bonds for which $\lambda_S<\Delta l_D$ are broken. In the continuous limit (for $N\gg 1$), the fraction of broken sacrificial bonds can thus be written as

$$\label{dist}
\phi(\epsilon)=\int_0^{\Delta l_D} P(\lambda_S)\textrm{d}\lambda_D  \; .$$

Here, we will consider a Gaussian threshold distribution with mean $\langle\lambda_S\rangle$ and standard deviation $\delta\lambda$. For this case, it follows from equations \[forcebalance\] and \[dist\] that at the moment of macroscopic failure (where $\Delta l_M=\lambda_M$), the fraction of broken sacrificial bonds is given by

$$\label{dist2}
\phi_f=\frac{1}{2}\left[1+\textrm{erf}\left(\Delta\tilde{\alpha}\right)\right]  \; ,$$

with

$$\label{Datilde}
\Delta\tilde{\alpha}=\frac{\Delta\alpha\langle\lambda_S\rangle}{\delta\lambda\sqrt{2}}  \; ,$$

a normalized parameter to quantify the distance to the BDT, which takes into account the disorder in the thresholds. The fraction of broken sacrificial bonds thus increases gradually from 0 to 1 along the BDT and the parameter $\Delta\tilde{\alpha}$ sets the steepness of this transition. With increasing disorder, the transition becomes more gradual. The strain at break follows from equations \[forcebalance\] and \[strain\]:

$$\label{eq:failureStrain}
\epsilon_f=\lambda_M\left(\frac{\mu_M+\phi_f\mu_S}{\mu_M+\mu_S}\right)  \;.$$

In the brittle regime ($\Delta\tilde{\alpha}\ll-1$, $\phi_f\approx0$) we find $\epsilon_f=\mu_M\lambda_M/(\mu_M+\mu_S)$, while in the ductile regime ($\Delta\tilde{\alpha}\gg1$, $\phi_f\approx1$) we have $\epsilon_f=\lambda_M$.

Random Spring Network Model
---------------------------

Being a 1D model, the multi-spring model cannot account for stress heterogeneity and the resulting localization of stresses. We therefore consider a random spring network model 