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Nonaffine displacements in crystalline solids in the harmonic limit

Saswati Ganguly1, Surajit Sengupta1,2, Peter Sollich3, and Madan Rao4,5

  • 1Indian Association for the Cultivation of Science, 2A & 2B Raja S. C. Mullick Road, Jadavpur, Kolkata 700032, India
  • 2TIFR Centre for Interdisciplinary Sciences, 21 Brundavan Colony, Narsingi, Hyderabad 500075, India
  • 3King's College London, Department of Mathematics, Strand, London WC2R 2LS, United Kingdom
  • 4Raman Research Institute, C.V. Raman Avenue, Bangalore 560080, India
  • 5National Centre for Biological Sciences (TIFR), Bellary Road, Bangalore 560 065, India

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Phys. Rev. E 87, 042801 – Published 2 April, 2013

DOI: https://doi.org/10.1103/PhysRevE.87.042801

Abstract

A systematic coarse graining of microscopic atomic displacements generates a local elastic deformation tensor D as well as a positive definite scalar χ measuring nonaffinity, i.e., the extent to which the displacements are not representable as affine deformations of a reference crystal. We perform an exact calculation of the statistics of χ and D and their spatial correlations for solids at low temperatures, within a harmonic approximation and in one and two dimensions. We obtain the joint distribution P(χ,D) and the two-point spatial correlation functions for χ and D. We show that nonaffine and affine deformations are coupled even in a harmonic solid, with a strength that depends on the size of the coarse-graining volume Ω and dimensionality. As a corollary to our work, we identify the field hχ conjugate to χ and show that this field may be tuned to produce a transition to a state where the ensemble average 〈χ〉 and the correlation length of χ diverge. Our work should be useful as a template for understanding nonaffine displacements in realistic systems with or without disorder and as a means for developing computational tools for studying the effects of nonaffine displacements in melting, plastic flow, and the glass transition.

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Outline

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Phys. Rev. E 87, 042801– Published 2 April, 2013
  • Received 26 December 2012

©2013 American Physical Society

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