Yielding behavior of glasses under asymmetric cyclic deformation
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Yielding behavior of glasses under asymmetric cyclic deformation
Monoj Adhikari,1 Muhittin Mungan,2 and Srikanth Sastry1, ∗
1
Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkar Campus, 560064 Bengaluru, India
2
Institut für angewandte Mathematik, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
We consider the yielding behaviour of a model glass subjected to asymmetric cyclic shear de-
formation, wherein the applied strain varies between 0 and a maximum value γmax , and study its
dependence on the degree of annealing of the glass and system size. The yielding behaviour of well
annealed glasses (unlike poorly annealed glasses) display striking differences from the symmetric
case, with the emergence of an intermediate strain regime with substantial plasticity but no yield-
ing. The observed behaviour is satisfactorily captured by a recently proposed model. For larger
system sizes, the intermediate strain regime narrows, leading to a remarkable reversal of yield strain
arXiv:2201.06535v2 [cond-mat.soft] 19 Jan 2022
with annealing.
The response to applied mechanical stresses is a funda- dynamics is diffusive, as observed in non-Brownian col-
mental characteristic of solids that is of central relevance loidal suspensions, glasses, related systems and models
to their use as structural materials. For large enough ap- thereof [36–41]. Memory formation in models of suspen-
plied stresses or deformations, plastic deformations con- sions and glasses have also been a subject of considerable
tribute significantly to such response, leading eventually interest [42–45].
to yielding and flow. In the case of amorphous solids, In many of these works, particularly when related to
ranging from hard glasses such as oxide glasses to soft yielding, the cyclic deformations protocols have been
solids, these plastic deformations are relevant for under- symmetric, i.e. the applied strain of the system is varied
sym sym
standing their yielding behaviour and rheology [1, 2]. Re- through a cycle as 0 → γmax → 0 → −γmax → 0, where
sym
cent years have witnessed significant activity in develop- γmax is the amplitude of shear [19, 21, 23, 24, 29, 35].
ing a statistical mechanical description of these phenom- Given that significant structural change is observed be-
ena [2–5]. Yielding behaviour in model amorphous solids low yielding for poorly annealed cases but not for well
has been investigated experimentally [6–8], through com- annealed cases, one may expect that the choice of range
puter simulations [9–26], and theoretical investigations from γmin to γmax over which the strain is varied cycli-
including the study of elastoplastic models and corre- cally may significantly influence the plasticity and yield-
sponding mean field theories [4, 13, 27–33]. These inves- ing behaviour. Indeed, such dependence is of practical
tigations have largely focused on the response to uniform importance in determining the characteristics of fatigue
shear, but several investigations have explored yielding and fatigue failure [1, 46], which in turn dictate the scope
behaviour under cyclic deformation [11, 12, 19, 21, 23– and limits of operability of such materials in real-life ap-
26, 29, 31–35]. plications.
A specific issue that has received considerable atten- With the aim of investigating the dependence of the na-
tion recently is the role of annealing of the glasses that ture of plasticity and yielding on particular cyclic defor-
are subjected to deformation, in determining the nature mation protocols, here we consider the response to totally
1 1
of yielding. Indeed, under both uniform shear and cyclic asymmetric cycles of shear, 0 → γmax → 0 → γmax ....
shear, it has been demonstrated that a qualitative change Specifically, we simulate a model glass employing the
occurs in the yielding behaviour when the degree of an- athermal quasistatic (AQS) shear protocol, and study
nealing of the glasses increases. Under cyclic deformation the response of samples with a widely differing degree
[23–26, 29, 31–33], poorly annealed glasses display signif- of annealing, system size, and subjected to a range of
1
icant mechanically induced annealing, and converge to a strain amplitudes γmax .
