We construct a linear response theory of applying shear deformations from boundary walls in the film geometry in Kubo’s theoretical scheme. Our method is applicable to any solids and fluids. For glasses, we assume quasi-equilibrium around a fixed inherent state. Then, we obtain linear-response expressions for any variables including the stress and the particle displacements, even though the glass interior is elastically inhomogeneous. In particular, the shear modulus can be expressed in terms of the correlations between the interior stress and the forces from the walls. It can also be expressed in terms of the inter-particle correlations, as has been shown in the previous literature. Our stress relaxation function includes the effect of the boundary walls and can be used for inhomogeneous flow response. We show the presence of long-ranged, long-lived correlations among the fluctuations of the forces from the walls and the displacements of all the particles in the cell. We confirm these theoretical results numerically in a two-dimensional model glass. As an application, we describe emission and propagation of transverse sounds after boundary wall motions using these time-correlation functions. We also find resonant sound amplification when the frequency of an oscillatory shear approaches that of the first transverse sound mode.
REFERENCES
1.
K.
Maeda
and S.
Takeuchi
, “Atomistic process of plastic deformation in a model amorphous metal
,” Philos. Mag. A
44
, 643
(1981
).2.
R.
Yamamoto
and A.
Onuki
, “Dynamics of highly supercooled liquids: Heterogeneity, rheology, and diffusion
,” Phys. Rev. E
58
, 3515
–3529
(1998
).3.
D. L.
Malandro
and D. J.
Lacks
, “Relationships of shear-induced changes in the potential energy landscape to the mechanical properties of ductile glasses
,” J. Chem. Phys.
110
, 4593
(1999
).4.
C.
Maloney
and A.
Lemaître
, “Amorphous systems in athermal, quasistatic shear
,” Phys. Rev. E
74
, 016118
(2006
).5.
M.
Tsamados
, A.
Tanguy
, C.
Goldenberg
, and J.-L.
Barrat
, “Local elasticity map and plasticity in a model Lennard-Jones glass
,” Phys. Rev. E
80
, 026112
(2009
).6.
A.
Widmer-Cooper
, H.
Perry
, P.
Harrowell
, and D. R.
Reichman
, “Localized soft modes and the supercooled liquid’s irreversible passage through its configuration space
,” J. Chem. Phys.
131
, 194508
(2009
).7.
M. L.
Manning
and J.
Liu
, “Vibrational modes identify soft spots in a sheared disordered packing
,” Phys. Rev. Lett.
107
, 108302
(2011
).8.
T.
Kawasaki
and A.
Onuki
, “Slow relaxations and stringlike jump motions in fragile glass-forming liquids: Breakdown of the Stokes-Einstein relation
,” Phys. Rev. E
87
, 012312
(2013
).9.
D. R.
Squire
, A. C.
Holt
, and W. G.
Hoover
, “Isothermal elastic constants for argon. Theory and Monte Carlo calculations
,” Physica
42
, 388
(1969
).10.
J. R.
Ray
, “Elastic constants and statistical ensembles in molecular dynamics
,” Comput. Phys. Rep.
8
, 109
–151
(1988
).11.
J. F.
Lutsko
, “Generalized expressions for the calculation of elastic constants by computer simulation
,” J. Appl. Phys.
65
, 2991
(1989
).12.
S.
Hess
, M.
Kröger
, and W. G.
Hoover
, “Shear modulus of fluids and solids
,” Physica A
239
, 449
–466
(1997
).13.
K.
Yoshimoto
, G. J.
Papakonstantopoulos
, J. F.
Lutsko
, and J. J.
de Pablo
, “Statistical calculation of elastic moduli for atomistic models
,” Phys. Rev. B
71
, 184108
(2005
).14.
M.
Parrinello
and A.
Rahman
, “Strain fluctuations and elastic constants
,” J. Chem. Phys.
76
, 2662
–2666
(1982
).15.
A. A.
Gusev
, M. M.
Zehnder
, and U. W.
Suter
, “Fluctuation formula for elastic constants
,” Phys. Rev. B
54
, 1
(1996
).16.
S.
Sengupta
, P.
Nielaba
, M.
Rao
, and K.
Binder
, “Elastic constants from microscopic strain fluctuations
,” Phys. Rrev. E
61
, 1072
–1080
(2000
).17.
C.
Maloney
and A.
Lemaître
, “Universal breakdown of elasticity at the onset of material failure
,” Phys. Rev. Lett.
93
, 195501
(2004
).18.
A.
Lemaître
and C.
Maloney
, “Sum rules for the quasi-static and visco-elastic response of disordered solids at zero temperature
,” J. Stat. Phys.
123
, 415
–453
(2006
).19.
S. R.
Williams
and D. J.
Evans
, “The rheology of solid glass
,” J. Chem. Phys.
132
, 184105
(2010
).20.
S. R.
Williams
, “Communication: Broken-ergodicity and the emergence of solid behaviour in amorphous materials
,” J. Chem. Phys.
