an accurate description of the stress distribution in the material. We therefore consider a more realistic double random spring model that does allow for a heterogeneous stress distribution in the material (Fig. \[fig:ModelOverview\](b)).

 Multi-spring model
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![Minimal models for elastic double networks and their reaction to failure. The springs represent elements of the sacrificial network (blue) and matrix network (red). (a) Multi-springs (MS) model. (b) Random Spring Network (RSN) model. Simulations are performed for networks of $50\times50$ nodes.[]{data-label="fig:ModelOverview"}](figure2.eps)

In the multi-spring model (MS), the double network is modeled as a series of elements, each consisting of two parallel springs, one representing the sacrificial network (with spring constant $\mu_S$) and one representing the matrix (with spring constant $\mu_M$) (see Fig. \[fig:ModelOverview\](a)). Each spring is assumed to be a linear Hookean spring, so that the force acting on it is given by $F_i=\mu_i\Delta l_i$, with $\Delta l_i$ the extension of spring $i$. Removal of a sacrificial spring in an element leads to the formation of a microcrack, which is bridged by the remaining matrix spring. As the elements are connected in series, the load on each element is equal. This force balance between the elements implies

$$\label{forcebalance}
\sigma=(\mu_M+\mu_S)\Delta l_D = \mu_M\Delta l_M 