Simulation of micro-indentation hardness of FCC single crystals by mechanism-based strain gradient crystal plasticity. (English) Zbl 1426.74079
Summary: The size effect observed in the micro-indentation of FCC single crystal copper is modelled by the employment of mechanism-based strain gradient crystal plasticity (MSG-CP). The total slip resistance in each active system is assumed to be due to a mixed population of forest obstacles arising from both statistically stored and geometrically necessary dislocations. The MSG-CP constitutive model is implemented into the Abaqus/Standard FE platform by developing the User MATerial subroutine UMAT. The simulation of micro-indentation hardness on (0 0 1) and (1 1 1) single crystal copper, with a conical indenter having a sharp tip, a conical indenter with a spherical tip and a three-sided Berkovich indenter, is undertaken. The phenomena of pile-up and sink-in have been observed in the simulation and dealt with by appropriate use of the contact analysis function in Abaqus. These phenomena have been taken into account in the determination of the contact areas and hence the average indentation depth for anisotropic single crystals. The depth dependence of the micro-indentation hardness, the size effect, is calculated. The micro-hardness results from the simulation are compared with those of the published experimental ones in the literature and a good agreement is found.
MSC:
| 74C20 | Large-strain, rate-dependent theories of plasticity |
| 74E15 | Crystalline structure |
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 74M15 | Contact in solid mechanics |
References:
| [1] | Acharya, A.; Bassani, J. L.: Lattice incompatibility and a gradient theory of crystal plasticity, Journal of the mechanics and physics of solids 48, 1565-1595 (2000) · Zbl 0963.74010 · doi:10.1016/S0022-5096(99)00075-7 |
| [2] | Acharya, A.; Beaudoin, A. J.: Grain-size effect in viscoplastic polycrystals at moderate strains, Journal of the mechanics and physics of solids 48, 2213-2230 (2000) · Zbl 0958.74013 · doi:10.1016/S0022-5096(00)00013-2 |
| [3] | Al-Rub, R. K. A.: Prediction of micro and nano-indentation size effect from conical or pyramidal indentation, Mechanics of materials 39, 787-802 (2007) |
| [4] | Al-Rub, R. K. A.: Interfacial gradient plasticity governs scale-dependent yield strength and strain hardening rates in micro/nano structured metals, International journal of plasticity 24, 1277-1306 (2008) · Zbl 1388.74077 |
| [5] | Al-Rub, R. K. A.; Voyiadjis, G. Z.: Analytical and experimental determination of the material intrinsic length scale of strain gradient plasticity theory from micro- and nano-indentation experiments, International journal of plasticity 20, 1139-1182 (2004) |
| [6] | Al-Rub, R. K. A.; Voyiadjis, G. Z.: A physically based gradient plasticity theory, International journal of plasticity 22, 654-684 (2006) · Zbl 1190.74003 · doi:10.1016/j.ijplas.2005.04.010 |
| [7] | Anand, L.; Ames, N. M.: On modeling the micro-indentation response of an amorphous polymer, International journal of plasticity 22, 1123-1170 (2006) · Zbl 1176.74038 · doi:10.1016/j.ijplas.2005.07.006 |
| [8] | Arsenlis, A.; Parks, D. M.: Crystallographic aspects of geometrically-necessary and statistically-stored dislocation density, Acta materialia 47, 1597-1611 (1999) |
| [9] | Asaro, R. J.; Needleman, A.: Texture development and strain hardening in rate dependent polycrystals, Acta metallurgica et materialia 33, 923-953 (1985) |
