# Size Effects During Nanoindentation: Molecular Dynamics Simulation

## Abstract

In this chapter, the molecular dynamics (MD) simulation of nanoindentation experiment is revisited. The MD simulation provides valuable insight into the atomistic process occurring during nanoindentation. First, the simulation details and methodology for MD analysis of nanoindentation are presented. The effects of boundary conditions on the nanoindentation response are studied in more detail. The dislocation evolution patterns are then studied using the information provided by atomistic simulation. Different characteristics of metallic sample during nanoindentation experiment, which have been predicted by theoretical models, are investigated. Next, the nature of size effects in samples with small length scales are studied during nanoindentation. The results indicate that the size effects at small indentation depths cannot be modeled using the forest hardening model, and the source exhaustion mechanism controls the size effects at the initial stages of nanoindentation. The total dislocation length increases by increasing the dislocation density which reduces the material strength according to the exhaustion hardening mechanisms. The dislocation interactions with each other become important as the dislocation content increases. Finally, the effects of grain boundary (GB) on the controlling mechanisms of size effects are studied using molecular dynamics.

## Keywords

Nanoindentation Molecular dynamics Size effects Dislocation Grain boundary## References

- A.H. Almasri, G.Z. Voyiadjis, Nano-indentation in FCC metals: experimental study. Acta Mech.
**209**, 1–9 (2010)CrossRefMATHGoogle Scholar - R.K.A. Al-Rub, G.Z. Voyiadjis, 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)CrossRefGoogle Scholar - M.I. Baskes, Modified embedded-atom potentials for cubic materials and impurities. Phys. Rev. B
**46**, 2727 (1992)CrossRefGoogle Scholar - S.G. Corcoran, R.J. Colton, E.T. Lilleodden, W.W. Gerberich, Anomalous plastic deformation at surfaces: nanoindentation of gold single crystals. Phys. Rev. B
**55**, 16057–16060 (1997)CrossRefGoogle Scholar - C.F.O. Dahlberg, Y. Saito, M.S. Öztop, J.W. Kysar, Geometrically necessary dislocation density measurements associated with different angles of indentations. Int. J. Plast.
**54**, 81–95 (2014)CrossRefGoogle Scholar - M.S. Daw, M.I. Baskes, Embedded-atom method: derivation and application to impurities, surfaces, and other defects in metals. Phys. Rev. B
**29**, 6443–6453 (1984)CrossRefGoogle Scholar - E. Demir, D. Raabe, N. Zaafarani, S. Zaefferer, Investigation of the indentation size effect through the measurement of the geometrically necessary dislocations beneath small indents of different depths using EBSD tomography. Acta Mater.
**57**, 559–569 (2009)CrossRefGoogle Scholar - E. Demir, D. Raabe, F. Roters, The mechanical size effect as a mean-field breakdown phenomenon: example of microscale single crystal beam bending. Acta Mater.
**58**, 1876–1886 (2010)CrossRefGoogle Scholar - K. Durst, B. Backes, M. Göken, Indentation size effect in metallic materials: correcting for the size of the plastic zone. Scr. Mater.
**52**, 1093–1097 (2005)CrossRefGoogle Scholar - J.A. El-Awady, Unravelling the physics of size-dependent dislocation-mediated plasticity. Nat. Commun.
**6**, 5926 (2015)CrossRefGoogle Scholar - J.A. El-Awady, M. Wen, N.M. Ghoniem, The role of the weakest-link mechanism in controlling the plasticity of micropillars. J. Mech. Phys. Solids
**57**, 32–50 (2009)CrossRefMATHGoogle Scholar - D. Faken, H. Jonsson, Systematic analysis of local atomic structure combined with 3D computer graphics. Comput. Mater. Sci.
**2**, 279–286 (1994)CrossRefGoogle Scholar - J.R. Greer, Nano and Cell Mechanics: Fundamentals and Frontiers. Wiley, Chichester, pp 163–190 (2013)Google Scholar
- A. Hasnaoui, P.M. Derlet, H. Van Swygenhoven, Interaction between dislocations and grain boundaries under an indenter – a molecular dynamics simulation. Acta Mater.
