×

Nonconvex energy minimization and dislocation structures in ductile single crystals. (English) Zbl 0964.74012

From the summary: Plastically deformed crystals are often observed to develop intricate dislocation patterns such as the labyrinth, mosaic, fence and carpet structures. In this paper, we give an energetic interpretation of such dislocation structures with the aid of direct methods of the calculus of variations. We formulate the theory in terms of deformation fields and of the dislocations as manifestations of the incompability of the plastic deformation gradient field. Within this framework, we show that the incremental displacements of inelastic solids follow as minimizers of a suitably defined pseudoelastic energy function. In addition, we show that a characteristic length scale can be built into the theory by taking into account the self-energy of dislocations. The extended theory leads to scaling laws which are in good qualitative and quantitative argreement with observations.

MSC:

74E15 Crystalline structure
74G65 Energy minimization in equilibrium problems in solid mechanics
74C15 Large-strain, rate-independent theories of plasticity (including nonlinear plasticity)
82D25 Statistical mechanics of crystals
PDFBibTeX XMLCite
Full Text: DOI

References:

[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.; 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.; 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.; 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.; Asaro, R.J., Rice, J.R., 1977. Strain localization in ductile single crystals. Journal of the Mechanics and Physics of Solids 25, 309. · Zbl 0375.73097
[5] Ball, J.M., Chu, C., James, R.D., 1995. Hysteresis during stress-induced variant rearrangement. Journal de Physique IV, 5, C245-C8-251.; 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.; Ball, J.M., James, R.D., 1987. Fine phase mixtures as minimizers of energy. Archive for Rational Mechanics and Analysis 100, 13-52. · Zbl 0629.49020
[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.; 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. · Zbl 0731.73020
[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.; 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. · Zbl 0731.73021
[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.; 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.; 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.; Bhattacharya, K., 1992. Self-accommodation in martensite. Archive for Rational Mechanics and Analysis 120, 201-244. · Zbl 0771.73007
[12] Bilde-Sorensen, J.B., 1986. Deformation bands in 〈120〉 grains in coarse-grained aluminum. Materials Science and Engineering 81, 211-216.; 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.; 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.; 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.; Carter, P., Martin, J.B., 1976. Work bounding functions for plastic materials. Journal of Applied Mechanics 98, 434-438. · Zbl 0357.73038
[16] Chaikin, P.M., Lubensky, T.C., 1995. Principles of Condensed Matter Physics. Cambridge University Press, Cambridge, U.K.; 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.; 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.; Chipot, M., Kinderlehrer, D., 1988. Equilibrium configurations of crystals. Archive for Rational Mechanics and Analysis 103, 237-277. · Zbl 0673.73012
[19] Crandall, G., Pazy, A., 1969. Semi-groups of nonlinear contractions and dissipative sets. Journal of Functional Analysis 6, 376-418.; Crandall, G., Pazy, A., 1969. Semi-groups of nonlinear contractions and dissipative sets. Journal of Functional Analysis 6, 376-418. · Zbl 0182.18903
[20] Crandall, G., Pazy, A., 1970. On accretive sets in Branch spaces. Journal of Functional Analysis 5, 204-217.; Crandall, G., Pazy, A., 1970. On accretive sets in Branch spaces. Journal of Functional Analysis 5, 204-217. · Zbl 0198.18503
[21] Cuitiño, A.M., Ortiz, M., 1992. Computational modelling of single crystals. Modelling and Simulation in Materials Science and Engineering 1, 255-263.; Cuitiño, A.M., Ortiz, M., 1992. Computational modelling of single crystals. Modelling and Simulation in Materials Science and Engineering 1, 255-263.
[22] Cuitiño, A.M., Ortiz, M., 1993. Constitutive modeling of L \(1_2\); Cuitiño, A.M., Ortiz, M., 1993. Constitutive modeling of L \(1_2\)
[23] Dacorogna, B., 1989. Direct Methods in the Calculus of Variations. Springer-Verlag, New York.; Dacorogna, B., 1989. Direct Methods in the Calculus of Variations. Springer-Verlag, New York. · Zbl 0703.49001
[24] Dickson, J.I., Boutin, J., LEspeŕance, G., 1986a. An explanation of labyrinth walls in fatigued FCC metals. Acta Metallurgica 34, 1505-1514.; 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.; 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.; Ericksen, J.L., 1980. Some phase transitions in crystals. Archive for Rational Mechanics and Analysis 73, 99-124. · Zbl 0429.73007