common threshold energy, before yielding takes place. In The observed behaviour is found to be markedly differ-
contrast, well annealed glasses (with energies below the ent from the case of symmetric cyclic shear. For poorly
threshold energy), do not display any change in proper- annealed glasses (with initial energies above the thresh-
ties with increasing amplitude of shear until the yield- old energy) the yielding behaviour follows the symmetric
ing amplitude is reached. The subsequent yield event is case, with a rescaling of the strain amplitudes, as we
accompanied by a discontinuous change in energy and discuss below. For the well-annealed samples, for the
stress, the amount of which depends on the degree of an- smaller system sizes considered, we find an intermediate
1
nealing. Under uniform shear corresponding responses range of γmax values over which the stress decreases from
are observed [20, 23]. the maximum value attained, i.e. beyond the stress peak,
Apart from the context of yielding, response to cyclic but no diffusive behavior is present. The onset of diffu-
1
shear has been investigated in order to understand the re- sive behavior, at a larger γmax value, is identified with
versible to irreversible transition, i.e. the transition from yielding [11, 19, 21, 23]. In order to better understand
a dynamics towards an absorbing state to one where the our results, we consider a recently proposed mesostate2
model [31] that we adapt to the asymmetric shear pro- and γdiff , respectively. Note that both for symmetric and
tocol. We show that this model qualitatively captures asymmetric shear, significant plastic rearrangements oc-
the observed behaviour, thereby shedding light on the cur before an absorbing steady state is established.
underlying relaxation mechanisms. We investigate the
dependence on system size N , and find that the interme- -6.90
(a) (b)
1 Einit = -6.89
diate window of γmax narrows with system size for well 0.8
annealed case. Our results suggest, but cannot conclu- -6.92
0.4
E
sively demonstrate, that the intermediate strain window
σ
will vanish as N → ∞, but the narrowing itself leads to a -6.94 0.0
0.060
remarkable conclusion: Under asymmetric shear, well an- 0.095 -0.4
-6.96 0.150
nealed glasses will yield at smaller strain amplitudes than 0.170 -0.8
poorly annealed glasses, reversing the trend observed in 0.200
-6.98 0 0.04 0.08 0.12 0.16 0.2 0 0.04 0.08 0.12 0.16 0.2
the case of symmetric shear protocols. γ γ
Simulations. We perform AQS simulations of a three
dimensional model glass former, the 80 : 20 Kob- FIG. 1. Variation of the energy and shear stress through one
Andersen binary Lennard-Jones mixture (KA-BMLJ), cycle of strain in the steady state for a single poorly-annealed
in which particles interact with a Lennard-Jones po- glass sample with N = 4000 and initial energy Einit = −6.89.
tential, employing a quadratic cutoff (with details pro- The different curves correspond to different driving ampli-
1
tudes γmax , as indicated in the legend of panel (b).
vided in the Supplemental Material (SM) [47]). The
results presented here were obtained from the simula-
tions with system size N = 200(25), 400(15), 800(15), Poorly-annealed glasses. We first consider the steady
4000(15), 8000(3), 32000(2), and 64000(1) particles, states attained under asymmetric shear for a poorly an-
where the numbers in parentheses indicate the num- nealed glass for N = 4000, with Einit = −6.89. In
ber of independent samples. With V being the sam- Fig. 1 we consider a single sample and show how the
ple volume, we equilibrate the system at fixed den- energy (left panel) and stress (right panel), as a func-
sity ρ = N/V = 1.2 in the liquid state and at re- tion of the applied strain γ, change over a driving cy-
duced temperatures T = 1.0, 0.60, 0.466, 0.40, 0.37 via cle, once a steady-state has been reached (Data averaged
a constant temperature molecular dynamics simulation. over samples is shown in SM Fig. S1). The curves shown
1
The energy of equilibrated configurations are mini- correspond to different driving amplitudes γmax , as in-
mized to obtain inherent structure or glass configura- dicated in (b). The energy vs. strain curves display a
tions, which have average energies per particle Einit = minimum Emin at a non-zero strain value γEmin which
1
−6.89, −6.92, −6.98, −7.03, −7.05, respectively, for the we denote as the plastic strain. For γmax ≤ 0.15, γEmin
1
T values indicated. Energy minimization is performed increases approximately as γEmin = γmax /2 (more be-
1
using the conjugate gradient algorithm. The molecu- low). For γmax > 0.15, the energy curves display two
lar dynamics and AQS simulations are performed using well-separated minima and the response ceases to be pe-
LAMMPS [48]. riodic. These two observations can be explained by a
Shear deformation protocol. The inherent structures, or shift in the plastic strain: during the transient leading
glasses, are subjected to cyclic shear deformation in the to periodic response, the plastic strain evolves towards
1
xz plane using the AQS protocol, which involves the ap- γEmin = γmax /2, so that the asymmetric shear defor-
1
plication of strain by small increments (dγ = 2 × 10−4 ) mation protocol 0 → γmax → 0, effectively becomes a
1
followed by energy minimization (further details in the symmetric one around the plastic strain γmax /2 with am-
1
SM). We apply asymmetric shear cycles that follow the plitude γmax /2. In fact, the observation of yielding for
1 1 1
sequence: 0 → γmax → 0, where γmax is the amplitude of γmax > 0.15, when taking into account the shift of γEmin ,
deformation, continuing until a steady state is reached, implies an effective yielding amplitude of about 0.075 un-
in which the system either reaches an absorbing state so der a symmetric shear protocol, which is consistent with
that the same sequence of configurations is visited during the observed value γysym ≈ 0.075 [23].
