Abstract
An extended crystal plasticity theory that accounts for the length-scale effects in plastic strain gradient fields is presented. First, foundations and kinematics of crystal plasticity theory is reviewed. Then, experimental evidences for the size-effects in small-sized bent single crystals are presented. Total amounts of apparent strain hardening, which were experimentally observed, are decomposed into isotropic and kinematic hardening components. Physically-based models are formulated to describe the size-dependent isotropic and kinematic hardening behaviors, utilizing possible micromechanical information with respect to dislocations and their motions. Roles of the geometrically necessary dislocations (GNDs) in strain hardening behavior are studied in detail. Furthermore, some aspects of numerical computations of the extended size-dependent crystal plasticity theory are presented. The developed theory involves extra boundary conditions for crystallographic slips and/or the GND densities. Effects of these extra boundary conditions are demonstrated through numerical simulations for some basic boundary value problems. Finally, a phenomenological strain gradient plasticity theory is revisited, based on the knowledge from the present size-dependent crystal plasticity theory.
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Bibliography
A. Acharya. A model of crystal plasticity based on the theory of continuously distributed dislocations. Journal of the Mechanics and Physics of Solids, 49:761–784, 2001.
A. Acharya and J.L. Bassani. Lattice incompatibility and a gradient theory of crystal plasticity. Journal of the Mechanics and Physics of Solids, 48: 1565–1595, 2000.
E.C. Aifantis. On the microstructural origin of certain inelastic models. Journal of Engineering Materials and Technology, 106:326–330, 1984.
A. Arsenlis, D.M. Parks, R. Becker, and V.V. Bulatov. On the evolution of crystallographic dislocation density in non-homogeneously deforming crystals. Journal of the Mechanics and Physics of Solids, 52:1213–1246, 2004.
R.J. Asaro and A. Needleman. Texture development and strain hardening in rate dependent polycrystals. Acta Metallurgica, 33:923–953, 1985.
M.F. Ashby. The deformation of plastically non-homogeneous materials. Philosophical Magazine, 21:399–424, 1970.
C.J. Bayley, W.A.M. Brekelmans, and M.G.D. Geers. A comparison of dislocation induced back stress formulations in strain gradient crystal plasticity. International Journal of Solids and Structures, 43:7268–7286, 2006.
P. Cermelli and M.E. Gurtin. On the characterization of geometrically necessary dislocations in finite plasticity. Journal of the Mechanics and Physics of Solids, 29:1531–1568,, 2001.
A.H. Cottrell. Dislocations and plastic flow in crystals. Oxford University Press, London, 1953.
D.M. Dimiduk, M.D. Uchic, and T.A. Parthasarathy. Size-affected singleslip behavior of pure nickel microcrystals. Acta Materialia, 53:4065–4077, 2005.
B. Ehrler, X.D. Hou, T.T. Zhu, K.M.Y. Png, C.J. Walker, A.J. Bushby, and D.J. Dunstan. Grain size and sample size interact to determine strength in a soft metal. Philosophical Magazine, 25:3043–3050, 2008.
U. Essmann and H. Mughrabi. Annihilation of dislocations during tensile and cyclic deformation and limits of dislocation densities. Philosophical Magazine, A40:731–756, 1979.
L.P. Evers, W.A.M. Brekelmans, and M.G.D. Geers. Non-local crystal plasticity model with intrinsic SSD and GND effects. Journal of the Mechanics and Physics of Solids, 52:2379–2401, 2004.
N.A. Fleck and J.W. Hutchinson. A reformulation of strain gradient plasticity. Journal of the Mechanics and Physics of Solids, 49:2245–2271, 2001.
N.A. Fleck, G.M. Muller, M.F. Ashby, and J.W. Hutchinson. Strain gradient plasticity: theory and experiment. Acta Metallurgica et Materialia, 42: 475–487, 1994.
M.G.D. Geers, W.A.M. Brekelmans, and C.J. Bayley. Second-order crystal plasticity: internal stress effects and cyclic loading. Modelling and Simulation in Materials Science and Engineering, 16:S133–S145, 2007.
I. Groma, F.F. Csikor, and M. Zaiser. Spatial correlations and higher-order gradient terms in a continuum description of dislocation dynamics. Acta Materialia, 51:1271–1281, 2003.
