CONTENTS I . I N T R O D U C T I O N .

1. Defini t ion of the subject 2. H i s to r i ca l out l ine . 3. Phys ica l background 4. The stress-strain curve

I I . F O U N D A T I O N S O F T H E T H E O R Y 1. The ideal plast ic body 2. The cr i ter ion o f y ie ld ing 3. Strain-hardening 4. The complete stress-strain relations 6. The L e v y - M i s e s and Reuse equations 6. The H e n c k y stress-strain equations 7. Other theories

I I I . G E N E R A L T H E O R E M S . 1. The plast ic potential 2. Uniqueness of a stress dis t r ibut ion under given boundary conditions 3. E x t r e m u m and var ia t iona l principles

I V . T H E S O L U T I O N O F P L A S T I C - E L A S T I C P R O B L E M S . I 1. In t roduct ion . . . . . 2. Theory o f Hohenemser's experiment 3. Combined torsion a n d tension of a thin-walled tube 4. Combined torsion and tension o f a cy l indr ica l bar . 5. Compression under conditions o f plane strain 6. Bend ing under conditions o f plane s t ra in . 7. Bending o f a prismatic beam 8. Torsion o f a pr ismat ic bar . . . . 9. Tors ion o f a bar of non-uniform section

V . T H E S O L U T I O N O F P L A S T I C - E L A S T I C P R O B L E M S . I I 1. The expansion o f a spherical shell . 2. The expansion of a cy l indr ica l tube 3. Theory o f the autofrettage process . . . . 4. Expans ion o f a cy l indr ica l cav i ty in an infinite medium

V I . P L A N E P L A S T I C S T R A I N A N D T H E T H E O R Y O F T H E S L I P L I N E F I E L D . . . . . .

1. Assumpt ion o f a plast ic-r igid material 2. The plane strain equations referred to Cartesian coordinates

1 1 2 4 8

14 14 16 23 33 38 45 48

50 δ0 63 60

70 70 71 74 76 77 79 81 84 94

97 97 106 114 125

. 128 128 129

T i i i C O N T E N T S

3. T h e plane strain, equations referred to the slip-lines 4- Geometry of the slip-line field δ. The numerical calculat ion o f s l ip-l ine fields. 6. The numerical calculat ion o f the veloci ty d is t r ibut ion 7. A n a l y t i c integration of the plane strain equations . 8. Discontinuit ies i n the stress

. 132

. 136

. 140

. 149

. 151

. 157

V I I . T W O - D I M E N S I O N A L P R O B L E M S O F S T E A D Y M O T I O N . 161 1. F o r m u l a t i o n of the problem . . . . . 161 2. Sheet-drawing . . . . . . . 163 3- I roning o f a thin-walled cup . . . . . 178 4. Sheet-extrusion . . . . . . . 181 5. P i e r c ing . . . . . . . . 186 6. S t r ip- ro l l ing . . . . . . . 188 7. Mach in ing . . . . . . . . 206 8. F l o w through a converging channel . . . . 209

V I I I . N O N - S T E A D Y M O T I O N P R O B L E M S I N T W O D I M E N S I O N S . I 213 1. Geometr ic s imi la r i ty and the un i t diagram . . . . 213 2. Wedge-indentat ion. . . . . . . 215 3-. Compression o f a wedge by a flat die . . . . 221 4. E x p a n s i o n of a semi-cyl indrical c av i t y i n a surface . . · 223 5. Compression of a b lock between rough plates . . . 226

D C . N O N - S T E A D Y M O T I O N P R O B L E M S I N T W O D I M E N S I O N S . H 237 1. In t roduct ion . . . . . . . 237 2. F o r m u l a t i o n of the problem . . . . 238 3. Y i e l d i n g of notched bars under tension . , . . 245 4. Plas t ic yie lding round a c a v i t y . . . . . 252 5. Indentat ion and the theory of hardness tests . . . 254

X . A X I A L S Y M M E T R Y . . . . . . 262 1. Fundamenta l equations . . . . . . 262 2. Ex t rus ion from a contract ing cy l indr ica l container . . . 263 3. Compression of a cyl inder under certain distr ibuted loads . 265 4. Cy l indr i ca l tube under ax ia l tension and internal pressure . . 267 5. Tube-sinking . . . . . . . 269 6. Stress dis t r ibut ion i n the neck of a tension specimen . . 272 7. Compression of a cyl inder between rough plates . . . 277 8. Relat ions a long slip-lines and flow-lines. . . . . 278

