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Crack Initiation in Viscoelastic Materials

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In viscoelastic materials, individually short-lived bonds collectively result in a mechanical resistance which is long-lived but finite, as ultimately cracks appear. Here we provide a microscopic mechanism by which cracks emerge from the nonlinear local bond dynamics. This mechanism is different from crack initiation in solids, which is governed by a competition between elastic and adhesion energy. We provide and numerically verify analytical equations for the dependence of the critical crack length on the bond kinetics and applied stress.
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Crack initiation in viscoelastic materials
Yuval Mulla1, Giorgio Oliveri2, Johannes T.B. Overvelde2, Gijsje H. Koenderink1
1Living Matter Department, AMOLF, Science Park 104, 1098 XG Amsterdam and
2Designer Matter Department, AMOLF, Science Park 104, 1098 XG Amsterdam
In viscoelastic materials, individually short-lived bonds collectively result in a mechanical
resistance which is long-lived but finite, as ultimately cracks appear. Here we provide a
microscopic mechanism by which cracks emerge from the nonlinear local bond dynamics.
This mechanism is different from crack initiation in solids, which is governed by a competition
between elastic and adhesion energy. We provide and numerically verify analytical equations
for the dependence of the critical crack length on the bond kinetics and applied stress.
Liquids cannot fracture, but solids can. We consider
the intermediate case: viscoelastic materials. These ma-
terials are made of filaments or particles interconnected
by short-lived bonds. This design theme of transient net-
works is commonly used in both natural and man-made
materials such as cytoskeletal polymer networks in cells
[1], physical gels [2], associative and telechelic polymers
[3], and colloidal gels [4].
The molecular dynamics of transient networks lead to
interesting macroscopic mechanics: at times shorter than
the bond lifetime the material behaves like a solid [1],
while on longer time scales the bonds reorganize and
the material deforms viscoelastically [1, 5, 6]. As a re-
sult, transient networks are much more deformable than
permanent networks [7]. However, viscoelastic materials
can resist mechanical stress only for a limited time, af-
ter which the system suddenly loses its mechanical per-
colation, a process which is known as fracturing [3, 8–
10]. This raises the question how we can design transient
networks such that the robustness against stress is op-
timized, which requires an understanding of the mecha-
nism by which transient networks fracture.
Fracturing of viscoelastic materials is often explained
by the Griffith theory of crack initiation in brittle solids
[3, 8, 11–14]. The Griffith theory predicts that beyond a
critical stress, initial defects will grow into macroscopic
cracks as the elastic energy released by the crack dom-
inates the surface energy required for separation [15].
However, this framework was originally developed for
solids, and assumes defects are either static or growing.
This assumption is clearly invalid for viscoelastic mate-
rials, as defects are not static entities but instead con-
tinuously appear and heal [16]. Therefore, viscoelastic
materials require a framework which takes into account
the reversible bond dynamics.
The seminal work of Bell on cellular adhesion provides
such a framework of reversible bond dynamics under force
[17], and has received considerable attention in studying
small-scale systems such as protein clusters which provide
cellular adhesion [18, 19], fracturing of a single colloidal
strand [9] and protein unfolding [20].
Corresponding author:
In all of these works, force is assumed to be homoge-
neously distributed across all bonds, which appears to be
a realistic assumption for microscopically small systems.
Contrarily, in the context of viscoelastic materials, theo-
retical [21, 22] and experimental work on both synthetic
gels [23] and biopolymer networks [24] has revealed non-
affine deformations upon application of global stress [25].
Indeed, imaging of various networks under stress showed
inhomogeneities of the local force which are strongly cor-
related in space [26–28]. These inhomogeneities might be
negligible when considering bulk properties such as the
average bond lifetime under stress [29, 30], but likely play
a key role in crack initiation. In situ imaging of stressed
viscoelastic materials revealed that fracturing occurs via
well-defined cracks [3, 11, 31, 32] rather than via the
homogeneous degradation expected from the Bell model
[17–20, 29, 30, 33], suggesting local rather than global
load sharing. At which length scale does the global load
sharing assumption become inaccurate? And what deter-
mines the fracturing properties of a system of reversible
bonds under load beyond this length scale?
To answer these questions we developed a minimal
model that includes reversible bond dynamics (figure
1a) in the simplest possible ’material’ that is capable of
exhibiting spatial inhomogeneity required for studying
crack initiation: bonds distributed over a 1D-space, sub-
ject to mechanical stress. To account for inhomogeneous
load sharing, we assume a force distribution that depends
on the local bond spacing (figure 1b). We show that this
minimal model system exhibits spontaneous crack initi-
ation and subsequent fracture, in a manner that is con-
sistent with experimental observations in wide range of
viscoelastic materials [3, 11, 31, 32]. We verify our re-
sults by comparison with a mechanical model. We study
the process of crack initiation in more detail by locally
’ablating’ bonds (figure 1c), which reveals a critical crack
length beyond which fracturing occurs. We provide an-
alytical equations describing the process of crack initia-
tion on basis of the nonlinear bond dynamics, and predict
the dependence of the critical crack length on both bond
properties and applied stress. Our work reveals that the
process of crack initiation in viscoelastic materials is fun-
damentally different from that in traditional solids, as a
consequence of the reversible bond dynamics.
