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Encyclopedia of Complexity and Systems Science pp 1–25Cite as

Elastic Percolation Networks

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Definition of the Subject

Materials or structures with sufficiently low connectivity are floppy and have very low elastic moduli, while at high connectivity, they are rigid and have relatively high elastic moduli. Elastic percolation networks describe the transition from floppy to rigid that occurs as the network connectivity increases. The percolative geometry and elastic behavior near percolation are of particular interest. Conventional percolative geometries describe some experimental systems however the elastic critical behavior falls into several different universality classes. Moreover, distinct percolative geometries occur in systems with only central forces or which have soft torsional forces, and in these cases, both the geometry and elastic behavior may be distinct from conventional percolation. Granular media manifest a further distinct elastic percolation network, with the concept of an isostatic network underlying elastic behavior near jamming. This rich fundamental...

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Abbreviations

Boson peak:

The Boson peak is an excess of low-frequency modes observed in glasses, as manifested, for example, in inelastic neutron scattering data. In rigidity percolation, the Boson peak is related to the number of floppy modes.

Constraint:

Edges in a graph constrain the degrees of freedom of the nodes in the graph. If edges are independent, each edge acts as one constraint.

Degrees of freedom:

In d dimensions, a point object has d degrees of freedom, while a body has d(d + 1)/2 degrees of freedom due to rotations and translations.

Floppy mode:

A floppy mode is a deformation of a structure which is soft and in ideal models is treated as a zero energy deformation.

Generic rigidity:

A network is generic if none of its edges are dependent due to the particular geometric arrangement of the nodes in the network. With high probability random networks are generic, while regular lattices are nongeneric.

Isostatic network:

An isostatic network is rigid but has no redundant bonds. Isostatic networks are marginally rigid as removal of any edge induces a floppy mode. Ideal generic granular media are isostatic at jamming.

Redundant bond:

A redundant bond is not essential to the rigidity of a structure. Generic networks which contain redundant bonds are overconstrained and internally stressed.

Rigidity percolation threshold:

The rigidity threshold marks the transition from floppy networks which have zero elastic moduli to rigid networks with finite elastic moduli.

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  1. Michigan State University, East Lansing, USA

    Phillip M. Duxbury

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Correspondence to Phillip M. Duxbury .

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  1. Ramtech Ltd., Larkspur, California, USA

    Robert A. Meyers

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Duxbury, P.M. (2013). Elastic Percolation Networks. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-3-642-27737-5_170-3

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  • DOI: https://doi.org/10.1007/978-3-642-27737-5_170-3

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