Structural evolution of granular systems: theory
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ORIGINAL PAPER
Structural evolution of granular systems: theory Clara C. Wanjura1,2 · Paula Gago3 · Takashi Matsushima4 · Raphael Blumenfeld2 Received: 12 July 2020 © The Author(s) 2020
Abstract A general theory is developed for the evolution of the cell order (CO) distribution in planar granular systems. Dynamic equations are constructed and solved in closed form for several examples: systems under compression; dilation of very dense systems; and the general approach to steady state. We find that all the steady states are stable and that they satisfy a detailed balance-like condition when the CO≤ 6 . Illustrative numerical solutions of the evolution are shown. Our theoretical results are validated against an extensive simulation of a sheared system. The formalism can be readily extended to other structural characteristics, paving the way to a general theory of structural organisation of granular systems. Keywords Granular dynamics · Structural evolution · Cell order distribution · Non-equilibrium detailed balance
1 Introduction Modelling self-organisation of dense granular matter (DGM) is essential to many natural phenomena and technological applications. This is a key problem because both the dense flow dynamics and the large-scale properties of consolidated DGM, e.g. permeability [1], catalysis, heat exchange, fuel cell functionality [2], depend strongly on the particle-scale structure [1, 3–7]. Here, we address this issue and develop structural evolution equations for quasi-static two-dimensional (2D) particulate systems. Structural evolution of DGM proceeds via continual making and breaking of intergranular contacts, modifying the intergranular force transmission, which in turn drives the
evolution. The contact network is a graph containing cells, which are the smallest voids enclosed by grains (aka irreducible loops), and they are characterised by the number of grains enclosing it—its order. We develop here a theory for the evolution of the cell order distribution (COD), which has been argued [8] and shown [9, 10] to converge to a universal form. This theory can be extended to model other structural descriptors. We construct the evolution equations and solve them, under some assumptions, first analytically for both very dense closed systems and closed systems approaching a steady state, and then numerically for general cases. A comparison of our predictions with simulations of sheared systems supports the theory.
2 The evolution equations Electronic supplementary material The online version of this article (https://doi.org/10.1007/s10035-020-01056-4) contains supplementary material, which is available to authorized users. * Raphael Blumenfeld [email protected] Clara C. Wanjura [email protected] 1
Ulm University, Albert‑Einstein‑Allee 11, 89081 Ulm, Germany
2
Cavendish Laboratory, Cambridge University, JJ Thomson Avenue, Cambridge CB3 0HE, UK
3
Imperial College London, London SW7 2AZ, UK
4
University of Tsukuba, Tsukuba, Japan
In the following, we focus on 2D systems of convex parti
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