We initialize a one-dimensional (1D) network with N
arXiv:1802.04017v1 [cond-mat.soft] 12 Feb 2018
equally spaced binding sites using periodic boundary con-
ditions, each bond having a probability Kto start in a
closed state. Next we model the dynamics of the bonds
with a kinetic Monte Carlo scheme [34] using the follow-
ing bond dynamics:
kon +koff,0
where kon is the rate of bond closing and koff,0 the rate
of bond opening in the absence of force (figure 1a). We
normalize time by the on-rate, kon. The off-rate increases
exponentially with the applied force fon the bond in
keeping with the Bell model [17]:
koff(fi) = koff,0 ·exp( fi
where f1/e is the force where the off-rate has fallen to 1/e
of koff,0. We calculate the force per bond fivia
where σis the stress on the system and αis a yet to
be defined stress intensity factor per bond. In global
load sharing, the applied stress is equally divided over all
bonds. To investigate the effect of inhomogeneous force
distribution as present in any network under stress [26–
28], we investigate a local load sharing model. In this
model, we assume that the force distribution is depen-
dent on the distance liof a bond to its nearest neighbor
on both sides (figure 1b). Explicitly, we define a stress
intensity factor αon a closed bond at site iby:
ΣiniGlobal (4)
where niequals 1 when the bond is closed and 0 when the
bond is open. Note that in both modes of load sharing
the total amount of force is independent of the bound
fraction and normalized by the system size,Pifi
N=σ. We
normalize the applied stress by the bond force sensitivity
f1/e . After calculating the force on all bonds, we employ
a kinetic Monte Carlo step to either open or close a bond
stochastically. We repeat this process of stochastic bond
removal/addition until all bonds are removed.
As shown in figure 2a, the fraction of closed bonds fluc-
tuates over time, until it drops precipitously to zero at
a certain moment that we denote as the rupture time.
The rupture time is exponentially distributed - indica-
tive of the stochastic nature of fracturing [35]. To test
the sensitivity of the average rupture time to the ap-
plied stress, we perform simulations at a fixed network
size and fixed bond affinity for different levels of stress
(N= 20,K= 0.9,σ= 0.5...2). For both local and global
FIG. 1. Schematic of model a) Bonds switch from an open
to a closed state with rate kon and reverse with a rate of koff,0
in the absence of force. b) Bonds in a closed state share an
applied load σ, where the load distribution depends on the
distance lifrom bond ito its nearest neighbors. c) We per-
form bond ablation experiments by opening all bonds in lablate
adjacent positions to investigate the critical length required
for triggering fracturing. Periodic boundary conditions are
used to prevent edge effects from influencing the results.
load sharing, we find that the average rupture time shows
two distinct regimes with a transition around <trupt>1
(figure 2b). As we will explain later on, these regimes
correspond to a metastable network at low stress and
an unstable network at high stress. Importantly, net-
works with local load sharing are markedly less robust
than globally load sharing networks, with smaller aver-
age rupture times at all stresses.
To test how the system size influences the average rup-
ture time, we perform simulations for networks with N
varying between 5 and 100 (figure 2c). In case of global
load sharing, we see that the average rupture time mono-
tonically increases with system size, as the relative fluctu-
ations of the fraction of closed bonds ( Σini(t)
N) decreases
[9, 17–19]. In case of local load sharing, we find simi-
lar rupture times as compared to globally load sharing
networks for small N. But strikingly, beyond a crit-
ical length (around N= 12 for these conditions), we
find that only in case of local load sharing the rupture
time decreases with increasing system size, according to
<trupt>N1(inset of figure 2c). This dependence sug-
gests a constant crack initiation rate for every 12 bonds at
this particular stress. Indeed, kymographs of simulations
using local load sharing reveal that fracturing proceeds
via cracks rather than homogeneous degradation (figure
To understand what sets this critical length for crack
initiation, we performed ’ablation experiments’ (figure
1c): first we equilibrate the network under stress, next
we remove all bonds in lablate adjacent positions, then
we study whether bond ablation triggered network frac-
turing. We chose the system size N=lablate·10, such that
0 20 40 60
0 1 2
<trupt >
025 50
<trupt >
10 11 0 2
10 1
10 4
¿rupt-200 ¿rupt
Simulation step
Bond position
FIG. 2. Stochastic rupture of simulated 1D transient
networks subject to a mechanical stress. a) Typical
example of the fraction of closed bonds in time upon appli-
cation of stress, after t=trupt, spontaneous fracture occurs
(K= 0.9,N= 20,σ= 0.7, global load sharing). b) Stress
dependence of rupture time. Although quantitatively differ-
ent, global and local load sharing show qualitatively similar
behavior with two exponential regimes with a cross over at
around <trupt>1 (K= 0.9,N= 20). c) The system size
dependence of the rupture time reveals a qualitative difference
between global and local load sharing: whereas the rupture
time increases with system size for global load sharing, local
load sharing shows an optimum in strength at a well-defined
system size (K= 0.9,σ= 0.7). Inset: same data on a log-log
scale, showing that after a critical system size <trupt>N1
for local load sharing. d) Kymograph of crack initiation under
local load sharing (white=closed, black=open). Plotting the
bond state as a function of position (y-axis) versus simulation
step (x-axis) clearly reveals how bond opening proceeds via
a well-defined crack. The x-axis shows simulation step rather
than time, as the crack propagation is orders of magnitude
faster than the crack initiation (figure 2a).
