Numerical investigation of nanoindentation size effect using micropolar theory. (English) Zbl 1326.74105

Summary: We investigate the properties of micropolar continua in the conical indentation of elastic-plastic material. The micropolar formulation is derived in axisymmetric condition, and a finite element model is implemented for the elastic-plastic contact solution of the indentation problem. It has been shown that micropolar description of the material is consistent with the Nix-Gao indentation size effect model at high indentation depth. The size effects obtained from micropolar continua are governed by hardening rules and yield function rather than the material elasticity. The numerical simulations have been verified by the experimental results coming from the open literature. Moreover, it has been shown that the proposed micropolar model allows the prediction of indentation size effect for both micro- and nanoindentations at the same time.


74M20 Impact in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
74M25 Micromechanics of solids
74A35 Polar materials
74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
Full Text: DOI


[1] Gouldstone, A.; Chollacoop, N.; Dao, M.; Li, J.; Minor, A.M.; Shen, Y.-L., Indentation across size scales and disciplines: recent developments in experimentation and modeling, Acta Mater., 55, 4015-4039, (2007)
[2] Pharr, G.M.; Herbert, E.G.; Gao, Y., The indentation size effect: a critical examination of experimental observations and mechanistic interpretations, Ann. Rev. Mater. Res., 40, 271-292, (2010)
[3] Nix, W.D.; Gao, H., Indentation size effects in crystalline materials: a law for strain gradient plasticity, J. Mech. Phys. Solids, 46, 411-425, (1998) · Zbl 0977.74557
[4] Ma, Q.; Clarke, D.R., Size dependent hardness of silver single crystals, J. Mater. Res., 10, 853-863, (1995)
[5] McElhaney, K.; Vlassak, J.; Nix, W., Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments, J. Mater. Res., 13, 1300-1306, (1998)
[6] Lim, Y.Y.; Chaudhri, M.M., The effect of the indenter load on the nanohardness of ductile metals: an experimental study on polycrystalline work-hardened and annealed oxygen-free copper, Philos. Mag. A, 79, 2979-3000, (1999)
[7] Swadener, J.; George, E.; Pharr, G., The correlation of the indentation size effect measured with indenters of various shapes, J. Mech. Phys. Solids, 50, 681-694, (2002) · Zbl 1116.74300
[8] Feng, G.; Nix, W.D., Indentation size effect in mgo, Scripta Mater., 51, 599-603, (2004)
[9] Huang, Y.; Zhang, F.; Hwang, K.; Nix, W.; Pharr, G.; Feng, G., A model of size effects in nano-indentation, J. Mech. Phys. Solids, 54, 1668-1686, (2006) · Zbl 1120.74658
[10] Qiao, X.G.; Starink, M.J.; Gao, N., The influence of indenter tip rounding on the indentation size effect, Acta Mater., 58, 3690-3700, (2010)
[11] Abu Al-Rub, R.K.; Voyiadjis, G.Z., Analytical and experimental determination of the material intrinsic length scale of strain gradient plasticity theory from micro- and nano-indentation experiments, Int. J. Plast., 20, 1139-1182, (2004)
[12] Abu Al-Rub, R.K., Prediction of micro and nanoindentation size effect from conical or pyramidal indentation, Mech. Mater., 39, 787-802, (2007)
[13] Fleck, N.; Hutchinson, J., Strain gradient plasticity, Adv. Appl. Mech., 33, 295-361, (1997) · Zbl 0894.73031
[14] Fleck, N.; Hutchinson, J., A reformulation of strain gradient plasticity, J. Mech. Phys. Solids, 49, 2245-2271, (2001) · Zbl 1033.74006
[15] Danas, K.; Deshpande, V.S.; Fleck, N., Size effects in the conical indentation of an elasto-plastic solid, J. Mech. Phys. Solids, 60, 1605-1625, (2012)
[16] Guha, S., Sangal, S., Basu, S.: Finite element studies on indentation size effect using a higher order strain gradient theory. Int. J. Solids Struct. 50(6), 863-875 (2013)
[17] Gao, H.; Huang, Y.; Nix, W.; Hutchinson, J., Mechanism-based strain gradient plasticity—I. theory, J. Mech. Phys. Solids, 47, 1239-1263, (1999) · Zbl 0982.74013
[18] Saha, R.; Xue, Z.; Huang, Y.; Nix, W.D., Indentation of a soft metal film on a hard substrate: strain gradient hardening effects, J. Mech. Phys. Solids, 49, 1997-2014, (2001) · Zbl 1093.74501