135
, 131102
(2011
).21.
H.
Yoshino
, “Replica theory of the rigidity of structural glasses
,” J. Chem. Phys.
136
, 214108
(2012
).22.
A.
Tanguy
, J. P.
Wittmer
, F.
Leonforte
, and J.-L.
Barrat
, “Continuum limit of amorphous elastic bodies: A finite-size study of low-frequency harmonic vibrations
,” Phys. Rev. B
66
, 174205
(2002
).23.
H.
Mizuno
, S.
Mossa
, and J.-L.
Barrat
, “Measuring spatial distribution of the local elastic modulus in glasses
,” Phys. Rev. E
87
, 042306
(2013
).24.
A.
Zaccone
and E.
Scossa-Romano
, “Approximate analytical description of the nonaffine response of amorphous solids
,” Phys. Rev. B
83
, 184205
(2011
).25.
A.
Zaccone
and E. M.
Terentjev
, “Disorder-assisted melting and the glass transition in amorphous solids
,” Phys. Rev. Lett.
110
, 178002
(2013
).26.
J. P.
Wittmer
, H.
Xu
, P.
Polińska
, F.
Weysser
, and J.
Baschnagel
, “Shear modulus of simulated glass-forming model systems: Effects of boundary condition, temperature, and sampling time
,” J. Chem. Phys.
138
, 12A533
(2013
).27.
J. P.
Wittmer
, H.
Xu
, O.
Benzerara
, and J.
Baschnagel
, “Fluctuation-dissipation relation between shear stress relaxation modulus and shear stress autocorrelation function revisited
,” Mol. Phys.
113
, 2881
–2893
(2015
).28.
I.
Fuereder
and P.
Ilg
, “Influence of inherent structure shear stress of supercooled liquids on their shear moduli
,” J. Chem. Phys.
142
, 144505
(2015
).29.
S.
Saw
and P.
Harrowell
, “Rigidity in condensed matter and its origin in configurational constraint
,” Phys. Rev. Lett.
116
, 137801
(2016
).30.
K.
Yoshimoto
, T. S.
Jain
, K.
Van Workum
, P. F.
Nealey
, and J. J.
de Pablo
, “Mechanical heterogeneities in model polymer glasses at small length scales
,” Phys. Rev. Lett.
93
, 175501
(2004
).31.
H.
Mizuno
, L. E.
Silbert
, and M.
Sperl
, “Spatial distributions of local elastic moduli near the jamming transition
,” Phys. Rev. Lett.
116
, 068302
(2016
).32.
S.
Sastry
, P. G.
Debenedetti
, and F. H.
Stillinger
, “Signatures of distinct dynamical regimes in the energy landscape of a glass-forming liquid
,” Nature
393
, 554
–557
(1998
).33.
A.
Heuer
, “Exploring the potential energy landscape of glass-forming systems: From inherent structures via metabasins to macroscopic transport
,” J. Phys.: Condens. Matter
20
, 373101
(2008
).34.
S.
Abraham
and P.
Harrowell
, “The origin of persistent shear stress in supercooled liquids
,” J. Chem. Phys.
137
, 014506
(2012
).35.
S.
Chowdhury
, S.
Abraham
, T.
Hudson
, and P.
Harrowell
, “Long range stress correlations in the inherent structures of liquids at Rest
,” J. Chem. Phys.
144
, 124508
(2016
).36.
R.
Zwanzig
, “Time-correlation functions and transport coefficients in statistical mechanics
,” Annu. Rev. Phys. Chem.
16
, 67
–101
(1965
).37.
D. J.
Evans
and G. P.
Morriss
, Statistical Mechanics of Nonequilibrium Liquids
(Academic
, London
, 1990
).38.
39.
A.
Onuki
, Phase Transition Dynamics
(Cambridge University Press
, Cambridge
, 2002
).40.
L.
Onsager
, “Reciprocal relations in irreversible processes. I.
,” Phys. Rev.
37
, 405
–426
(1931
).41.
M. S.
Green
, “Markoff random processes and the statistical mechanics of time-dependent phenomena. II. Irreversible processes in fluids
,” J. Chem. Phys.
22
, 398
–413
(1954
).42.
L. P.
Kadanoff
and P. C.
Martin
, “Hydrodynamic equations and correlation functions
,” Ann. Phys.
24
, 419
–469
(1963
).43.
R.
Kubo
, “Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems
,” J. Phys. Soc. Jpn.
12
, 570
–586
(1957
).44.
L.
Bocquet
and J.-L.
Barrat
, “Flow boundary conditions from nano- to micro-scales
,” Soft Matter
3
, 685
–693
(2007
).45.
V. A.
Levashov
, J. R.
Morris
, and T.
Egami
, “Anisotropic stress correlations in two-dimensional liquids
,” Phys. Rev. Lett.
106
, 115703
(2011
).46.
A.
Lemaître
, “Structural relaxation is a scale-free process
,” Phys. Rev. Lett.
113
, 245702
(2014
).47.
M.
