Research Papers

Mesoscale Analysis of Homogeneous Dislocation Nucleation

[+] Author and Article Information
Akanksha Garg

Carnegie Mellon University,
Pittsburgh, PA 15213
e-mail: a.garg29@gmail.com

Asad Hasan

Carnegie Mellon University,
Pittsburgh, PA 15213
e-mail: asadhasan32@gmail.com

Craig E. Maloney

Carnegie Mellon University,
Pittsburgh, PA 15213
e-mail: c.maloney@northeastern.edu

1Present address: The Dow Chemical Company, Freeport, TX.

2Present address: DeepCortex.ai, Arlington, VA 22201.

3Present address: Northeastern University, Boston, MA 02115.

Contributed by the Applied Mechanics Division of ASME for publication in the Journal of Applied Mechanics. Manuscript received March 2, 2019; final manuscript received May 25, 2019; published online June 27, 2019. Assoc. Editor: A. Amine Benzerga.

J. Appl. Mech 86(9), 091005 (Jun 27, 2019) (11 pages) Paper No: JAM-19-1102; doi: 10.1115/1.4043885 History: Received March 02, 2019; Accepted May 25, 2019

We perform atomistic simulations of dislocation nucleation in two-dimensional (2D) and three-dimensional (3D) defect-free hexagonal crystals during nanoindentation with circular (2D) or spherical (3D) indenters. The incipient embryo structure in the critical eigenmode of the mesoregions is analyzed to study homogeneous dislocation nucleation. The critical eigenmode or dislocation embryo is found to be localized along a line (or plane in 3D) of atoms with a lateral extent, ξ, at some depth, Y, below the surface. The lowest energy eigenmode for mesoregions of varying radius, rmeso, centered on the localized region of the critical eigenmode is computed. The energy of the lowest eigenmode, λmeso, decays very rapidly with increasing rmeso and λmeso ≈ 0 for rmesoξ. The analysis of a mesoscale region in the material can reveal the presence of incipient instability even for rmesoξ but gives reasonable estimate for the energy and spatial extent of the critical mode only for rmesoξ. When the mesoregion is not centered at the localized region, we show that the mesoregion should contain a critical part of the embryo (and not only the center of embryo) to reveal instability. This scenario indicates that homogeneous dislocation nucleation is a quasilocal phenomenon. Also, the critical eigenmode for the mesoscale region reveals instability much sooner than the full system eigenmode. We use mesoscale analysis to verify the scaling laws shown previously by Garg and Maloney in 2D [2016, “Universal Scaling Laws for Homogeneous Dissociation Nucleation During Nano-Indentation,” J. Mech. Phys. Solids, 95, pp. 742–754.] for the size, ξ, and depth from the surface, Y*, of the dislocation embryo with respect to indenter radius, R, in full 3D simulations.

Copyright © 2019 by ASME
Your Session has timed out. Please sign back in to continue.


