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Slow Fatigue and Highly Delayed Yielding via Shear Banding in Oscillatory Shear
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Slow Fatigue and Highly Delayed Yielding via Shear Banding in Oscillatory Shear
James O. Cochran , Grace L. Callaghan , Miles J. G. Caven, and Suzanne M. Fielding
Department of Physics, Durham University, Science Laboratories, South Road, Durham DH1 3LE, United Kingdom
(Received 21 November 2022; revised 21 November 2023; accepted 15 March 2024; published 17 April 2024)
We study theoretically the dynamical process of yielding in cyclically sheared amorphous materials,
within a thermal elastoplastic model and the soft glassy rheology model. Within both models we find aninitially slow accumulation, over many cycles after the inception of shear, of low levels of damage in the
form strain heterogeneity across the sample. This slow fatigue then suddenly gives way to catastrophic
yielding and material failure. Strong strain localization in the form of shear banding is key to the failuremechanism. We characterize in detail the dependence of the number of cycles N
/C3before failure on the
amplitude of imposed strain, the working temperature, and the degree to which the sample is annealed prior
to shear. We discuss our finding with reference to existing experiments and particle simulations, andsuggest new ones to test our predictions.
DOI: 10.1103/PhysRevLett.132.168202
Amorphous materials [1–3]include soft solids such
as emulsions, colloids, gels, and granular materials, and
harder metallic and molecular glasses. Unlike crystallinesolids, they lack order in the arrangement of their con-
stituent microstructures (droplets, grains, etc.). Under-
standing their rheological properties is thus a majorchallenge. Typically, they behave elastically at low loads
then yield plastically at larger loads. Much effort has been
devoted to understanding the dynamics of yielding follow-ing the imposition of a shear stress σor strain rate ˙γ, which
is held constant after switch-on. This often involves the
formation of shear bands [4], which can slowly heal away
to leave homogeneous flow in complex fluids [5–15],
or trigger catastrophic failure in solids [16,17] . In many
applications, however, materials are subject to a cyclically
repeating deformation or load. Cyclic shear is also impor-
tant fundamentally in revealing key fingerprints of a
material ’s nonlinear rheology, with large amplitude oscil-
latory shear intensely studied [18–34].
The response of an amorphous material to an oscillatory
shear strain depends strongly on the strain amplitude γ
0
relative to a threshold γc[35–55].F o r γ0<γc, a material
typically settles into deep regions of its energy landscape,
showing reversible response from cycle to cycle (aftermany cycles), via an absorbing state transition. The number
of cycles to settle, however, diverges as γ
0→γ?c.F o r
γ0>γc, a material instead yields into a state of higherenergy that is chaotically irreversible from cycle to cycle,
and often shear banded [32,33,55 –57].
Indeed, the process of repeatedly straining or loading a
material over many cycles typically leads to the gradual
accumulation of microstructural damage. While the earlysignatures of such fatigue are often difficult to detect, its
slow buildup can eventually undermine material stability
and precipitate catastrophic failure. Understanding theaccumulation of microstructural fatigue and identifying
the microscopic precursors that prefigure failure is thus
central to the prediction of material stability and lifetime,and the development of strategies to improve them.
In hard materials, the buildup of microstructural damage
is often interpreted in terms of the formation of micro-
cracks. Far less well understood in soft materials, it remains
the topic of intense study, as recently reviewed [58].
Colloidal gels in oscillatory stress [59–61]display an
intricate, multistage yielding process in which the sample
remains solidlike for many cycles, before slipping at therheometer wall, then forming coexisting solid-fluid bulk
shear bands and finally fully fluidizing [61]. The number of
cycles before yielding increases dramatically at low stressamplitudes [60,61] . Particle [62] and fiber bundle [63]
simulations likewise show increasing yielding delay with
decreasing cyclic load amplitude. Metallic glass simula-tions show an increasing number of cycles to shear band
formation with decreasing γ
0[64]. Particle simulations
[36,38] and experiments on colloidal glass [47] show a
number of strain cycles to attain a yielded steady state
diverging as γ0→γ?c.
