1°. Ep itrochoid al section 6022°. Bo oth's lemniscate 604

3°. The loop of Bernou lli's lemniscate 605

4°. Confocal ellipses. Eccen tric circles 607

§ 135. Ex tens ion by long itudinal forces 607

§ 136. Ben ding by couples, applied to the ends 608

§ 137. Bending by transve rse forces 612

§ 138. On the solution of prob lem s of bending for different

cross-sections 618§ 138a. Exa m ple. B ending of a circular cylinder or tu be . . 619

CHAPT ER 23. TORSION OF BARS CONSISTING OF DIFF ER EN T

MATERIALS.

§ 139. General formulae 621

§ 140. Solution by means of integ ral equ ation s 626

§ 140a. Applications..

1°. Torsion of a circu lar cylinde r, reinforced by a

long itudina l rou nd ba r of a different m ate rial . . . 630

2°. Torsion of a rectangular bar, consisting of two

different rectan gular pa rts 635

CHAPTER 24. EX TE NSIO N AND BEND ING OF BARS, CONSISTING OF

DIFFERENT MATERIALS WITH UNIFORM POISSON'S RATIO.

§ 141. N otatio n 640§ 142. Exten sion 642

§ 143. Bending by a couple 642

§ 144. Ben ding by a transve rse force 643



CONTENTS XXXI

§ 144a. Exam ple . Bending of a compound circular tub e b y a

tran sve rse force, applied to one of its ends 647

CHAPTER 25 . EXTE NSION AND BENDING FOR DIFFERE NT P O I S -SON'S RATIOS.

§ 145. An auxil iary problem of plane deformation . . . . 650

§ 146. The problem of extension and of bend ing by a couple 652

§ 147. Pa rticu lar cases.

1°. Extension of a bar, having an axis of sym m etry 662

2°. B ar with plane of sym m etry, ben t by a couple . . 663

§ 148. Prin cipal axis of exten sion and princip al planes of

bending 664§ 149. Ap plication of complex repres entatio n. Exam ples . . 670

§ 150. Problem of bending by a tran sve rse force 675

APPENDIX 1. On th e concept of a tenso r 682

A P P E N D I X 2. On th e determ inatio n of functions from the ir perfect

differentials in m ul tip ly conn ected regions 697

APPENDIX 3 . Dete rm ina tion of a function of a complex variab le from

its real part. Indefinite integrals of holomorphicfunctions 708

AUTHORS INDEX AND REFEREN CES 713

SUPPLEMENTARY REFE REN CES 727

SUBJECT INDEX 729

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