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Topology determines force distributions in one-dimensional random spring networks

Knut M. Heidemann1, Andrew O. Sageman-Furnas1, Abhinav Sharma2,3, Florian Rehfeldt2, Christoph F. Schmidt2,*, and Max Wardetzky1,†

  • 1Institute for Numerical and Applied Mathematics, University of Goettingen, 37083 Goettingen, Germany
  • 2Third Institute of Physics - Biophysics, University of Goettingen, 37077 Goettingen, Germany
  • 3Leibniz Institute of Polymer Research Dresden, 01069 Dresden, Germany

  • *cfs@physik3.gwdg.de
  • †wardetzky@math.uni-goettingen.de
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Phys. Rev. E 97, 022306 – Published 15 February, 2018

DOI: https://doi.org/10.1103/PhysRevE.97.022306

Abstract

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.

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Phys. Rev. E 97, 022306– Published 15 February, 2018
  • Received 10 July 2017
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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