[27] Fonseca, I., 1987. Variational methods for elastic crystals. Archive for Rational Mechanics and Analysis 97, 189-220.; Fonseca, I., 1987. Variational methods for elastic crystals. Archive for Rational Mechanics and Analysis 97, 189-220. · Zbl 0611.73023
[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.; Fonseca, I., 1988. The lower quasiconvex envelope of the stored energy function for an elastic crystal. J. Math. Pures et Appl. 67, 175-195. · Zbl 0718.73075
[29] Fourie, J.T., Wilsdorf, H., 1959. A study of slip lines in \(α\); Fourie, J.T., Wilsdorf, H., 1959. A study of slip lines in \(α\)
[30] Franciosi, P., Berveiller, M., Zaoui, A., 1980. Latent hardening in copper and aluminium single crystals. Acta Metallurgica 28, 273.; 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.; 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.; 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.; Gear, C.W., 1971. Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-Hall, Englewood Cliffs, NJ. · Zbl 1145.65316
[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.; 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. · Zbl 0832.73051
[35] Gurtin, M., 1981. An Introduction to Continuum Mechanics. Academic Press, New York.; Gurtin, M., 1981. An Introduction to Continuum Mechanics. Academic Press, New York. · Zbl 0559.73001
[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.; 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.; 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.; Havner, K.S., 1973. On the mechanics of crystalline solids. Journal of the Mechanics and Physics of Solids 21, 383. · Zbl 0277.73041
[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.; 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.; 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. · Zbl 0331.73048
[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.; 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. · Zbl 0254.73031
[42] Hirth, J.P., Lothe, J., 1968. Theory of Dislocation. McGraw-Hill, New York.; Hirth, J.P., Lothe, J., 1968. Theory of Dislocation. McGraw-Hill, New York. · Zbl 0096.45303
[43] Holt, D. L., 1970. Dislocation cell formation in metals. Journal of Applied Physics 41, 3197-3201.; 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.; 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.; 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.; 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.; 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.; 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.; 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.; 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.; Kohn, R., 1991. The relaxation of a double-well energy. Continuum Mechanics and Thermodynamics 3, 193-236. · Zbl 0825.73029
[52] Kohn, R.V., Müller, S., 1992. Branching of twins near an austenite-twinned-martensite interface. Philosophical Magazine A66, 697-715.; 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.; 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. · Zbl 0609.49008
[54] Kuhlmann-Wilsdorf, D., 1989. Theory of plastic deformation : properties of low energy dislocation structures. Materials Science and Engineering A113, 1.; 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.; 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.; 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.; 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.; 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.; Lee, E.H., 1969. Elastic-plastic deformation at finite strains. Journal of Applied Mechanics 36, 1. · Zbl 0179.55603
[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.; 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.; 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.; 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.; 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.; Lubliner, J., 1972. On the thermodynamic foundations of non-linear solid mechanics. International Journal of Non-Linear Mechanics 7, 237-254. · Zbl 0265.73005
[65] Lubliner, J., 1973. On the structure of the rate equations of materials with internal variables. Acta Mechanica 17, 109-119.; Lubliner, J., 1973. On the structure of the rate equations of materials with internal variables. Acta Mechanica 17, 109-119. · Zbl 0279.73002
[66] Lubliner, J., 1990. Plasticity Theory. Macmillan Publishing Company, New York.; Lubliner, J., 1990. Plasticity Theory. Macmillan Publishing Company, New York. · Zbl 0745.73006
[67] Maier, G., 1969. Some theorems for plastic strain rates and plastic strains. Journal de Mécanique 8, 5.; Maier, G., 1969. Some theorems for plastic strain rates and plastic strains. Journal de Mécanique 8, 5. · Zbl 0176.25901
[68] Mandel, J., 1972. Plasticité classique et viscoplasticité. Technical Report, Lecture Notes, Int. Centre for Mech. Sci., Udine, Springer, Berlin.; 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.; 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.; 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.; 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.; Modica, L., 1987. Gradient theory of phase transitions and minimal interface criterion. Archive for Rational Mechanics and Analysis 98, 123-142. · Zbl 0616.76004
