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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: g.koenderink@amolf.nl

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

2

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:

K=kon

kon +koﬀ,0

(1)

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

f1/e

)(2)

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

fi=αi·σ(3)

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:

αi=(N·li

ΣiliLocal

N

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

Open

Closed

kon

koff,0

N

σ

a)

b)

c)

li

lablate

i

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>∼N−1(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

2d).

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

3

0 20 40 60

t

0.0

0.5

1.0

§ni=N

trupt

a)

0 1 2

¾

10-1

101

103

<trupt >

Global

Local

b)

025 50

N

0

50

100

<trupt >

c)

10 11 0 2

10 1

10 4

¿rupt-200 ¿rupt

Simulation step

100

0

Bond position

d)

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>∼N−1

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):

2·koﬀ(1

/2·σ·

∗

L)≈kon ·

∗

L(5)

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

K−1))

−σ(6)

Lunstable(σ, K )≈2·W−1(−σ·(1

K−1))

−σ(7)

where Wis the Lambert W function with W0the main

branch and W−1the 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+e−lcrit−lablate 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

[35].

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,

4

101102103

lablate

0.0

0.5

1.0

Árupt

0.03

0.30

¾

a)

0 100 200

lablate

0.0

0.5

1.0

Árupt

0.4

0.9

K

b)

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

koff;0

kon

koff

a)

0.5 1.0 1.5

lablate=Lunstable

0.0

0.5

1.0

Árupt

b)

0.0 0.4 0.8

¾

100

102

104

Lcrit

Theory

Ablation data

Optimal N

c)

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.

ACKNOWLEDGEMENTS

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:

αi=Ui

PiUi

N(8)

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:

Ui=Z{s}T

i{de}i=1

2[sxx syy sxy ]i

exx

eyy

exy

i

(9)

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.

7

h

σFEA

Transient bonds

Solid body

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

0

1

2

3

4

5

®

Global

Local

FEA

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

N

0

60

120

<trupt >

b)

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.

8

0 50 100 150 200

trupt

0.00

0.02

0.04

Frequency

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.