X I . M I S C E L L A N E O U S T O P I C S . . . . 282 1. Deep-drawing . . . . . . . 282 2. General theory o f sheet-bending . . . . . 287

C O N T E N T S ix

3. Plane strain of a general plastic material . 4. The theory of plane plastic stress, with applications 6. Completely plastic states of stress in a prismatic bar

X I I . P L A S T I C A N I S O T R O P Y -1. The yield criterion . 2. Relations between stress and strain-increment 3. Plastic anisotropy of rolled sheet . 4. Length changes in a twisted tube 5. The earing of deep-drawn cups 6. Variation of the anisotropic parameters during cold-work 7. Theory of plane strain for anisotropic metals

A P P E N D I X E S . . - . . -

294 300 313

317 318 320 321 325 328 332 334

341

A U T H O R I N D E X

S U B J E C T I N D E X

361

364

AUTHOR INDEX Author»' name» are followed by the number» of page» on which paper» or book» are explicitly mentioned. Where essentially distinct part» of the tame source are quoted on different page», all page number» are given. If the same material i» quoted more than once, only the page

where it first occur» it given.

Aldous, C. W., 267. Allans. W. E . . 27. Allen, D . N . de G. , 126, 246, 307. Ansoff, H . I., 162, 172. Aul , E . L . , 287. Baldwin, W . M . , Jr . . 17, 269, 271, 282,

317, 323, 330. Baranski, G. , 78. Barrett, C. S., 4, 329. Bauer, F . B. , 68, 64, 66. Becker, A . J . , 23. Belayev, Ν . Μ . , 124. Beltrami, E . , 20. Bishop, R . F . , 104, 127, 218. Bitter, F . , 326. Bland, D . R. , 189, 197, 201. Boas, W. , 4. Bohlen, E . C., 330. Bourne, L . , 323, 330. Bouwinexq, J . , 19, 135. Bragg, W. L . , 14. Brick, R. M . , 329. Bridgman, P. W., 12, 16, 77, 273, 334. Briggs, G. C , 168. Burghoff, H . L . , 330. CairnB, W. J . , 14, 251. Caratheodory, C., 153. Carrier, G . ¥., 160, 162, 172. Chow, C. C, 334. ChriHtianovitch, S., 143. ChriKtophertion, D. G., 92. Coffin, L . F., Jr., 125. Cook, G., 23, 119, 123, 124, 125. Cook, M . , 199, 277, 317, 322, 329. Cottrell, A . H . , 4, 11. Coulomb. C. Α. , 19. Cox, H . L. , 23. Cunningham, D. M . , 32.

Dana, A . W., 334. Davidenkov, Ν . Ν . , 273. Davies, R. M . , 268. Davis, Ε . Α . , 22, 32, 45, 172. Davis, Η . Ε . , 32. Dehlinger, U . , 23. Dokos, S. J . , 172. Dorn, J . E . , 14, 32, 320, 333. Drucker, D. C , 47. Eddy, R . P., 96. Edwards, S. H . , 22. Eichinger, Α . , 23, 29, 32, 171.

Eiabein, W., 185. Elam, C. F . , 4, 14. Emat, H . , 206, 207.

Fangmeier, E . , 187. Fisher, J . C , 125. Ford, H . , 78, 189, 197, 201, 258. Foster, P. Field, 213. Fraenkel, S. J . , 32, 35, 45. Francis, E . L . , 178. Fried, M . L . , 248. Galin, L . Α . , 262. Geiringer, H . , 88, 136, 167, 233. Gensaroer, M . , 32. Gleyzal, Α . , 48. Goldberg, Α. . 14. Greenberg, H . J . , 58, 64, 66, 67. Guest, J . J . , 23. Hear, Α. , 67, 280. Haigh, B. P. , 17, 20. Handelman, G. H . , 33, 45. Hankirtf, G. Α. , 257. Hanson, D., 27. Hencky, H . , 20, 45, 135, 281. Hermito, R. L . , 218. Heyer, R. H . , 261. H i l l , R., 41, 59, 64, 67, 77, 100, 115, 119,

125, 142, 163, 178, 181, 182, 203, 215, 218, 219, 221, 223, 228, 242, 248, 256, 263, 265, 278, 284, 287, 312, 318, 323, 329, 330, 334.