| [10] | Balint, D. S.; Deshpande, V. S.; Needleman, A.; Vander Giessen, E.: Discrete dislocation plasticity analysis of the wedge indentation of films, Journal of the mechanics and physics of solids 54, 2281-2303 (2006) · Zbl 1120.74615 · doi:10.1016/j.jmps.2006.07.004 |
| [11] | Busso, E. P.; Meissonnier, F. T.; O’dowd, N. P.: Gradient-dependent deformation of two-phase single crystal, Journal of the mechanics and physics of solids 48, 2333-2361 (2000) · Zbl 0994.74012 · doi:10.1016/S0022-5096(00)00006-5 |
| [12] | Chen, J. Y.; Wei, Y.; Huang, Y.; Hutchinson, J. W.; Hwang, K. C.: The crack tip fields in strain gradient plasticity: the asymptotic and numerical analyses, Engineering fracture mechanics 64, 625-648 (1999) |
| [13] | Cheong, K. S.; Busso, E. P.; Arsenlis, A.: A study of microstructural length scale effects on the behaviour of FCC polycrystals using strain gradient concepts, International journal of plasticity 21, 1797-1814 (2005) · Zbl 1114.74371 · doi:10.1016/j.ijplas.2004.11.001 |
| [14] | Choi, Y. S.; Parthasarathy, T. A.; Uchic, D. M.; Dimiduk, M. D.: Numerical study of the flow responses and the geometric constraint effects in ni-base two-phase single crystals using strain gradient plasticity, Materials science and engineering 397, 69-83 (2005) |
| [15] | Cordill, M. J.; Moody, N. R.; Gerberich, W. W.: The role of dislocation walls for nanoindentation to shallow depths, International journal of plasticity 25, 281-301 (2009) · Zbl 1419.74005 |
| [16] | Dai, H., Parks, D.M., 1997. Geometrically-Necessary Dislocation Density in Continuum Crystal Plasticity Theory and FEM Implementation and Applications. MIT, Ph.D. Dissertation. |
| [17] | Evers, L. P.; Parks, D. M.; Brekelmans, W. A. M.; Geers, M. G. D.: Crystal plasticity model with enhanced hardening by geometrically necessary dislocation accumulation, Journal of the mechanics and physics of solids 50, 2403-2424 (2002) · Zbl 1100.74531 · doi:10.1016/S0022-5096(02)00032-7 |
| [18] | Fivel, M. C.; Robertson, C. F.; Canova, G. R.; Boulanger, L.: Three-dimensional modeling of indent-induced plastic zone at a mesoscale, Acta materialia 46, 6183-6194 (1998) |
| [19] | Fleck, N. A.; Hutchinson, J. W.: A phenomenological theory for strain gradient effects in plasticity, Journal of the mechanics and physics of solids 41, 1825-1857 (1993) · Zbl 0791.73029 · doi:10.1016/0022-5096(93)90072-N |
| [20] | Fleck, N. A.; Hutchinson, J. W.: Strain gradient plasticity, Advances in applied mechanics 33, 95-361 (1997) · Zbl 0894.73031 |
| [21] | Fleck, N. A.; Hutchinson, J. W.: A reformulation of strain gradient plasticity, Journal of the mechanics and physics of solids 49, 2245-2271 (2001) · Zbl 1033.74006 · doi:10.1016/S0022-5096(01)00049-7 |
| [22] | Fleck, N. A.; Muller, G. M.; Ashby, M. F.; Hutchinson, J. W.: Strain gradient plasticity: theory and experiments, Acta metallurgica et materialia 42, 475-487 (1994) |
| [23] | Fleck, N. A.; Willis, J. R.: A mathematical basis for strain gradient plasticity theory part I: Scale plastic multiplier, Journal of the mechanics and physics of solids 57, 161-177 (2009) · Zbl 1195.74020 · doi:10.1016/j.jmps.2008.09.010 |
| [24] | Fornell, J.; Concustell, A.; Surinach, S.; Li, W. H.; Cuadrado, N.; Gebert, A.; Baró, M. D.; Sort, J.: Yielding and intrinsic plasticity of ti – zr – ni – cu – be bulk metallic Glass, International journal of plasticity 25, 1540-1559 (2009) · Zbl 1272.74088 |
| [25] | Gao, H.; Huang, Y.; Nix, W. D.: Modeling plasticity at the micro-meter scale, Naturwissenschaften 86, 507-515 (1999) |
| [26] | Gao, H.; Huang, Y.; Nix, W. D.; Hutchinson, J. W.: Mechanism-based strain gradient plasticity – I theory, Journal of the mechanics and physics of solids 47, 1239-1263 (1999) · Zbl 0982.74013 · doi:10.1016/S0022-5096(98)00103-3 |