Foss, C. A., Hornyak, G. L., Stockert, J. A., and Martin, C. R., 1994, “Template-Synthesized Nanoscopic Gold Particles: Optical Spectra and the Effects of Particle Size and Shape,” J. Phys. Chem., 98(11), pp. 2963–2971. [CrossRef]
Whitney, T. M., Jiang, J. S., Searson, P. C., and Chien, C. L., 1993, “Fabrication and Magnetic Properties of Arrays of Metallic Nanowires,” Science 261(5126), pp. 1316–1319. [CrossRef] [PubMed]
Wu, Y., Xiang, J., Yang, C., Lu, W., and Lieber, C. M., 2004, “Single-Crystal Metallic Nanowires and Metal/Semiconductor Nanowire Heterostructures,” Nature, 430, pp. 61–65. [CrossRef] [PubMed]
Choi, J. H., and Korach, C. S., 2009, “Nanoscale Defect Generation in CMP of Low-k/Copper Interconnect Patterns,” J. Electrochem. Soc., 156(12), pp. H961–H970. [CrossRef]
Srivastava, G., and Higgs, C. F., III, 2015, “An Industrial-Scale, Multi-Wafer CMP Simulation Using the PAML Modeling Approach”, ECS J. Solid State Sci. Technol., 4(11), pp. P5088–P5096. [CrossRef]
Srivastava, G., and Higgs, C. F., III, 2015, “A Full Wafer-Scale PAML Modeling Approach for Predicting CMP,” Tribol. Lett., 59(2), p. 32. [CrossRef]
Zhang, J., Zhang, J. Y., Liu, G., Zhao, Y., and Sun, J., 2009, “Competition Between Dislocation Nucleation and Void Formation as the Stress Relaxation Mechanism in Passivated Cu Interconnects,” Thin Solid Films, 517(9), pp. 2936–2940. [CrossRef]
Lee, J. H., and Gao, Y., 2011, “Mixed-Mode Singularity and Temperature Effects on Dislocation Nucleation in Strained Interconnects,” Int. J. Solids Struct., 48(7–8), pp. 1180–1190. [CrossRef]
Gerberich, W. W., Nelson, J. C., Lilleodden, E. T., Anderson, P., and Wyrobek, J. T., 1996, “Indentation Induced Dislocation Nucleation: The Initial Yield Point,” Acta Mater., 44(9), pp. 3585–3598. [CrossRef]
Corcoran, S. G., Colton, R. J., Lilleodden, E. T., and Gerberich, W. W., 1997, “Anomalous Plastic Deformation at Surfaces: Nanoindentation of Gold Single Crystals,” Phys. Rev. B, 55(24), p. R16057. [CrossRef]
Kelchner, C. L., Plimpton, S. J., and Hamilton, J. C., 1998, “Dislocation Nucleation and Defect Structure During Surface Indentation,” Phys. Rev. B, 58(17), p. 11085. [CrossRef]
Tadmor, E. B., Miller, R., Phillips, R., and Ortiz, M., 1999, “Nanoindentation and Incipient Plasticity,” J. Mater. Res., 14(6), pp. 2233–2250. [CrossRef]
Zimmerman, J. A., Kelchner, C. L., Klein, P. A., Hamilton, J. C., and Foiles, S. M., 2001, “Surface Step Effects on Nanoindentation,” Phys. Rev. Lett., 87(16), p. R16057. [CrossRef]
Lilleodden, E. T., Zimmerman, J. A., Foiles, S. M., and Nix, W. D., 2003, “Atomistic Simulations of Elastic Deformation and Dislocation Nucleation During Nanoindentation,” J. Mech. Phys. Solids, 51(5), pp. 901–920. [CrossRef]
Van Vliet, K. J., Li, J., Zhu, T., Yip, S., and Suresh, S., 2003, “Quantifying the Early Stages of Plasticity Through Nanoscale Experiments and Simulations,” Phys. Rev. B, 67(10), p. 104105. [CrossRef]
Mason, J. K., Lund, A. C., and Schuh, C. A., 2006, “Determining the Activation Energy and Volume for the Onset of Plasticity During Nanoindentation,” Phys. Rev. B, 73(5), p. 054102. [CrossRef]
Schall, P., Cohen, I., Weitz, D. A., and Spaepen, F., 2006, “Visualizing Dislocation Nucleation by Indenting Colloidal Crystals,” Nature, 440(7082), pp. 319–323. [CrossRef] [PubMed]