**52**, 2251–2258 (2004)CrossRefGoogle Scholar - H. Jang, D. Farkas, Interaction of lattice dislocations with a grain boundary during nanoindentation simulation. Mater. Lett.
**61**, 868–871 (2007)CrossRefGoogle Scholar - C.L. Kelchner, S.J. Plimpton, J.C. Hamilton, Dislocation nucleation and defect structure during surface indentation. Phys. Rev. B
**58**, 11085–11088 (1998)CrossRefGoogle Scholar - C.C. Koch, I.A. Ovid’ko, S. Seal, S. Veprek,
*Structural Nanocrystalline Materials: Fundamentals and Applications*(Cambridge University Press, Cambridge, 2007)CrossRefGoogle Scholar - M. de Koning, R.J. Kurtz, V.V. Bulatov, C.H. Henager, R.G. Hoagland, W. Cai, M. Nomura, Modeling of dislocation–grain boundary interactions in FCC metals. J. Nucl. Mater.
**323**, 281–289 (2003)CrossRefGoogle Scholar - O. Kraft, P. Gruber, R. Mönig, D. Weygand, Plasticity in confined dimensions. Annu. Rev. Mater. Res.
**40**, 293–317 (2010)CrossRefGoogle Scholar - Y. Kulkarni, R.J. Asaroa, D. Farkas, Are nanotwinned structures in fcc metals optimal for strength, ductility and grain stability? Scr. Mater.
**60**, 532–535 (2009)CrossRefGoogle Scholar - J.W. Kysar, C.L. Briant, Crack tip deformation fields in ductile single crystals. Acta Mater.
**50**, 2367–2380 (2002)CrossRefGoogle Scholar - J.W. Kysar, Y.X. Gan, T.L. Morse, X. Chen, M.E. Jones, High strain gradient plasticity associated with wedge indentation into face-centered cubic single crystals: geometrically necessary dislocation densities. J. Mech. Phys. Solids
**55**, 1554–1573 (2007)CrossRefGoogle Scholar - Y. Lee, J.Y. Park, S.Y. Kim, S. Jun, Atomistic simulations of incipient plasticity under Al (111) nanoindentation. Mech. Mater.
**37**, 1035–1048 (2005)CrossRefGoogle Scholar - J. Li, K.J. Van Vliet, T. Zhu, S. Yip, S. Suresh, Atomistic mechanisms governing elastic limit and incipient plasticity in crystals. Nature
**418**, 307–310 (2002)CrossRefGoogle Scholar - S.N. Medyanik, S. Shao, Strengthening effects of coherent interfaces in nanoscale metallic bilayers. Comput. Mater. Sci.
**45**, 1129–1133 (2009)CrossRefGoogle Scholar - M.A. Meyers, A. Mishra, D.J. Benson, Mechanical properties of nanocrystalline materials. Prog. Mater. Sci.
**51**, 427–556 (2006)CrossRefGoogle Scholar - Y. Mishin, D. Farkas, M.J. Mehl, D.A. Papaconstantopoulos, Interatomic potentials for monoatomic metals from experimental data and ab initio calculations. Phys. Rev. B
**59**, 3393–3407 (1999)CrossRefGoogle Scholar - A.K. Nair, E. Parker, P. Gaudreau, D. Farkas, R.D. Kriz, Size effects in indentation response of thin films at the nanoscale: a molecular dynamics study. Int. J. Plast.
**24**, 2016–2031 (2008)CrossRefMATHGoogle Scholar - W.D. Nix, H.J. Gao, Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids
**46**, 411–425 (1998)CrossRefMATHGoogle Scholar - D.M. Norfleet, D.M. Dimiduk, S.J. Polasik, M.D. Uchic, M.J. Mills, Dislocation structures and their relationship to strength in deformed nickel microcrystals. Acta Mater.
**56**, 2988–3001 (2008)CrossRefGoogle Scholar - T.A. Parthasarathy, S.I. Rao, D.M. Dimiduk, M.D. Uchic, D.R. Trinkle, Contribution to size effect of yield strength from the stochastics of dislocation source lengths in finite samples. Scr. Mater.
**56**, 313–316 (2007)CrossRefGoogle Scholar - P. Peng, G. Liao, T. Shi, Z. Tang, Y. Gao, Molecular dynamic simulations of nanoindentation in aluminum thin film on silicon substrate. Appl. Surf. Sci.
**256**, 6284–6290 (2010)CrossRefGoogle Scholar - S. Plimpton, Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys.
**117**, 1–19 (1995)CrossRefMATHGoogle Scholar - N.M. Pugno, A general shape/size-effect law for nanoindentation. Acta Mater.
**55**, 1947–1953 (2007)CrossRefGoogle Scholar - S.I. Rao, D.M. Dimiduk, M. Tang, T.A. Parthasarathy, M.D. Uchic, C. Woodward, Estimating the strength of single-ended dislocation sources in micron-sized single crystals. Philos. Mag.
**87**, 4777–4794 (2007)CrossRefGoogle Scholar - S.I. Rao, D.M. Dimiduk, T.A. Parthasarathy, M.D. Uchic, M. Tang, C. Woodward, Athermal mechanisms of size-dependent crystal flow gleaned from three-dimensional discrete dislocation simulations. Acta Mater.