Despite this rapid experimental progress, the dynamics
of yielding in cyclic shear remains poorly understoodtheoretically. An insightful recent study of athermal mate-
rials captured delayed yielding after a number of cycles that
increases at low strain amplitude [65]. In being mean field,Published by the American Physical Society under the terms of
theCreative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article ’s title, journal citation,
and DOI.PHYSICAL REVIEW LETTERS 132, 168202 (2024)
Editors' Suggestion
0031-9007 =24=132(16) =168202(7) 168202-1 Published by the American Physical Societyhowever, this work necessarily neglects the development of
damage in the form of strain heterogeneity and shear bandsthat are key to understanding yielding.
In this Letter, we study theoretically the yielding of
amorphous materials in oscillatory shear strain. Our con-tributions are fourfold. First, we predict a slow accumu-lation, over many cycles, of initially low levels of damagein the form of strain heterogeneity across the sample.
Second, we show that this early fatigue later gives way to
catastrophic material failure, after a number of cycles N
/C3.
Third, we show that the formation of shear bands is key tothe failure mechanism, as seen experimentally. Finally, we
characterize the dependence of N
/C3on the strain amplitude
γ0, the working temperature T, and the degree of sample
annealing prior to shear.
Models. —To gain confidence that our predictions are
generic across a wide range of amorphous materials,independent of specific constitutive modeling assumptions,we study numerically two different widely used models ofelastoplastic rheology: the soft glassy rheology (SGR)
model [66] and a thermal elastoplastic (TEP) model [1].
The SGR model comprises an ensemble of elastoplastic
elements, each corresponding to a mesoscopic region of
material large enough to admit a local continuum shear
strain land stress Gl, with modulus G. Under an imposed
shear rate ˙γ, any element strains at rate ˙l?˙γ. Elemental
stresses are, however, intermittently released via local
plastic yielding events, occurring stochastically at rate
r?τ
?1
0minf1;exp???E?1
2Gl2?=T/C138g, with τ0a micro-
scopic attempt time, Ea local energy barrier, and T
temperature. Upon yielding, any element resets its strain,
l→0, and selects a new yield energy from a distribution
ρ?E??exp??E=T g?=Tg. The model captures a glass
transition at temperature T?Tgand predicts rheological
aging at low loads in its glass phase, Tscopic elastoplastic stress σis the average of the elemental
stresses. The total stress Σ?σ?η˙γincludes a Newtonian
contribution of viscosity η.
The TEP model is defined likewise, except each element
has the same yield energy E, and after yielding selects its
newlfrom a Gaussian of width lh. Both models thus
combine the basic ingredients of elastic deformation
punctuated by plastic rearrangements and stress propaga-
tion. But whereas SGR incorporates disorder in thematerial ’s energy landscape via ρ?E?to capture glassy
behavior, yet neglects frustrated local stresses, the TEP
model conversely neglects glassiness, but captures frus-
trated local stresses via the posthop ldistribution.
To capture catastrophic yielding, it is crucial to allow for
strain localization and shear banding. Accordingly, in each
model the elastoplastic elements are arranged across S
streamlines stacked in the flow gradient direction y, with M
elements per streamline. The imposed shear rate, averaged
across streamlines, is ?˙γ?t?. The local shear rate can,
however, vary across streamlines: at uniform total stressΣin creeping flow we have Σ?t??σ?y; t??η˙γ?y; t??
?σ?t??η?˙γ?t?, with ya streamline ’s flow gradient coordi-
nate. After any local yielding event with stress drop
of magnitude Δlwe furthermore pick three random
elements on each neighboring streamline and adjust theirl→l?wΔl??1;?2;?1?. We thus implement 1D Eshelby
stress propagation [67] and stress diffusion [68], which are
key to shear banding.
Protocol. —We study oscillatory shear strain ?γ
?t??
γ0sin?ωt?, imposed for all times t>0. Prior to shear,
the sample is prepared via ageing or annealing. Within
SGR, we perform a sudden deep quench at time t??tw
from infinite temperature to a working temperature Tin the glass phase, then age the sample for a waiting time tw.
Within TEP, we first equilibrate the sample to a temperature
T0, then suddenly at time t?0quench to a working
temperature Tcorresponds to better annealing.
