Modeling crack propagation in heterogeneous materials: Griffith's law, intrinsic crack resistance, and avalanches

Various kinds of heterogeneity in solids, including atomistic discreteness, affect the fracture strength as well as the failure dynamics remarkably. Here we study the effects of an initial crack in a discrete model for fracture in heterogeneous materials, known as the fiber bundle model. We find three distinct regimes for fracture dynamics depending on the initial crack size. If the initial crack is smaller than a certain value, it does not affect the rupture dynamics and the critical stress, while for a larger initial crack, the growth of the crack leads to breakdown of the entire system, and the critical stress depends on the crack size in a power-law manner with a nontrivial exponent. The exponent, as well as the limiting crack size, depend on the strength of heterogeneity and the range of stress relaxation in the system.

Figure 1

The figures show the stress profile of a 1D FBM of size $L$ under a local load sharing scheme (LLS) with a tensile force $F$ applied on it. A crack of length $l(=3)$ (shown by the dotted red lines) is inserted in the middle. This is equivalent to a crack of length $la$ (or $l$ if we consider $a=1$). The local stress of all fibers is ${\sigma }_e(=F/L)$, except for the two fibers at the crack tip that carry a stress $5{\sigma }_e/2$ each [see Eq. (2)].

Figure 2

(a) Variation of $\rho$ with $B$ for various $l$ values and for $\beta =1.0$. The red dots stand for a maximum value ${\rho }_m$ of the patch density. (b) Variation of ${\rho }_m$ with $l$ for $\beta =1.0$. For $l>{\xi }^{*}, {\rho }_m$ falls to $1/L$ (pure nucleating). For $l<\xi , {\rho }_m$ remains constant independent of l. (c) Variation of $P_n$ and $P_{\mathrm{scn}}$ with $l$ for $\beta =1.0$. Three distinct regions I, II, and III are seen. Region I: $P_{\mathrm{scn}}$ is zero and $P_n$ is small, indicating that the effect of the preexisting crack is not observed here. Region II: $P_n$ is unity but $P_{\mathrm{scn}}$ is low, indicating that many cracks are observed here but the preexisting crack nucleates to create global failure. Region III: Both $P_n$ and $P_{\mathrm{scn}}$ have assumed the value unity, indicating only the preexisting crack propagates here prior to global failure. (d) Average crack size $s_{av}$ and maximum crack size $s_{\max }$, just before global failure, are plotted against $l$ for $\beta =1.0$. For $l<\xi$, both $s_{av}$ and $s_{\max }$ are independent of $l$. Above $l>\xi , s_{\max }\sim l$, suggesting coalescence of the cracks to the preexisting crack leading to fracture of the bundle. Beyond $l>{\xi }^{*}, s_{av}=s_{\max }\sim l$, suggesting nucleation of the preexisting crack only.

Figure 3

(a) The figure shows the variation of the critical stress ${\sigma }_c$ with $l$ for LLS stress redistribution scheme and for $\beta =1.0$. For $l<\xi , {\sigma }_c$ is independent of $l$. For $l>\xi , {\sigma }_c\sim l^{-\alpha }$, where $\alpha \simeq 0.85$ is an exponent. (b) Avalanche size distribution $P\left(\mathrm{\Delta }\right)$ vs $\mathrm{\Delta }$ for different lengths $l$ of preexisting cracks for LLS fiber bundle model at $\beta =1.0$. The inset shows the exponential decrease of $P\left(\mathrm{\Delta }\right)$ with $\mathrm{\Delta }$ for small avalanche size $\mathrm{\Delta }$ in log-normal scale. (c) The average size $\langle {\mathrm{\Delta }}_f\rangle$ of the large avalanches is plotted against $l$. (d) The characteristic length ${\mathrm{\Delta }}_0$ which defines the exponential fall of the distribution is plotted against $l$.

Figure 4

(a) The figure shows the variation of $\xi$ and ${\xi }^{*}$ with $\beta$ for LLS FBM. We observe three regions: I, II, and III, depending on the spatial correlation during failure. Region I: Conventional LLS limit where the crack does not play any role. Region II: Multiple cracks but only the preexisting crack nucleates and creates global failure. Region III: Only a single crack, the preexisting one, propagates prior to global failure. No other crack originates within the bundle. (b) Variation of $\alpha$ with $\beta$ for LLS FBM. $\alpha$ has a high value, close to 1, for low $\beta$ and then decreases with $\beta$. For $\beta >0.5, \alpha$ remains constant at around 0.85 independent of the disorder strength.

Figure 5

(a) Variation of ${\sigma }_c$ with $l$ for $\gamma =0.5$ ($<{\gamma }_c$) and $\gamma =4.0$ ($>{\gamma }_c$) and for $\beta =1.0$. The behavior matches with Eq. (7) (LLS FBM) for $\gamma$ is 4.0. For $\gamma =0.5$ the critical stress is ${10}^{\beta }/2\beta eln10$ at low $l$ and decreases when $l$ is high. (b), (c) and (d) respectively show the variation of $\xi , {\xi }^{*}$, and $\alpha$ with $\gamma$. High $\gamma$ corresponds to the LLS limit. On the other hand, when $\gamma$ is low, both $\xi$ and ${\xi }^{*}$ increase with decreasing $\gamma$. This is due to the fact that in low $\gamma$, the local stress concentration does not play a crucial role, and the preexisting crack will be visible to the model only when the length of the crack is large. $\alpha$ decreases with decreasing $\gamma$, suggesting less effect of the crack on the critical stress.

Figure 6

(a) ${\sigma }_c$ vs $l$ for system sizes ranging between ${10}^3$ and ${10}^5$. For $l<\xi , {\sigma }_c$ is independent of $l$ but decreases with increasing $L$ [see inset (i)]. The system size effect is not visible in the region $l>\xi$ [see inset (ii)]. (b) Variation of characteristic length $\xi$ with disorder strength $\beta$ for system sizes ranging in between ${10}^3$ and ${10}^5$. $\xi$ is not observed to change with size of the system. We set $\beta =1.0$.