Propagating mode-I fracture in amorphous materials using the continuous random network model

We study propagating mode-I fracture in two-dimensional amorphous materials using atomistic simulations. We use the continuous random network model of an amorphous material, creating samples using a two-dimensional analog of the Wooten-Winer-Weaire Monte Carlo algorithm. For modeling fracture, molecular-dynamics simulations were run on the resulting samples. The results of our simulations reproduce the main experimental features. In addition to achieving a steady-state crack under a constant driving displacement (which has not yet been achieved by other atomistic models for amorphous materials), the runs show microbranching, which increases with driving, transitioning to macrobranching for the largest drivings. In addition to the qualitative visual similarity of the simulated cracks to experiment, the simulation also succeeds in reproducing qualitatively the experimentally observed oscillations of the crack velocity.

Figure 1

(a) Image of a steady-state crack in the CRN model. We see (in the circle) the origin of microbranching. The crack reaches a “tough zone” and breaks parallel bonds, but it does not have enough energy to create a visible microbranch. $\eta =2$ and $\Delta /{\Delta }_G=2.7$. (b) At higher driving ($\eta =1.5$ and $\Delta /{\Delta }_G=3.1$), when the crack reaches the same “tough zone” (displaced in this figure to enable showing its downstream development) the crack now has enough energy to create two branches. Eventually one branch wins and continues as the main crack.

Figure 2

Steady-state propagating crack in our simulation, in which there are approximately $44000$ particles [zoomed-out picture of Fig. 1a]. $\eta =2$ and $\Delta /{\Delta }_G=2.7$.

Figure 3

Velocity of the crack, normalized to the Rayleigh velocity $c_R$, as a function of driving displacement $\Delta$, normalized to the critical Griffith displacement for crack propagation ${\Delta }_G$ in two amorphous CRNs with different widths [82 (black circles) and 162 (red squares) atoms in the $\widehat{y}$ direction, correspondingly] and in a lattice (blue diamonds) system. The dashed line represents the critical velocity of the lattice system. The amorphous systems have no well-defined threshold for microbranching and exhibit microbranching for $\Delta /{\Delta }_G$ well below the critical velocity of the lattice system. In all of these runs $\eta =1.5$.

Figure 4

Average total length of microbranches (number of bonds broken off the main branch) as a function of the average crack velocity, normalized to the Rayleigh velocity $c_R$. The width of the system was 162 atoms. The number of realizations varied from 15 to 20 for each point.

Figure 5

(a) Image of microbranching in PMMA. The figure is taken from Ref. [3]. (b) and (c) Images of macrobranching in experiments in (b) Homalite-100 (glassy polymer) and taken from Ref. [31] and (c) PMMA and taken from Ref. [5]. (d) Image of microbranching fracture in our CRN simulation for $\eta =1.5$ and $\Delta /{\Delta }_G=3.1$. (e) In higher driving ($\eta =1.5$ and $\Delta /{\Delta }_G=4$) macrobranching begins to appear in our CRN simulation; both branches of the crack reach the end of the sample.

Figure 6

(a) Rate of change of the fracture surface $\dot{L}$ normalized to the Rayleigh wave speed $c_R$ as a function of time for three different drivings. For small driving $\Delta /{\Delta }_G=2.7$ the curve is quite flat (black, bottom curve). At larger drivings $\Delta /{\Delta }_G=3.1$ (red, middle curve) the curve starts to oscillate due to the appearance of microbranching or macrobranching, with the amplitude larger at the highest $\Delta /{\Delta }_G=4$ (blue, top curve). (b) Correlation test to the crack’s velocity oscillations. The oscillations of the electrical resistance (solid black curve) behave like $\dot{L}\left(t\right)$ (dashed red curve) and not like the main crack velocity (dash-dotted blue curve). Here $\eta =1.5$ and $\Delta /{\Delta }_G=3.5$. For both panels, the width is 162 atoms.

Figure 7

Amplitude of the rms fluctuations in the oscillations of the electrical resistance during the course of the simulation as a function of the normalized driving $\Delta /{\Delta }_G$. The wide system is 162 atoms wide (red squares) and the narrow system is 82 atoms wide (black circles). In both cases $\eta =1.5$.

Figure 8

(a) Regular honeycomb lattice. (b) Section of a CRN prepared by a 2D analog to the WWW Monte Carlo scheme.

Figure 9

(a) Spread of the radial distance between the neighboring atoms. (b) Spread of the angles between the neighboring atoms.

Figure 10

(a) Experimental radial distribution function of amorphous silicon taken from Ref. [36] (black circles) along with the CRN RDF produced by two 2D Monte Carlo simulations at low temperature (red diamonds) and high temperature (blue squares) (with an adjustment to three dimensions, i.e., multiplied by a factor $r^3/r^2$, normalized to the amorphous silicon units). (b) RDF of a hexagonal lattice in lattice units.