Nonconvex energy minimization and dislocation structures in ductile single crystals

Journal of the Mechanics and Physics of Solids

Volumn 47, Issue 2, 1999, Pages 397-462

  Ortiz, M ^a   Repetto, E A ^a  

^a CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

A Dislocations; B Constitutive behavior; B Crystal plasticity; C Energy methods; Principles

Indexed keywords

CRYSTAL MICROSTRUCTURE; DISLOCATIONS (CRYSTALS); DUCTILITY; PLASTIC DEFORMATION;

DEFORMATION MICROSTRUCTURE; DISLOCATION PATTERNS; NONCONVEX ENERGY MINIMIZATION;

SINGLE CRYSTALS;

EID: 0033076749     PISSN: 00225096     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0022-5096(97)00096-3     Document Type: Article

References (112)

1. Ackermann, F., Kubin, L., Lepinoux, J., Mugharbi, H., 1984. The dependence of dislocation microstructure on plastic strain amplitude in cyclically strained copper single crystals. Acta Metallurgica 32, 715- 725.
2. Asaro, R.J., 1979. Geometrical effects in the inhomogeneous deformation of ductile single crystals. Acta Metallurgica 27, 445-453.
3. Asaro, R.J., 1983. Micromechanics of crystals and polycrystals. Advances in Applied Mechanics 23, 1-115.
4. Asaro, R.J., Rice, J.R., 1977. Strain localization in ductile single crystals. Journal of the Mechanics and Physics of Solids 25, 309.
5. Ball, J.M., Chu, C., James, R.D., 1995. Hysteresis during stress-induced variant rearrangement. Journal de Physique IV, 5, C245-C8-251.
6. Ball, J.M., James, R.D., 1987. Fine phase mixtures as minimizers of energy. Archive for Rational Mechanics and Analysis 100, 13-52.
7. Bassani, J.L., Wu, T Y., 1991a. Latent hardening in single crystals. 1. Theory and experiments. Proceedings of the Royal Society of London A435, 1-19.
8. Bassani, J.L., Wu, T.Y., 1991b. Latent hardening in single crystals. 2. Analytical characterization and predictions. Proceedings of the Royal Society of London A435, 21-41.
9. Bassim, M., Klassen, R., 1986. Variation in dislocation cell size with local strain in a low alloy steel. Materials Science and Engineering 81, 163-167.
10. Bhattacharya, K., 1991. Wedge-like microstructure in martensites. Acta Metallurgica et Materialia 39, 2431-2444.
11. Bhattacharya, K., 1992. Self-accommodation in martensite. Archive for Rational Mechanics and Analysis 120, 201-244.
12. Bilde-Sorensen, J.B., 1986. Deformation bands in 〈120〉 grains in coarse-grained aluminum. Materials Science and Engineering 81, 211-216.
13. Boulanger, L., Bisson, A., Tavassoli, A.A., 1985. Labyrinth structure and persistent slip bands in fatigued 316 stainless steel. Philosophical Magazine A51, L5-L11.
14. Boutin, J., 1983. Ph.D. thesis, Êcole Polytechnique de Montréal, Canada.
15. Carter, P., Martin, J.B., 1976. Work bounding functions for plastic materials. Journal of Applied Mechanics 98, 434-438.
16. Chaikin, P.M., Lubensky, T.C., 1995. Principles of Condensed Matter Physics. Cambridge University Press, Cambridge, U.K.
17. Charsley, P., 1981. Dislocation arrangements in polycrystalline copper alloys fatigued to saturation. Materials Science and Engineering 47, 181-185.
18. Chipot, M., Kinderlehrer, D., 1988. Equilibrium configurations of crystals. Archive for Rational Mechanics and Analysis 103, 237-277.
19. Crandall, G., Pazy, A., 1969. Semi-groups of nonlinear contractions and dissipative sets. Journal of Functional Analysis 6, 376-418.
20. Crandall, G., Pazy, A., 1970. On accretive sets in Branch spaces. Journal of Functional Analysis 5, 204-217.
21. Cuitiño, A.M., Ortiz, M., 1992. Computational modelling of single crystals. Modelling and Simulation in Materials Science and Engineering 1, 255-263.
22. 2 intermetallic crystals. Materials Science and Engineering A170, 111-123.
23. Dacorogna, B., 1989. Direct Methods in the Calculus of Variations. Springer-Verlag, New York.
24. Dickson, J.I., Boutin, J., LEspeŕance, G., 1986a. An explanation of labyrinth walls in fatigued FCC metals. Acta Metallurgica 34, 1505-1514.
25. Dickson, J.I., Handfield, L., LEpeŕance, G., 1986b. Geometrical factors influencing the orientations of dipolar dislocation structures produced by cyclic deformation of fcc metals. Materials Science and Engineering 81, 477-492.