the system is large compared to the number of ablated
bonds, yet small enough to allow for equilibration with-
out spontaneous crack initiaton. Figure 3a shows how
fracturing becomes more likely upon increasing the abla-
tion size lablate, and that the required ablation size lablate
to initiate fracturing decreases with the applied stress
σ. Figure 3b shows that an increase of bond affinity K
increases the critical ablation size lablate required for trig-
gering fracture.
To quantitatively understand the ablation data, we de-
fine crack length Las the largest bond distance liin the
system. In case of global load sharing, the force on bonds
at the edge of the crack stays independent of Las long as
LN, so ablation does not induce fracture. By contrast,
in case of local load sharing, the force on the bond at the
edge of the crack is f=σ·L·1
/2(the factor 1
/2is because
the load on the hole is shared by the bonds at both ends).
Thus, koff exponentially grows with the crack size due to
a linearly increasing force, whereas the chance of rebind-
ing increases only linearly due to a larger area in which
rebinding can occur. As a result, for large enough lablate ,
bond unzipping will occur for any system under stress.
We are interested in the length Lunstable at which the
crack becomes unstable. As a first order approximation,
we can find the fixed points of crack length
Lby calcu-
lating the length at which the rates of bond opening and
closing are equal (figure 4a):
L)kon ·
This condition is met at the average bond distance at
equilibrium, Lstable, and the bond distance at the unsta-
ble point, Lunstable:
Lstable(σ, K )2·W0(σ·(1
Lunstable(σ, K )2·W1(σ·(1
where Wis the Lambert W function with W0the main
branch and W1the second branch [36]. Note that
the network transitions from metastable to unstable at
Lstable=Lunstable (seen as a change in slope in figure 2b
at around <trupt>1). For local load sharing, the transi-
tion from a metastable to an unstable network occurs be-
yond a critical bond-to-bond distance Lunstable, whereas
for global load sharing this transition occurs beyond a
critical fraction of open bonds and therefore explains the
continuous increase of rupture time as function of time
[9, 17–19].
To test equation 7, we show in figure 4b that all ab-
lation data can be successfully collapsed onto a single
master curve using a normalized ablation size lablate
Lunstable .
To compare equation 7 with both the ablation data and
the typical length scale observed in figure 2b, we first
define a critical ablation length, lcrit, which we obtain
by fitting the size dependence of the rupture probabil-
ity φrupt to a sigmoidal function φrupt=1
1+elcritlablate at
each applied stress and at bond affinity K= 0.9. We
can now combine the critical ablation length lcrit from
figure 3a with the optimal system size Nfound in figure
2c and conclude that all these data are well-described by
equation 7 (figure 4c).
Up to now, theoretical work on transient network frac-
turing has been limited to the assumption of global load
sharing [9, 17–19]. We find that < trupt >is insensi-
tive to this assumption for microscopic systems (up to
approximately 10-100 bonds, see equation 7 and figure
2c). However, fracturing of larger system follows funda-
mentally different rules, in which the notion of local load
sharing becomes important. Our study investigates the
idealized limit of fully localized load sharing, but we ob-
serve similar behavior of a typical fracture length when
the load distribution is simulated via a mechanical model
Our model predicts features which are different from
global load sharing, but consistent with experimental ob-
servations on a wide range of viscoelastic materials. First,
0 100 200
FIG. 3. Characterization of critical crack length in lo-
cal load sharing Ablation experiments were performed by
first equilibrating the system under stress until t= 1, next
bonds were ablated: ni= 0 for i=0...lablate. After the abla-
tion, the network was studied up to t= 2. This experiment
was repeated 30x per condition, and the fraction of observed
ruptures φrupt was recorded. We plot the ablation size lablate
versus the fraction of observed ruptures φrupt for different
values of a) applied stress σat K= 0.9or b) bond affinity
Kat σ= 0.1. Control ablation experiments using the same
parameter with global load sharing never showed fracturing.
Lstable Lunstable
0.5 1.0 1.5
0.0 0.4 0.8
Ablation data
Optimal N
FIG. 4. Comparison between theory and simulation
a) As a function of crack size, the on-rate increases linearly,
whereas the off-rate increase exponentially. As a result, the
crack becomes unstable after Lunstable. b) All data from figure
3 can be collapsed obtained onto a single master curve by nor-
malizing according to lablate/(Lunstable (σ, K)). c) Equation 7
quantitatively predicts both the critical length for ablation,
and the width at which the maximal rupture time is observed
in figure 2.
rupturing of macroscopic viscoelastic materials proceeds
via spontaneous crack initiation, different from the ho-
mogeneous failure predicted by global load sharing mod-
els [9, 17–19]. This prediction is borne out by experi-
mental observations of a wide range of viscoelastic ma-
terials [3, 11, 31, 32]. Second, the model of global load
sharing predicts that the rupture time strongly increases
with the system size. As a result, delayed fracturing
(<trupt>kon ) would only be experimentally observable
very close to the critical stress for any macroscopic sys-
tem. Instead, delayed fracture is experimentally observed
for many different viscoelastic materials over a wide range
of stresses [3, 8–10]. We find that in case of local load
sharing, the dependence of < trupt >on σdoes not di-
verge upon increasing system size N. Thus, our model
for the first time explains why delayed fracturing is read-
ily observable on laboratory timescales over a wide range
of stresses in experiments.
