the Mathematical Theory of Plasticity

CONTENTS I. I N T R O D U C T I O N . 1. Definition of the subject 2. Historical outline . 3. Physical background 4. T h e stress-strain curve II.

1 1 2 4 8 14 14 16 23 33 38 45 48

FOUNDATIONS OF T H E T H E O R Y 1. The ideal plastic body 2. T h e criterion o f yielding 3. Strain-hardening 4. The complete stress-strain relations 6. T h e L e v y - M i s e s a n d Reuse equations 6. The H e n c k y stress-strain equations 7. Other theories

III. G E N E R A L T H E O R E M S . 1. The plastic potential 2. Uniqueness of a stress distribution under given boundary conditions 3. E x t r e m u m a n d variational principles

50

IV. T H E SOLUTION O F PLASTIC-ELASTIC PROBLEMS. 1. Introduction . . . . . 2. Theory o f Hohenemser's experiment 3. Combined torsion a n d tension of a thin-walled tube 4. Combined torsion a n d tension o f a cylindrical bar . 5. Compression under conditions o f plane strain 6. B e n d i n g under conditions o f plane strain . 7. B e n d i n g o f a prismatic beam 8. Torsion o f a prismatic bar . . . . 9. Torsion o f a bar of non-uniform section

I

70 70 71 74 76 77 79 81 84 94

V. T H E SOLUTION OF PLASTIC-ELASTIC PROBLEMS. 1. T h e expansion o f a spherical shell . 2. T h e expansion of a cylindrical tube 3. Theory o f the autofrettage process . . . 4. E x p a n s i o n o f a cylindrical cavity i n an infinite medium

II

97 97 106 114 125

.

δ0

63 60

VI. P L A N E PLASTIC STRAIN A N D T H E T H E O R Y OF T H E SLIP . 128 LINE FIELD . . . . . . 128 1. Assumption o f a plastic-rigid material 129 2. T h e plane strain equations referred to Cartesian coordinates

Tiii

CONTENTS 3. 4δ. 6. 7. 8.

T h e plane strain, equations referred t o the slip-lines Geometry of the slip-line field T h e numerical calculation o f slip-line fields. T h e numerical calculation o f the velocity distribution A n a l y t i c integration of the plane strain equations . Discontinuities i n the stress

. . . . . .

132 136 140 149 151 157

VII. 1. 2. 34. 5. 6. 7. 8.

TWO-DIMENSIONAL PROBLEMS OF STEADY F o r m u l a t i o n of the problem . . . Sheet-drawing . . . . . Ironing o f a thin-walled c u p . . . Sheet-extrusion . . . . . Piercing . . . . . . Strip-rolling . . . . . Machining . . . . . . F l o w through a converging channel . .

.

161 161 163 178 181 186 188 206 209

VIII. 1. 2. 3-. 4. 5.

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 Geometric similarity a n d the unit diagram . . . . 213 Wedge-indentation. . . . . . . 215 Compression o f a wedge b y a flat die . . . . 221 E x p a n s i o n of a semi-cylindrical c a v i t y i n a surface . . · 223 Compression of a b l o c k between rough plates . . . 226

MOTION . . . . . . . . . . . . . . .

.

DC. 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 1. Introduction . . . . . . . 2. F o r m u l a t i o n of the problem . . . . 3. Y i e l d i n g of notched bars under tension . , . . 4. Plastic yielding round a c a v i t y . . . . . 5. Indentation a n d the theory of hardness tests . . .

237 237 238 245 252 254

X. AXIAL SYMMETRY . . . . 1. F u n d a m e n t a l equations . . . . . 2. E x t r u s i o n from a contracting cylindrical container . 3. Compression of a cylinder under certain distributed loads 4. Cylindrical tube under axial tension a n d internal pressure 5. Tube-sinking . . . . . . 6. Stress distribution i n the neck of a tension specimen 7. Compression of a cylinder between rough plates . 8. Relations along slip-lines a n d flow-lines. . . .

.

262 262 263 265 267 269 272 277 278

.

. 282 282 287

X I . MISCELLANEOUS TOPICS 1. Deep-drawing . . . 2. General theory o f sheet-bending .

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. . . .

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.

.

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CONTENTS

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 XII. 1. 2. 3. 4. 5. 6. 7.

PLASTIC ANISOTROPY The yield criterion . Relations between stress and strain-increment Plastic anisotropy of rolled sheet . Length changes in a twisted tube The earing of deep-drawn cups Variation of the anisotropic parameters during cold-work Theory of plane strain for anisotropic metals

APPENDIXES

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294 300 313 317 318 320 321 325 328 332 334 341

AUTHOR INDEX

361

SUBJECT I N D E X

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. Hill, R., 41, 59, 64, 67, 77, 100, 115, 125, 142, 163, 178, 181, 182, 203, 218, 219, 221, 223, 228, 242, 248, 263, 265, 278, 284, 287, 312, 318, 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, 48Ishlinsky, 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.

119, 215, 256, 323,

352

AUTHOR

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. Lin, 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.

INDEX 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.

AUTHOR Underwood, L . R., 195, 197. Voce, E . , 13. Warren, A. 0., 121, 124. Webster, D. Α . , 277. Weiss, L . , 178. Westergaard, Η . Μ . , 17.

INDEX 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.

353

INDEX 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, 30713; expansion in surface of semicylindrical, 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 deepdrawing, 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 plasticrigid 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;

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. I V , 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 .

SUBJECT 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 workhardening 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.

INDEX

355

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