Elasticity and Response in Nearly Isostatic Periodic Lattices

Abstract

The square and kagome lattices with nearest neighbor springs of spring constant $k$ are isostatic with a number of zero-frequency modes that scale with their perimeter. We analytically study the approach to this isostatic limit as the spring constant $k^{\prime }$ for next-nearest-neighbor bonds vanishes. We identify a characteristic frequency ${\omega }^{\ast }\sim \sqrt{k^{\prime }}$ and length $l^{\ast }\sim \sqrt{k/k^{\prime }}$ for both lattices. The shear modulus $C_{44}=k^{\prime }$ of the square lattice vanishes with $k^{\prime }$, but that for the kagome lattice does not.

4 pages, 5 figures, revised text as published in PRL

References in corpus (5)

• Jamming at Zero Temperature and Zero Applied Stress: the Epitome of Disorder
• Critical scaling in linear response of frictionless granular packings near jamming
• Excess Vibrational Modes and the Boson Peak in Model Glasses
• Nonaffine rubber elasticity for stiff polymer networks
• Energy transport in jammed sphere packings

Cited by in corpus (5)

• Elasticity of Filamentous Kagome Lattice
• Mechanical instability at finite temperature
• Patterns on a Roll: A Method for Continuous Feed Nanoprinting
• Critical temperatures of the three- and four-state Potts models on the kagome lattice
• Phonons and elasticity in critically coordinated lattices