The model makes several concrete predictions that can
be tested experimentally by applying shear stress on vis-
coelastic materials. Firstly, we predict that the aver-
age rupture time measured at constant stress will be in-
versely proportional to the system size (figure 2c) as the
crack initiation rate is constant per volume. Secondly,
the presence of a critical crack length can be measured
directly by performing laser ablation on viscoelastic ma-
terials under stress, a technique that is common in bio-
physical studies of cell and tissue tension [37]. Thirdly,
the dependence of the critical crack length on the ap-
plied stress and bond kinetics (Equation 7) can be tested
experimentally. The bond kinetics can for instance be
experimentally controlled by changing the temperature
in cross linked actin networks [38] or salt conditions in
polyelectrolyte gels [39].
Our framework for understanding the crack initiation
process in viscoelastic materials can be used to rationally
design more robust materials. We have considered evenly
distributed bonds. For future work, it would be inter-
esting to investigate the effect of inhomogeneity under
local load sharing. It is interesting to note that cellu-
lar adhesion proteins are not randomly distributed but
clustered with a well-defined size [40]. Simulations have
shown that an intermediate degree of clustering is opti-
mal for preventing fracturing [41, 42], although the na-
ture of this optimum remained poorly understood. We
speculate that this optimal clustering density is related
to the critical length scale for crack initiation and that
this strategy of clustering bonds is an interesting design
principle for synthetic materials.
We thank Pieter Rein ten Wolde, Chase Broedersz,
David Brueckner and Mareike Berger for fruitful discus-
sions. This work is part of the research program of the
Netherlands Organisation for Scientific Research (NWO).
We gratefully acknowledge financial support from an
ERC Starting Grant (335672-MINICELL).
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Supplementary Information
To validate the behavior of cracks in viscoelastic ma-
terials obtained with the local load sharing assump-
tion, we implemented a 2D mechanical model (figure S5)
based on Finite Element Analysis (FEA) [43]. The tran-
sient bonds, modeled by a linear elastic material with a
Young’s modulus Eand Poisson’s ratio ν, are modeled
as elastic bodies fixed at their bottom and attached to
an upper body with the same material properties. Both
the bonds and the elastic body are discretized using bi-
linear square elements consisting of four nodes. The elas-
tic body consists of h·Nelements, where his the height of
the elastic body expressed in terms of the in number of el-
ements and Nis the number of bonds. Periodic boundary
conditions are applied to the left and right boundaries.
To apply tension to the bonds, a vertical displacement is
applied to the upper boundary of the solid part, until a
force σFEA is reached. A Monte Carlo scheme similar to
the one used for the local load sharing model is applied
to determine the transient behavior of the bonds, using
equation 2 and 3 from the main text, and we define the
stress intensity factor value αifor FEA as:
where Uiis the bonds’ elastic strain energy density which
can be found by integrating the stress vector on the bond
{s}ifor the strain vector {e}iaccording to:
2[sxx syy sxy ]i
Figure S6 shows how the elastic body redistributes the
applied force on the bonds. Importantly, we observe a
stress intensity distribution for FEA that is comparable
to that of the local load sharing assumption using the
settings of h=Nand Ebody =Ebond. For future work
it will be interesting to vary hand/or Ebond/Ebody and
test the effect on the stress distribution and subsequent
fracturing behavior.
FIG. 5. Finite element model Finite element model rep-
resentation of an elastic body (blue) of total thickness h
with random distribution of bonds (green) at rest (left) and
stretched (right). The deformation, due to the applied force
σFEA on the elastic body, is exaggerated for clarity.
3 6 9 11 13 14 21 23 25 27 28
Bond position i
FIG. 6. Comparison of load distribution Stress intensity
factor comparison of different modes of load sharing (global,
local, FEA) for the bond distribution shown in Figure S5..
0 20 40
<trupt >
FIG. 7. FEA verifies main qualitative difference be-
tween local and global load sharing Unlike global load
sharing but similar to local load sharing (main text figure 2c),
the average rupture time in FEA is peaked for intermediate
system size N.
0 50 100 150 200
FIG. 8. Rupture times are exponentially distributed
We performed fracturing simulations under local load shar-
ing for 1D networks (see main text figure 1) using identical
parameters ( σ= 0.7,K= 0.9,N= 20,1000 repeats) and
recorded τrupt. The distribution of rupture times is exponen-
tial, suggesting a stochastic process.
... initiation in viscoelastic materials, which is the rate-limiting step of rupturing (Extended Data Fig. 5) 14,15 . We used idealized Bell-Evans force-dependent unbinding kinetics to capture the catch or slip bond behaviour ( Fig. 1b and equation (1)), and allowed for unbound linkers to rebind at a random new location 16 . ...