| [27] | Gerberich, W. W.; Mook, W. M.; Cordill, M. J.; Carter, C. B.; Perrey, C. R.; Heberlein, J. V.; Girshick, S. L.: Reverse plasticity in single crystal silicon nanospheres, International journal of plasticity 21, 2391-2405 (2005) · Zbl 1101.74319 · doi:10.1016/j.ijplas.2005.03.001 |
| [28] | Ghoniem, N. M.; Huang, J.; Wang, Z.: Affine covariant-contravariant vector forms for the elastic field of parametric dislocation in isotropic crystals, Philosophical magazine letters 82, 55-63 (2002) |
| [29] | Gurtin, M. E.: On the plasticity of single crystals: free energy, microforces, plastic-strain gradients, Journal of the mechanics and physics of solids 48, 989-1036 (2000) · Zbl 0988.74021 · doi:10.1016/S0022-5096(99)00059-9 |
| [30] | Gurtin, M. E.: 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) · Zbl 1043.74007 · doi:10.1016/S0022-5096(01)00104-1 |
| [31] | Haj-Ali, R.; Kim, H. K.; Koh, S. W.; Saxena, A.; Tummala, R.: Nonlinear constitutive models from nanoindentation tests using artificial neural networks, International journal of plasticity 24, 371-396 (2008) · Zbl 1220.74014 · doi:10.1016/j.ijplas.2007.02.001 |
| [32] | Han, C. S.; Gao, H. J.; Huang, Y. G.; Nix, W. D.: Mechanism-based strain gradient crystal plasticity (I): theory, Journal of the mechanics and physics of solids 53, 1188-1203 (2005) · Zbl 1120.74356 · doi:10.1016/j.jmps.2004.08.008 |
| [33] | Han, C. S.; Gao, H. J.; Huang, Y. G.; Nix, W. D.: Mechanism-based strain gradient crystal plasticity (II): analysis, Journal of the mechanics and physics of solids 53, 1204-1222 (2005) · Zbl 1120.74357 · doi:10.1016/j.jmps.2005.01.004 |
| [34] | Segerstad, P. Hard Af; Toll, S.; Larsson, R.: Computational modelling of dissipative open-cell cellular solids at finite deformations, International journal of plasticity 25, 802-821 (2009) · Zbl 1419.74010 |
| [35] | Hill, R.: Generalized constitutive relations for incremental deformation of metal crystals by multi-slip, Journal of the mechanics and physics of solids 15, 79-95 (1967) |
| [36] | Hill, R.; Rice, J. R.: Constitutive analysis of elastic – plastic crystals at arbitrary strain, Journal of the mechanics and physics of solids 20, 401-413 (1972) · Zbl 0254.73031 · doi:10.1016/0022-5096(72)90017-8 |
| [37] | Huang, Y.; Chen, J. Y.; Guo, T. F.; Zhang, L.; Hwang, K. C.: Analytic and numerical studies on mode I and mode II fracture inelastic – plastic materials with strain gradient effects, International journal of fracture 100, 1-27 (1999) |
| [38] | Huang, Y.; Gao, H.; Nix, W. D.; Hutchinson, J. W.: Mechanism-based strain gradient plasticity – II analysis, Journal of the mechanics and physics of solids 48, 99-128 (2000) · Zbl 0990.74016 · doi:10.1016/S0022-5096(99)00022-8 |
| [39] | Huang, Y.; Xue, Z.; Gao, H.; Nix, W. D.; Xia, Z. C.: A study of micro-indentation hardness tests by mechanism-based strain gradient plasticity, Journals of material research 15, 1786-1796 (2000) |
| [40] | Huang, Y.; Zhang, F.; Hwang, K. C.; Nix, W. D.; Pharrd, G. M.; Feng, G.: A model of size effects in nano-indentation, Journal of the mechanics and physics of solids 54, 1668-1686 (2006) · Zbl 1120.74658 · doi:10.1016/j.jmps.2006.02.002 |
| [41] | Hutchinson, J. W.: Bounds and self-consistent estimates for creep of polycrystalline materials, Proceedings of the physical society of London 348, 101-127 (1976) · Zbl 0319.73059 · doi:10.1098/rspa.1976.0027 |
| [42] | Kalidindi, S. R.; Bronkhorst, C. A.; Anand, L.: Crystallographic texture evolution in bulk deformation processing of FCC metals, Journal of the mechanics and physics of solids 40, 537-569 (1992) |