About an initially uniform state, tiny levels of hetero-
geneity are seeded naturally via MandSbeing finite. In
SGR we also test the effect of adding a small initial pertur-
bation to the well depths E→E?1?δcos2πy?. That we
observe the same physics in both cases shows that ourresults are robust to small initial randomicity.
In response to the imposed strain, we measure the shear
stress Σ?t?and report its root mean square Σ
rmsover each
cycle vs cycle number N. We also define the degree of shear
banding Δ˙γ?t?via the standard deviation of the strain rate
across streamlines, normalized by ˙γ0?γ0ω, and report its
mean over each cycle, hΔ˙γi?N?. When this quantity is high,
the strain rate profile is significantly shear banded acrossthe flow gradient direction.
Parameters. —Both models have as parameters the mean
local yield energy hEi, attempt time τ
0, temperature T,
number of streamlines S, elements per streamline M,
Newtonian viscosity η, and stress diffusion w. The degree
of annealing is prescribed by the waiting time tw(SGR)
or prequench temperature T0(TEP). The imposed shear
has amplitude γ0and frequency ω. We choose units
τ0?1,G?1,hEi?1. We set η?0.05,w?0.05,
lh?0.05,δ?0.01, suited to the Newtonian viscosity,
stress diffusivity, and initial heterogeneity being small. We
set the numerical parameters S?25,M?10 000 , having
checked for robustness to variations in these. For computa-
tional efficiency, we set ω?0.1in SGR, but checked
that our findings also hold for ω?0.01. In TEP we set
ω?0.01. We then explore yielding as a function of strain
amplitude γ0, working temperature T, and degree of
annealing before shear.
SGR results. —The key physics that we report is exem-
plified by Figs. 1(a) and1(b). These show that yielding
comprises two distinct stages as a function of cyclenumber N. In the first stage, the sample remains nearly
homogeneous, with only low level material fatigue (small
strain heterogeneity hΔ˙γi) slowly accumulating from cyclePHYSICAL REVIEW LETTERS 132, 168202 (2024)
168202-2to cycle, and the stress remaining high. After a delay that
increases dramatically with decreasing imposed strainamplitude γ
0in curve sets left to right, a second stage
ensues: the stress drops quickly, the strain becomes highlylocalized into shear bands, and the sample fails cata-strophically.
To quantify the delay during which fatigue slowly
accumulates before the sample catastrophically fails, wedefine the cycle at failure N
/C3as that in which Σrmsfirst falls
below1
2?Σmax?Σmin?, where ΣmaxandΣminare the global
maximum and minimum of Σrmsversus N[69]. We further
define the magnitude of yielding via the normalized stressdropΔΣ??Σ
max?Σmin?=ΣSS, where ΣSSis the steady
state stress as N→∞, and the extent to which strain
becomes localized via the final degree of shear bandinghΔ˙γi
f?limN→∞hΔ˙γi?N?. These three quantities are plot-
ted vs γ0in Figs. 1(c)–1(e).
Clearly apparent is a transition at strain amplitude
γ0?γc≈1.4, below which the stress drop ΔΣand degree
of strain localization hΔ˙γifbecome negligible: for γ0<γc,
no appreciable yielding occurs. For γ0>γc, we see a range
ofγ0, increasing with increasing tw, over which yielding is
both strongly apparent and heavily delayed. The delayincreases dramatically with decreasing γ
0, although N/C3
shows no apparent divergence over the window of strains
for which yielding is appreciable.
The dependence of yielding on the degree of ageing prior
to shear, tw, is further explored in Fig. 2. Panels (a) and (b)
again reveal the two stage yielding just described, withcurve sets left to right showing a longer delay withincreasing t
w, with N/C3?tαw(panel c). Importantly, there-
fore, ultra annealed samples tw→∞are predicted to show
an indefinite delay before suddenly failing.
So far, we have characterized the dependence of yielding
on the strain amplitude γ0and waiting time twseparately.
Its dependence on both parameters is summarized in Fig. 3.
