Electrical breakdown in a fuse network with random, continuously distributed breaking strengths

We investigate the breakdown properties of a random resistor-fuse network in which each network element behaves as a linear resistor if the voltage drop is less than a threshold value, but then ‘‘burns out’’ and changes irreversibly to an insulator for larger voltages. We consider a fully occupied network in which each resistor has the same resistance (in the linear regime), and with the threshold voltage drop uniformly distributed over the range $v_{\mathrm{-}}$=1-w/2 to $v_{+}$=1+w/2 (0<w≤2). The breakdown properties of this model depend crucially on w, and also on L, the linear dimension of the network. For sufficiently small w, ‘‘brittle’’ fracture occurs, in which catastrophic breaking is triggered by the failure of a vanishingly small fraction of bonds in the network. In this regime, the average voltage drop per unit length required to break the network, 〈$v_b$〉, varies as $v_{\mathrm{-}}$+O(1/$L^2$), and L→∞, and the distribution of breakdown voltages decays exponentially in $v_b$. By probabilistic arguments, we also establish the existence of a transition between this brittle regime and a ‘‘ductile’’ regime at a critical value of w=$w_c$(L), which approaches 2, as L→∞. This suggests that the fuse network fails by brittle fracture in the thermodynamic limit, except in the extreme case where the distribution of bond strengths includes the value zero. The ductile regime, w>$w_c$(L), is characterized by crack growth which is driven by increases in the external potential, before the network reaches the breaking point. For this case, numerical simulations indicate that the average breaking potential decreases as 1/(lnL${)}^y$, with y≤0.8, and that the distribution of breakdown voltages has a double experimental form. Numerical simulations are also performed to provide a geometrical description of the details of the breaking process as a function of w.