Soft Modes and Elasticity of Nearly Isostatic Lattices: Randomness and Dissipation

Xiaoming Mao, Ning Xu, and T. C. Lubensky
Phys. Rev. Lett. 104, 085504 – Published 24 February 2010

Abstract

The square lattice with nearest neighbor central-force springs is isostatic and does not support shear. Using the coherent potential approximation (CPA), we study how the random addition, with probability P=(z4)/4 (z=average number of contacts), of next-nearest-neighbor (NNN) springs restores rigidity and affects phonon structure. The CPA effective NNN spring constant κ˜m(ω), equivalent to the complex shear modulus G(ω), obeys the scaling relation, κ˜m(ω)=κmh(ω/ω*), at small P, where κm=κ˜m(0)P2 and ω*P, implying nonaffine elastic response at small P and the breakdown of plane-wave states beyond the Ioffe-Regel limit at ωω*. We identify a divergent length l*P1, and we relate these results to jamming.

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  • Received 14 September 2009

DOI:https://doi.org/10.1103/PhysRevLett.104.085504

©2010 American Physical Society

Authors & Affiliations

Xiaoming Mao1, Ning Xu1,2,3, and T. C. Lubensky1

  • 1Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
  • 2The James Frank Institute, University of Chicago, Chicago, Illinois 60637, USA
  • 3Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong

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Issue

Vol. 104, Iss. 8 — 26 February 2010

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