Abstract.
We consider single-crystal plasticity in the limiting case of infinite latent hardening, which signifies that the crystal must deform in single slip at all material points. This requirement introduces a nonconvex constraint, and thereby induces the formation of fine-scale structures. We restrict attention throughout to linearized kinematics and deformation theory of plasticity, which is appropriate for monotonic proportional loading and confers the boundary value problem of plasticity a well-defined variational structure analogous to elasticity.
We first study a scale-invariant (local) problem. We show that, by developing microstructures in the form of sequential laminates of finite depth, crystals can beat the single-slip constraint, i.e., the macroscopic (relaxed) constitutive behavior is indistinguishable from multislip ideal plasticity. In a second step, we include dislocation line energies, and hence a length scale, into the model. Different regimes lead to several possible types of microstructure patterns. We present constructions which achieve the various optimal scaling laws, and discuss the relation with experimentally known scalings, such as the Hall-Petch law.
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References
Ambrosio, L., Coscia, A., Dal Maso, G.: Fine properties of functions with bounded deformation. Arch. Ration. Mech. Anal. 139, 201–238 (1997)
Anzellotti, G., Giaquinta, M.: Existence of the displacements field for an elasto-plastic body subject to Hencky’s law and von Mises yield condition. Manuscripta Math. 32, 101–136 (1980)
Anzellotti, G., Giaquinta, M.: On the existence of the fields of stresses and displacements for an elasto-perfectly plastic body in static equilibrium. J. Math. Pures Appl. 61, 219–244 (1982)
Aubry, S., Ortiz, M.: The mechanics of deformation-induced subgrain dislocation structures in metallic crystals at large strains. Proc. R. Soc. Lond. A 459, 3131–3158 (2003)
Bassani, J.L., Wu, T.Y.: Latent hardening in single crystals, 2, analytical characterization and predictions. Proc. Roy. Soc. London A 435, 21–41 (1991)
Bassim, M.N., Klassen, R.J.: Variation in dislocation cell size with local strain in a low alloy steel. Mat. Sci. Eng. 81, 163–167 (1986)
Ben Belgacem, H., Conti, S., DeSimone, A., Müller, S.: Energy scaling of compressed elastic films. Arch. Ration. Mech. Anal. 164, 1–37 (2002)
Braides, A., Defranceschi, A.: Homogenization of multiple integrals. Claredon Press, Oxford, 1998
Braides, A., Defranceschi, A., Vitali, E.: A relaxation approach to Hencky’s plasticity. Appl. Math. Optim. 35, 45–68 (1997)
Choksi, R., Kohn, R.V.: Bounds on the micromagnetic energy of a uniaxial ferromagnet. Comm. Pure Appl. Math. 51, 259–289 (1998)
Choksi, R., Kohn, R.V., Otto, F.: Domain branching in uniaxial ferromagnets: a scaling law for the minimum energy. Comm. Math. Phys. 201, 61–79 (1999)
Choksi, R., Kohn, R.V., Otto, F.: Energy minimization and flux domain structure in the intermediate state of a type-I superconductor. J. Nonlinear Sci. 14, 119–171 (2004)
Conti, S.: Branched microstructures: scaling and asymptotic self-similarity. Comm. Pure Appl. Math. 53, 1448–1474 (2000)
Conti, S., Faraco, D., Maggi, F.: A new approach to counterexamples to L1 estimates: Korn’s inequality, geometric rigidity, and regularity for gradients of separately convex functions. MPI-MIS Preprint 93/2003. To appear in Arch. Ration. Mech. Anal. 175, 287–300 (2005)
Dal Maso, G.: An introduction to Γ-convergence. Birkhäuser, Boston, 1993
Franciosi, P., Berveiller, M., Zaoui, A.: Latent hardening in copper and aluminium single crystals. Acta Metall. 28, 273–283 (1980)
Franciosi, P., Zaoui, A.: Glide mechanisms in bcc crystals: An investigation of the case of α-iron through multislip and latent hardening tests. Acta Metall. 31, 1331 (1983)
Goffman, C., Serrin, J.: Sublinear functions of measures and variational integrals. Duke Math. J. 31, 159–178 (1964)
Grosskreutz, J.C., Mughrabi, H.: Description of the Work-Hardened Structure at Low Temperature in Cyclic Deformation. In: Consitutive Equations in Plasticity (Cambridge, Mass.) (A. S. Argon, ed.), MIT Press, 1975, pp. 251–326
Hall, E.O.: The deformation and ageing of mild steel: III discussion of results. Proc. Phys. Soc. Lond. B 64, 747–753 (1951)
Hansen, N., Hughes, D.A.: Analysis of large dislocation populations in deformed metals. Phys. Stat. Sol. A 149, 155–172 (1995)
Hardt, R., Kinderlehrer, D.: Elastic plastic deformation. Appl. Math. Optim. 10, 203–246 (1983)
Hirth, J.P., Lothe, J.: Theory of dislocations. McGraw-Hill, 1968