... Methods Minimal 1D crosslinker model. To investigate the effect of molecular catch bonding on the strength of cytoskeletal filament networks, we use a computational model we recently developed to predict the failure of transient networks 14,15 , using a Gillespie algorithm to model stochastic linker binding and unbinding. The detailed motivation behind the design of the model, including a discussion of its assumptions and limitations, are presented in Supplementary Note 1. ...
... As the actin concentration is much larger than the crosslinker concentration both in our reconstituted networks (48.00 versus 0.48 μM, respectively) and in living cells (~100 versus ~1 μM, respectively 32 ), we consider tenfold more binding sites than crosslinkers to prevent competition for actin-binding sites. For control simulations where the linkers are immobile (Extended Data Fig. 6c), we only allow for rebinding in the same place where the crosslinkers are unbound 14 . ...
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Molecular catch bonds are ubiquitous in biology and essential for processes like leucocyte extravasion¹ and cellular mechanosensing². Unlike normal (slip) bonds, catch bonds strengthen under tension. The current paradigm is that this feature provides ‘strength on demand³’, thus enabling cells to increase rigidity under stress1,4–6. However, catch bonds are often weaker than slip bonds because they have cryptic binding sites that are usually buried7,8. Here we show that catch bonds render reconstituted cytoskeletal actin networks stronger than slip bonds, even though the individual bonds are weaker. Simulations show that slip bonds remain trapped in stress-free areas, whereas weak binding allows catch bonds to mitigate crack initiation by moving to high-tension areas. This ‘dissociation on demand’ explains how cells combine mechanical strength with the adaptability required for shape change, and is relevant to diseases where catch bonding is compromised7,9, including focal segmental glomerulosclerosis¹⁰ caused by the α-actinin-4 mutant studied here. We surmise that catch bonds are the key to create life-like materials.
... In contrast to the main assumption of homogenous force distribution across bonds within a cluster, imaging of tissues under stress has revealed inhomogeneous force distribution across connections [82]. A recent theoretical study demonstrated that a nonhomogeneous distribution of forces across bonds influences the critical size for crack initiation, giving rise to a critical cluster size above which the lifetime of the cluster decreases [83]. Despite tremendous progress in modeling cell-cell adhesion, we know little about the effects of unequally spaced bonds subjected to nonhomogeneous and non-constant forces. ...
... Mechanotransductory signaling within cells may recruit proteins to sites exposed to larger forces, thus resulting in nonhomogeneously spaced bonds (e.g., [41,84]). It has been speculated that such clustering density has been optimized in relation to the critical length for crack initiation [83]. Further, as mentioned previously, two types of junctions simultaneously play a crucial role in maintaining tissue integrity, each of which possesses different dynamics, thus resulting in a complex stochastic behavior that has not yet been investigated theoretically. ...
During development and in adult physiology, living tissues are continuously subjected to mechanical stresses originating either from cellular processes intrinsic to the tissue or from external forces. As a consequence, rupture is a constant risk and can arise as a result of excessive stresses or because of tissue weakening through genetic abnormalities or pathologies. Tissue fracture is a multiscale process involving the unzipping of intercellular adhesions at the molecular scale in response to stresses arising at the tissue or cellular scale that are transmitted to adhesion complexes via the cytoskeleton. In this review we detail experimental characterization and theoretical approaches for understanding the fracture of living tissues at the tissue, cellular, and molecular scales.
... (e) Example of a fibre network model that includes bending interactions along fibres and a more detailed description of inter-fibre bonds. Reprinted from Ref. [60] with permission from Elsevier. (f) Example of Kremer-Grest type model for a polymer network. ...
... To explore the tunability of delayed fracture we studied the effect of load redistribution on delayed fracture in a one-dimensional fibre bundle model 59 similar to Ref. [60] as shown in Figure 7.7. To capture the stochastic nature of fracture in the presence of thermal fluctuations, rupture and rupture time of the fibres is determined by a Gillespie algorithm 61 . ...
... We provide a detailed perspective for tuning the attachment strength of a soft probabilistic fastener via load-sharing rules known from rupture theory in fiber bundle models. 12,13 ...
... As a result, the lost load during the detachment of an individual pillar From rupture studies in fiber bundle models, this so-called "local load-sharing" is known to lead to the catastrophic detachment of the entire paired region. 13,27 Such effects of load-sharing have been exploited experimentally in gecko-inspired elastomeric microfibrillar adhesives, by Song et al., 28 where the maximum adhesion force was found to be 14 times larger than the adhering membrane in local load-sharing case. We suggest that this mechanism is present in the mechanical interlocking based elastomeric adhesive as well, attributing the lower value of the individual pillar strength for samples having small interpillar distances, i.e., for with higher feature density. ...
