Realizations of Isostatic Material Frameworks

Abstract

This article studies the set of equivalent realizations of isostatic frameworks and algorithms for finding all such realizations. It is shown that an isostatic framework has an even number of equivalent realizations that preserve edge lengths and connectivity. The complete set of equivalent realizations for a toy framework with pinned boundary in two dimensions is enumerated and the impact of boundary length on the emergence of these realizations is studied. To ameliorate the computational complexity of finding a solution to a large multivariate quadratic system corresponding to the constraints, alternative methods—based on constraint reduction and distance‐based covering map or Cayley parameterization of the search space—are presented. The application of these methods is studied on atomic clusters, a model of 2D glasses and jamming.

Viscoelastic Scaling Regimes for Marginally Rigid Fractal Spring Networks

“…Discussion.-It has been shown that the design of isostatic networks [52] fractal up to an arbitrarily-large length is possible, and that for the Sierpinski trianglebased spring networks considered here, power-law viscoelastic scaling was observed with an exponent ∆ ′ that can be theoretically derived. That this scaling survives above a cut-off frequency for systems into the solid phase indicates relevance to real-world applications utilising post-gelled materials.…”

Viscoelastic scaling regimes for marginally-rigid fractal spring networks

“…Qualitatively similar results have been observed for elastic sphere packings under compression and elastic beams under shear [49,52], but it is unclear if there is any relationship between the exponents in these systems. Discussion.-It has been shown that the design of isostatic networks [53] fractal up to an arbitrarily-large length is possible, and that for the Sierpinski trianglebased spring networks considered here, power-law viscoelastic scaling was observed with an exponent ∆ that can be theoretically derived. That this scaling survives above a cut-off frequency for systems into the solid phase indicates relevance to real-world applications utilising post-gelled materials.…”

Coupler curves of moving graphs and counting realizations of rigid graphs

“…Bounds on the number of realizations of such graphs have been investigated in [1-4, 8, 11, 24]. Algorithms and theory for computing the precise number of realizations for minimally rigid graphs have been investigated in [5,14,21].…”