Abstract
The theory of mechanism-based strain gradient (MSG) plasticity involves two material length parameters, namely the intrinsic material length land the mesoscale cell size l ε, which are on the order of a few microns and 0.1 μm, respectively. Prior studies suggest that l εhas essentially no effect on the macroscopic quantities, but it may affect the local stress distribution. We demonstrate in this paper that there is a boundary layer effect associated with l εin MSG plasticity, and the thickness of the boundary layer is on the order of l 2 ε big/l. By neglecting this boundary layer effect, a stress-dominated asymptotic field around a crack tip in MSG plasticity is obtained. This asymptotic field is valid at a distance to the crack tip between l εand l(i.e., from 0.1 μm to a few microns). The stress in this asymptotic field has an approximate singularity of r −2/3, which is more singular than not only the HRR field in classical plasticity but also the classical elastic Kfield (r −1/2). The stress level in this asymptotic field is two to three times higher than the HRR field, which provides an alternative mechanism for cleavage fracture in ductile materials observed in experiments.
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References
Acharya, A. and Bassani, J.L. (2000). Lattice incompatibility and a gradient theory of crystal plasticity. Journal of Mechanics and Physics of Solids 48,1565–1595.
Acharya A. and Beaudoin A.J. (2000). Grain-size effect in viscoplastic polycrystals at moderate strains. Journal of Mechanics and Physics of Solids 48,2213-2230.
Aifantis, E.C. (1984). On the microstructural origin of certain inelastic models.Journal of Engineering Materials and Technology 106,326–330.
Aifantis, E.C. (1992). On the role of gradients in the localization of deformation and fracture. International Journal of Engineering and Sciences 301279–1299.
Arsenlis, A. and Parks, D.M. (1999). Crystallographic aspects of geometrically-necessary and statistically-stored dislocation density.Acta Materials 47,1597–1611.
Ashby, M.F. (1970). The deformation of plastically non-homogeneous alloys.Philosophical Magazine 21,399–424.
Bagchi, A., Lucas, G.E., Suo, Z. and Evans, A.G. (1994). A new procedure for measuring the decohesion energy of thin ductile films on substrates.Journal of Materials Research 9,1734–1741.
Bagchi, A. and Evans, A.G. (1996). The mechanics and physics of thin film decohesion and its measurement. Interface Science 3, 169–193.
Begley, M.R. and Hutchinson, J.W. (1998). The mechanics of size-dependent indentation. Journal of Mechanics and Physics of Solids 46,2049–2068.
Beltz, G.E., Rice, J.R., Shih, C.F. and Xia, L. (1996). A self-consistent model for cleavage in the presence of plastic flow.Acta Materials 44,3943–3954.
Beltz G.E. and Wang, J.S. (1992). Crack direction effects along copper sapphire interfaces.Acta Metallurgica et Materialia 40,1675–1683.
Chen, S.H. and Wang, T.C. (2000). A new hardening law for strain gradient plasticity.Acta Materials 48,3997–4005.
Chen, J.Y., Wei, Y., Huang, Y., Hutchinson, J.W. and Hwang, K.C. (1999). The crack tip fields in strain gradient plasticity: the asymptotic and numerical analyses.Engeneering Fracture Mechanics 64,625–648.
Cleveringa, H.H.M., VanderGiessen, E. and Needleman, A. (1997). Comparison of discrete dislocation and continuum plasticity predictions for a composite material.Acta Materials 45,3163–3179.
Cleveringa, H.H.M., VanderGiessen, E. and Needleman, A. (1998). Discrete dislocation simulations and size dependent hardening in single slip.Journal of PhysicsIV,8,83–92.
Cleveringa, H.H.M., VanderGiessen, E. and Needleman, A. (1999a). A discrete dislocation analysis of bending.Internationals Journal of Plasticity 15,837–868.
Cleveringa, H.H.M., VanderGiessen, E. and Needleman, A. (1999b). A discrete dislocation analysis of residual stresses in a composite material.Philosophical Magazine A79,893–920.
Cleveringa, H.H.M., VanderGiessen, E. and Needleman, A. (2000). A discrete dislocation analysis of mode I crack growth. Journal of Mechanics and Physics of Solids,48,1133–1157.
Cottrell, A.H. (1964). The Mechanical Properties of Materials,J. Willey, New York, 277.