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Probabilistic fasteners are known to provide strong attachment onto their respective surfaces. Examples are Velcro® and the "3M dual lock" system. However, these systems typically only function using specific counter surfaces and are often destructive to other surfaces such as fabrics. Moreover, the design parameters to optimize their functionality are not obvious. Here, we present a surface patterned with soft micrometric features inspired by the mushroom shape showing a nondestructive mechanical interlocking and thus attachment to fabrics. We provide a scalable experimental approach to prepare these surfaces and quantify the attachment strength with rheometric and video-based analysis. In these "probabilistic fasteners," we find that higher feature densities result in higher attachment force; however, the individual feature strength is higher on a low feature density surface. We interpret our results via a load-sharing principle common in fiber bundle models. Our work provides new handles for tuning the mechanical attachment properties of soft patterned surfaces that can be used in various applications including soft robotics.
... Based on Eq. (7) in Ref. [75] and Eq. (4) in Ref. [147], we introduce a nonuniform force load by making the substitution h → C P i h i δ σ i ;−1 = N c ðfσ i gÞ in Eq. (2), where C ≡ Nh= P i h i is a normalization constant such that initially, i.e., when all bonds are closed, the total force load is h. The load on bond i, denoted as h i , is given by ...
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We illuminate the many-body effects underlying the structure, formation, and dissolution of cellular adhesion domains in the presence and absence of forces. We consider mixed Glauber-Kawasaki dynamics of a two-dimensional model of nearest-neighbor-interacting adhesion bonds with intrinsic binding affinity under the action of a shared pulling or pushing force. We consider adhesion bonds that are immobile due to being anchored to the underlying cytoskeleton, as well as adhesion molecules that are transiently diffusing. Highly accurate analytical results are obtained on the pair-correlation level of the Bethe-Guggenheim approximation for the complete thermodynamics and kinetics of adhesion clusters of any size, including the thermodynamic limit. A new kind of dynamical phase transition is uncovered-the mean formation and dissolution times per adhesion bond change discontinuously with respect to the bond-coupling parameter. At the respective critical points, cluster formation and dissolution are the fastest, while the statistically dominant transition path undergoes a qualitative change-the entropic barrier to a completely bound or unbound state is rate-limiting below, and the phase transition between dense and dilute phases above the dynamical critical point. In the context of the Ising model, the dynamical phase transition reflects a first-order discontinuity in the magnetization-reversal time. Our results provide a potential explanation for the mechanical regulation of cell adhesion and suggest that the quasistatic and kinetic responses to changes in the membrane stiffness or applied forces is largest near the statical and dynamical critical points, respectively.
... Originally designed to study fluid streamlines in laminar flow 48 , the Hele-Shaw cell has since the 1950s been used to study hydrodynamic instabilities, most notably viscous fingering (VF), where the interface between a less viscous fluid invading a more viscous defending fluid develops undulations that grow into finger-like invasion patterns [49][50][51] . VF has been studied in different types of complex rheologies 52 , including shear thinning [53][54][55] , shear thickening 56 , yield stress 57 , gels 58 , viscoelastic fluids 8,9,[59][60][61][62][63][64][65] , and saturated granular media [66][67][68] . ...
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Recent theoretical and experimental work suggests a frictionless-frictional transition with increasing inter-particle pressure explains the extreme solid-like response of discontinuous shear thickening suspensions. However, analysis of macroscopic discontinuous shear thickening flow in geometries other than the standard rheometry tools remain scarce. Here we use a Hele-Shaw cell geometry to visualise gas-driven invasion patterns in discontinuous shear thickening cornstarch suspensions. We plot quantitative results from pattern analysis in a volume fraction-pressure phase diagram and explain them in context of rheological measurements. We observe three distinct pattern morphologies: viscous fingering, dendritic fracturing, and system-wide fracturing, which correspond to the same packing fraction ranges as weak shear thickening, discontinuous shear thickening, and shear-jammed regimes. The microscopic mechanisms underlying the discontinuous shear thickening transition in dense granular systems are still under debate. Here, the authors explore this transition by characterizing the shape of invasion patterns in Hele-Shaw cell experiments with confined cornstarch suspensions.
... The sudden relaxation mechanism characterized by heterogeneous dynamics likely results from local depinning of contractile actin network from anchoring surfaces as well as local nucleation and propagation of ruptures within the material, events that we also previously identified in experiments employing fluorescence microscopy 33 and that were also seen in computational studies of contractile gels. 52,88,89 These rupture events likely gradually weaken the network, eventually triggering macroscopic contraction. Prior studies have investigated dynamic precursors to catastrophic failure events across a broad range of systems, including earthquakes, 90 avalanches, 91 and gels that fracture under the influence of an external mechanical load. ...
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Cells and tissues have the remarkable ability to actively generate the forces required to change their shape. This active mechanical behavior is largely mediated by the actin cytoskeleton, a crosslinked network of actin filaments that is contracted by myosin motors. Experiments and active gel theories have established that the length scale over which gel contraction occurs is governed by a balance between molecular motor activity and crosslink density. By contrast, the dynamics that govern the contractile activity of the cytoskeleton remain poorly understood. Here we investigate the microscopic dynamics of reconstituted actin–myosin networks using simultaneous real-space video microscopy and Fourier-space dynamic light scattering. Light scattering reveals different regimes of microscopic dynamics as a function of sample age. We uncover two dynamical precursors that precede macroscopic gel contraction. One is characterized by a progressive acceleration of stress-induced rearrangements, while the other consists of sudden, heterogeneous rearrangements. Intriguingly, our findings suggest a qualitative analogy between self-driven rupture and collapse of active gels and the delayed rupture of passive gels observed in earlier studies of colloidal gels under external loads.
