= \frac{\mu_M\lambda_M}{\mu_S + \mu_M}  \;.$$

More generally, we can define a parameter $\Delta\alpha$ to quantify the distance from the BDT:

$$\label{Da}
\Delta\alpha=\frac{\mu_M\lambda_M}{(\mu_M+\mu_S)\lambda_S}-1  \; .$$

If $\Delta\alpha=0$ the system is at the BDT. Brittle failure occurs for $\Delta\alpha<0$; in this case only one microcrack is enough to cause global failure so the fraction of broken sacrificial bonds at failure is $\phi_f=1/N$ with $N$ the system size. Ductile failure occurs for $\Delta\alpha>0$, where all sacrificial bonds break before the system fails globally, so $\phi_f=1$. It follows from this analysis that for ductile failure we need $\lambda_S<\lambda_M$, which explains why the creation of tough, ductile networks requires the sacrificial network to be much weaker than the matrix.

For a perfectly homogeneous system, the BDT in this model is an abrupt transition. Experiments, however, show that the descriptors of failure vary in a continuous manner through the BDT [@Matsuda2016]. We therefore introduce disorder by assuming that the thresholds of the sacrificial network vary according to a certain distribution $P(\lambda_S)$. All other parameters are taken the same. In this case, the sacrificial bonds will fail progressively, with the weakest bonds breaking first and the stronger bonds breaking later. At a given strain $\epsilon$, all sacrificial 