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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.

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Figures

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.

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