| [43] | Lele, S. P.; Anand, L.: A large-deformation strain-gradient theory for isotropic viscoplastic materials, International journal of plasticity 25, 420-453 (2009) · Zbl 1277.74009 |
| [44] | Liu, K., Melkote, S.N., 2004. A strain gradient based finite element model for micro/meso-scale orthogonal cutting process. In: Proceedings of 2004 JUSFA, 2004 Japan-USA Symposium on Flexible Automation, Denver, Colorado. |
| [45] | Liu, Y.; Wang, B.; Yoshino, M.; Roya, S.; Lu, H.; Komanduri, R.: Combined numerical simulation and nanoindentation for determining mechanical properties of single crystal copper at mesoscale, Journal of the mechanics and physics of solids 53, 2718-2741 (2005) · Zbl 1120.74398 · doi:10.1016/j.jmps.2005.07.003 |
| [46] | Ma, Q.; Clarke, D. R.: Size dependent hardness of silver single crystal, Journals of material research 10, 853-863 (1995) |
| [47] | Mcelhaney, K. W.; Vlassak, J. J.; Nix, W. D.: Determination of indentation tip geometry and indentation contact area for depth-sensing indentation experiments, Journals of material research 13, 1300-1306 (1998) |
| [48] | Meissonier, F. T.; Busso, E. P.; O’dowd, N. P.: Finite element implementation of a generalized non-local rate-dependent crystallographic formulation for finite strains, International journal of plasticity 17, 601-640 (2001) · Zbl 1052.74054 · doi:10.1016/S0749-6419(00)00064-4 |
| [49] | Menzel, A.; Steinmann, P.: On the continuum formulation of higher gradient plasticity for single and polycrystals, Journal of the mechanics and physics of solids 48, 1777-1796 (2000) · Zbl 0999.74029 · doi:10.1016/S0022-5096(99)00024-1 |
| [50] | Mindlin, R. D.: Micro-structure in linear elasticity, Archive rational mechanics analysis 16, 51-78 (1964) · Zbl 0119.40302 · doi:10.1007/BF00248490 |
| [51] | Mindlin, R. D.: Second gradient of strain and surface tension in linear elasticity, International journal of solids and structures 1, 417-438 (1965) |
| [52] | Nair, A. K.; Parker, E.; Gaudreau, P.; Farkas, D.; Kriz, R. D.: Size effects in indentation response of thin films at the nanoscale: a molecular dynamics study, International journal of plasticity 24, 2016-2031 (2008) · Zbl 1147.74035 · doi:10.1016/j.ijplas.2008.01.007 |
| [53] | Nix, W. D.: Mechanical properties of thin films, Materials transactions 20, 2217-2245 (1989) |
| [54] | Nix, W. D.; Gao, H. J.: Indentation size effects in crystalline materials: a law for strain gradient plasticity, Journal of the mechanics and physics of solids 46, 411-425 (1998) · Zbl 0977.74557 · doi:10.1016/S0022-5096(97)00086-0 |
| [55] | Ohashi, T.: Crystal plasticity analysis of dislocation emission from micro voids, International journal of plasticity 21, 2071-2088 (2005) · Zbl 1330.74035 |
| [56] | Okumura, D.; Higashi, Y.; Sumida, K.; Ohno, N.: A homogenization theory of strain gradient single crystal plasticity and its finite element discretization, International journal of plasticity 23, 1148-1166 (2007) · Zbl 1294.74055 |
| [57] | Oliver, W. C.; Pharr, G. M.: An improved technique for determining hardness and elastic modulus using loading and displacement sensing indentation, Journal of materials research 7, 1564-1583 (1992) |
| [58] | Qin, J.; Huang, Y.; Xiao, J.; Hwang, K. C.: The equivalence of axisymmetric indentation model for three-dimensional indentation hardness, Journals of material research 24, 776-783 (2009) |
| [59] | Qu, S.; Huang, Y.; Pharr, G. M.; Hwang, K. C.: The indentation size effect in the spherical indentation of iridium: a study via the conventional theory of mechanism-based strain gradient plasticity, International journal of plasticity 22, 1265-1286 (2006) · Zbl 1161.74342 · doi:10.1016/j.ijplas.2005.07.008 |