Hitchcock, J . H . , 189. Hodge, P . G. , 64, 89, 118, 220. Hoff, H . , 325. Hohenemser, K . , 72. Hollomon, J . H . , 14, 29, 30. Howald, T. S., 271, 282, 323, 330. Howard, J . V . , 16. Huber, Μ . Τ., 20. Hume-Rothery, W., 4. Ilyushin, Α . Α . , 46, 47, 48-Ishlinsky, Α . , 281. Iterson, F. K . Th. van., 162, 181. Jackson, K . L . , 287. Jackson, L . R., 30, 320, 333. Jacobs, J . Α . , 252. Karman, Th. von., 67, 198, 280. Klingler, L . J . , 172, 318, 322, 323. Kochendorfer, Α . , 26.

352 A U T H O R I N D E X Korber, F . , 29, 171, 256, 325. Kdeter, W., 317. Hotter, F. , 136, 299.

Lankford, W . T., 32, 320, 333. Larke, E . C , 199, 277. Lee, Ε . Η . , 115, 119, 142, 215, 228. Lessells, J . M . , 22. Levy, M . , 38. L in , C. C , 33, 45. Linicus, W., 177. Lode, W., 18, 22, 44. 45. Low, J . R., 32. Lubahn, J . D., 14, 181, 287. Ludwik, P., 12, 14. 29. Lueg, W., 201. Lunt, R. W., 176, 177. McAdam, D. J . , Jr. , 16. MacGregor, C. W., 12, 22, 125. MacLellan, Q. D . S., 176, 177. Malaval, M . P. , 29. Mandel, J . , 297, 299. Marin, J . , 22. Markov, Α . Α . , 68. Mason, W., 23. Maxwell, J . Clerk, 20. Mebs, R . W. , 15. Melan, E . , 34, 57. Merchant, Μ . Ε . , 207, 208. Meyer, Ο. Ε . , 261. Miller, C. P., 22. Misee, R . von, 20, 21, 38, 51, 141. Mohr, 0., 296. Morkovin, D. , 23. Morrieon, J . L . M . , 23, 72, 94, 123, 124. Mott, N . F . , 104, 127, 218. Nadai, Α . , 20, 32, 41, 45, 46, 76, 84, 87, 93,

94, 117, 122, 136, 192, 197, 211, 230, 234, 253, 255, 256, 269, 308.

Nye, J . F . , 14, 228, 251. Odquist, F . K . G. , 30. Orowan, E . , 14, 189, 192, 201, 202, 261. Oegood, W. R. , 32. Palm. J . H . , 13. Palmer, E . W., 322. Parasyuk, O. S., 253. Parker, E . R. , 32. Pearson, C. E . . 185. Pearson, K . , 38. Philippine. A . H . , 67. Piispanen, V . , 208. Polanyi, M . , 16. Pope, J . Α . , 23. Prager, W., 33, 41, 45, 48, 49, 64, 65, 72,

88, 93, 136, 139, 142, 157, 158. Prandtl, L . , 39, 41, 86, 233, 235, 255, 256,

297. Pumphrey, S. L . , 276. Putnam, W. J . , 23. Quinney. H . . 22, 27, 45, 171.

Reiner, M . , 13. Reuse, Α . , 39, 41, 100. Robertson, Α. , 23. Roderick, J . W., 84. Roe, M . , 23, 32. Roeenhain, W., 206. Roes, A . W., 323, 330. Saehe, G., 14, 23, 171, 172, 177, 181, 186,

248, 255, 269, 287, 318, 322, 323, 334. Sadowsky, Μ . Α . , 67, 135. Saint-Venant, B . de, 38, 117, 131. Sangdahl, G. S., Jr., 287. Scheu, Α . , 29. Schleioher, F. , 21. Schmid, E . . 4, 16. Schmidt, E . , 163. Schmidt, R „ 27, 46. Schroeder, W., 277. Scoble, W. Α. , 23. Seely, F . B. , 23. Seitz, F . , 4. Shaw, F . S., 92, 96. Shepherd, W . M . , 29, 72, 123. Shevchenko, Κ . Η . , 162, 172. Shoji, H . , 14. Sidebotham, 0., 23. Siebel, E . , 162, 181, 187, 201, 256, 277. Siniteky, A . K . , 124. Smith, C. S., 322. Smith, K . F . , 320, 333. Smith, S. L . , 15. Sneddon, I. N „ 253. Snitko, Ν . Κ . , 23. Sokolnikoe, I. S., 83, 84, 86. Sokoloveky, W. W., 17, 48, 84, 90, 93, 96,

124, 141, 162, 169, 172, 235, 250, 256, 258, 299, 303, 304, 307.