Wagner, R. J., Ma, L., Tavazza, F., and Levine, L. E., 2008, “Dislocation Nucleation During Nanoindentation of Aluminum,” J. Appl. Phys., 104(11), p. 114311. [CrossRef]
Morris, J. R., Bei, H., Pharr, G. M., and George, E. P., 2011, “Size Effects and Stochastic Behavior of Nanoindentation Pop In,” Phys. Rev. Lett., 106(16), p. 165502. [CrossRef] [PubMed]
Agrawal, V., and Dayal, K., 2015, “A Dynamic Phase-Field Model for Structural Transformations and Twinning: Regularized Interfaces With Transparent Prescription of Complex Kinetics and Nucleation. Part I: Formulation and One-Dimensional Characterization,” J. Mech. Phys. Solids, 85, pp. 270–290. [CrossRef]
Miller, R., and Rodney, D., 2008, “On the Nonlocal Nature of Dislocation Nucleation During Nanoindentation,” J. Mech. Phys. Solids, 56(4), pp. 1203–1223. [CrossRef]
Delph, T. J., Zimmerman, J. A., Rickman, J. M., and Kunz, J. M., 2009, “A Local Instability Criterion for Solid-State Defects,” J. Mech. Phys. Solids, 57(1), pp. 67–75. [CrossRef]
Delph, T. J., and Zimmerman, J. A., 2010, “Prediction of Instabilities at the Atomic Scale,” Modell. Simul. Mater. Sci. Eng., 18(4), p. 045008. [CrossRef]
Garg, A., Acharya, A., and Maloney, C., 2014, “A Study of Conditions for Dislocation Nucleation in Coarser-Than-Atomistic Scale Models,” J. Mech. Phys. Solids, 75, pp. 76–92. [CrossRef]
Hill, R., 1962, “Acceleration Waves in Solids,” J. Mech. Phys. Solids, 10(1), pp. 1–16. [CrossRef]
Gouldstone, A., Van Vliet, K. J., and Suresh, S., 2001, “Nanoindentation: Simulation of Defect Nucleation in a Crystal,” Nature, 411(6838), pp. 656–656. [CrossRef] [PubMed]
Li, J., Van Vliet, K. J., Zhu, T., Yip, S., and Suresh, S., 2002, “Atomistic Mechanisms Governing Elastic Limit and Incipient Plasticity in Crystals,” Nature, 418(6895), pp. 307–310. [CrossRef] [PubMed]
Van Vliet, K. J., Li, J., Zhu, T., Yip, S., and Suresh, S., 2003, “Quantifying the Early Stages of Plasticity Through Nanoscale Experiments and Simulations,” Phys. Rev. B, 67(10), p. 104105. [CrossRef]
Garg, A., 2014, “Homogeneous Dislocation Nucleation,” Ph.D. dissertation, Carnegie Mellon University, Pittsburgh, PA, Paper 401.
Garg, A., and Maloney, C. E., 2016, “Mechanical Instabilities in Perfect Crystals: From Dislocation Nucleation to Buckling Like Modes,” ASME J. Appl. Mech., 83(12), p. 121006. [CrossRef]
Garg, A., and Maloney, C., 2016, “Universal Scaling Laws for Homogeneous Dissociation Nucleation During Nano-Indentation,” J. Mech. Phys. Solids, 95, pp. 742–754. [CrossRef]
Plimpton, S., 1995, “Fast Parallel Algorithms for Short-Range Molecular Dynamics,” J. Comput. Phys., 117(1), pp. 1–19. [CrossRef]
Miller, R., 2004, “A Stress-Gradient Based Criterion for Dislocation Nucleation in Crystals,” J. Mech. Phys. Solids, 52(7), pp. 1507–1525. [CrossRef]
Jones, E., Oliphant, T., and Peterson, P., 2001, “SciPy: Open Source Scientific Tools for Python,” Online, Accessed August 28, 2014.
Hasan, A., and Maloney, C. E., 2012, “Mesoscale Harmonic Analysis of Homogenous Dislocation Nucleation,” arXiv e-print 1205.1700.
Acharya, A., 2001, “A Model of Crystal Plasticity Based on the Theory of Continuously Distributed Dislocations,” J. Mech. Phys. Solids, 49(4), pp. 761–784. [CrossRef]
Acharya, A., 2010, “New Inroads in an Old Subject: Plasticity, From Around the Atomic to the Macroscopic Scale,” J. Mech. Phys. Solids, 58(5), pp. 766–778. [CrossRef]