M.E. Gurtin. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. Journal of the Mechanics and Physics of Solids, 50:5–32, 2002.
M.E. Gurtin. The burgers vector and the flow of screw and edge dislocations in finite-deformation single-crystal plasticity. Journal of the Mechanics and Physics of Solids, 54:1882–1898, 2006.
M.E. Gurtin. A finite deformation, gradient theory of single-crystal plasticity with free energy dependent on densities of geometrically necessary dislocations. International Journal of Plasticity, 24:702–725, 2008.
M.E. Gurtin and L. Anand. A gradient theory for single-crystal plasticity. Modelling and Simulation in Materials Science and Engineering, 15: S263–S270, 2007.
M.E. Gurtin and L. Anand. Thermodynamics applied to gradient theories involving the accumulated plastic strain: The theories of aifantis and fleck and hutchinson and their generalization. Journal of the Mechanics and Physics of Solids, 57:405–421, 2009.
C.-S. Han, H. Gao, Y. Huang, and W.D. Nix. Mechanism-based strain gradient crystal plasticityi. theory. Journal of the Mechanics and Physics of Solids, 53:1188–1203, 2005.
M.A. Haque and M.T.A. Saif. Strain gradient effect in nanoscale thin films. Acta Materialia, 51:30533061, 2003.
I. Hayashi, M. Sato, and M. Kuroda. Strain hardening in bent copper foils. Journal of the Mechanics and Physics of Solids, 59:1731–1751, 2011.
T.J.R. Hughes. Generalization of selective integration procedures to anisotropic and nonlinear media. International Journal for Numerical Methods in Engeering, 15:1413–1418, 1980.
S. Ikawa, M. Asano, M. Kuroda, and K. Yoshida. Effects of crystal orientation on bendability of aluminum alloy sheet. Materials Science and Engineering, A 528:40504054, 2011.
D. Kiener, W. Grosinger, G. Dehm, and R. Pippan. A further step towards an understanding of size-dependent crystal plasticity: In situ tension experiments of miniaturized single-crystal copper samples. Acta Materialia, 56:580–592, 2008.
E. Kröner. Allegmeine kontinuumstheorie der versetzungen und eigenspannungen. Arch. Rational Mech. Anal., 4:273–334, 1960.
L.P. Kubin, G. Canova, M. Condat, B. Devincre, V. Pontikis, and Y. Brechet. Dislocation microstructures and plastic flow: a 3d simulation. Solid State Phenomena, 23-24::445–472, 1992.
M. Kuroda. On large-strain finite element solutions of higher-order gradient crystal plasticity. International Journal of Solids and Structures, 48: 3382–3394, 2011.
M. Kuroda and V. Tvergaard. Shear band development predicted by a non-normality theory of plasticity and comparison to crystal plasticity predictions. International Journal of Solids and Structures, 38:8945–8960, 2001.
M. Kuroda and V. Tvergaard. Studies of scale dependent crystal viscopalsticity models. Journal of the Mechanics and Physics of Solids, 54: 1789–1810, 2006.
M. Kuroda and V. Tvergaard. Effects of texture on shear band formation in plane strain tension/compression and bending. Internaitonal Journal of Plasticity, 23:244–272, 2007.
M. Kuroda and V. Tvergaard. On the formulations of higher-order strain gradient crystal plasticity. Journal of the Mechanics and Physics of Solids, 56:1591–1608, 2008 a.
M. Kuroda and V. Tvergaard. A finite deformation theory of higher-order gradient crystal plasticity. Journal of the Mechanics and Physics of Solids, 56:2573 2584, 2008 b.
M. Kuroda and V. Tvergaard. Effects of microscopic boundary conditions on plastic deformations of small-sized single crystals. International Journal of Solids and Structures, 46:4396–4408, 2009.
M. Kuroda and V. Tvergaard. An alternative treatment of phenomenological higher-order strain plasticity theory. International Journal of Plasticity, 26:507–515, 2010.
M. Kuroda, V. Tvergaard, and T.Ohashi. Simulations of micro-bending of thin foils using a scale dependent crystal plasticity model. Modelling and Simulation in Materials Science and Engineering, 15:S13–S22, 2007.
J.W. Kysar, Y. Saito, M.S. Oztop, D. Lee, and W.T. Huh. Experimental lower bounds on geometrically necessary dislocation density. International Journal of Plasticity, 26:1097–1123, 2010.