Sopwith, D . G., 23, 125. Southwell, R. V. , 83, 84, 92, 97, 131, 245,

307. Spiridonova, N . I., 273. Stanley, R . L . . . 22. Steveneon, A. 0., 253. Sturney, A . O , 206. Swainger, Κ . Η . , 48. Swift, H . W., 94, 168, 269, 282, 286, 328. Symonds, P. S., 141, 250, 263. Tabor, D. , 258, 260. Taylor, G. I., 22, 24, 26, 27, 45, 52, 171,

307. Thompeon, F . C., 178. Thomson, E . G. , 32. Tietz, Τ. Ε . , 14. Timoshenko, S., 83, 84, 94, 97, 131, 190,

258. Todhunter, I., 38. Tracy, D. P., 181. Trefltz, E . , 92. Treeca, H . , 19. Trinke, W., 197. Teolikov. A . T.. 195. Tupper, S. J . , 115, 119, 142, 163, 215, 228.

I Turner, L. B. , 110, 121.

A U T H O R I N D E X 353

Underwood, L . R. , 195, 197. Voce, E . , 13.

Warren, A . 0. , 121, 124. Webster, D. Α . , 277. Weiss, L . , 178. Westergaard, Η . Μ . , 17.

Wheeler, Μ . Α . , 27. White, Ο. Ν . , Jr., 118. Williams, Η . Α. , 84. Wilson, F . H . , 329. Winzer, Α . , 48, 160. Wistreinh, J . O., 178.

Zener, C , 14, 29, 30.

Anisotropy, 29, 44, 45, Chap. X I I . Annealing, 5. Autofrettaged tube, axial expansion of,

121-2; axial stress distribution in, 116, 120; under plane strain, 115-18; with closed ends, 118-24; with open ends, 124.

Bauschinger effect, 8, 16, 24, 30, 72, 121. Bending, of a sheet, 79-81, 287-92; of a

beam, 81-4. — under tension, 292-4.

Cavity, expansion in plate of circular, 307-13; expansion in surface of semi-cylindrical, 223-6; expansion of cylindri­cal, 125-7, 252; expansion of spherical, 103-6.

Channel, flow of plastic material through, 209-12.

Characteristics, theory of, 345-8; in deep-drawing, 285; in expansion of a tube, 113; in plane strain, 132, 296-7, 336; in plane stress, 300-1; in torsion, 89, 95-6.

Compressibility during plastic distortion, 26-7.

Compression, of block between rough plates, 226-36; of block between smooth plates, 77-9; of cylinder between rough plates, 277; of cylinder under dis­tributed load, 265-7 ; of wedge by a flat die, 221-2, 351.

Consistent strain-increment for plastic-rigid body, 59.

Constraint factor in notched-bar test, 250. Creep, physical mechanism of, 5. Criterion of yielding, definition of, 15;

dependence on cold-work of, 23-32; experimental investigations of, 19,22-3; for anisotropic metals, 318; geometrical representation of, 17, 295, 301; influence of hydrostatic stress on, 16.

Crystal, lattice structure of, 4; plastic glide in, 6.

Cycloidal slip-line field. 232. Cylindrical cavity in infinite medium,

125-7, 252. — tube, under internal pressure, 115-25;

under tension and internal pressure, 267-9; under tension and torsion, 71-5.

Deep-drawing, 282-7. 328-32. Deviatoric, strain-increment, 27 ; stress, 16. Discontinuity, in stress, 93, 157; in stress

gradient, 133; in stress rate, 55, 239; in velocity, 150, 160; in velocity gradient, 134.

Drawing, deformation in, 171; die pres­sure and load in, 167, 172-3; efficiency of, 170; influence of back-pull in, 175;

INDEX of sheet, 163-76; of wire, 176-8; stand­ing wave in, 168.

