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Delayed fracture in gels
Xiao Wanga and Wei Hong*ab
Received 9th March 2012, Accepted 21st May 2012
DOI: 10.1039/c2sm25553g
Subject to a subcritical load, a swollen polymeric gel may hesitate for a prolonged period of time
without showing any macroscopic symptom, and then break suddenly. Such a phenomenon is usually
referred to as the delayed fracture in gels. In this paper, we present a possible mechanism for the delayed
fracture in gels from the continuum and fracture mechanics point of view. Using a continuum
visco-poroelastic model for polymeric gels, we calculate the evolution of the inhomogeneous stress field
around a pre-existing crack, in consequence of the coupled viscoelastic creep and solvent migration. We
invoke the instantaneous energy release rate as the local driving force for a crack, and find it to be an
increasing function of time. With the dissipation from viscoelastic creep and solvent migration
excluded, the criterion for crack advancing is that the instantaneous energy release rate equals the
intrinsic fracture energy of the polymer. The fracture delay could thus be attributed to the time needed
for viscoelastic creep and solvent migration to bring the instantaneous energy release rate to the level of
the intrinsic fracture energy. For most swollen gels, solvent migration is the limiting process, and
therefore the delay time depends on the size of a pre-existing crack in a similar way as common
diffusion-limited processes. Finally, by assuming a specific size distribution of microcracks, we provide
a simple statistical analysis towards the lifetime prediction of a swollen gel.
1 Introduction
Because of their biodegradability and mechanical and chemical
compatibility with natural tissues, polymeric gels are believed to
be the future materials for tissue scaffolding and replacement.1
However, common synthetic gels suffer from the brittleness
which accompanies their natural softness.2–4 The mechanical
strength of regular polymeric gels is insufficient for in vivo
loading and manipulation. The increasing interest in developing
strong and tough gels5 has heightened the need for understanding
the fundamentals of their failure processes.6–11
The simplest mechanical test for a gel is to load it with a dead
weight. An interest phenomenon known as delayed fracture has
been observed when a dead load just below the ultimate strength
is applied.12–14 Instead of instantaneous or continuous crack
growth, the gel would hesitate for some time without showing
any sign of failure macroscopically, and then undergo a dynamic
fracture. The corresponding delay time tf has several distinct
characteristics as revealed by experiments: (1) a significant
statistical variance exists in the results obtained from samples of
identical materials; (2) the average delay time t^f is highly sensitive
to the stress level, i.e. a small decrease of stress s may increase t^f
by orders of magnitude, from seconds to hours; and (3) the shape
a
Department of Materials Science and Engineering, Iowa State University,
Ames, IA 50011, USA
b
Department of Aerospace Engineering, Iowa State University, Ames, IA
50011, USA. E-mail: whong@iastate.edu; Fax: +1-515-294-3262; Tel:
+1-515-294-8850
This journal is ª The Royal Society of Chemistry 2012
of the s t^f relation depends on material type and swelling
condition. The technical importance of understanding the
mechanism of delayed fracture could be exemplified with applications such as tissue scaffolding. A gel scaffold must function
and keep its structural integrity under specific load; no macroscopic crack initiation should occur during the entire service
period for cell growth and differentiation.1,15
Current understanding of delayed fracture mechanism
involves the mechano-chemical damage of stress carrying chains
in the gel.12–14,16–18 The applied stress lowers the activation energy
barrier of chain breakage, increases the chain dissociation rate,
and consequently facilitates nucleation and subcritical growth of
microcracks. Since similar dynamic chain breakage and recombination processes also lead to the viscoelastic deformation of
a physical gel, the observation that gels with similar viscosity can
have drastically different delay time12 has put such explanations
into question.
On the other hand, all gels are known to be poroelastic –
solvent migrates in a gel if the deformation is inhomogeneous.19,20 In this paper, we will investigate the contribution from
the poroelasticity of a gel to the propagation of microcracks. For
simplicity in description, we will focus specifically on polymeric
gels, while a similar analysis may also be applicable to other types
of gels. The concurrent deformation of the polymer network and
migration of interstitial solvent driven by the inhomogeneous
stress field around a crack will be captured using a recently
developed continuum model,21 as briefed in Section 2. The
driving force to advance a microcrack is the instantaneous energy
Soft Matter, 2012, 8, 8171–8178 | 8171
release rate at the crack tip, besides additional energy dissipation
through solvent migration and viscoelasticity. Due to the viscoporoelastic deformation around the crack, the instantaneous
energy release rate is a function of time as evaluated in Section 3.
