Kertész line of thermally activated breakdown phenomena

Based on a fiber bundle model we substantially extend the phase-transition analogy of thermally activated breakdown of homogeneous materials. We show that the competition of breaking due to stress enhancement and due to thermal fluctuations leads to an astonishing complexity of the phase space of the system: varying the load and the temperature a phase boundary emerges, separating a Griffith-type regime of abrupt failure analogous to first-order phase transitions from disorder dominated fracture where a spanning cluster of cracks emerges. We demonstrate that the phase boundary is the Kertész line of the system along which thermally activated fracture appears as a continuous phase transition analogous to percolation. The Kertész line has technological relevance setting the boundary of safe operation for construction components under high thermal loads.

Figure 1

(Color online) Final stable states of the time evolution of a fiber bundle of size $L=256$ obtained at different temperatures and external loads: (a) $T=0.02$, $\sigma =0.2$; (b) $T=0.04$, $\sigma =0.03$; (c) $T=0.05$, $\sigma =0.019$; and (d) $T=0.05$, $\sigma =0.01$. The color code represents the local load of fibers: deep blue (dark)—broken fibers with zero load; red—highest load (lighter colors indicate higher load).

Figure 2

(Color online) The average size of the largest cluster $⟨S_{max}⟩$ as a function of the external load $\sigma$ in a system of size $L=1024$ varying the temperature in a broad range. Equation (1) provides a good quality fit only for high loads, which characterize the Griffith regime of the system. The inset presents the same data replotted as a function of the control parameter $\varphi$. Equation (3) provides an excellent fit in the percolated regime.

Figure 3

(Color online) Phase diagram of thermally activated breakdown of homogeneous FBMs. The blue (dashed) and red (continuous) curves indicate ${\sigma }^{\ast }\left(T\right)$ and ${\sigma }_K\left(T\right)$, respectively. The curve of ${\sigma }_K\left(T\right)$ defines the Kertész line of the system. Black squares indicate the position of the maxima of $⟨S⟩$ in Fig. 4b.

Figure 4

(Color online) (a) The fraction of broken fibers $⟨N_b⟩/N$ obtained at the temperature and load values of Fig. 2 as a function of $\varphi$. (b) Average cluster size $⟨S⟩$ as a function of load $\sigma$ for several temperatures $T$. (c) Size distribution of clusters for a single temperature value $T=0.1$ varying the load $\sigma$ both below and above ${\sigma }_K\left(T\right)$. (d) The evolution of $⟨S⟩$ for a single sample below, above, and at the Kertész line.