\; ,$$

where $\Delta l_D$ and $\Delta l_M$ represent the extension of the network in the intact double network and in the matrix spanning the microcrack, respectively, and where $\sigma$ denotes the stress. The overall strain of the system $\epsilon$ can be written as

$$\label{strain}
\epsilon=\Delta l_D(1-\phi)+\Delta l_M \phi  \; ,$$

where $\phi$ denotes the fraction of broken sacrificial bonds (i.e. the fraction of microcracks).

Rupture of a bond is assumed to occur instantaneously when the extension of a spring $\Delta l_i$ exceeds its threshold $\lambda_i$. In the absence of disorder, all sacrificial bonds have the same threshold $\lambda_S$ and all matrix bonds have the same threshold $\lambda_M$. Brittle failure after formation of a microcrack occurs when the bridging matrix bond reaches its threshold before the other sacrificial bonds, while ductile failure occurs when the sacrificial bonds reach their threshold first. From equation \[forcebalance\] it immediately follows that the brittle-to-ductile transition occurs when $(\mu_S+\mu_M)\lambda_S=\mu_M\lambda_M$. We can therefore predict which system parameters are required for a system at the BDT. For example, the sacrificial bond threshold $\lambda_S^{*}$ for which the rupture force of a (matrix-reinforced) sacrificial bond is equal to the rupture force of a matrix bond is given by

$$\lambda_S^{*} 