The classical criterion that a crack advances when the energy
release rate exceeds the intrinsic fracture energy G0 of the polymer network will be used to determine the delay time: G(tf) ¼ G0.
In the case of a relatively large microcrack, the delay phenomenon is limited by solvent migration, and the time evolution of
G follows the typical scaling law of a diffusion limited kinetic
process. The size and stress dependency of fracture delay time is
reduced to a single master curve in Section 4.1. Finally, the
statistical nature of delay fracture events is accounted for by
considering the defect size distribution, and a statistical lifetime
theory of gels is presented in Section 4.2.
operator Vx, as well as the solvent flux j, is defined with respect to
the deformed geometry in the current state. They are related to
their nominal counterparts by the simple geometric relations:
V ¼ FT$Vx and F$J ¼ jdet F.
Finally, we prescribe the stress–strain relation using a freeenergy function. Adopting the Flory–Rehner model,22 we write
the total Helmholtz free energy per unit reference volume, U, into
the sum of the elastic energy of stretching Un and the energy of
mixing the solvent with polymer chains Um, U ¼ Un + Um.
Without specific material information, we take the neo-Hookean
form for Un, and the Flory–Huggins form for Um:19,22
h
i
1
Un ðFÞ ¼ NkT F : F 3 2lnðdet FÞ ;
(5)
2
Um ðCÞ ¼ 2 Material model for a visco-poroelastic gel
In order to make a comparison between the contributions from
viscoelastic deformation and poroelastic solvent migration,
a continuum model for the coupled visco-poroelastic behavior of
polymeric gels will be used. The theory and the governing
equations are briefly summarized as follows, while the detailed
formulation and derivation could be found in ref. 21.
With the dry state as the reference, the deformation of the
polymer network is traced by the current position x(X,t) of the
material particle located at X in the reference state. We assume
the molecular incompressibility and insist that the deformation
gradient F ¼ Vx and the nominal solvent concentration C are
related as19
UC ¼ det F 1,
(1)
where U is volume occupied by each solvent molecule. The
conservation of solvent molecules further relates the time rate of
change in C to the nominal diffusion flux J as
C_ + V$J ¼ 0.
(2)
Both solvent migration and viscoelastic deformation are irreversible processes, through which the system free energy is
dissipated. Following the usual approach of non-equilibrium
thermodynamics, we assume linear kinetics for both dissipative
processes. The true flux of solvent migration j is related to its
driving force, the gradient of chemical potential m:
j ¼ mVxm,
(3)
with m being the mobility of solvent molecules. Similarly, the
viscous shear stress s, as a driving force for viscous flow, is
related to the velocity gradient Vxv as
" #
1
1
T
Vx v þ ðVx vÞ
s¼h
ðVx $vÞ1 ;
(4)
2
3
with h being the viscosity, the superscript T indicating the tensor
transpose, and 1 representing the identity tensor. Eqn (4)
assumes that the viscous behavior of the gel is similar to that of
an incompressible Newtonian fluid. Here, the differential
8172 | Soft Matter, 2012, 8, 8171–8178
kT
1
c
UC ln 1 þ
þ
:
U
UC
1 þ UC
(6)
Here kT is the temperature in the unit of energy, and c is the
Flory–Huggins parameter for the enthalpy of mixing. With N
being the number of polymer chains per unit volume, the product
NkT is the modulus of dry polymer in the reference state. With
the contributions from the polymer network, the viscous deformation, and the solvent included, the nominal stress takes the
form
vUn
det F dUm
s¼
þ s$F T det F þ
m F T :
(7)
U
vF
dC
The first term on the right-hand side of eqn (7) is the elastic stress
from the network, the second term is the viscous stress, and the
last term is the osmotic stress from the solvent.
Even in the case of fast crack propagation, the velocity and
acceleration of a material particle in gel are small. We will thus
neglect any inertial effect, and assume the system to be in
mechanical equilibrium:
V$ s¼ 0.
(8)
Substituting eqn (1) and (3) into (2), and (7) into (8), we obtain
the governing equations of the model, a set of PDEs for the fields
x(X,t) and m(X,t). In the case when the energy is mainly dissipated through solvent migration, the viscous stress can be
omitted from eqn (7), and the current model reduces to that of
a poroelastic gel.19 The theory described above and in ref. 21 is
mostly based on the nominal quantities measured with respect to
the geometry in the reference state. However, for the fracture of
a polymeric gel, it is often more convenient to consider quantities
with respect to the initial state at t ¼ 0. In equilibrium with the
solvent of chemical potential m0, the gel swells isotropically and
freely with linear swelling ratio l, given by the nonlinear
equation:23
1
kT
1
c m
NkT l þ
ln 1 3 l3 þ 1 þ 3 0 l3 ¼ 0: (9)
U
l
Ul
l
l
In the following discussion, unless otherwise stated, all dimensions refer to those in the initial state, and all physical parameters
and fields are measured with respect to this state. For example,
the modulus of the gel in the initial state is related to NkT as E ¼
NkT/l.