Dai, H. and Parks, D.M. (2001). Geometrically-necessary dislocation density in continuum crystal plasticity theory and FEM implementation (unpublished manuscript).
de Borst R. and Mühlhaus H.-B. (1991). Computational strategies for gradient continuum models with a view to localization of deformation. In Proceedings of International Conference on Nonlinear Engineering Computation. (Edited by N. Bićanićc, P. Marovićc, D.R.J. Owen, V. Jović and A. Mihanović), Prineridge Press, Swansea, 239–260.
de Borst R. and Mühlhaus H.-B. (1992). Gradient-dependent plasticity: formulation and algorithmic aspects.International Journal for Numerical Methods in Engineering 35,521–539.
de Guzman, M.S., Neubauer, G., Flinn, P. and Nix, W.D. (1993). The role of indentation depth on the measured hardness of materials.Materials Research Symposium Proceedings 308,613–618.
Elssner, G., Korn, D. and Ruehle, M. (1994). The influence of interface impurities on fracture energy of UHV diffusion bonded metal-ceramic bicrystals.Scripta Metallurgica et Materialia 31,1037–1042.
Fleck, N.A. and Hutchinson, J.W. (1993).A phenomenological theory for strain gradient effects in plasticity. Journa of Mechanics and Physics of Solids 41,1825–1857.
Fleck, N.A. and Hutchinson, J.W. (1997). Strain gradient plasticity.Advances in Applied Mechanics(Edited by J.W. Hutchinson, and T.Y. Wu), Vol 33,Academic Press, New York,295–361.
Fleck, N.A., Muller, G.M., Ashby, M.F. and Hutchinson, J.W. (1994). Strain gradient plasticity: theory and experiments.Acta Metallurgica et Materialia 42,475–487.
Gao, H., Huang, Y. and Nix, W.D. (1999a). Modeling plasticity at the micrometer scale.Naturwissenschaften 86,507–515.
Gao, H., Huang, Y., Nix, W.D. and Hutchinson, J.W. (1999b). Mechanism-based strain gradient plasticity-I. Theory.Journal of Mechanics and Physics of Solids 47,1239–1263.
Gurtin, M.E. (2000). On the plasticity of single crystals: free energy, microforces, plastic-strain gradients. Journal of Mechanics and Physics of Solids 48,989–1036.
Huang, Y., Gao, H., Nix, W.D. and Hutchinson, J.W. (2000a). Mechanism-based strain gradient plasticity-II. Analysis.Journal of Mechanics and Physics of Solids 48,99–128.
Huang, Y., Xue, Z., Gao, H. and Xia, Z.C. (2000b). A study of micro-indentation hardness tests by mechanismbased strain gradient plasticity.Journal of Materials Research 15,1786–1796.
Huang, Y., Zhang, L., Guo, T.F. and Hwang, K.C. (1995). Near-tip fields for cracks in materials with strain-gradient effects. Proceedings of IUTAM Symposium on Nonlinear Analysis of Fracture(Edited by J.R. Willis), Kluwer Academic Publishers, Cambridge, England, 231–242.
Huang, Y., Zhang, L., Guo, T.F. and Hwang, K.C. (1997). Mixed mode near-tip fields for cracks in materials with strain-gradient effects.Journal of Mechancis and Physics of Solids 45,439–465.
Hutchinson, J.W. (1968). Singular behavior at the end of a tensile crack in a hardening material.Journal of Mechanics and Physics of Solids 16,13–31.
Hutchinson, J.W. (1997). Linking scales in mechanics. Advances in Fracture Research(Edited by B.L. Karihaloo, Y.W. Mai, M.I. Ripley and R.O. Ritchie),Pergamon Press, Amsterdam,1–14.
Jiang, H., Huang, Y., Zhuang, Z. and Hwang, K.C. (2001). Fracture in mechanism-based strain gradient plasticity.Journal of Mechanics and Physics of Solids 49,979–993.
Korn D., Elssner, G., Fischmeister, H.F., et al. (1992). Influence of interface impurities on the fracture energy of UHV bonded niobium sapphire bicrystals.Acta Metallurgica et Materialia 40,S355-S360 Suppl. S.
Lasry D. and Belytschko, T. (1988). Localization limiters in transient problems. International Journal of Solids and Structures 24,581–597.
Ma, Q. and Clarke, D.R. (1995). Size dependent hardness of silver single crystals.Journal of Materials Research 10,853–863.
McElhaney, K.W., Vlassak, J.J. and Nix, W.D. (1998). Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments.Journal of Materials Research 13, 1300–1306.
Mühlhaus H.B. and Aifantis E.C. (1991). A variational principle for gradient plasticity.Interntional Journal of Solids and Structures 28,845–857.
