Prestrain-induced contraction in one-dimensional random elastic chains

Ihusan Adam, Franco Bagnoli, Duccio Fanelli, L. Mahadevan, and Paolo Paoletti

Phys. Rev. E 105, 065002 – Published 21 June 2022

Abstract

Prestrained elastic networks arise in a number of biological and technological systems ranging from the cytoskeleton of cells to tensegrity structures. Motivated by this observation, we here consider a minimal model in one dimension to set the stage for understanding the response of such networks as a function of the prestrain. To this end we consider a chain [one-dimensional (1D) network] of elastic springs upon which a random, zero mean, finite variance prestrain is imposed. Numerical simulations and analytical predictions quantify the magnitude of the contraction as a function of the variance of the prestrain, and show that the chain always shrinks. To test these predictions, we vary the topology of the chain, consider more complex connectivity and show that our results are relatively robust to these changes.

Received 6 October 2021
Revised 28 March 2022
Accepted 4 May 2022

DOI:https://doi.org/10.1103/PhysRevE.105.065002

Physics Subject Headings (PhySH)

Collective behavior in networks Elastic deformation Network structure

1-dimensional systems

Network Models

NetworksNonlinear Dynamics

Authors & Affiliations

Ihusan Adam ^1,2, Franco Bagnoli ^2,3, Duccio Fanelli^2,3, L. Mahadevan⁴, and Paolo Paoletti ⁵

¹Department of Information Engineering, University of Florence, Florence 50019, Italy
²Department of Physics and Astronomy, and CSDC, University of Florence, Sesto Fiorentino 50019, Italy
³INFN, Florence Section, Sesto Fiorentino 50019, Italy
⁴School of Engineering and Applied Sciences, Department of Physics, and Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, Massachusetts 02138, USA
⁵School of Engineering, University of Liverpool, L69 3GH Liverpool, United Kingdom

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Vol. 105, Iss. 6 — June 2022

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Images

Figure 1
Comparison between theoretical predictions and numerical results. The solid black lines show histograms of $x_{f_{i}} / x_{0_{i}}$ plotted for all nodes of the network. For each of the values of maximum noise $δ_{\max}$ shown, a new network was generated and tested according to the method outlined in the text. Each of the tested networks has $N = 2000$ masses and have been created according to the Watts-Strogratz strategy [24] with rewiring probability $p = 0.7$ . The dotted green line represents the expected shift per node, $1 / ψ$ , against $δ_{\max}$ . The circle markers show the mean of each of the histograms. A top view of the plot is shown in the inset.
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Figure 2
(a) Variation of the relative shrinkage $Δ R g^{2} < 0$ against the maximum prestrain $δ_{\max}$ for networks of $N = 500$ masses while (b) and (c) show similar results of tests done for $N = 2000$ and $N = 4000$ , respectively. The dashed green line represents the expectation calculated using Eq. (12) and the solid lines refer to numerical simulations. Networks of different average coordination numbers, $ρ_{a v}$ are shown by the different markers. Each marker shows the mean of six trials with separate realisations of networks, generated according to the Watts-Strogatz (rewiring probability $p = 0.7$ ). The error bars represent the Standard Error. The inset shows the difference between the expected relative shrinkage and the shrinkage from the numerical simulations $Δ R g^{2} - 〈 Δ R g^{2} 〉$ .
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Figure 3
Distribution (mean and standard error) of relative shrinkage $〈 Δ R g^{2} 〉$ against the rewiring probability $p$ of the generated Watts-Strogatz networks. 50 trials were tested for each rewiring probability by generating a new network of $N = 1000$ (shown in blue or circle markers) and $N = 2000$ (shown in magenta or diamond markers). The prestrain level was set to the maximum $δ_{\max} = 0.9$ and the dotted green line is the expectation given by the theoretical analysis. The scatter of the data is shown underneath as circles or diamonds of a paler color, to show the overall distribution. The red points (first two sets of points) refer to tests done with rewiring $p = 0$ and have been added on to the same axis for comparison for both $N = 1000$ (red left-hand side or circle markers) and $N = 2000$ (red right-hand side or diamond markers).
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Figure 4
Distribution (mean and standard error) of relative shrinkage $〈 Δ R g^{2} 〉$ against the degree exponent $γ$ of the generated scale-free networks. 50 trials were tested for each $γ$ by generating a new realisation of a network of $N = 2000$ for each trail. The prestrain strength was set to the maximum $δ_{\max} = 0.9$ and dashed green line is the expectation given by the theory. $N = 2000$ is shown in magenta (or diamond markers) and the results of $N = 1000$ is shown in blue (or circle markers). The pale dots show the scatter of the data. Notice that the imposed vertical scale differs from that employed in Fig. 3.
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Figure 5
(a) Sketch of of an End-Hub network of size $N = 8$ with $H = 1$ hub on either side. (b) Distribution (mean and standard error) of relative shrinkage $〈 Δ R g^{2} 〉$ against the rewiring probability of the generated End-Hub networks with $H = 3$ hubs per side of the network. 35 trials were tested for each rewiring probability by generating a new network of $N = 1000$ nodes (shown in blue or circle markers) and 16 trials have been tested for $N = 2000$ (shown in magenta or diamond markers). The prestrain strength was set to $δ_{\max} = 0.9$ and dashed green line is the expectation given by the theory. Pale color dots show the scatter of the data. The red points (first two sets of points) refer to tests done with rewiring $p = 0$ and have been added on to the same axis for comparison for both $N = 1000$ (red left-hand side or circle markers) and $N = 2000$ (red right-hand side or diamond markers). Notice that the imposed vertical scale differs from that employed in Figs. 3 and 4.
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