Athermal Fracture of Elastic Networks: How Rigidity Challenges the Unavoidable Size-Induced Brittleness

By performing extensive simulations with unprecedentedly large system sizes, we unveil how rigidity influences the fracture of disordered materials. We observe the largest damage in networks with connectivity close to the isostatic point and when the rupture thresholds are small. However, irrespective of network and spring properties, a more brittle fracture is observed upon increasing system size. Differently from most of the fracture descriptors, the maximum stress drop, a proxy for brittleness, displays a universal nonmonotonic dependence on system size. Based on this uncommon trend it is possible to identify the characteristic system size $L^{*}$ at which brittleness kicks in. The more the disorder in network connectivity or in spring thresholds, the larger $L^{*}$. Finally, we speculate how this size-induced brittleness is influenced by thermal fluctuations.

Figure 1

Fracture depends on connectivity and thresholds. (a) When deforming diluted elastic networks, aligned sets of load-bearing bonds, called force chains, appear. Line thickness quantifies spring deformation. (b) Example of stress $\sigma$—strain $\epsilon$ curve and fracture descriptors. (c) Fraction of broken bonds at failure $n_f$ and (d) hidden length exploited in the fracture process, as a function of the connectivity parameter $p$ for different rupture threshold $\lambda$ and fixed system size $L=128$. Results are for 2D triangular networks.

Figure 2

The role of system size. (a) Size scaling of number of broken bonds $N_f$ for triangular networks with $p=0.65$, $\lambda =0.03$. Black circles indicate single-run values, red squares are averages, and the dashed line is the power-law fit used to extract $d_f$. (b) Damage fractal dimension $d_f$ as a function of $p$ and $\lambda$. (c) Single-run stress-strain curve for increasing system size $L$, showing that response is abrupt for large $L$. (d) Size scaling of the maximum stress ${\sigma }_c$ (log-log plot). Symbols correspond to averages and the line is the best fit. Data follow a power-law decay. Results are for 2D triangular networks.

Figure 3

Regimes based on fracture abruptness. (a) Size scaling of fracture abruptness $\mathrm{\Delta }{\sigma }_{\max }$ for networks close to the isostatic point with small $\lambda$. A minimum in the trend is evident. (b) Size scaling of maximum stress (empty circles) and fracture abruptness (filled squares) for diluted networks ($p=0.75$) with all springs with same ${\lambda }_i=0.10$ (solid lines), and with additional disorder in the spring thresholds ${\lambda }_{\mathrm{i}}$ uniformly distributed in the two intervals indicated in the legend (dashed and dotted lines). The additional threshold disorder shifts the minimum to larger $L$. (c) For $p=0.90$ and $\lambda =0.30$, a third region is probed for large $L$ where the abruptness scales as the maximum stress. (d) Size scaling of abruptness close to the isostatic point for different $\lambda$, color coded with the associated damage fractal dimension $d_f$ (inset). (e) Universal trend for rescaled abruptness (see also Supplemental Material [37]): at small $L$ force chains dominate the mechanical response but beyond $L^{*}$ stress concentration leading to abrupt fracture is unavoidable. Results are for 2D triangular networks.

Figure 4

Athermal limit and macroscopic cracks. Main graph: Size scaling of broken bonds for simulations performed with different energy minimization tolerance $F_{\mathrm{RMS}}$. Larger $L$ requires lower $F_{\mathrm{RMS}}$ to perform simulations in the athermal limit, where fracture occurs due to a single macrocrack (see postmortem snapshot on the left, where only broken bonds are shown). For larger $F_{\mathrm{RMS}}$ and huge system sizes, multiple macrocracks are observed (right snapshot) and the network response shows a large ductile interval $\mathrm{\Delta }{\epsilon }_{\mathrm{duct}}$ (bottom right).