A new form of M-integral associated with time dependence parameters, is presented herein for viscoelastic materials. Based on the equivalent domain integral method, this time-dependent M-integral is numerically implemented as an effective and accepted fracture mechanical parameter for damage induced by crack growth in viscoelastic materials. Based on the linear viscoelastic model defined through Prony series, the conservation of the time-dependent M-integral for viscoelasticity is verified by applying user defined Python scripts. The results show that the newly proposed time-dependent M-integral can be successfully calculated numerically. Furthermore, numerical examples are given to demonstrate the validity and relevance of the time-dependent M-integral in viscoelastic material. In particular, the variations of the time-dependent M-integral for different loading rates are considered, which show that the crack growth behavior over a period of time in viscoelastic material can be evaluated based on the value of the time-dependent M-integral. In addition, the time-dependent M-integral is calculated to assess the damage degree induced by two collinear cracks.
Biological function requires cell-cell adhesions to tune their cohesiveness; for instance, during the opening of new fluid-filled cavities under hydraulic pressure. To understand the physical mechanisms supporting this adaptability, we develop a stochastic model for the hydraulic fracture of adhesive interfaces bridged by molecular bonds. We find that surface tension strongly enhances the stability of these interfaces by controlling flaw sensitivity, lifetime, and optimal architecture in terms of bond clustering. We also show that bond mobility embrittles adhesions and changes the mechanism of decohesion. Our study provides a mechanistic background to understand the biological regulation of cell-cell cohesion and fracture.
Reversible crosslinking is a design paradigm for polymeric materials, wherein they are microscopically reinforced with chemical species that form transient crosslinks between the polymer chains. Besides the potential for self-healing, recent experimental work suggests that freely diffusing reversible crosslinks in polymer networks, such as gels, can enhance the toughness of the material without substantial change in elasticity. This presents the opportunity for making highly elastic materials that can be strained to a large extent before rupturing. Here, we employ Gaussian chain theory, molecular simulation, and polymer self-consistent field theory for networks to construct an equilibrium picture for how reversible crosslinks can toughen a polymer network without affecting its elasticity. Maximisation of polymer entropy drives the reversible crosslinks to bind preferentially near the permanent crosslinks in the network, leading to local molecular reinforcement without significant alteration of the network topology. In equilibrium conditions, permanent crosslinks share effectively the load with neighbouring reversible crosslinks, forming multi-functional crosslink points. The network is thereby globally toughened, while the linear elasticity is left largely unaltered. Practical guidelines are proposed to optimise this design in experiment, along with a discussion of key kinetic and timescale considerations.
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We propose a theoretical framework for dealing with a transient polymer network undergoing small deformations, based on the rate of breaking and re-forming of network crosslinks and the evolving elastic reference state. In this framework, the characteristics of the deformed transient network at microscopic and macroscopic scales are naturally unified. Microscopically, the breakage rate of the crosslinks is affected by the local force acting on the chain. Macroscopically, we use the classical continuum model for rubber elasticity to describe the structure of the deformation energy, whose reference state is defined dynamically according to when crosslinks are broken and formed. With this, the constitutive relation can be obtained. We study three applications of the theory in uniaxial stretching geometry: for the stress relaxation after an instantaneous step strain is imposed, for the stress overshoot and subsequent decay in the plastic regime when a strain ramp is applied, and for the cycle of stretching and release. We compare the model predictions with experimental data on stress relaxation and stress overshoot in physically bonded thermoplastic elastomers and in vitrimer networks.
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The ability to precisely tune the mechanical properties of polymeric composites is vital for harnessing these materials in a range of diverse applications. Polymer-grafted nanoparticles (PGNs) that are cross-linked into a network offer distinct opportunities for tailoring the strength and toughness of the material. Within these materials, the free ends of the grafted chains form bonds with the neighboring chains, and tailoring the nature of these bonds could provide a route to tailoring the macroscopic behavior of the composite. Using computational modeling, we simulate the behavior of three-dimensional PGN networks that encompass both high-strength “permanent” bonds and weaker, more reactive labile bonds. The labile connections are formed from slip bonds and biomimetic “catch” bonds. Unlike conventional slip bonds, the lifetime of the catch bonds can increase with an applied force, and hence, these bonds become stronger under deformation. With our 3D model, we examined the mechanical response of the composites to a tensile deformation, focusing on samples that encompass different numbers of permanent bonds, different bond energies between the labile bonds, and varying numbers of catch bonds. We found that at the higher energy of the labile bonds (Ul = 39kBT), the mechanical properties of the material could be tailored by varying both the number of permanent bonds and catch bonds. Notably, as much as a 2-fold increase in toughness could be achieved by increasing the number of permanent bonds or catch bonds in the sample (while the keeping other parameters fixed). In contrast, at the lower energy of the labile bonds considered here (Ul = 33kBT), the permanent bonds played the dominant role in regulating the mechanical behavior of the PGN network. The findings from the simulations provide valuable guidelines for optimizing the macroscopic behavior of the PGN networks and highlight the utility of introducing catch bonds to tune the mechanical properties of the system.