Hubert, A.: Zur Theorie der zweiphasigen Domänenstrukturen in Supraleitern und Ferromagneten. Phys. Stat. Sol 24, 669–682 (1967)
Hughes, D.A., Dawson, D.B., Korellis, J.S., Weingarten, L.I.: Near-surface microstructures developing under large sliding loads. J. Mat. Engin. Perform. 3, 459–475 (1994)
Hughes, D.A., Hansen, N.: Microstructural evolution in nickel during rolling from intermediate to large strains. Met. Trans. A 24, 2021–2037 (1993)
Hughes, D.A., Nix, W.D.: Strain hardening and substructural evolution in Ni-Co solid solutions at large strains. Materials Science and Engineering A122, 153–172 (1989)
Jin, N.Y., Winter, T.: Cyclic deformation of copper single crystals oriented for double slip. Acta Metall. 32, 989–995 (1984)
Jin, W., Sternberg, P.: Energy estimates of the von Kármán model of thin-film blistering. J. Math. Phys. 42, 192–199 (2001)
Kirchheim, B., Müller, S., Šverák, V.: Studying nonlinear pde by geometry in matrix space. In: Geometric analysis and nonlinear partial differential equations (S. Hildebrandt & H. Karcher, eds.), Springer-Verlag, 2003, pp. 347–395
Kocks, U.F.: Latent hardening and secondary slip in aluminum and silver. Trans. Metall. Soc. AIME 230, 1160 (1964)
Kohn, R.V., Müller, S.: Branching of twins near an austenite-twinned-martensite interface. Phil. Mag. A 66, 697–715 (1992)
Kohn, R.V., Müller, S.: Surface energy and microstructure in coherent phase transitions. Comm. Pure Appl. Math. 47, 405–435 (1994)
Landau, L.D.: The intermediate state of supraconductors. Nature 141, 688 (1938)
Landau, L.D.: On the theory of the intermediate state of superconductors. J. Phys. USSR 7, 99 (1943)
Martin, J.B.: Plasticity: Fundamentals and general results. MIT Press, Cambridge, USA, 1975
Mughrabi, H.: Description of the Dislocation Structure after Unidirectional Deformation at Low Temperatures. In: Constitutive Equations in Plasticity (Cambridge, Mass.) (A. S. Argon, ed.), MIT Press, 1975, pp. 199–250
Müller, S.: Variational models for microstructure and phase transitions. In: Calculus of variations and geometric evolution problems (F. Bethuel et al., eds.), Springer Lecture Notes in Math. 1713, Springer-Verlag, 1999, pp. 85–210
Mura, T.: Micromechanics of defects in solids. Kluwer Academic Publishers, Boston, 1987
Nye, J.F.: Some geometrical relations in dislocated crystals. Acta Metall. 1, 153–162 (1953)
Ornstein, D.A.: A non-inequality for differential operators in the L1-norm. Arch. Ration. Mech. Anal. 11, 40–49 (1962)
Ortiz, M., Repetto, E.A.: Nonconvex energy minimization and dislocation structures in ductile single crystals. J. Mech. Phys. Solids 47, 397–462 (1999)
Ortiz, M., Repetto, E.A., Stainier, L.: A theory of subgrain dislocation structures. J. Mech. Phys. Solids 48, 2077–2114 (2000)
Petch, N.J.: The cleavage strength of polycrystals. J. Iron Steel Inst. 174, 25–28 (1953)
Pierce, D., Asaro, R.J., Needleman, A.: An analysis of nonuniform and localized deformation in ductile single crystals. Acta Metall. 30, 1087–1119 (1982)
Piercy, G.R., Cahn, R.W., Cottrell, A.H.: A study of primary and conjugate slip in crystals of alpha-brass. Acta Metall. 3, 333–338 (1955)
Privorotskii, I.: Thermodynamic theory of domain structures. Wiley, New York, 1976
Rasmussen, K.V., Pedersen, O.B.: Fatigue of copper polycrystals at low plastic strain amplitudes. Acta Metall. 28, 1467–1478 (1980)
Reshetnyak, Yu.G.: Weak convergence of completely additive vector functions on a set. Siberian Math. J. 9, 1039–1045 (1968)
Saimoto, S.: Low temperature tensile deformation of copper single crystals oriented for multiple slip. Ph.D. thesis, M. I. T., Cambridge, Mass., 1963
Suquet, P.: Existence et régularité des solutions des équations de la plasticité. C. R. Acad. Sc. Paris 286, 1201–1204 (1978)
Temam, R.: Mathematical problems in plasticity. Bordas, Paris, 1985
Temam, R., Strang, G.: Functions of bounded deformation. Arch. Ration. Mech. Anal. 75, 7–21 (1980)
Wu, T.Y., Bassani, J.L., Laird, C.: Latent hardening in single crystals, 1, theory and experiments. Proc. Roy. Soc. London A 435, 1–19 (1991)
Zimmer, W.H., Hecker, S.S., Rohr, D.L., Murr, L.E.: Large strain plastic deformation of commercially pure nickel. Metal Science 17, 198–206 (1983)
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Conti, S., Ortiz, M. Dislocation Microstructures and the Effective Behavior of Single Crystals. Arch. Rational Mech. Anal. 176, 103–147 (2005). https://doi.org/10.1007/s00205-004-0353-2
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DOI: https://doi.org/10.1007/s00205-004-0353-2