subsequent cycles, or a diffusive state with no periodicity Well-annealed glasses. The picture changes dramatically
but statistically stationary properties. when we consider well-annealed glasses. In the case of
For symmetric cyclic shear [11, 19], the yield strain symmetric shear, a well-annealed glass with N = 4000
amplitude γysym has been identified as the strain value and Einit = −7.05 shows, under increasing strain ampli-
sym
marking the onset of the diffusive state, which also co- tude γmax , no change in the minimum Energy Emin and
incides with a discontinuous stress drop from the largest plastic strain of the cylic response until yielding, which
sym
stress value attained just prior yielding [11, 19]. As de- occurs at γmax = 0.105 [23]. Fig. 2 (a) shows Emin as
scribed below, for asymmetric shear, the location of the a function of cycle number Ncycles for selected asymmet-
1
stress peak and the onset of diffusion differ in general, and ric shear amplitudes γmax . As in the case of symmet-
1
we will therefore distinguish these strain values as γpeak ric shear, for small γmax = 0.060, 0.095 the energy does3
(a) (b) (c)
0
-6.96 0.08 10
MSD
min
Emin
γE
0.060
-7.00 0.095
0.04 0.130 10
-3
0.150
0.160
-7.04 0.170
0.200
0.00 -6
0 1 2 3 0 1 2 3 10 0 1 2
10 10 10
Ncycles 10 10 10 10
Ncycles 10 10 10 Ncycles 10
(d) 0.12 Einit = -6.89 (e) 1.60 (f)
-6.92 Einit = -6.92
1 /2
Einit = -6.98 x
=γn ma 1.20
γ E mi
-6.96 0.08 Einit = -7.03
σmax
min
Emin
Einit = -7.05
γE
-7.00 0.80
0.04
-7.04 γy γy 0.40 γy
0
0.04 0.08 1 0.12 0.16 0.2 0.04 0.08 1 0.12 0.16 0.2 0.04 0.08 1 0.12 0.16 0.2
γ max γ max γ max
FIG. 2. Behavior of the minimum energy Emin of a cycle (a), the plastic strain γEmin at which it is attained (b), and the
mean-squared displacement (MSD) from the initial configuration (c), as a function of cycles of strain Ncycles . The data shown
1
is for a well-annealed glass with N = 4000, initial energy Einit = −7.05, and driven by a range of asymmetric shear strain
1
amplitudes γmax , as indicated in the legend of (b). (d) – (f): The yielding diagram showing how steady states properties (for
1
N = 4000) depend on asymmetric strain amplitude γmax and the degree of annealing, as indicated in the legend of (e). The
1
steady state values shown are Emin , γEmin and shear stress σmax at maximum strain γmax . The yield strain amplitude γy , below
which the MSD curves in (c) show zero slopes, is marked by red arrows in panels (d) – (f).
not change with Ncycles . However, different from the re- The yielding diagram. We summarise the results for the
sponse under symmetric cyclic shear, for larger values full range of annealing of the glasses we considered in Fig.
