We construct a mean-field elastoplastic description of the dynamics of amorphous solids under arbitrary time-dependent perturbations, building on the work of Lin and Wyart [Phys. Rev. X 6, 011005 (2016)] for steady shear. Local stresses are driven by power-law distributed mechanical noise from yield events throughout the material, in contrast to the well-studied Hébraud–Lequeux model where the noise is Gaussian. We first use a mapping to a mean first passage time problem to study the phase diagram in the absence of shear, which shows a transition between an arrested and a fluid state. We then introduce a boundary layer scaling technique for low yield rate regimes, which we first apply to study the scaling of the steady state yield rate on approaching the arrest transition. These scalings are further developed to study the aging behavior in the glassy regime for different values of the exponent μ characterizing the mechanical noise spectrum. We find that the yield rate decays as a power-law for 1 < μ < 2, a stretched exponential for μ = 1, and an exponential for μ < 1, reflecting the relative importance of far-field and near-field events as the range of the stress propagator is varied. A comparison of the mean-field predictions with aging simulations of a lattice elastoplastic model shows excellent quantitative agreement, up to a simple rescaling of time.

1.
A.
Nicolas
,
E. E.
Ferrero
,
K.
Martens
, and
J.-L.
Barrat
, “
Deformation and flow of amorphous solids: Insights from elastoplastic models
,”
Rev. Mod. Phys.
90
,
045006
(
2018
).
2.
D.
Bonn
,
M. M.
Denn
,
L.
Berthier
,
T.
Divoux
, and
S.
Manneville
, “
Yield stress materials in soft condensed matter
,”
Rev. Mod. Phys.
89
,
035005
(
2017
).
3.
P.
Hébraud
and
F.
Lequeux
, “
Mode-coupling theory for the pasty rheology of soft glassy materials
,”
Phys. Rev. Lett.
81
,
2934
(
1998
).
4.
G.
Picard
,
A.
Ajdari
,
F.
Lequeux
, and
L.
Bocquet
, “
Elastic consequences of a single plastic event: A step towards the microscopic modeling of the flow of yield stress fluids
,”
Eur. Phys. J. E
15
,
371
(
2004
).
5.
A.
Lemaître
and
C.
Caroli
, arXiv:0705.3122 [cond-mat] (
2007
).
6.
A.
Lemaître
and
C.
Caroli
, “
Plastic response of a two-dimensional amorphous solid to quasistatic shear: Transverse particle diffusion and phenomenology of dissipative events
,”
Phys. Rev. E
76
,
036104
(
2007
).
7.
J.
Lin
and
M.
Wyart
, “
Mean-field description of plastic flow in amorphous solids
,”
Phys. Rev. X
6
,
011005
(
2016
).
8.
J.
Lin
and
M.
Wyart
, “
Microscopic processes controlling the Herschel-Bulkley exponent
,”
Phys. Rev. E
97
,
012603
(
2018
).
9.
R.
Chacko
,
P.
Sollich
, and
S.
Fielding
, “
Slow coarsening in jammed athermal soft particle suspensions
,”
Phys. Rev. Lett.
123
,
108001
(
2019
).
10.
M.
Agarwal
and
Y. M.
Joshi
, “
Signatures of physical aging and thixotropy in aqueous dispersion of carbopol
,”
Phys. Fluids
31
,
063107
(
2019
).
11.
P.
Lidon
,
L.
Villa
, and
S.
Manneville
, “
Power-law creep and residual stresses in a carbopol gel
,”
Rheol. Acta
56
,
307
(
2017
).
12.
E. H.
Purnomo
,
D.
van den Ende
,
S. A.
Vanapalli
, and
F.
Mugele
, “
Glass transition and aging in dense suspensions of thermosensitive microgel particles
,”
Phys. Rev. Lett.
101
,
238301
(
2008
).
13.
E. H.
Purnomo
,
D. v. d.
Ende
,
J.
Mellema
, and
F.
Mugele
, “
Linear viscoelastic properties of aging suspensions
,”
Europhys. Lett.
76
,
74
(
2006
).
14.
E. H.
Purnomo
,
D.
van den Ende
,
J.
Mellema
, and
F.
Mugele
, “
Rheological properties of aging thermosensitive suspensions
,”
Phys. Rev. E
76
,
021404
(
2007
).
15.
P.
Sollich
,
J.
Olivier
, and
D.
Bresch
, “
Aging and linear response in the Hébraud-Lequeux model for amorphous rheology
,”
J. Phys. A: Math. Theor.
50
,
165002
(
2017
).
16.