This journal is ª The Royal Society of Chemistry 2012
3 Numerical simulation of delayed fracture
The model of visco-poroelastic gel is implemented into a finiteelement method21 to enable numerical calculations. To exemplify
poroelastic effects in gel fracture, let us consider the plane-strain
problem of an edge crack of length L in a large sample of size
W [ L, as sketched in Fig. 1a. A uniform tensile stress s in the
Y-direction is applied remotely to the crack, symmetrically on
upper and lower edges of the sample. The edge crack is in pure
mode I. A symmetric boundary condition is applied on the right
edge of the sample. Both the left edge of the sample and the crack
faces are assumed to be traction free. Neglecting solvent evaporation throughout the duration of the processes involved, we
assume all boundaries to be impermeable to solvent.
Following Lake and Thomas,24 we divide the fracture energy
of a gel into two contributions: the intrinsic fracture energy G0
needed to cleave polymer chains and form new surfaces, and the
energy dissipation from the irreversible deformation in the
region near the crack tip. The intrinsic fracture energy could be
regarded as a material parameter, while the energy dissipation
per unit crack area is highly dependent on the detailed deformation process. Due to the rate dependency of visco-poroelastic
deformation, which could be measured using tradition fracture
mechanics tests,2 the traditional definition of energy release rate
does not give a unique value. We may still define the instantaneous energy release rate G as the reduction in total potential
energy upon advancing of unit area of crack followed by an
elastic relaxation. Such a definition excludes the viscous relaxation and the solvent transportation in response to the crack
propagation, as if the gel is incompressible and inviscid. Therefore, the instantaneous energy release rate is the direct driving
force of polymer chain cleavage. The definition of the instantaneous energy release rate G gives rise to the criterion for crack
extension G $ G0. Such a fracture criterion implies the
assumption that the crack opening due to the cleavage of
a polymer chain is much faster than viscous relaxation or solvent
migration.
During irreversible deformation, the states of all material
particles surround a crack change, so that the instantaneous
energy release rate of a crack would vary with time. As will be
shown by our numerical results, G(t) is a monotonically
increasing function of time for a static crack under a constant
remote stress. We propose the following mechanism for delayed
fracture. Immediately after a subcritical load is applied, the
instantaneous energy release rate is below the intrinsic fracture
energy, G(0) < G0. After a certain delay time, tf, determined from
the criterion G(tf) ¼ G0, the crack starts to propagate. Once the
propagation starts, G increases with the crack length and the
crack becomes unstable. The swollen gel usually fails like a brittle
solid. Therefore, rendering the complete fracture process of an
advancing crack may be unnecessary, and we will only focus on
the stage prior to crack propagation, 0 < t < tf. Dimensional
analysis dictates that the instantaneous energy release rate
depends on various parameters as:
G
tE tD s EU
m
¼g
; 2; ;
; c; 0 ;
kT
EL
h L E kT
Fig. 1 (a) Sketch of a mode I edge crack in a gel, remotely loaded by
stress s. Crack size L is much smaller than sample size W, and at the same
time much larger than the cohesive zone size P. (b) The spring constant of
the cohesive zone k plotted as a function of the distance from the crack
tip, X.
This journal is ª The Royal Society of Chemistry 2012
(10)
where g is a dimensionless function.
The system contains two time scales: the characteristic time for
viscoelastic relaxation, tv ¼ h/E, and that of poroelastic solvent
migration, tp ¼ L2/D. Here D is the diffusion coefficient of
solvent in the gel, which relates to the mobility through the
Einstein relation D ¼ mkTdet F/(det F 1), and is almost
independent of the swelling ratio for a highly swollen gel.19 Using
the self-diffusion coefficient of water at room temperature, D ¼ 1
109 m2 s1,20,25 we estimate the typical time for solvent
migration over the heterogeneity created by defects of size 1 mm
to 1 mm to be tp ¼ 103 to 103 s. On the other hand, the characteristic time for viscoelastic deformation strongly depends on
composition and environment parameters. In the current paper,
we will focus on relatively large cracks with tp [ tv to elucidate
the role of poroelasticity. Although tv and tp may overlap in
problems concerning very short cracks or highly viscous gels,
a complete discussion of a coupled problem is beyond the scope
of this paper.
