Topology determines force distributions in one-dimensional random spring networks

Networks of elastic fibers are ubiquitous in biological systems and often provide mechanical stability to cells and tissues. Fiber-reinforced materials are also common in technology. An important characteristic of such materials is their resistance to failure under load. Rupture occurs when fibers break under excessive force and when that failure propagates. Therefore, it is crucial to understand force distributions. Force distributions within such networks are typically highly inhomogeneous and are not well understood. Here we construct a simple one-dimensional model system with periodic boundary conditions by randomly placing linear springs on a circle. We consider ensembles of such networks that consist of $N$ nodes and have an average degree of connectivity $z$ but vary in topology. Using a graph-theoretical approach that accounts for the full topology of each network in the ensemble, we show that, surprisingly, the force distributions can be fully characterized in terms of the parameters $(N,z)$. Despite the universal properties of such $(N,z)$ ensembles, our analysis further reveals that a classical mean-field approach fails to capture force distributions correctly. We demonstrate that network topology is a crucial determinant of force distributions in elastic spring networks.

Figure 1

(a) An example network on the circle, with $N=5$ and $z=2.4$. (b) Graph representation of the network in (a). The spring orientations are depicted by black arrows. The network contains two fundamental cycles, for example, $\{l_1,l_2,l_3,l_4\}$ and $\{l_4,l_5,l_6\}$. After choosing arbitrary orientations for both cycles (gray arrows), we construct linear constraints that fix their winding numbers [Eq. (1)]—here: $l_1+l_2+l_3-l_4=-1$ (winds around circle once) and $l_4+l_5+l_6=0$ (contractible). (c) The abstract cycle graph ($z=2$) with $N=5$ (left) and three realizations on the circle with distinct topologies (same graphs but different winding numbers $g$). Top and bottom row show initial and corresponding relaxed configurations, respectively. Note that, for visualization purposes, overlapping springs are drawn with a slight offset.

Figure 2

(a) The complete graph ($z=N-1$) for $N=9$ vertices, here shown as undirected graph for clarity. (b) A spanning tree of a directed version of the complete graph in (a). The chosen spanning tree is based at the vertex at the center and, from there, reaches out to all $N-1$ other vertices. The edge orientations of the spanning tree edges (black) are chosen such that they have opposite orientation with respect to a connecting cycle (see, e.g., the cycle formed by the gray edge).

Figure 3

Normalized conditional variance $Var\left(\mathbf{\Delta }\mathit{l}\right|\overline{\mathit{l}}=\overline{\ell })/{\mathit{E}}_{\overline{\mathit{l}}}\left[Var\left(\mathbf{\Delta }\mathit{l}\right|\overline{\mathit{l}})\right]$ as a function of $\overline{\ell }$ for graphs with $N=100$ and varying $z$ values, corresponding to a cycle graph (a), intermediate regime graphs (b), and a complete graph (c). For each value of $z$, data points correspond to ensemble averages (repeated simulations) with $4.95\times {10}^6$ springs in total. We use local linear regression with $3\times {10}^4$ nearest neighbors to estimate the variance for different values of $\overline{\ell }$. The solid lines correspond to the analytically derived expressions for the cycle and complete graph. In the intermediate regime of connectivity, the variance shows a continuous transition between the two extreme cases.

Figure 4

Conditional probability density $p_{\mathbf{\Delta }\mathit{l}|\overline{\mathit{l}}=\overline{\ell }}\left(\mathrm{\Delta }\ell \right)$ for spring networks with $N=100$ and varying $z$, conditioned on different $\overline{\ell }$ values. For each value of $z=2.0$ (a), $z=2.2$ (b), $z=3.0$ (c), $z=8.0$ (d), data points correspond to ensemble averages (repeated simulations) with $4.95\times {10}^6$ springs in total. Solid lines correspond to best-fit normal distributions. The cycle graph ($z=2$) is close to being normally distributed—as proven for $N\to \infty$. Whereas for $z=2.2$, there are still deviations from a normal distribution, for $z=3$ and larger, the densities rapidly approach a normal distribution.

Figure 5

Probability density $p_{{\mathit{l}}^{*}}\left({\ell }^{*}\right)$ for the final spring lengths for networks with $N=1000$ and varying $z$, corresponding to $z=3$ (a), $z=4$ (b), $z=6$ (c), and $z=16$ (d). Solid lines show the analytic expression for $p_{{\mathit{l}}^{*}}\left({\ell }^{*}\right)$ Eq. (C1), both, using the exact expressions for $\mathit{E}\left(\mathbf{\Delta }\mathit{l}|\overline{\mathit{l}}=\overline{\ell }\right)$ and $E_{\overline{\mathit{l}}}\left[Var\left(\mathbf{\Delta }\mathit{l}\right|\overline{\mathit{l}})\right]$ (solid black line), as well as using their mean-field approximations (solid red line). The data points correspond to averages over 50 simulations; the error bars correspond to the standard deviation. The initial uniform spring length distribution $p_{\overline{\mathit{l}}}\left(\overline{\ell }\right)$ is shown as a gray dashed line.

Figure 6

Mean-field approach (here, $z=4$): The edge $\overline{l}$ connects two nodes $({\mathrm{n}}_1,{\mathrm{n}}_2)$. The initial node forces (gray arrows) are given by $f_{{\mathrm{n}}_1}={\sum }_{i=1}^{z-1}{\overline{l}}_{{\mathrm{n}}_1,i}+\overline{l}$ and $f_{{\mathrm{n}}_2}={\sum }_{i=1}^{z-1}{\overline{l}}_{{\mathrm{n}}_2,i}-\overline{l}$. The spring $\overline{l}$ contributes to the forces with different signs because its length is measured from ${\mathrm{n}}_1$ to ${\mathrm{n}}_2$ (depicted by the black triangle). While displacing an individual node by $u_{{\mathrm{n}}_{\mathrm{k}}}=z^{-1}{f}_{{\mathrm{n}}_{\mathrm{k}}}$ introduces force balance at that node, this approach neglects that the nodes are coupled, i.e., force balance has to be established at all nodes simultaneously, as in Eq. (1).

Figure 7

Comparison of graph-theoretical (black) and mean-field (gray) variances as a function of average degree $z$. Shown are the unconditional variance $Var\left(\mathbf{\Delta }\mathit{l}\right)$ as well as the expected conditional variance ${\mathit{E}}_{\overline{\mathit{l}}}\left[Var\left(\mathbf{\Delta }\mathit{l}\right|\overline{\mathit{l}})\right]$ as derived in the text. The graph-theoretically derived expected variance exhibits a maximum at $z=4$ (filled circle), while the corresponding mean-field expected variance monotonically decreases from its value at $z=2$.