Importantly, these color maps suggest the possibility of
long delayed (large N/C3) and catastrophic (large ΔΣ)
yielding even at large strain amplitudes, provided thesample age prior to shear is large enough. The strain γ
0
at yielding onset in panel (b) roughly coincides with the end
of the linear regime, in which the viscoelastic spectra G0
andG00are constant functions of γ0[33].(a)
rms(c)
(b)
(d)
(e)
FIG. 1. SGR model. (a) Root mean square stress and (b) mean
degree of shear banding over each cycle versus cycle number N
for strain amplitudes γ0?1.00;1.25;…;2.75in curve sets with
drops in (a) and rises in (b) right to left. Each curve within a set
corresponds to a different random initial condition. tw?107,
T?0.3. (c) Cycle number at failure N/C3, (d) magnitude of stress
dropΔΣ, and (e) final degree of shear banding h˙γifvs strain
amplitude γ0for waiting times tw?102;103;…;107in curves
bottom to top. Panel (c) only shows samples with ΔΣ>0.1.N/C3,
ΔΣ,h˙γifaveraged over initial condition.(a)
(b)rms(c)
(d)
(e)
FIG. 2. SGR model. (a) Root mean square stress and (b) mean
degree of shear banding over each cycle as a function of cycle
number Nfor waiting time tw?101;102;…;107in curve sets
with drops in (a) and rises in (b) left to right. γ0?1.5,T?0.3.
(c) Cycle number at failure N/C3, (d) magnitude of stress drop ΔΣ,
and (e) final degree of shear banding h˙γifvs waiting time, tw.
Strain amplitude γ0?1.125;1.250;1.375;…;2.250 in curves
blue to orange, i.e., top to bottom in (c), bottom to top at right of(d), and with γ
0?1.125;1.25…1.375bottom up and 1.5,...2.25
top down in (e). Panel (c) only shows cases for which ΔΣ>0.1.
(a) (b)
FIG. 3. SGR model. (a) cycle number at failure N/C3and (b) stress
dropΔΣas a function of waiting time twand strain amplitude γ0.
In the white region, no yielding occurs.PHYSICAL REVIEW LETTERS 132, 168202 (2024)
168202-3TEP results. —We now show the same physics to obtain
in the TEP model, thereby increasing confidence that it willbe generic across many amorphous materials. Figures 4(a),
4(b), and 5(a)–5(d) again show a two-stage yielding
process, with strain heterogeneity slowly accumulatingand the stress barely declining, before catastrophic failurein which the stress suddenly drops and shear bands form.The number of cycles N
/C3before failure again increases
dramatically with decreasing imposed strain γ0, as seen for
several prequench temperatures T0in Fig. 4(c)and working
temperatures Tin (d). An interesting difference between
TEP and SGR is also apparent. In SGR, recall that N/C3
increases rapidly with decreasing γ0, but with no apparent
divergence before the magnitude of yielding becomesnegligible [Figs. 1(c)–1(e)]. In TEP, N
/C3diverges at a
nonzero γ0for which yielding is still strongly apparent
[Figs. 4(c)–4(d)]. Whether this constitutes a fundamental
difference between the models or is simply due to our TEPresults being for lower Tand stronger annealing than are
computationally accessible in SGR is unclear.
We now consider the way in which yielding depends in
TEP on the degree to which the sample is annealed prior toshear. In Fig. 5(a)and5(b), a collection of yielding curves
for decreasing annealing temperature T
0in curves left to
right demonstrates a dramatically increasing delay before
yielding with increasing sample annealing (lower T0). The
number of cycles before yielding is fit to the Boltzmannform N
/C3?Aexp?B=T 0?in Fig. 5(e). Ultra annealedsamples ( T0→0) are thus predicted in TEP to show
indefinitely delayed yielding N/C3→∞, in close analogy
with the corresponding limit tw→∞in SGR.
We explore finally the dependence of yielding on
working temperature Tin TEP. A collection of yielding
curves left to right in Figs. 5(c) and 5(d) shows a
dramatically increasing delay before yielding with decreas-ingT. The number of cycles before yielding is fit to the
Boltzmann form N
/C3?Aexp?B=T?in Fig. 5(f). Accor-
dingly, then, TEP predicts infinitely delayed yielding in theathermal limit of zero working temperature T→0at fixed
strain amplitude γ
0and prequench temperature T0.