We consider here a full local stress relaxation. The post-yield local stress may however also be modeled as drawn from a distribution of residual stress other than a Dirac function at zero. Previous studies have shown this not to change the behaviour qualitatively.38 

17.
J. D.
Eshelby
, “
The determination of the elastic field of an ellipsoidal inclusion, and related problems
,”
Proc. R. Soc., A
241
,
376
(
1957
).
18.
I.
Fernández Aguirre
and
E. A.
Jagla
, “
Critical exponents of the yielding transition of amorphous solids
,”
Phys. Rev. E
98
,
013002
(
2018
).
19.
E. E.
Ferrero
and
E. A.
Jagla
, “
Criticality in elastoplastic models of amorphous solids with stress-dependent yielding rates
,”
Soft Matter
15
,
9041
(
2019
).
20.

This will change quantitative aspects such as the location of the phase diagram; however, the scaling forms below are determined by the asymptotic power-law regime for small |δσ|, making the precise form of the upper cutoff irrelevant.

21.
L.
Bocquet
,
A.
Colin
, and
A.
Ajdari
, “
Kinetic theory of plastic flow in soft glassy materials
,”
Phys. Rev. Lett.
103
,
036001
(
2009
).
22.
A. A.
Dubkov
,
B.
Spagnolo
, and
V. V.
Uchaikin
, “
Lévy flight superdiffusion: An introduction
,”
Int. J. Bifurcation Chaos
18
,
2649
(
2008
).
23.
A.
Zoia
,
A.
Rosso
, and
M.
Kardar
, “
Fractional Laplacian in bounded domains
,”
Phys. Rev. E
76
,
021116
(
2007
).
24.

The exponent μ/2 corresponds to the pseudogap exponent in Ref. 7.

25.
S. V.
Buldyrev
,
S.
Havlin
,
A. Y.
Kazakov
,
M. G. E.
da Luz
,
E. P.
Raposo
,
H. E.
Stanley
, and
G. M.
Viswanathan
, “
Average time spent by Lévy flights and walks on an interval with absorbing boundaries
,”
Phys. Rev. E
64
,
041108
(
2001
).
26.
E.
Agoritsas
,
E.
Bertin
,
K.
Martens
, and
J.-L.
Barrat
, “
On the relevance of disorder in athermal amorphous materials under shear
,”
Eur. Phys. J. E: Soft Matter Biol. Phys.
38
,
71
(
2015
).
27.
A.
Giménez
,
F.
Morillas
,
J.
Valero
, and
J. M.
Amigó
, “
Stability and numerical analysis of the Hébraud–Lequeux model for suspensions
,”
Discrete Dyn. Nat. Soc.
2011
,
415921
.
28.

Although we have not studied in detail the aging of asymmetric distributions in the current model, for the HL model,15 the same asymptotic decay and scalings are shown to hold also for the asymmetric case under generic assumptions.

29.

For lattices of different sizes, the large δσ details of the list of propagator elements are identical. In order to run Gillespie simulations with an arbitrary number of sites, we proceed by extrapolating the small δσ power law in order to obtain the desired number of propagator elements.

30.
M. L.
Falk
and
J. S.
Langer
, “
Dynamics of viscoplastic deformation in amorphous solids
,”
Phys. Rev. E
57
,
7192
(
1998
).
31.
P.
Sollich
,
F.
Lequeux
,
P.
Hébraud
, and
M. E.
Cates
, “
Rheology of soft glassy materials
,”
Phys. Rev. Lett.
78
,
2020
(
1997
).
32.
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
).
33.
R.
Candelier
,
O.
Dauchot
, and
G.
Biroli
, “
Dynamical facilitation decreases when approaching the granular glass transition
,”
Europhys. Lett.
92
,
24003
(
2010
).
34.
J.-P.
Bouchaud
,
S.
Gualdi
,
M.
Tarzia
, and
F.
Zamponi
, “
Spontaneous instabilities and stick-slip motion in a generalized Hébraud-Lequeux model
,”
Soft Matter
12
,
1230
(
2016
).
35.
M.
Popović
,
T. W. J.
de Geus
,
W.
Ji
, and
M.
Wyart
, arXiv:2009.04963 [cond-mat] (
2020
).
36.
M.
Warren
and
J.
Rottler
, “
Quench, equilibration, and subaging in structural glasses
,”
Phys. Rev. Lett.
110
,
025501
(
2013
).
37.
C.
Liu
,
E. E.
Ferrero
,
K.
Martens
, and
J.-L.
Barrat
, “
Creep dynamics of athermal amorphous materials: A mesoscopic approach
,”
Soft Matter
14
,
8306
(
2018
).
38.
E.
Agoritsas
and
K.
Martens
, “
Non-trivial rheological exponents in sheared yield stress fluids
,”
Soft Matter
13
,
4653
(
2017
).
You do not currently have access to this content.