Grahic Jump Location
Fig. 1

(a) Elastic energy, U, stored in the crystal as a function of the indenter depth, D, LJ crystal, L = 40, R = 40. (b) Corresponding load, F, on the indenter in the vertical direction as a function of the indenter depth, D. All lengths such as Lx, L, R, and C are measured in units of the lattice constant, (a) energy and forces are measured in LJ units.

Grahic Jump Location
Fig. 2

Schematic of the setup. All the relevant geometrical parameters are shown: width, Lx; thickness, L; indenter radius, R; contact length, C. We indicate the crystallographic orientation with respect to the indentation axis as O1, O2, O3, and O4, respectively.

Grahic Jump Location
Fig. 3

Lowest four energy eigenvalues, λ, for L = 160, R = 40, O1, and LJ crystal, as a function of δD = DDc. The cyan line is a critical energy eigenmode along which the system is driven to instability. (Color version online.)

Grahic Jump Location
Fig. 4

(a) Lowest eigenmode at δD = DDc ≈ 10−6 for L = 160, R = 40, O1, and LJ crystal. (b) Corresponding transverse mode gradient, Ω/Ωmax.

Grahic Jump Location
Fig. 5

Top: Characteristic λminrmeso curves for an undeformed crystal and a deformed crystal close to nucleation (δD ∼ 10−6). Bottom: Mesoregion eigenmodes corresponding to the lowest energy eigenvalue of the mesoregion with radius rmeso = 8: from an undisturbed crystal (left) and a configuration close to dislocation nucleation (right).

Grahic Jump Location
Fig. 6

Structure of the mesomode for three different rmeso from the system with L = 160, R = 120, and δD ≈ 10−6. rmeso = ∞ corresponds to the entire system. Note, how quickly the mesomode captures the structure of the incipient defect.

Grahic Jump Location
Fig. 7

ln(λmin) versus rmeso curves for various indenter radii, R, and system size, L = 160

Grahic Jump Location
Fig. 8

Ω(s) curves on the slip plane computed from lowest eigenmodes of various scale regions for R = 120 and L = 160

Grahic Jump Location
Fig. 9

(a) ξmeso as a function of rmeso for various indenter radii, R, and system size, L = 160. Inset shows that the curves can be made to collapse by rescaling the axes with their plateau value ξ. (b) Collapsed ξmeso versus rmeso curves (rescaled by ξ) for indenter radius, R = 160, and different system sizes, L.

Grahic Jump Location
Fig. 10

Structure of the lowest energy eigenmode for rmeso = 8 and indenter radius, R = 40, when the region is moved along the slip plane. The “red cross” corresponds to the embryo center and the “green cross” corresponds to the mesoregion center. (Color version online.)

Grahic Jump Location
Fig. 11

For fixed rmeso = 40 and various indenter radii, R, the lowest mode can detect the embryo inside the solid curve as predicted by Eq. (6). When the mesoregion is centered at the “red” atoms, the lowest mode can detect the embryo. The colors represent maximum Ω for the lowest mode. The black solid curve shows the prediction based on Eq. (6) for δc. (a) R = 40. (b) R = 80. (c) R = 120. (d) R = 160. (e) R = 225. (f) R = 270. (Color version online.)

Grahic Jump Location
Fig. 12

Schematic diagram used to derive the geometrical relation between δc, rmeso, and x as in Eq. (5). δc is the critical distance from the “Hot atom” upto which the critical mesomode has a lower energy than the long wavelength mode. x is the critical portion of the embryo required to capture the defect.

Grahic Jump Location
Fig. 13

Analysis when the mesoregion is moved along the slip plane. (a) Schematic diagram for Eq. (7) and (b) x, the portion of the embryo necessary for the lowest mode to be the critical mode versus rmeso for different indenter radii, R.

Grahic Jump Location
Fig. 14

Left: The difference between the indenter depth at which the meso-mode shows signature of incipient dislocation, Dmeso, and Dc, δDmeso vs. rmeso for various indenter radius. Right: Corresponding to Dmeso, the indenter force at which the meso-mode detects incipient dislocation nucleation, Fmeso vs. rmeso.

Grahic Jump Location
Fig. 15

Critical mesomode centered at the embryo core. Different views of the embryo are shown. The colors show magnitude of the mode at each atom.

Grahic Jump Location
Fig. 16

Top: Ω versus s along n1 and n4. Bottom: Ω versus s along n1, n2, and n3.

Grahic Jump Location
Fig. 17

Left: The embryo size, ξ, scaled by R0.5 as a function of R. Right: The embryo core depth from the surface, Y*, scaled by R0.75 as a function of R.



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In