Boundary layers in constrained plastic flow: comparison of nonlocal and discrete dislocation plasticity. (English) Zbl 1015.74004

Summary: We analyze the simple shear of a constrained strip by using discrete dislocation plasticity and strain gradient crystal plasticity theory. Both single slip and symmetric double slip are considered. The loading is such that for a local continuum description of plastic flow, the deformation state is one of homogeneous shear. In the discrete dislocation formulation, the dislocations are all of edge character and are modeled as line singularities in elastic material. Dislocation nucleation, the lattice resistance to dislocation motion, and dislocation annihilation are incorporated into the formulation through a set of constitutive rules. A complementary solution that enforces the boundary conditions is obtained via finite element method. The discrete dislocation solutions give rise to boundary layers in the deformation field and in the dislocation distributions. The back-extrapolated flow strength for symmetric double slip increases with decreasing strip thickness, so that we observe a size effect. The strain gradient plasticity theory used here is also found to predict a boundary layer and a size effect. Nonlocal material parameters can be chosen to fit some, but not all, of the features of the discrete dislocation results. Additional physical insight into the slip distribution across the strip is provided by simple models for an array of mode II cracks.


74C10 Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity)
74A60 Micromechanical theories
74E15 Crystalline structure
Full Text: DOI


[1] Acharya, A.; Bassani, J.L., Incompatibility and crystal plasticity, J. mech. phys. solids, 48, 1565-1595, (2000) · Zbl 0963.74010
[2] Aifantis, E.C., On the microstructural origin of certain inelastic models, Trans. ASME J. engng. mater. technol., 106, 326-330, (1984)
[3] Armstrong, R.; Codd, I.; Douthwaite, R.M.; Petch, N.J., The plastic deformation of polycrystalline aggregates, Philos. mag., 7, 45-57, (1962)
[4] Ashby, M.F., The deformation of plastically non-homogeneous materials, Philos. mag., 21, 399-424, (1970)
[5] Bassani, J.L.; Needleman, A.; Van der Giessen, E., Plastic flow in a composite: a comparison of nonlocal continuum and discrete dislocation predictions, Int. J. solids struct., 38, 833-853, (2001) · Zbl 1004.74006
[6] Busso, E.P.; Meissonnier, F.T.; O’Dowd, N.P., Gradient-dependent deformation of two-phase single crystals, J. mech. phys. solids, 48, 2333-2361, (2000) · Zbl 0994.74012
[7] Cleveringa, H.H.M.; Van der Giessen, E.; Needleman, A., Comparison of discrete dislocation and continuum plasticity predictions for a composite material, Acta mater., 45, 3163-3179, (1997)
[8] Cleveringa, H.H.M.; Van der Giessen, E.; Needleman, A., A discrete dislocation analysis of bending, Int. J. plast., 15, 837-868, (1999) · Zbl 0976.74048
[9] Cleveringa, H.H.M.; Van der Giessen, E.; Needleman, A., A discrete dislocation analysis of mode I crack growth, J. mech. phys. solids, 48, 1133-1157, (2000) · Zbl 0984.74076
[10] Fleck, N.A., Brittle fracture due to an array of microcracks, Proc. roy. soc. London A, 432, 55-76, (1991) · Zbl 0714.73059
[11] Fleck, N.A.; Hutchinson, J.W., A phenomenological theory for strain gradient plasticity, J. mech. phys. solids, 41, 1825-1857, (1993) · Zbl 0791.73029
[12] Fleck, N.A.; Hutchinson, J.W., Strain gradient plasticity, Adv. appl. mech., 33, 295-361, (1997) · Zbl 0894.73031
[13] Fleck, N.A.; Muller, G.M.; Ashby, M.F.; Hutchinson, J.W., Strain gradient plasticity: theory and experiment, Acta metall. mater., 42, 475-487, (1994)
[14] Freund, L.B., The mechanics of dislocations in strained-layer semiconductor-materials, Adv. appl. mech., 30, 1-66, (1994) · Zbl 0804.73049
[15] Gao, H.; Huang, Y.; Nix, W.D.; Hutchinson, J.W., Mechanism-based strain gradient plasticity - I. theory, J. mech. phys. solids, 47, 1239-1263, (1999) · Zbl 0982.74013
[16] Gulluoglu, A.N.; Srolovitz, D.J.; LeSar, R.; Lomdahl, P.S., Dislocation distributions in two dimensions, Scripta metall., 23, 1347-1352, (1989)
[17] Gurtin, M., On plasticity of crystals: free energy, microforces, plastic strain gradients, J. mech. phys. solids, 48, 989-1036, (2000) · Zbl 0988.74021
[18] Hansen, N., Polycrystalline strengthening, Metall. trans., 16A, 2167-2190, (1985)
[19] Hirth, J.P.; Lothe, J., Theory of dislocations, (1968), McGraw-Hill New York
[20] Huang, Y.; Gao, H.; Nix, W.D.; Hutchinson, J.W., Mechanism-based strain gradient plasticity - II. analysis, J. mech. phys. solids, 48, 99-128, (2000) · Zbl 0990.74016
[21] Hughes, D.A.; Hansen, N., Microstructural evolution in nickel during rolling from intermediate to large strains, Metall. trans., A 24, 2021-2037, (1993)
[22] Hutchinson, J.W., Plasticity at the micron scale, Int. J. solids struct., 37, 225-238, (2000) · Zbl 1075.74022
[23] Kubin, L.P.; Canova, G.; Condat, M.; Devincre, B.; Pontikis, V.; Bréchet, Y., Dislocation microstructures and plastic flow: a 3D simulation, Solid state phenom., 23 & 24, 455-472, (1992)
[24] Mughrabi, H., Dislocation wall and cell structures and long-range internal stresses in deformed metal crystals, Acta metall., 31, 1367-1379, (1983)
[25] Muskhelishvili, N.I., Some basic problems of the mathematical theory of elasticity, (1953), Noordhoff Leyden · Zbl 0052.41402
[26] Nabarro, F.R.N., Theory of crystal dislocations, (1967), Oxford University Press Oxford · Zbl 0046.44804
[27] Ortiz, M.; Repetto, E.A.; Stainier, L., A theory of subgrain dislocation structures, J. mech. phys. solids, 48, 2077-2114, (2000) · Zbl 1001.74007
[28] Petch, N.J., The cleavage strength of polycrystals, J. iron steel inst. London, 174, 25-28, (1953)
[29] Rao, S.I., Hazzledine, P.M., Dimiduk, D.M., 1995. Interfacial strengthening in semi-coherent metallic multilayers. Proceedings of MRS Symposium, vol. 362, pp. 67-77.
[30] Shen, Y.-L.; Suresh, S.; He, M.Y.; Bagchi, A.; Kienzle, O.; Rühle, M.; Evans, A.G., Stress evolution in passivated thin films of cu on silica substrates, J. mater. res., 13, 1928-1937, (1998)
[31] Shu, J.Y.; Fleck, N.A., Strain gradient crystal plasticity: size-dependent deformation of bicrystals, J. mech. phys. solids, 47, 297-324, (1999) · Zbl 0956.74006
[32] Sun, S.; Adams, B.L.; King, W.E., Observations of lattice curvature near the interface of a deformed aluminium bicrystal, Philos. mag. A, 80, 9-25, (2000)
[33] Van der Giessen, E.; Needleman, A., Discrete dislocation plasticity: a simple planar model, Model. simul. mater. sci. engng., 3, 689-735, (1995)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.