26. Ericksen, J.L., 1980. Some phase transitions in crystals. Archive for Rational Mechanics and Analysis 73, 99-124.
27. Fonseca, I., 1987. Variational methods for elastic crystals. Archive for Rational Mechanics and Analysis 97, 189-220.
28. Fonseca, I., 1988. The lower quasiconvex envelope of the stored energy function for an elastic crystal. J. Math. Pures et Appl. 67, 175-195.
29. Fourie, J.T., Wilsdorf, H., 1959. A study of slip lines in α-brass as revealed by the electron microscope. Acta Metallurgica 7, 339-349.
30. Franciosi, P., Berveiller, M., Zaoui, A., 1980. Latent hardening in copper and aluminium single crystals. Acta Metallurgica 28, 273.
31. Franciosi, P., Zaoui, A., 1982. Multislip in F.C.C. crystals. A theoretical approach compared with experimental data. Acta Metallurgica 30, 1627.
32. Frank, F.C., 1950. Proceedings of the Pittsburgh symposium on the plastic deformation of crystalline solids. Technical Report, Office of Naval Research, Washington, DC.
33. Gear, C.W., 1971. Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-Hall, Englewood Cliffs, NJ.
34. Gioia, G., Ortiz, M., 1994. The morphology and folding patterns of buckling-driven thin-film blisters. Journal of the Mechanics and Physics of Solids 42, 531-559.
35. Gurtin, M., 1981. An Introduction to Continuum Mechanics. Academic Press, New York.
36. Hansen, N., 1969. Microstructure and flow stress of aluminum and dispersion strengthened aluminum aluminum-oxide products drawn at room-temperature. Transactions of the Metallurgical Society of the AIME 245, 2061-2067.
37. Hansen, N., Kuhlmann-Wilsdorff, D., 1986. Low energy dislocation structures due to unidirectional deformation of low temperatures. Materials Science and Engineering 81, 141-161.
38. Havner, K.S., 1973. On the mechanics of crystalline solids. Journal of the Mechanics and Physics of Solids 21, 383.
39. Higashida, K., Takamura, J., Narita, N., 1986. The formation of deformation bands in f.c.c. crystals. Materials Science and Engineering 81, 239-258.
40. Hill, R., Hutchinson, J.W., 1975. Bifurcation phenomena in the plane strain tension test. Journal of the Mechanics and Physics of Solids 23, 239-264.
41. Hill, R., Rice, J.R., 1972. Constitutive analysis of elastic-plastic crystals at arbitrary strains. Journal of the Mechanics and Physics of Solids 20, 401.
42. Hirth, J.P., Lothe, J., 1968. Theory of Dislocation. McGraw-Hill, New York.
43. Holt, D. L., 1970. Dislocation cell formation in metals. Journal of Applied Physics 41, 3197-3201.
44. Jin, N., 1987. Dislocation structures in cyclically deformed [011] copper crystals. Philosophical Magazine Letters 56, 23-28.
45. Jin, N.Y., 1983. Dislocation-structures in fatigued copper single-crystals oriented for double-slip. Philosophical Magazine 48, L33-L38.
46. Jin, N.Y., Winter, A.T., 1984a. Dislocation structures in cyclically deformed [001] copper crystals. Acta Metallurgica 32, 1173-1176.
47. Jin, N.Y., Winter, T., 1984b. Cyclic deformation of copper single crystals oriented for double slip. Acta Metallurgica 32, 989-995.
48. Kocks, U.F., 1960. Polyslip in single crystals. Acta Metallurgica 8, 345-352.
49. Kocks, U.F., 1964. Latent hardening and secondary slip in aluminum and silver. Transactions of the Metallurgical Society of the AIME 230, 1160.
50. Kocks, U.F., 1966. A statistical theory of flow stress and work-hardening. Philosophical Magazine 13, 541.
51. Kohn, R., 1991. The relaxation of a double-well energy. Continuum Mechanics and Thermodynamics 3, 193-236.
52. Kohn, R.V., Müller, S., 1992. Branching of twins near an austenite-twinned-martensite interface. Philosophical Magazine A66, 697-715.
53. Kohn, R.V., Strang, G., 1986. Optimal design and relaxation of variational problems, I, II and III. Communications on Pure and Applied Mathematics 39, 113-137, 139-182, 353-377.
54. Kuhlmann-Wilsdorf, D., 1989. Theory of plastic deformation : properties of low energy dislocation structures. Materials Science and Engineering A113, 1.
55. Kuhlmann-Wilsdorff, D., 1970. A critical test on theories of work hardening for the case of drawn iron wires. Metallurgical Transactions 1, 3173-3179.