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Brittle materials propagate opening cracks under tension. When stress increases beyond a critical magnitude, then quasistatic crack propagation becomes unstable. In the presence of several precracks, a brittle material always propagates only the weakest crack, leading to catastrophic failure. Here, we show that all these features of brittle fracture are fundamentally modified when the material susceptible to cracking is bonded to a hydrogel, a common situation in biological tissues. In the presence of the hydrogel, the brittle material can fracture in compression and can hydraulically resist cracking in tension. Furthermore, the poroelastic coupling regularizes the crack dynamics and enhances material toughness by promoting multiple cracking.
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The creep deformation and eventual breaking of polymeric samples under a constant tensile load F is investigated by molecular dynamics based on a particle representation of the fiber bundle model. The results of the virtual testing of fibrous samples consisting of 40000 particles arranged on Nc=400 chains reproduce characteristic stages seen in the experimental investigations of creep in polymeric materials. A logarithmic plot of the bundle lifetime τ versus load F displays a marked curvature, ruling out a simple power-law dependence of τ on F. A power law τ∼F−4, however, is recovered at high load. We discuss the role of reversible bond breaking and formation on the eventual fate of the sample and simulate a different type of creep testing, imposing a constant stress rate on the sample up to its breaking point. Our simulations, relying on a coarse-grained representation of the polymer structure, introduce new features into the standard fiber bundle model, such as real-time dynamics, inertia, and entropy, and open the way to more detailed models, aiming at material science aspects of polymeric fibers, investigated within a sound statistical mechanics framework.
Transient molecular networks, a class of adaptive soft materials with remarkable application potential, display complex, and intriguing dynamic behavior. By performing dynamic light scattering on a wide angular range, we study the relaxation dynamics of a reversible network formed by DNA tetravalent nanoparticles, finding a slow relaxation mode that is wave vector independent at large q and crosses over to a standard q-2 viscoelastic relaxation at low q. Exploiting the controlled properties of our DNA network, we attribute this mode to fluctuations in local elasticity induced by connectivity rearrangement. We propose a simple beads and springs model that captures the basic features of this q0 behavior.
We investigate the nucleation and propagation of cracks in self-assembled viscoelastic fluids, which are made of surfactant micelles reversibly linked by telechelic polymers. The morphology of the micelles can be continuously tuned, from spherical to rod-like to wormlike, thus producing transient double networks when the micelles are sufficiently long and entangled, and transient single networks otherwise. For a single network, we show that cracks nucleate when the sample deformation rate involved is comparable to the relaxation time scale of the network. For a double network, by contrast, significant rearrangements of the micelles occur as a crack nucleates and propagates. We show that birefringence develops at the crack tip over a finite length, $\xi$, which corresponds to the length scale over which micelles alignment occur. We find that $\xi$ is larger for slower cracks, suggesting an increase of ductility.
The linear response of homogeneous and isotropic two-dimensional networks subjected to an applied strain at zero temperature was studied. A dimensionless scalar quantity, being a combination of the material length scales that specified to which regime a given network belonged was identified. It was found that the elastic modulii vanished for network densities at a rigidity percolation threshold. By a direct geometric measure, it was also shown that the degree of affinity under the strain correlated with the distinct elastic regimes.
Many soft materials are classified as viscoelastic. They behave mechanically neither quite fluid-like nor quite solid-like - rather a bit of both. Biomaterials are often said to fall into this class. Here, we argue that this misses a crucial aspect, and that biomechanics is essentially damage mechanics, at heart. When deforming an animal cell or tissue, one can hardly avoid inducing the unfolding of protein domains, the unbinding of cytoskeletal crosslinkers, the breaking of weak sacrificial bonds, and the disruption of transient adhesions. We classify these activated structural changes as inelastic. They are often to a large degree reversible and are therefore not plastic in the proper sense, but they dissipate substantial amounts of elastic energy by structural damping. We review recent experiments involving biological materials on all scales, from single biopolymers over cells to model tissues, to illustrate the unifying power of this paradigm. A deliberately minimalistic yet phenomenologically very rich mathematical modeling framework for inelastic biomechanics is proposed. It transcends the conventional viscoelastic paradigm and suggests itself as a promising candidate for a unified description and interpretation of a wide range of experimental data. This article is part of a Special Issue entitled: Mechanobiology. Copyright © 2015 Elsevier B.V. All rights reserved.
We examine the bond-breaking dynamics of transiently cross-linked semiflexible networks using a single filament model in which that filament is peeled from an array of cross-linkers. We examine the effect of quenched disorder in the placement of the linkers along the filament and the effect of stochastic bond-breaking (assuming Bell model unbinding kinetics) on the dynamics of filament cross-linker dissociation and the statistics of ripping events. We find that bond forces decay exponentially away from the point of loading and that bond breaking proceeds sequentially down the linker array from the point of loading in a series of stochastic ripping events. We compare these theoretical predictions to the observed trajectories of large beads in a cross-linked microtubule network and identify the observed jumps of the bead with the linker rupture events predicted by the single filament model.