1
γmax ≥ 0.130, Emin increases with Ncycles and saturates 2 (d)-(f). Fig. 2 (d) shows the steady state energies as a
1 1
at values that grow with γmax . Interestingly, the plastic function of Einit and the strain amplitude γmax . For the
strain γEmin exhibits a similar behavior, Fig. 2(b): γEmin poorly-annealed glasses with Einit = −6.89, −6.92, the
1
remains fixed at 0 for γmax = 0.060, 0.095, but grows with variation of the energies is similar to the case of symmet-
1 1
Ncycles as well as γmax for the larger γmax values. More- ric shear, displaying a non-monotonic change in energy
1
over, the asymptotic behavior γEmin ≈ γmax /2 is attained across the yielding amplitude. In sharp contrast, for the
1 1
only when γmax > 0.150, while for γmax = 0.130, 0.150 well-annealed glasses, Einit = −7.03, −7.05, the energies
1 1 1
the asymptotic values of γmax lie between 0 and γmax /2. remain constant up to values of γmax that turn out to be
1
Nevertheless, for γmax > 0.095, substantial plastic de- close to the yield amplitudes that had been established
formations appear to lead to finite γEmin values, even for the symmetric case as γysym = 0.098, 0.105, respec-
1
though, as shown in Fig. 2 (c), a diffusive steady-state is tively [23]. For values of γmax beyond this, Emin increases
1 1
reached only for γmax > 0.150. with γmax until reaching a value of Emin ≈ −6.985, which
was identified as the threshold energy in [23] across which
The corresponding results for the poorly annealed the character of yielding changes in the symmetric case.
glasses with Einit = −6.89 are shown in the SM, dis- The intermediate case of Einit = −6.98 displays an inter-
playing a gradual change of both Emin and γEmin with esting non-monotonic behaviour.
1
Ncycles as well as γmax , but with yielding occurring only
1 1
when γmax > 0.15, as in the well annealed case. The SM Irrespective of the degree of annealing, for γmax > 0.15
also contains results to show the absence of anisotropy of the energies Emin all increase along a common curve, and
the sheared glasses. we identify γdiff ≈ 0.155 as the onset of the diffusive4
steady-state regime. A rationalisation of these results given by E,γmin (γ) = − + κ2 (γ − γmin )2 . The√stability
can be found in the behaviour of the plastic strain γEmin range of each mesostate is given by γ ± = γmin ± . Upon
1
shown in Fig. 2 (e). The extent to which γEmin ≈ γmax /2 variation of strain when a given mesostate (, γmin ) be-
is achieved can be seen as indicative of the plastic de- comes unstable, a new state (0 , γmin0
) is drawn randomly
formations that have taken place before a steady state subject to the condition E0 ,γmin 0 (γ) < E,γmin (γ). We
was reached. Clearly, for Einit = −7.03, −7.05, little de- study the model here subject to the additional range re-
formation occurs until γmax 1
≈ 0.1, whereas for higher strictions 0 ∈ (−δ, +δ), γmin
0
∈ (γmin −δγ, γmin +δγ).
1
γmax the plastic strain shifts, along with the energies, Note that the restrictions imposed on 0 and γmin 0
(though
until configurations are reached which are stable under not the exact values δ and δγ) are found to be important
the imposed cyclic strain. For higher Einit , such reorga- for reproducing the simulation results qualitatively, and
1
nizations occur for all γmax , and to a greater degree for we choose δ = 0.05, δγ = 0.1.
larger Einit . Finally, in Fig. 2(f) we consider the variation As shown in Fig. 3, the qualitative aspects of the ob-
1
of shear stress σmax evaluated at γmax . Once again, for servations from particle simulations, Fig. 2(d) and (e),
Einit = −6.89, −6.92, we observe a monotonic increase are remarkably well reproduced by the model. This sug-
of σmax before yielding, but for lower Einit , we observe a gests, on the one hand, that the mesostate model has
highly unusual non-monotonic change of σmax , well be- the right ingredients to describe yielding in amorphous
fore the yield point. The maximum stress values obtained solids, and on the other hand, that the phenomena we
1
when γmax = γpeak = 0.095 are comparable to the yield observe are robust and generic. The fact that only some
stress values in the symmetric cyclic shear case [23]. specific choices of model parameters reproduce the sim-
Unlike the symmetric shear case, the location of the ulation results, in particular the evolution of the plastic
stress maximum and the onset of diffusive behaviour do strain, offers guidance for how such modeling may be pur-
not coincide for asymmetric shearing of well annealed sued to faithfully capture behaviour of amorphous solids
glasses, as seen in Fig. 2(f), (c), with γpeak (location of subjected to deformation.