Maier
, A.
Zippelius
, and M.
Fuchs
, “Emergence of long-ranged stress correlations at the liquid to glass transition
,” Phys. Rev. Lett.
119
, 265701
(2017
).48.
L.
Klochko
, J.
Baschnagel
, J. P.
Wittmer
, and A. N.
Semenov
, “Long-range stress correlations in viscoelastic and glass-forming fluids
,” Soft Matter
14
, 6835
(2018
).49.
B. U.
Felderhof
, “Fluctuations of polarization and magnetization in dielectric and magnetic media
,” J. Chem. Phys.
67
, 493
–500
(1977
).50.
K.
Takae
and A.
Onuki
, “Fluctuations of local electric field and dipole moments in water between metal walls
,” J. Chem. Phys.
143
, 154503
(2015
).51.
A. W.
Lees
and S. F.
Edwards
, “The computer study of transport processes under extreme conditions
,” J. Phys. C: Solid State Phys.
5
, 1921
–1929
(1972
).52.
X.
Jia
, C.
Caroli
, and B.
Velicky
, “Ultrasound propagation in externally stressed granular media
,” Phys. Rev. Lett.
82
, 1863
(1999
).53.
E.
Somfai
, J.-N.
Roux
, J. H.
Snoeijer
, M.
van Hecke
, and W.
van Saarloos
, “Elastic wave propagation in confined granular systems
,” Phys. Rev. E
72
, 021301
(2005
).54.
H.
Shiba
and A.
Onuki
, “Plastic deformations in crystal, polycrystal, and glass in binary mixtures under shear: Collective yielding
,” Phys. Rev. E
81
, 051501
(2010
);H.
Shiba
and A.
Onuki
, “Jammed particle configurations and dynamics in high-density Lennard-Jones binary mixtures in two dimensions
,” Prog. Theor. Phys. Suppl.
184
, 232
(2010
).55.
J. H.
Irving
and J. G.
Kirkwood
, “The statistical mechanical theory of transport processes. IV. The equations of hydrodynamics
,” J. Chem. Phys.
18
, 817
–829
(1949
).56.
A.
Widmer-Cooper
and P.
Harrowell
, “How reproducible are dynamic heterogeneities in a supercooled liquid?
,” Phys. Rev. Lett.
93
, 135701
(2004
);
[PubMed]
A.
Widmer-Cooper
and P.
Harrowell
, “Predicting the long-time dynamic heterogeneity in a supercooled liquid on the basis of short-time heterogeneities
,” Phys. Rev. Lett.
96
, 185701
(2006
).
[PubMed]
57.
H. R.
Schober
and G.
Ruocco
, “Size effects and quasilocalized vibrations
,” Philos. Mag.
84
, 1361
–1372
(2006
).58.
Y.
Matsuoka
, H.
Mizuno
, and R.
Yamamoto
, “Acoustic wave propagation through a supercooled liquid: A normal mode analysis
,” J. Phys. Soc. Jpn.
81
, 124602
(2012
).59.
S. N.
Taraskin
and S. R.
Elliott
, “Propagation of plane-wave vibrational excitations in disordered systems
,” Phys. Rev. B
61
, 12017
–12030
(2000
).60.
S.
Gelin
, H.
Tanaka
, and A.
Lemaître
, “Anomalous phonon scattering and elastic correlations in amorphous solids
,” Nat. Mater.
15
, 1177
–1181
(2016
).61.
M.
Doi
and S. F.
Edwards
, The Theory of Polymer Dynamics
(Clarendon Press
, Oxford
, 1986
).62.
R.
Zwanzig
and R. D.
Mountain
, “High-frequency elastic moduli of simple fluids
,” J. Chem. Phys.
43
, 4464
–4471
(1965
).63.
A.
Kushima
, X.
Lin
, J.
Li
, J.
Eapen
, J. C.
Mauro
, X.
Qian
, P.
Diep
, and S.
Yip
, “Computing the viscosity of supercooled liquids
,” J. Chem. Phys.
130
, 224504
(2009
).64.
65.
T.
Kawasaki
and A.
Onuki
, “Acoustic resonance in periodically sheared glass
,” e-print arXiv:1708.03166.66.
J.
Bastide
and L.
Leibler
, “Large-scale heterogeneities in randomly cross-linked networks
,” Macromolecules
21
, 2647
(1988
).67.
D. A.
Head
, A. J.
Levine
, and E. C.
MacKintosh
, “Deformation of cross-linked semiflexible polymer networks
,” Phys. Rev. Lett.
91
, 108102
(2003
).68.
B. A.
DiDonna
and T. C.
Lubensky
, “Nonaffine correlations in random elastic media
,” Phys. Rev. E
72
, 066619
(2005
).69.
W. G.
Ellenbroek
, Z.
Zeravcic
, W.
van Saarloos
, and M.
van Hecke
, “Non-affine response: Jammed packings vs. spring networks
,” Europhys. Lett.
87
, 34004
(2009
).© 2019 Author(s).
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