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Research Papers

# Mechanical Instabilities in Perfect Crystals: From Dislocation Nucleation to Bucklinglike Modes

[+] Author and Article Information
Akanksha Garg

Carnegie Mellon University,
Pittsburgh, PA 15213

Craig E. Maloney

Carnegie Mellon University,
Pittsburgh, PA 15213

1Present address: Research Division, Center for Property Risk Solutions, FM Global, Norwood, MA 02062.

2Present address: 360 Huntington Ave., Northeastern University, Boston, MA 02115.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received July 14, 2016; final manuscript received August 25, 2016; published online September 14, 2016. Assoc. Editor: A. Amine Benzerga.

J. Appl. Mech 83(12), 121006 (Sep 14, 2016) (9 pages) Paper No: JAM-16-1357; doi: 10.1115/1.4034564 History: Received July 14, 2016; Revised August 25, 2016

## Abstract

We perform atomistic simulations of nanoindentation on Lennard–Jones 2D hexagonal crystals. In this work, we find a new spatially extended buckling-like mode of instability, which competes with the previously known instability governed by dislocation-dipole nucleation. The geometrical parameters governing these instabilities are the lattice constant, a, the radius of curvature of the indenter, R, and the thickness of the indenter layer, Ly. Whereas dislocation nucleation is a saddle-node bifurcation governed by R/a, the buckling-like instability is a pitchfork bifurcation (like classical Euler buckling) governed by $R/Ly$. The two modes of instability exhibit strikingly different behaviors after the onset of instability. The dislocation nucleation mode results in a stable final configuration containing a surface step and a stable dislocation at some depth beneath the surface, while the buckling modes are always followed immediately by subsequent nucleation of many dislocation dipoles. We show that this subsequent dislocation nucleation is also observed immediately after buckling in free standing rods, but only for rods which are of sufficiently wide aspect ratio, while thinner rods exhibit stable buckling followed only later by dislocation nucleation in the buckled state. Finally, we study the utility of several recently proposed local and quasi-local stability criteria in detecting the buckling mode. We find that the so-called Λ criterion, based on the stability of a representative homogeneously deformed lattice, is surprisingly useful in detecting the transition from dislocation-type instability to buckling-type instability.

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## Figures

Fig. 1

Schematic of two different orientations of crystal with respect to indenter axis. The red atoms correspond to the crystal, and the blue atoms correspond to the rigid base. (a) O1 and (b) O2.

Fig. 2

Load, F (in L–J units) of the L–J crystal, O2, Ly = 120, as a function of indenter depth, D: (a) R = Ly, (b) R=2Ly, and (c) R=3Ly

Fig. 3 Left

The critical eigenmode just before instability as a function of indenter radius, R. Right: Ω for the corresponding mode on the left as a function of indenter radius, R, for O2: (a) and (b) R = Ly, (c) and (d) R=2Ly, and (e) and (f) R=3Ly.

Fig. 4

Depth of indenter, D*, in the last stable configuration scaled by film thickness, Ly, as a function of R/Ly for orientation O1 and O2, Ly = 120

Fig. 5

Lowest four energy eigenvalues, λ, for O2, (a) R = Ly, (b) R = 2Ly and (c) R = 3Ly, LJ crystal, as function of δD=D−Dc. The cyan line is a critical energy eigenmode shown on the left (Ω field) along which the system is driven to instability. (a) and (b) R = Ly, (c) and (d) R = 2 Ly, and (e) and (f) R = 3 Ly.

Fig. 6

(a) Potential energy, U, as a function of ϵyy for an atomistic rod with Lx/Ly  = 0.2. (b)–(d) The sequence of rod configuration as it is compressed: (b) configuration just before buckling at “red cross” in (a), (c) configuration just after buckling, and (d) the buckled configuration emits dislocations from edges when compressed further at “black cross” in (a) as shown.

Fig. 7

Potential energy, U, as a function of ϵyy : (a) Lx/Ly  = 0.2, (b) Lx/Ly  = 0.33, and (c) Lx/Ly  = 0.5. As the crystal film thickness increases, the buckled configuration emits dislocations sooner. For large enough Lx/Ly as in (c), the buckling accompanies nucleation of dislocations.

Fig. 8

Λ (given by Eq. (7)) criterion for (a) R = Ly, (b) R = 2 Ly, and (c) R = 3 Ly. The arrows in the figures represent the critical eigenmode.

Fig. 9

Critical eigenmodes just before instability for buckling and dislocation-dipole nucleation as shown in (a) R=3Ly and (b) R = Ly. Corresponding η (given by Eq. (13)) calculated using linear stability of FDM, for LJ crystal, O2: (c) R=3Ly and (d) R = Ly. η is orders of magnitudes higher for dislocation-dipole nucleation as compared to the buckling.

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