the stress maximum) < γdiff (onset of the diffusive steady System size analysis. In order to interrogate better the
state). These results are surprising in the context of 1
intermediate regime γpeak < γmax < γdiff , that emerges
cyclic shear, since previous observations of plasticity be- for the well-annealed samples, we consider next the effect
fore yielding have invariably been associated with anneal- of system size. In Fig. 4 (a) and (b) we plot the behavior
ing (decrease of energy). The new non-trivial ingredient of the minimum energy Emin and the plastic strain γEmin
that is brought forth by the asymmetric shear results is 1
at steady state against γmax , for different system sizes,
dynamics induced by deformation along the plastic strain Einit = −7.05. A strong system size dependence is ap-
1
axis. We will return to the regime γpeak < γmax < γdiff parent: the smaller the system size, the larger the strain
when discussing the effect of system sizes. 1 1
γmax value beyond which γEmin = γmax /2. In Fig. 4 (c)
we show the corresponding evolution for σmax . For sys-
0 tem sizes larger than 4000, the stress maximum appears
(a) 0.8 (b)
around a common value γpeak = 0.1, but the subsequent
-0.2 0.6 1 /2 drop of stress becomes sharper with increasing system
ax
Emin
min
=γ m size. Although we find γmax 1
values for each N above
γE
0.4 γ E min
-0.4 γpeak but below γdiff (see SM for details), we expect that
γdiff → γpeak as N → ∞. This leads to the remarkable
0.2
-0.6 conclusion that for asymmetric shear, the yield valuue
γdiff will be smaller for well annealed glasses compared
0 0.4 0.8 1 1.2 0
1.6 0 0.4 0.8 1 1.2 1.6 to poorly annealed glasses, reversing the trend seen for
γ max
γ max
symmetric shear.
FIG. 3. The dependence of energy Emin and plastic strain In summary, we have investigated the yielding be-
γEmin as a function of initial energy and strain amplitude haviour of a model amorphous solid under asymmetric
for the mesostate model defined in [31], with a constraint on cyclic shear deformation. We show that such yielding
the magnitude of change in mesostate energy δ = 0.05, and behaviour displays striking new features not observed for
refernece strain δγ = 0.1. symmetric cyclic shear, including the emergence of an
intermediate window of strain amplitudes dominated by
Simulations of a mesostate model. We next consider significant plastic rearrangements and accompanied by a
whether the observed behaviour can be reproduced by decrease of stress for well annealed glasses. Such a win-
a mesostate model proposed to describe yielding under dow is expected to vanish for N → ∞, as our system
cyclic deformation, in [31]. The model is defined in terms size results indicate. Nevertheless, our results reveal the
of mesostates, each of which is characterised by a mini- central role played by the non-trivial evolution of plastic
mum energy −, which is attained at a plastic strain γmin , strain in the case of asymmetric cyclic shear in deter-
with the total energy of the state at a given strain being mining plastic response, in addition to the evolution of5
-6.92 200 (a) Einit=-7.05 (b) 1.5 (c)
400
800 1 /2
4000 0.10 = n ma
γ x
-6.96
σmax
min
8000 γ E mi 1
Emin
γE
32000
-7 64000
0.05
0.5
-7.04
0.00
0 0.06 0.12 1 0.18 0.24 0 0.06 0.12 1 0.18 0.24 0 0.06 0.12 1 0.18 0.24
γ max γ max γ max
FIG. 4. System size dependence of Emin , γEmin , σmax for the well annealed system. The red line corresponds to γEmin =
1 1
1/2γmax . The brown left triangles in (c) indicate the smallest γmax for which the sheared glasses exhibit diffusive behaviour.
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