E.H. Lee. Elastic-plastic deformation at finite strains. Journal of Applied Mechanics, 36:1–6, 1969.
R.M. McMeeking and J.R. Rice. Finite-element formulations for problems of large elastic-plastic deformation international. Journal of Solids and Structures, 11:601–616, 1975.
C. Motz, T. Schöberl, and R. Pippan. Mechanical properties of micro-sized copper bending beams machined by the focused ion beam technique. Acta Materialia, 53:42694279, 2005.
H. Mughrabi. On the current understanding of strain gradient plasticity. Materials Science and Engineering, A 387-389:209–213, 2004.
E. Nakamachi, C.L. Xie, and M. Harimoto. Drawability assessment of bcc steel sheet by using elastic/crystalline viscoplastic finite element analyses. International Journal of Mechanical Sciences, 43:631–652, 2001.
J.F. Nye. Some geometrical relations in dislocated solids. Acta Metallurgica, 1:153–162, 1953.
T. Ohashi, editor. A new model of scale dependent crystal plasticity analysis, volume Solid mechanics and its applications vol. 115, 97-106 of Proceedings of IUTAM Symposium on Mesoscopic Dynamics in Fracture Process and Strength of Materials, Osaka, Japan,, 2004. Kluwer Academic Publishers, Dordrecht.
T. Ohashi. Crystal plasticity analysis of dislocation emission from micro voids. International Journal of Plasticity, 21:2071–2088, 2005.
D. Peirce, R. J. Asaro, and A. Needleman. Material rate dependence and localized deformation in crystalline solids. Acta Metallurgica, 31:1951–1976, 1983.
P. Shrotriya, S.M. Allameh, J. Lou, T. Buchheit, and W.O. Soboyejo. On the measurement of the plasticity length scale parameter in liga nickel foils. Mechanics of Materials, 35:233243, 2003.
J.S. Stölken and A.G. Evans. A microbend test method for measuring the plasticity length scale. Acta Materialia, 45:5109–5115, 1998.
S. Sun, B.L. Adams, C.Q. Shet, S. Saigal, and W. King. Mesoscale investigation of the deformation field of an aluminum bicrystal. Scripta Materialia, 39:501–508, 1998.
S. Sun, B.L. Adams, and W. King. Observations of lattice curvature near the interface of a deformed aluminum bicrystal. Philosophical Magazine, A80:9–25, 2000.
K. Suzuki, Y. Matsuki, K. Masaki, M. Sato, and M. Kuroda. Tensile and microbend tests of pure aluminum foils with different thicknesses. Materials Science and Engineering, A 513-514:77–82, 2009.
G.I. Taylor. Plastic strain in metals. J. Inst. Metals, 62:307–325, 1938.
M.D. Uchic, D.M. Dimiduk, J.N. Florando, and W.D. Nix. Sample dimensions influence strength and crystal plasticity. Science, 305:986–989, 2004.
W.D.Nix and H.J.Gao. Indentation size effects in crystalline materials: A law for strain gradient plasticity. Journal of Mechanics and Physics of Solids, 46:411–425, 1998.
J. Weertman. Anomalous work hardening, non-redundant screw dislocations in a circular bar deformed in torsion, and non-redundant edge dislocations in a bent foil. Acta Materialia, 50:673–689, 2002.
K. Yamagishi, R. Takeda, and M. Takeda. The influence of grain size on the flex fatigue property of rolled copper foil. Copper and Copper Alloys, 45:27–30, 2006. (in Japanese with English abstract).
S. Yefimov, E. van der Giessen, and I. Groma. Bending of a single crystal: discrete dislocation and nonlocal crystal plasticity simulations. Modelling and Simulation in Materials Science and Engineering, 12:10691086, 2004 a.
S. Yefimov, I. Groma, and E. van der Giessen. A comparison of a statisticalmechanics based plasticity model with discrete dislocation plasticity calculations. Journal of the Mechanics and Physics of Solids, 52:279–300, 2004 b.
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Kuroda, M. (2014). On scale-dependent crystal plasticity models. In: Schröder, J., Hackl, K. (eds) Plasticity and Beyond. CISM International Centre for Mechanical Sciences, vol 550. Springer, Vienna. https://doi.org/10.1007/978-3-7091-1625-8_5
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