56. Kuhlmann-Wilsdorff, D., van der Merwe, J. H., 1982. Theory of dislocation cell sizes in deformed metals. Materials Science and Engineering 55, 79.
57. Laird, C., Charsley, P., Mugharbi, H., 1986. Low energy dislocation structures produced by cyclic deformation. Materials Science and Engineering 81, 433-450.
58. Laird, C., Wang, Z., Ma, B.T., Chai, H.F., 1989. Low energy dislocation structures produced by cyclic softening. Materials Science and Engineering A113, 245-257.
59. Lee, E.H., 1969. Elastic-plastic deformation at finite strains. Journal of Applied Mechanics 36, 1.
60. Lepistö, T.K., Kuokkala, V.T., Kettunen, P O., 1984. Dislocation arrangements in cyclically deformed copper single crystals. Materials Science and Engineering 81, 457-463.
61. Lepistö, T.K., Kuokkala, V.T., Kettunen, P.O., 1986. Dislocation arrangements in cyclically deformed copper single crystals. Materials Science and Engineering 81, 457-463.
62. LEsperance, G., Vogt, J.B., Dickson, J.L., 1986. The identification of labyrinth wall orientations in cyclically deformed AISI-SAE 316 stainless steel. Materials Science and Engineering 79, 141-147.
63. Lubarda, V.A., Blume, J.A., Needleman, A., 1993. An analysis of equilibrium dislocation distributions. Acta Metallurgica et Materialia 41, 625-642.
64. Lubliner, J., 1972. On the thermodynamic foundations of non-linear solid mechanics. International Journal of Non-Linear Mechanics 7, 237-254.
65. Lubliner, J., 1973. On the structure of the rate equations of materials with internal variables. Acta Mechanica 17, 109-119.
66. Lubliner, J., 1990. Plasticity Theory. Macmillan Publishing Company, New York.
67. Maier, G., 1969. Some theorems for plastic strain rates and plastic strains. Journal de Mécanique 8, 5.
68. Mandel, J., 1972. Plasticité classique et viscoplasticité. Technical Report, Lecture Notes, Int. Centre for Mech. Sci., Udine, Springer, Berlin.
69. Martin, J.B., Ponter, A.R.S., 1966. A note on a work inequality in linear viscoelasticity. Quarterly of Applied Mathematics 24, 161.
70. Mecke, K., Blochwitz, C., 1982. Saturation dislocation structures in cyclically deformed nickel single crystals of different orientations. Crystal Research and Technology 17, 743-758.
71. Mecking, H., Kocks, U.F., 1981. Kinetics of flow and strain hardening. Acta Metallurgica 29, 1865-1875.
72. Modica, L., 1987. Gradient theory of phase transitions and minimal interface criterion. Archive for Rational Mechanics and Analysis 98, 123-142.
73. Moreau, J., 1962. Décomposition orthogonale dans un espace Hilbertien selon deux cônes mutuellement polaires. C. R. Acad. Sci. Paris 255, 233-240.
74. Moreau, J., 1963. Fonctionnelles sous-différentiables. C. R. Acad. Sci. Paris 257, 4117-4119.
75. Moreau, J., 1965. Proximité et dualité dans un space Hilbertien. Bull. Soc. Math. France 93, 273-299.
76. Moreau, J., 1966. Convexity and duality. In Functional Analysis and Optimization, ed. ed. E. Caianello, pp. 145-169. Academic Press.
77. Moreau, J., 1967. Fonctionelles convexes. Séminaires sur les equations à dérivées partielles, Collége de France.
78. Moreau, J., 1970. Sur les lois de frottement, de plasticité et de viscosité. C. R. Acad. Sci. Paris 271, 608-611.
79. Moreau, J., 1971. Sur levolution dun système elasto-visco-plastique. C. R. Acad. Sci. Paris 273, 118-121.
80. Moreau, J., 1974. On unilateral constraints, friction and plasticity. In New Variational Techniques in Mathematical Physics, ed. G. Capriz and G. Stampacchia, pp. 175-322. Centro Internazionale Matematico Estivo, II Ciclo 1973, Edizioni Cremonese, Roma.
81. Moreau, J., 1976. Application of convex analysis to the treatment of elastoplastic systems. In Applications of Methods of Functional Analysis to Problems in Mechanics, ed. P. Germain and B. Nayroles. Springer-Verlag.
82. Mugharbi, H., Ackermann, F., Herz, K., 1979. Persistent slip bands in fatigued face-centered and body-centered cubic metals. Technical Report, American Society for Testing and Materials, Philadelphia.
83. Mugharbi, H., 1978. The cyclic hardening and saturation behaviour of copper single crystals. Materials Science and Engineering 33, 207-223.
