Temporal and Spacial Evolution of Bursts in Creep Rupture

We investigate the temporal and spacial evolution of single bursts and their statistics emerging in heterogeneous materials under a constant external load. Based on a fiber bundle model we demonstrate that when the load redistribution is localized along a propagating crack front, the average temporal shape of pulses has a right-handed asymmetry; however, for long range interaction a symmetric shape with parabolic functional form is obtained. The pulse shape and spatial evolution of bursts proved to be correlated, which can be exploited in materials’ testing. The probability distribution of the size and duration of bursts have power law behavior with a crossover to higher exponents as the load is lowered. The crossover emerges due to the competition of the slow and fast modes of local breaking being dominant at low and high loads, respectively.

Figure 1

(a) Bursting activity during the creep rupture of a fiber bundle of size $L=401$. Slowly damaging fibers (green-shade of gray in the upper right corner) trigger bursts of immediate breaking (random colors-shades of gray-different from green). The crack started in the upper right corner of the figure. (b) Advancement of the front of the same crack obtained such that fibers broken in a time interval have the same color. (c) Time series of bursts of (a) as a stochastic point process.

Figure 2

(a) Temporal and spatial evolution of a single burst of size $\Delta =3485$. The color code represents the normalized time $0\le u/W\le 1$. The burst starts from a small spot of broken fibers (blue-black) at the bottom left corner, then it gradually expands (green, yellow, red-lighter shades of gray) and finally stops again in a small spot (white) at the top right corner. (b) The size of subbursts ${\Delta }_s$ as a function of $u/W$ for a fixed duration $W=253$. The red curve corresponds to the burst of (a).

Figure 3

(a) Average pulse shapes for durations $W=50$, 100, 200, 300, 400, 500, 600 for LLS FBMs obtained at the load ${\sigma }_0/{\sigma }_c=0.01$. (b) Scaling collapse of average pulse shapes of different duration for LLS. The white continuous line represents the fit with the scaling function $f(x)$. The dashed black line presents the scaling function of the mean field limit of FBM for comparison. (c) Skewness $S$ of pulse shapes as a function of the duration $W$ at different loads. (d) The number of perimeter sites of bursts touching the crack front $L_p^{*}$ divided by the duration $W$ as a function of the corresponding skewness of bursts.

Figure 4

(a) Average size of bursts $⟨\Delta ⟩$ as a function of the duration $W$ for several load values ${\sigma }_0$. (b) The fraction of fibers breaking due to damage $n_d$ and to avalanches $n_b$ as function of ${\sigma }_0$.

Figure 5

Probability distribution of burst size $p(\Delta )$ (a) and duration $p(W)$ (b) varying the external load ${\sigma }_0$. The lines are fits with Eqs. (2). (c) Burst size distributions for different system sizes $L$ obtained at two load values ${\sigma }_0/{\sigma }_c=0.1$ and 0.001 for the upper and lower groups of curves, respectively. (d) High quality collapse of the distributions of (c) obtained by rescaling with $L$ using the scaling exponents $\xi =1.6$, $\tau =1.75$ and $\xi =2.4$, $\tau =2.4$ for the lower and upper curves, respectively. The straight lines represent power laws of slope $\tau$. In (a) and (b) the vertical axis, while in (d) both axis are arbitrarily rescaled to better see the data.