Nanoindentation and Indentation Size Effects: Continuum Model and Atomistic Simulation

  • Chi-Hua Yu
  • Kuan-Po Lin
  • Chuin-Shan Chen
Living reference work entry


Nanoindentation is one of the most widely used methods to measure the mechanical properties of materials at the nanoscale. For spherical indenters, when radius decreases, the hardness increases. The phenomenon is known as the indentation size effect (ISE). Nix and Gao developed a continuum model to explain the ISE in microindentation. However, the model overestimates the hardness at the nanoscale. The objective of this study is to develop proper methods to probe key quantities such as hardness and geometric necessary dislocation (GND) density from the quasi-static version of molecular dynamics (MD) simulations and to develop a mechanism-based model to elucidate the ISE phenomenon at the nanoscale. A reliable method is presented to extract the GND directly from dislocation length and the volume of plastic zone in the MD simulations. We conclude that the hardness determined directly from MD simulations matches well with the hardness determined from the Oliver–Pharr method. The ISE can be observed directly from the MD simulations without any free parameters. The model by Swadener et al. rooted from the Nix and Gao model underestimates the GND density at the nanoscale. However, this model can accurately predict the hardness size effects in nanoindentation if it uses the GND density directly calculated from the MD simulations.


Nanoindentation size effect Molecular dynamics simulations Geometrically necessary dislocation density Hardness 



This research was supported by the Ministry of Science and Technology in Taiwan. We are also grateful to the National Center for High-Performance Computing for providing the computational resources required for this study.


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Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.R&D divisionCoreTech. (Moldex3D)HsinchuTaiwan
  2. 2.Structural EngineeringUniveristy of CaliforiniaSan DiegoUSA
  3. 3.Department of Civil EngineeringNational Taiwan UniversityTaipeiTaiwan

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