Earing of deep-drawn cups, 328-32. Elastic limit, see Criterion of yielding. — moduli, influence of cold-work on, 6,15. Elastic-plastic boundary, conditions along,

55, 132, Chaps. IV , V , I X . Envelope of Mohr circles, 295-6. Equivalent, strain, 30; stress, 26. Extremum principles, for elastic body,

60-3; for plastic-elastic body, 63-6; for plastic-rigid body, 66-8.

Extrusion, deformation in, 185; direct, 186; efficiency of, 185; from contract­ing container, 263-5; inverted, 182; pressure of, 185.

Forward slip in rolling, 193.

Glide system, 6, 34, 50. Grain boundaries, 7.

Hardness test, significance of, 258-61. Hole, see Cavity. Hydrostatic stress, influence on necking,

12; influence on yielding, 16, 294-5. Hysteresis loop, 10.

Indentation, by cone, 218; by flat oylinder 281; by rectangular die, 254-8, 339-40; by sphere, 281; by wedge, 215-20.

Internal stress, 24, 35, 37, 50, 328. Invariants, 15, 30, 343. Ironing, 178-81. Isotropy, conditions for, 7.

Lattice imperfections, 6. Lode's variables, 18, 36, 44. Liiders bands, 10, 52, 255-6.

Machining, 206-9. Maximum plastic work, principle of, 66. Mild steel, yield-point in, 10, 23, 52, 123-4.

Necking in tension, of cylindrical bar, 272-7; of thin strip, 323-5.

Neutral loading, 33. — surface in bending, 79, 82, 288-91. Notched bar under tension, 245-52.

Orientation, preferred, 8, 317.

Piercing, 106, 186-8. Plane plastic strain, 77-8, Chap. V I .

stress, 300-7. Plastic potential, 50. Plastio-elaetic boundary, conditions along,

55, 132, Chaps. IV, V, I X . Plastic-rigid body, 38-9, 58.128, Chap. I X .

S U B J E C T I N D E X 355

Prismatic beam, bending of, 81-4; tension and torsion of, 75-6; torsion of, 84-94; torsion, tension, and bending of, 313-6.

Reduced stress, 16. Residual stresses in overstrained shell,

102-3, 121. Riemann method, 153-4. Rolling, efficiency of, 199; empirical

formulae in theory of, 199; influence of strip-tension on, 202; influence of work-hardening on, 200-1; pressure dis­tribution in, 199-201; roll-distortion in, 189-90.

Shear, deformation in, 326-7. Size effects, 23, 124. Slip bands, 6. Slip-lines, definition of, 134; Hencky'e

theorems on, 136-8; in anisotropic metals, 336; in axially-symmetric state, 278; numerical calculation of, 140-9; variation of curvatures of, 138-9; varia­tion of pressure along, 135; variation of velocity along, 136.

Soil mechanics, 294-300. Spherical cavity in infinite medium, 103-4,

106. Spherical shell, under internal pressure,

97-103, 104-5. Statically determined problems, 87, 100,

111, 131, 242-5. Strain, engineering, 9; equivalent, 30;

increment of, 26; logarithmic or natural, 9, 28.

Strain-hardening, see Work-hardening. Stress, deviatoric or reduced, 16; equiva­

lent, 26. Stress-strain curves, comparison of, 27-31;

significance of, 8-13. — relations, experimental, 44-5, 71-2;

geometrical representation of, 41, 301; inversion of, 68; theoretical, 38-40.

Suffix notation and summation convention, 342.

Thermal phenomena, 14. Torsion, membrane-roof analogy for, 86-7.

93; of anisotropic tube, 325-8; of annealed bar, 93; of non-uniform bar, 94-6; of uniform bar, 85-92; warping of section in, 88-9.

Tube drawing and sinking, 269-72.

Uniqueness theorems, for plastic-elastic body, 53-8, 242-5; for plastic-rigid body, 58-60.

Unit diagram, 213.

Variational principles, tee Extremum principles.

Viscous fluid, contrast to plastic solid, 38.

Work-hardening, dependence on plastic strain, 26-32; physical mechanism of, 5.

Yielding, tee Criterion of. Yield-point, in annealed mild steel, 10, 23,

52, 123-4.