Needleman, A. (2000).Computational mechanics at the mesoscale.Acta Materialia 48, 105–124.
Nix, W.D. (1989). Mechanical properties of thin films.Materials Transactions A 20A, 2217–2245.
Nix, W.D. (1997). Elastic and plastic properties of thin films on substrates: nanoindentation techniques. Materials Science and Engineering A-Struct. 234,37–44.
Nix, W.D. and Gao, H. (1998). Indentation size effects in crystalline materials: a law for strain gradient plasticity.Journal of Mechanics and Physics of Solids 46,411–425.
Nye, J.F. (1953). Some geometrical relations in dislocated crystals.Acta Metallurgica et Materialia 1, 153–162.
O'Dowd, N.P., Stout, M.G. and Shih, C.F. (1992). Fracture-toughness of alumina niobium interfaces – experiments and analyses.Philosophical Magazine A 66,1037–1064.
Oh, T.S., Cannon, R.M. and Ritchie, R.O. (1987). Subcritical crack growth along ceramic-metal interfaces. Jom-Journal of Minerals, Metals and Materials S 39,A57–A57.
Poole, W.J., Ashby, M.F. and Fleck, N.A. (1996). Micro-hardness of annealed and work-hardened copper polycrystals.Scripta Matererialia 34, 559–564.
Press, W.H., Flannery, B.P. Teukolsky, S.A. and Vetterling, W.T. (1986). Numerical Recipes, Cambridge University Press, Cambridge.
Rice, J.R. and Rosengren, G.F. (1968). Plane strain deformation near a crack tip in a power law hardening material.Journal of Mechanics and Physics of Solids 16,1–12.
Shi, M., Huang, Y., Gao, H. and Hwang, K.C. (2000). Non-existence of separable crack tip field in mechanismbased strain gradient plasticity.International Journal of Solids and Structures 37, 5995–6010.
Shu, J.Y. and Fleck, N.A. (1999). Strain gradient crystal plasticity: size-dependent deformation of bicrystals. Journal of Mechanics and Physics of Solids 47,297–324.
Sluys L.J., de Borst, R. and Mühlhaus H.B. (1993). Wave-propagation, localization and dispersion in a gradientdependent medium.International Journal of Solids and Structures 30,1153–1171.
Stelmashenko, N.A., Walls, A.G., Brown, L.M. and Milman, Y.V. (1993). Microindentation onWand Mo oriented single crystals: an STM study.Acta Metallurgica et Materialia 41, 2855–2865.
Stolken, J.S. and Evans, A.G. (1998). A microbend test method for measuring the plasticity length scale. Acta Materialia 46,5109–5115.
Suo, Z., Shih, C.F. and Varias A.G. (1993). A theory for cleavage cracking in the presence of plastic-flow. Acta Metallurgica et Materialia 411551–1557.
Suresh S., Nieh, T.G. and Choi, B.W. (1999). Nano-indentation of copper thin films on silicon substrates. Scripta Materialia 41,951–957.
Taylor, G.I. (1938). Plastic strain in metals.Journal of the Institute of Metals 62,307–324.
Wang, J.-S. and Anderson, P.M. (1991). Fracture behavior of embrittled FCC metal bicrystals and its misorientation dependence.Acta Metallurgica et Materialia 39,779–792.
Wei, Y. and Hutchinson, J.W. (1997). Steady-state crack growth and work of fracture for solids characterized by strain gradient plasticity.Journal of Mechanics and Physics of Solids 45,1253–1273.
Wei, Y. and Hutchinson, J.W. (1999).Models of interface separation accompanied by plastic dissipation at multiple scales.International Journal of Fracture 95,1–17.
Zbib H.M. and Aifantis E.C. (1988). On the localization and postlocalization behavior of plastic deformation. Part I. On the initiation of shear bands; Part II. On the evolution and thickness of shear bands. Part III. On the structure and velocity of Portevin-Le Chatelier bands.Res. Mech.,261–277, 279–292 and 293–305.
Zhang L., Huang Y., Chen J.Y. and Hwang K.C. (1998). The mode III full-field solution in elastic materials with strain gradient effects.International Journal of Fracture 92,325–348.
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Shi, M., Huang, Y., Jiang, H. et al. The boundary-layer effect on the crack tip field in mechanism-based strain gradient plasticity. International Journal of Fracture 112, 23–41 (2001). https://doi.org/10.1023/A:1013548131004
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DOI: https://doi.org/10.1023/A:1013548131004