| [60] | Rice, J. R.: Inelastic constitutive relations for solids: an internal-variable theory and its application to metal plasticity, Journal of the mechanics and physics of solids 19, 433-455 (1971) · Zbl 0235.73002 · doi:10.1016/0022-5096(71)90010-X |
| [61] | Shi, Z.; Feng, X.; Huang, Y.; Xiao, J.; Hwang, K. C.: The equivalent axisymmetric model for berkovich indenters in power-law hardening materials, International journal of plasticity 26, 141-148 (2010) · Zbl 1421.74079 |
| [62] | Shu, J. Y.; Fleck, N. A.: Strain gradient crystal plasticity: size-dependent deformation of bicrystals, Journal of the mechanics and physics of solids 47, 297-324 (1999) · Zbl 0956.74006 · doi:10.1016/S0022-5096(98)00081-7 |
| [63] | Siddiq, A.; Schmauder, S.; Huang, Y.: Fracture of bicrystal metal/ceramic interfaces: a study via the mechanism-based strain gradient crystal plasticity theory, International journal of plasticity 23, 665-689 (2007) · Zbl 1110.74049 · doi:10.1016/j.ijplas.2006.08.007 |
| [64] | Swadener, J. G.; George, E. P.; Pharr, G. M.: The correction of the indentation size effect measured with indenters of various shapes, Journal of the mechanics and physics of solids 50, 681-694 (2002) · Zbl 1116.74300 · doi:10.1016/S0022-5096(01)00103-X |
| [65] | Van Der Giessen, E.; Needleman, A.: Discrete dislocation plasticity: a simple planar model, Modelling and simulation in material science and engineering 3, 689-735 (1995) |
| [66] | Voyiadjis, G. Z.; Deliktas, B.: Mechanics of strain gradient plasticity with particular reference to decomposition of the state variables into energetic and dissipative components, International journal of engineering science 47, 1405-1423 (2009) · Zbl 1213.74068 · doi:10.1016/j.ijengsci.2009.05.013 |
| [67] | Voyiadjis, G. Z.; Deliktas, B.: Formulation of strain gradient plasticity with interface energy in a consistent thermodynamic framework, International journal of plasticity 25, 1997-2024 (2009) |
| [68] | Xue, Z.; Huang, Y.; Hwang, K. C.; Li, M.: The influence of indenter tip radius on the micro-indentation hardness, Journal of engineering materials and technology 124, 371-378 (2002) |
| [69] | Yefimov, S.; Van Der Giessen, E.: Size effects in single crystal thin films: non-local crystal plasticity simulations, European journal of mechanics – A/solids 24, 183-193 (2005) · Zbl 1069.74035 · doi:10.1016/j.euromechsol.2005.01.002 |
| [70] | Zbib, H. M.; Aifantis, E. C.: On the localization and post localization 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 Châtelier bands, Mechanica 23, No. 261-277, 279-292 (1988) |
| [71] | Zbib, H. M.; De La Rubia, T. Diaz: A multiscale model of plasticity, International journal of plasticity 18, 1133-1163 (2002) · Zbl 1062.74008 · doi:10.1016/S0749-6419(01)00044-4 |
| [72] | Zhang, F.; Saha, R.; Huang, Y.; Nix, W. D.; Hwang, K. C.; Qu, S.; Li, M.: Indentation of a hard film on a soft substrate: strain gradient hardening effects, International journal of plasticity 23, 25-43 (2007) · Zbl 1266.74032 |
| [73] | Zhang, K. S.: Microscopic heterogeneity and macroscopic mechanical behavior of polycrystalline materials, Acta mechanica sinica 36, 714-722 (2004) |
| [74] | Zhang, K. S.; Wu, M. S.; Feng, R.: Simulation of micro plasticity-induced deformation in uniaxially strained ceramics by 3-D Voronoi polycrystal modeling, International journal of plasticity 21, 801-834 (2005) · Zbl 1112.74347 · doi:10.1016/j.ijplas.2004.05.010 |
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