[73] Moreau, J., 1962. Décomposition orthogonale dans un espace Hilbertien selon deux cônes mutuellement polaires. C. R. Acad. Sci. Paris 255, 233-240.; Moreau, J., 1962. Décomposition orthogonale dans un espace Hilbertien selon deux cônes mutuellement polaires. C. R. Acad. Sci. Paris 255, 233-240. · Zbl 0109.08105
[74] Moreau, J., 1963. Fonctionnelles sous-différentiables. C. R. Acad. Sci. Paris 257, 4117-4119.; Moreau, J., 1963. Fonctionnelles sous-différentiables. C. R. Acad. Sci. Paris 257, 4117-4119. · Zbl 0137.31401
[75] Moreau, J., 1965. Proximité et dualité dans un space Hilbertien. Bull. Soc. Math. France 93, 273-299.; Moreau, J., 1965. Proximité et dualité dans un space Hilbertien. Bull. Soc. Math. France 93, 273-299. · Zbl 0136.12101
[76] Moreau, J., 1966. Convexity and duality. In Functional Analysis and Optimization, ed. ed. E. Caianello, pp. 145-169. Academic Press.; 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.; 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.; 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.; 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.; 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.; 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. · Zbl 0337.73004
[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.; 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.; 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.; Mura, T., 1987. Micromechanics of Defects in Solids. Kluwer Academic Publishers, Boston. · Zbl 0652.73010
[85] Nabarro, F.R.N., 1967. Theory of Crystal Dislocations. Oxford University Press, Oxford, U.K.; 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.; 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.; 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.; 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.; 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.; Pedregal, P., 1993. Laminates and microstructure. European Journal of Applied Mathematics 4, 121-149. · Zbl 0779.73050
[91] Pierce, D., Asaro, R., Needleman, A., 1982. An analysis of nonuniform and localized deformation in ductile single crystals. Acta Metallurgica 30, 1087-1119.; 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.; 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.; 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.; 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.; 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. · Zbl 0235.73002
[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.; 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.; Rockafellar, R., 1970. Convex Analysis. Princeton University Press, Princeton, NJ. · Zbl 0193.18401
[98] Saimoto, S., 1963. Low temperature tensile deformation of copper single crystals oriented for multiple slip. Ph.D. thesis, M.I.T., Cambridge, MA.; 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.; 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.; Silling, S., 1989. Phase changes induced by deformation in isothermal elastic crystals. Journal of the Mechanics and Physics of Solids 37, 293-316. · Zbl 0679.73060
[101] Soechting, J.F., Lance, R.H., 1969. A bounding principle in the theory of work-hardening plasticity. Journal of Applied Mechanics 36, 228.; Soechting, J.F., Lance, R.H., 1969. A bounding principle in the theory of work-hardening plasticity. Journal of Applied Mechanics 36, 228. · Zbl 0184.51601
[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.; 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.; 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.; Sternberg, P.. 1988. The effect of a singular perturbation on nonconvex variational problems. Archive for Rational Mechanics and Analysis 101, 209-260. · Zbl 0647.49021
[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.; 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.; Truskinovsky, L., Zanzotto, G., 1995. Fine-scale microstructures and metastability in one-dimensional elasticity. Meccanica 30, 557-589. · Zbl 0836.73061
[107] Truskinovsky, L., Zanzotto, G., 1996. Ericksens bar revisited : energy wiggles. Journal of the Mechanics and Physics of Solids 44, 1371-1408.; 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.; 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.; 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.; 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.; Yosida, K., 1965. Functional Analysis. Springer-Verlag, Berlin. · Zbl 0126.11504
[112] Yumen, L., 1989. Low-energy dislocation structure in cyclically deformed quench-tempered steel. Materials Science and Engineering A113, 237-244.; Yumen, L., 1989. Low-energy dislocation structure in cyclically deformed quench-tempered steel. Materials Science and Engineering A113, 237-244.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.