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Non-affine displacements in crystalline solids in the harmonic limit | | Saswati Ganguly
; Surajit Sengupta
; Peter Sollich
; Madan Rao
; | | Date: |
27 Dec 2012 | | Abstract: | A systematic coarse graining of microscopic atomic displacements generates a
local elastic deformation tensor ${mathsf D}$ as well as a positive definite
scalar $chi$ measuring non-affinity, 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 $chi$ and
${mathsf D}$ and their spatial correlations for solids at low temperatures,
within a harmonic approximation and in one and two dimensions. We obtain the
joint distribution $P(chi, {mathsf D})$ and the two point spatial correlation
functions for $chi$ and ${mathsf D}$. We show that non-affine and affine
deformations are coupled even in a harmonic solid, with a strength that depends
on the size of the coarse graining volume $Omega$ and dimensionality. As a
corollary to our work, we identify the field, $h_{chi}$, conjugate to $chi$
and show that this field may be tuned to produce a transition to a state where
the ensemble average, $<chi>$, and the correlation length of $chi$ diverge.
Our work should be useful as a template for understanding non-affine
displacements in realistic systems with or without disorder and as a means for
developing computational tools for studying the effects of non-affine
displacements in melting, plastic flow and the glass transition. | | Source: | arXiv, 1212.6377 | | Services: | Forum | Review | PDF | Favorites | |
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