[19] Hwang, K.; Jiang, H.; Huang, Y.; Gao, H.; Hu, N., A finite deformation theory of strain gradient plasticity, J. Mech. Phys. Solids, 50, 81-99, (2002) · Zbl 1043.74006
[20] Gao, H.; Huang, Y., Taylor-based nonlocal theory of plasticity, Int. J. Solids Struct., 38, 2615-2637, (2001) · Zbl 0977.74009
[21] Qu, S.; Huang, Y.; Pharr, G.; Hwang, K., The indentation size effect in the spherical indentation of iridium: A study via the conventional theory of mechanism-based strain gradient plasticity, Int. J. Plast., 22, 1265-1286, (2006) · Zbl 1161.74342
[22] Voyiadjis, G.Z.; Peters, R., Size effects in nanoindentation: an experimental and analytical study, Acta Mech., 211, 131-153, (2010) · Zbl 1397.74156
[23] Grammenoudis, P.; Tsakmakis, C., Isotropic hardening in micropolar plasticity, Arch. Appl. Mech., 79, 323-334, (2009) · Zbl 1178.74030
[24] Sulem, J.; Cerrolaza, M., Finite element analysis of the indentation test on rocks with microstructure, Comput. Geotech., 29, 95-117, (2002)
[25] Zhang, Z.; Liu, Z.; Liu, X.; Gao, Y.; Zhuang, Z., Wedge indentation of a thin film on a substrate based on micromorphic plasticity, Acta Mech., 221, 133-145, (2011) · Zbl 1398.74223
[26] Green, A.E.; Naghdi, P.M.; Osborn, R., Theory of an elastic-plastic Cosserat surface, Int. J. Solids Struct., 4, 907-927, (1968) · Zbl 0179.55602
[27] Lippmann, H., Eine Cosserat-theorie des plastischen fließens, Acta Mech., 8, 255-284, (1969) · Zbl 0188.58703
[28] Besdo, P.-D.D.-I.D., Ein beitrag zur nichtlinearen theorie des Cosserat-kontinuums, Acta Mech., 20, 105-131, (1974) · Zbl 0294.73003
[29] Muhlhaus, H.; Vardoulakis, I., The thickness of shear bands in granular materials, Geotechnique, 37, 271-283, (1987)
[30] Borst, R., A generalisation of J2-flow theory for polar continua, Comput. Methods Appl. Mech. Eng., 103, 347-362, (1993) · Zbl 0777.73014
[31] Forest, S.; Cailletaud, G.; Sievert, R., A Cosserat theory for elastoviscoplastic single crystals at finite deformation, Arch. Mech., 49, 705-736, (1997) · Zbl 0893.73023
[32] Sharbati, E.; Naghdabadi, R., Computational aspects of the Cosserat finite element analysis of localization phenomena, Comput. Mater. Sci., 38, 303-315, (2006)
[33] Grammenoudis, P.; Tsakmakis, C., Micropolar plasticity theories and their classical limits, part I: resulting model, Acta Mech., 189, 151-175, (2007) · Zbl 1117.74009
[34] Grammenoudis, P.; Sator, C.; Tsakmakis, C., Micropolar plasticity theories and their classical limits, part II: comparison of responses predicted by the limiting and a standard classical model, Acta Mech., 189, 177-191, (2007) · Zbl 1117.74008
[35] Altenbach, J.; Altenbach, H.; Eremeyev, V.A., On generalized Cosserat-type theories of plates and shells: a short review and bibliography, Arch. Appl. Mech., 80, 73-92, (2010) · Zbl 1184.74042
[36] Ramezani, S.; Naghdabadi, R.; Sohrabpour, S., An additive theory for finite elastic-plastic deformations of the micropolar continuous media, Acta Mech., 206, 81-93, (2009) · Zbl 1167.74007
[37] Khoei, A.; Yadegari, S.; Biabanaki, S., 3D finite element modeling of shear band localization via the micro-polar Cosserat continuum theory, Comput. Mater. Sci., 49, 720-733, (2010)
[38] Zhang, H.; Wang, H.; Wriggers, P.; Schrefler, B., A finite element model for contact analysis of multiple Cosserat bodies, Comput. Mech., 36, 444-458, (2005) · Zbl 1100.74059
[39] Zhang, H.; Xie, Z.; Chen, B.; Xing, H., A finite element model for 2D elastic-plastic contact analysis of multiple Cosserat materials, Eur. J. Mechanics-A/Solids, 31, 139-151, (2012) · Zbl 1278.74181
[40] Fischer-Cripps A.C.: Nanoindentation, vol. 1. Springer, Berlin (2011)
[41] Grammenoudis, P.; Tsakmakis, C., Predictions of microtorsional experiments by micropolar plasticity, Proc. R. Soc. A Math. Phys. Eng. Sci., 461, 189-205, (2005)
[42] Eringen A.C.: Nonlocal Continuum Field Theories. Springer, Berlin (2002) · Zbl 1023.74003
[43] Neff, P.; Chełmiński, K.; Müller, W.; Wieners, C., A numerical solution method for an infinitesimal elasto-plastic Cosserat model, Math. Models Methods Appl. Sci., 17, 1211-1239, (2007) · Zbl 1137.74012
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