Conclusions. —We have shown the yielding of amor-
phous materials in oscillatory shear to comprise a two-stageprocess. The first is one of slow fatigue, in which low levelsof strain heterogeneity gradually accumulate from cycle tocycle. In the second, the stress drops precipitously and thestrain strongly localizes into shear bands, leading to cata-strophic material failure. The number of cycles N
/C3before
failure increases dramatically with decreasing imposedstrain amplitude and increasing annealing. Finally, N
/C3
diverges in the limit of zero working temperature T→0,
showing that a small nonzero temperature is indispensableto ultra-delayed yielding.
In future, it would be interesting to consider how the slow
fatigue and catastrophic failure studied here ( “intercycle(a)
rms
(b) (d)(c)
FIG. 4. TEP model. (a) Root mean square stress and (b) mean
degree of shear banding over each cycle as a function of cyclenumber Nfor strain amplitudes γ
0?0.90;0.95;…;1.50in curve
sets with drops in (a) and rises in (b) right to left. T0?0.01,
T?0.007. Cycle number at yielding N/C3vs strain amplitude γ0
for (c) prequench temperatures T0?0.001;0.002;…;0.010 in
curves right to left at working temperature T?0.001 and
(d) working temperatures T?0.001;0.002;…;0.010in curves
right to left at prequench temperature T0?0.01. Solid lines in
(c)+(d) are fits to N/C3?A=?γ0?γc?. Insets show γc(symbols) fit
(lines) to (c) γc?B?C?????T0pand (d) γc?DT?E.(a)rms
(b)(c)
(d)(e)
(f)
FIG. 5. TEP model. (a) Root mean square stress and (b) mean
degree of shear banding over each cycle as a function of cyclenumber Nfor prequench temperatures T
0?0.001;0.002;…;
0.010in curves with drops in (a),(c) and rises in (b),(d) right to
left.γ0?1.15,T?0.001. (c)+(d) Counterpart curves for work-
ing temperatures T?0.001;0.002;…;0.010in curves turquoise
to magenta. γ0?1.05,T0?0.01. (e) Cycle number at yielding
N/C3vs prequench temperature T0for strain amplitudes γ0?1.10,
1.15, 1.17, 1.20, 1.22 in curves downward. T?0.001. Solid
lines: fits to N/C3?AeB=T 0. (f)N/C3vs working temperature Tfor
γ0?1.00, 1.05, 1.07, 1.10, 1.15 in curves downward. T0?0.01.
Solid lines: fits to N/C3?CeD=T.PHYSICAL REVIEW LETTERS 132, 168202 (2024)
168202-4yielding, ”over many cycles) relates to the alternating
“intracycle ”yielding (with shear banding formation) and
resolidification (with rehealing to homogeneous shear) thatarises in yield stress fluids once a state has been attained that
is invariant from cycle to cycle [28,29,32,33] .A n o t h e r
important challenge is to reconcile our divergent N
/C3in
the athermal limit T→0with a finite N/C3atT?0in the
mean field study of Ref. [65], which neglects banding. It
would also be interesting to model yielding in oscillatory
shear stress, as studied experimentally [59–61]. Indeed, any
fundamental similarities and differences between delayed
yielding in oscillatory shear and other protocols such as
creep should also be considered. A fuller exploration of thedistinction between ductile and brittle yielding is also
warranted [70].
Our predictions are directly testable experimentally.
Bulk rheological measurements of the cycle-to-cycle stress
can be compared with Figs. 1(a),1(d),2(a),2(d),3(b),4(a),
5(a), and 5(c). From these stress measurements, the number
of cycles to failure N
/C3can be extracted and compared with
Figs. 1(c),2(c),3(a),4(c),4(d),5(e), and 5(f). Ultrasound
imaging can be used to measure the velocity field [71],
from which the cycle-to-cycle degree of shear banding Δ˙γ
can be extracted as prescribed on p2 and compared with our
Figs. 1(b),1(f),2(b),2(f),4(b),5(b), and 5(d). All these
quantities can also be accessed directly in direct particle
simulations.
We thank Jack Parley and Peter Sollich for interesting
discussions. This project has received funding from the
European Research Council (ERC) under the European
Union ’s Horizon 2020 research and innovation programme
(Grant Agreement No. 885146). J. O. C. was supported
by the EPSRC funded Centre for Doctoral Training in
Soft Matter and Functional Interfaces (SOFI CDT —EP/
L015536/1).
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