**56**, 3245–3259 (2008)CrossRefGoogle Scholar - M.D. Sangid, T. Ezaz, H. Sehitoglu, I.M. Robertson, Energy of slip transmission and nucleation at grain boundaries. Acta Mater.
**59**, 283–296 (2011)CrossRefGoogle Scholar - S. Shao, S.N. Medyanik, Dislocation–interface interaction in nanoscale fcc metallic bilayers. Mech. Res. Commun.
**37**, 315–319 (2010)CrossRefMATHGoogle Scholar - W.A. Soer, J.T.M. De Hosson, Detection of grain-boundary resistance to slip transfer using nanoindentation. Mater. Lett.
**59**, 3192–3195 (2005)CrossRefGoogle Scholar - A. Stukowski, Structure identification methods for atomistic simulations of crystalline materials. Model. Simul. Mater. Sci. Eng.
**20**, 045021 (2012)CrossRefGoogle Scholar - A. Stukowski, Computational analysis methods in atomistic modeling of crystals. JOM
**66**, 399–407 (2014)CrossRefGoogle Scholar - A. Stukowski, K. Albe, Extracting dislocations and non-dislocation crystal defects from atomistic simulation data. Model. Simul. Mater. Sci. Eng.
**18**, 085001 (2010)CrossRefGoogle Scholar - A. Stukowski, K. Albe, D. Farkas, Nanotwinned fcc metals: strengthening versus softening mechanisms. Phys. Rev. B
**82**, 224103 (2010)CrossRefGoogle Scholar - A. Stukowski, V.V. Bulatov, A. Arsenlis, Automated identification and indexing of dislocations in crystal interfaces. Model. Simul. Mater. Sci. Eng.
**20**, 085007 (2012)CrossRefGoogle Scholar - S. Suresh, T.G. Nieh, B.W. Choi, Nanoindentation of copper thin films on silicon substrates. Scr. Mater.
**41**, 951–957 (1999)CrossRefGoogle Scholar - J.G. Swadener, E.P. George, G.M. Pharr, The correlation of the indentation size effect measured with indenters of various shapes. J. Mech. Phys. Solids
**50**, 681–694 (2002)CrossRefMATHGoogle Scholar - J. Tersoff, New empirical approach for the structure and energy of covalent systems. Phys. Rev. B
**37**, 6991–7000 (1988)CrossRefGoogle Scholar - T. Tsuru, Y. Kaji, D. Matsunaka, Y. Shibutani, Incipient plasticity of twin and stable/unstable grain boundaries during nanoindentation in copper. Phys. Rev. B
**82**, 024101 (2010)CrossRefGoogle Scholar - M.D. Uchic, P.A. Shade, D.M. Dimiduk, Plasticity of micrometer-scale single crystals in compression. Annu. Rev. Mater. Res. 39, 361--386 (2009)CrossRefGoogle Scholar
- G.Z. Voyiadjis, R.K.A. Al-Rub, Gradient plasticity theory with a variable length scale parameter. Int. J. Solids Struct.
**42**, 3998–4029 (2005)CrossRefMATHGoogle Scholar - G.Z. Voyiadjis, M. Yaghoobi, Large scale atomistic simulation of size effects during nanoindentation: dislocation length and hardness. Mater. Sci. Eng. A
**634**, 20–31 (2015)CrossRefGoogle Scholar - G.Z. Voyiadjis, M. Yaghoobi, Role of grain boundary on the sources of size effects. Comput. Mater. Sci.
**117**, 315–329 (2016)CrossRefGoogle Scholar - G.Z. Voyiadjis, M. Yaghoobi, Size and strain rate effects in metallic samples of confined volumes: dislocation length distribution. Scr. Mater.
**130**, 182–186 (2017)CrossRefGoogle Scholar - M. Yaghoobi, G.Z. Voyiadjis, Effect of boundary conditions on the MD simulation of nanoindentation. Comput. Mater. Sci.
**95**, 626–636 (2014)CrossRefGoogle Scholar - M. Yaghoobi, G.Z. Voyiadjis, Atomistic simulation of size effects in single-crystalline metals of confined volumes during nanoindentation. Comput. Mater. Sci.
**111**, 64–73 (2016a)CrossRefGoogle Scholar - M. Yaghoobi, G.Z. Voyiadjis, Size effects in fcc crystals during the high rate compression test. Acta Mater.
**121**, 190–201 (2016b)CrossRefGoogle Scholar - M. Yaghoobi, G.Z. Voyiadjis, Microstructural investigation of the hardening mechanism in fcc crystals during high rate deformations. Comp. Mater. Sci.
**138**, 10–15 (2017)CrossRefGoogle Scholar - N. Zaafarani, D. Raabe, F. Roters, S. Zaefferer, On the origin of deformation-induced rotation patterns below nanoindents. Acta Mater.
**56**, 31–42 (2008)CrossRefGoogle Scholar - T.T. Zhu, A.J. Bushby, D.J. Dunstan, Materials mechanical size effects: a review. Mater. Technol.
**23**, 193–209 (2008)CrossRefGoogle Scholar - J.A. Zimmerman, C.L. Kelchner, P.A. Klein, J.C. Hamilton, S.M. Foiles, Surface step effects on nanoindentation. Phys. Rev. Lett.
**87**, 165507 (2001)CrossRefGoogle Scholar