84. Mura, T., 1987. Micromechanics of Defects in Solids. Kluwer Academic Publishers, Boston.
85. Nabarro, F.R.N., 1967. Theory of Crystal Dislocations. Oxford University Press, Oxford, U.K.
86. Neumann, P., 1986. Low energy dislocation configurations : a possible key to the understanding of fatigue. Materials Science and Engineering 81, 465-475.
87. Nye, J.F., 1953. Some geometrical relations in dislocated crystals. Acta Metallurgica 1, 153-162.
88. Ortiz, M., 1981. Topics in constitutive theory for inelastic solids. Ph.D. thesis, University of California at Berkeley, Berkeley, California.
89. Otte, H.M., Hren, J.J., 1966. The observation of crystalline imperfections and their role in plastic deformation. Experimental Mechanics 6, 177-193.
90. Pedregal, P., 1993. Laminates and microstructure. European Journal of Applied Mathematics 4, 121-149.
91. Pierce, D., Asaro, R., Needleman, A., 1982. An analysis of nonuniform and localized deformation in ductile single crystals. Acta Metallurgica 30, 1087-1119.
92. Piercy, G.R., Cahn, R.W., Cottrell, A.H., 1955. A study of primary and conjugate slip in crystals of alpha-brass. Acta Metallurgica 3, 331-338.
93. Ramaswami, B., Kocks, U.F., Chalmers, B., 1965. Latent hardening in silver and Ag-Au alloy. Transactions of the Metallurgical Society of the AIME 233, 927.
94. Rasmussen, K., Pedersen, O., 1980. Fatigue of copper polycrystals at low plastic strain amplitudes. Acta Metallurgica 28, 1467-1478.
95. Rice, J.R., 1971. Inelastic constitutive relations for solids : an internal-variable theory and its applications to metal plasticity. Journal of the Mechanics and Physics of Solids 19, 433.
96. Rice, J.R., 1975. Continuum mechanics and thermodynamics of plasticity in relation to microscale deformation mechanisms. In Constitutive Equations in Plasticity, ed. A. Argon, pp. 23-79. MIT Press, Cambridge, MA.
97. Rockafellar, R., 1970. Convex Analysis. Princeton University Press, Princeton, NJ.
98. Saimoto, S., 1963. Low temperature tensile deformation of copper single crystals oriented for multiple slip. Ph.D. thesis, M.I.T., Cambridge, MA.
99. Schmid, E., Boas, N., 1961. Kristall. Plastizität. Springer, Berlin.
100. Silling, S., 1989. Phase changes induced by deformation in isothermal elastic crystals. Journal of the Mechanics and Physics of Solids 37, 293-316.
101. Soechting, J.F., Lance, R.H., 1969. A bounding principle in the theory of work-hardening plasticity. Journal of Applied Mechanics 36, 228.
102. Staker, M.R., Holt, D.L., 1972. The dislocation cell size and dislocation density in copper deformed at temperatures between 25 and 700°C. Acta Metallurgica 20, 569.
103. Steeds, J.W., 1966. Dislocation arrangement in copper single crystals as a function of strain. Proceedings of the Royal Society of London A292, 343-373.
104. Sternberg, P. 1988. The effect of a singular perturbation on nonconvex variational problems. Archive for Rational Mechanics and Analysis 101, 209-260.
105. Teodosiu, C., 1969. A dynamic theory of dislocations and its applications of the theory of the elastic-plastic continuum. In Conf. Fundamental Aspects of Dislocation Theory, ed. J. A. Simmons, Vol. 2, p. 837. National Bureau of Standards Special Publication, Washington.
106. Truskinovsky, L., Zanzotto, G., 1995. Fine-scale microstructures and metastability in one-dimensional elasticity. Meccanica 30, 557-589.
107. Truskinovsky, L., Zanzotto, G., 1996. Ericksens bar revisited : energy wiggles. Journal of the Mechanics and Physics of Solids 44, 1371-1408.
108. Wang, R., Mugharbi, H., 1984. Secondary cyclic hardening in fatigued copper monocrystals and polycrystals. Materials Science and Engineering 63, 147-163.
109. Winter, A.T., 1974. A model for the fatigue of copper at low plastic strain amplitudes. Philosophical Magazine 29, 719-738.
110. Woods, P., 1973. Low-amplitude fatigue of copper and copper-5 at. crystals. Philosophical Magazine 28, 155-193.
111. Yosida, K., 1965. Functional Analysis. Springer-Verlag, Berlin.
112. Yumen, L., 1989. Low-energy dislocation structure in cyclically deformed quench-tempered steel. Materials Science and Engineering A113, 237-244.