Evolution of lowest eigenvalues near dislocation nucleation

Fig 1: Plot of 4 lowest eigenvalues for 4 system geometries. The 4 'lowest' eigenvalues of the full system are plotted against indenter depth.

Fig 2: Fitted functions of the form y=-A*sqrt(B-x) - the cyan curves -to the tails of the eigenvalue ~ Depth plots (where they approach zero); A & B are constants. Except for x-triple, all systems show the existence of 1 critical mode whose eigenvalue seems to go to zero as the square root of (Dc - D) & for all systems, the parameter B was the critical depth. (in accord with expectations from a 'saddle node bifurcation' based nucleation mechanism).


Fig 3: The 4 lowest modes from the system x-triple corresponding to the last configuration before nucleation (roughly del_D ~ 1e-6). This image shows that more than 1 normal mode is active during nucleation partially accounting for different character of xtriple's eigenvalue plots. Mode 1 corresponds to the lowest eigenvalue (the one closest to zero).
Fig 4: The 4 lowest modes for the system x-quadruple corresponding a configuration with (Dc-D) ~ 1e-6. Notice the strong contrast with the modes of x-triple. Mode 1 is again the lowest.

Plotting Info: Used my numerical hessian routine to compute negative-definite hessian matrices. An ARPACK based Lanczos implementation built into matlab was used to diagonalize the hessian matrices. Only configurations with max_force <= 1e-8 were for the computation of hessians. And the minimizations were done with the pre-MR minimizer, ie. backtrack_quadratic.