Critical stress statistics and a fold catastrophe in intermittent crystal plasticity

The statistics and origin of the first discrete plastic event in a one-dimensional dislocation dynamics simulation are studied. This is done via a linear stability analysis of the evolving dislocation configuration up to the onset of irreversible plasticity. It is found that, via a fold catastrophe, the dislocation configuration prior to loading directly determines the stress at which the plastic event occurs and that between one and two trigger dislocations are involved. The resulting irreversible plastic strain arising from the instability is found to be highly correlated with these triggering dislocations.

Figure 1

(a) Plot of numerically simulated stress vs plastic strain evolution, beginning at zero load and ending with the arrest of the first discrete plastic strain event. Also shown is the prediction arising from the linearization of the dislocation equations of motions: Eq. (10). (b) Distribution of eigenvalues and (c) plot of the ten lowest eigenvalues of the dynamical matrix corresponding to the dislocation configuration prior to loading and prior to the plastic instability, with the lowest eigenvalue approaching zero as the instability is neared [see arrowed data in (c)]. (d) Participation number [Eq. (17)] of corresponding eigenvectors showing the approximate number of dislocations involved in each eigenmode.

Figure 2

(a) Plot of the six lowest eigenvalues of the dynamical matrix as a function of external stress. (b) Scatter plot of lowest (blue) eigenvalue prior to loading vs the critical stress of the first plastic event, derived from 1000 loading simulations. In (b) a scatter plot of the eigenvalue corresponding to the initial eigenmode with the highest overlap to the unstable eigenmode is shown in green.

Figure 3

(a) Analytical density of states (DOS) arising from an ordered array of dislocations with lattice constant ${\lambda }_0$ and $d=\infty$. (b) Numerical DOS for the $d=1028\mu \mathrm{m}$ system for an ordered array with lattice constant ${\lambda }_0$ (configuration 1) and $2{\lambda }_0$ (configuration 2—see text) with the DOS of Fig. 4.

Figure 4

(a) Plot of the average critical stress vs the inverse system size and its fit using an exponentially truncated power law. (b) The distribution of eigenvalues derived from the dynamical matrix of the linearized dislocation equations of motion for the largest and smallest considered system sizes. (c) Log-log and (d) linear scatter plot of lowest eigenvalues prior to loading vs the critical stress of the first plastic event for the considered system sizes. In both plots the solid line represents the prediction due to catastrophe theory for a fold instability, whereas the dashed line in (d) shows the prediction arising from an additional higher order term of the potential energy landscape giving the $O\left(5\right)$ fold [Eq. (23)].

Figure 5

(a) Plot of relevant potential energy landscape close to a fold catastrophe [Eq. (22)]. Also shown is the $O\left(5\right)$ order fold catastrophe energy [Eq. (23)]. (b) The fold ratio as a function of the external stress for the $O\left(5\right)$ fold.

Figure 6

(a) Average $i,i+\mathrm{\Delta }i$ element of linearized matrix indicating a matrix which is predominantly tridiagonal. (b) Normalized histograms of diagonal and off-diagonal bands. (c) Plot of $ij\mathrm{th}$ off-diagonal element value vs $x_i-x_j$. (d) Intradiagonal and intra-off-diagonal correlation functions.

Figure 7

(a) Histogram of overlap between the unstable eigenmode and the normalized plastic displacement vector characterizing the resulting irreversible plasticity. (b) Peak normalized histogram of the number of dislocations involved in the plastic event obtained from the participation number of the normalized plastic displacement vector [Eq. (17)]. Data are derived from approximately 2000 loading simulations for a system size of $d=1028\mu \mathrm{m}$.

Figure 8

(a) Scatter plot of the lowest dynamical matrix diagonal element vs the lowest eigenvalue. Those coloured blue (green) involve two (three) dislocations. (b) Low end distributions of the dynamical matrix diagonals and eigenvalues with predicted power law forms. Data are derived from $10000$ initial configurations for a system size of $d=1028\mu \mathrm{m}$.