For an elastic material, the energy release rate could be
calculated by evaluating the J-integral along an arbitrary
contour encircling the crack tip. In a visco-poroelastic gel, the
J-integral is non-conservative. To evaluate the instantaneous
energy release rate, an infinitesimal contour is needed to exclude
the visco-poroelastic dissipation. However, such an approach is
impractical due to the singularity at the crack tip and the
accompanying numerical error. Here this technical difficulty is
overcome by invoking a cohesive-zone model. As shown in
Fig. 1a, an array of nonlinear springs representing partially
Soft Matter, 2012, 8, 8171–8178 | 8173
damaged chains is placed on the prospective path ahead of
a crack, forming a cohesive zone of length P. The cohesive zone
physically corresponds to the region where the polymer chains
undergo a mechano-chemical process of recombination. It is
assumed that such a process is activated by extremely high
stresses, and is localized in a domain much smaller than that of
typical visco-poroelastic processes. For regular cohesive-zone
models, it has been shown that the actual form of the traction–
separation law is relatively unimportant.26,27 Here we take the
spring constant k to be a smoothed inverse Heaviside function of
the horizontal coordinate X, which varies from 0 at the crack tip
to infinity at the outer rim of the cohesive zone, X ¼ P, where the
material is intact. The cohesive-zone model reduces the singularity at the crack tip and enables the evaluation of the J-integral
over a contour just enclosing the nonlinear springs:
ðP
J ¼ 2 ty
0
vvþ
dX ;
vX
(11)
where ty is the vertical traction, and v+ the vertical displacement
of the upper surface of the cohesive zone. This method of
calculating the instantaneous energy release rate is validated
numerically on a hyperelastic material, of which the instantaneous energy release rate is indifferent from the regular energy
release rate. The J-integral is evaluated through three
approaches: via the contour over the cohesive zone, via regular
contours around the crack tip in the absence of cohesive zone,
and via regular contours around the crack tip when a cohesive
zone is present. The results plotted in Fig. 2 clearly show that the
introduction of the special cohesive-zone model does not affect
the energy release rate, and the integral over the cohesive zone is
sufficient.
To further validate this approach and to ensure accuracy, we
run numerical tests on various combinations of crack sizes and
cohesive zone sizes relative to the gel specimen. It is found that
the relative sizes W/L ¼ 10 and P/L ¼ 0.004 would already
provide repeatable results with acceptable accuracy.
Fig. 2 Energy release rate of a static crack in a neo-Hookean solid. The
value calculated with the current method, i.e. using the J-integral over
a cohesive zone, is compared to the J-integrals over regular contours,
with and without the presence of a cohesive zone. Four regular contours,
different contour sizes LC/L are taken in each case to ensure numerical
accuracy.
8174 | Soft Matter, 2012, 8, 8171–8178
4
Results and discussion
4.1 Poroelastic fracture of an isolated crack
In the numerical calculations, we use the stiffness of the gel
NkT ¼ 40 kPa, close to that estimated from experiment.12 In the
first example, we take the representative value of the Flory–
Huggins parameter, c ¼ 0.1. The dependence of the fracture
delay on polymer–solvent interaction will be discussed later.
Initially the gel is in equilibrium with pure solvent and swells with
stretch l ¼ 3.4. The elastic modulus of the gel in the initial state
can be calculated as E ¼ 12 kPa. Upon application of the remote
stress, the homogeneous initial state is perturbed by the highly
non-uniform deformation near the crack. Shortly after the load,
at t tp, when solvent molecules have hardly migrated, the gel
behaves just as an elastic rubber. The hydrostatic tension near
the crack tip results in a local decrease of chemical potential
(Fig. 3). The inhomogeneous field of chemical potential drives
a converging flux of solvent to the crack tip. The solvent transportation is localized at a length scale comparable to the crack
size L, while the bulk material away from crack tip is still unaffected (Fig. 4). The redistribution of solvent causes the local
swelling near the crack tip. The swelling rate decreases with
time and vanishes when a new equilibrium state is reached at
t [ tp.
A typical curve of the instantaneous energy release rate G(t) is
obtained with the method described in Section 3, and plotted in
Fig. 5. The material parameters are selected such that the two
time scales tv and tp are separated, and the corresponding twostage evolution of G is clearly shown. The monotonous increase
Fig. 3 The inhomogeneous field of chemical potential m/kT at the
vicinity of a crack. Snapshots are taken at dimensionless time tD/L2 of
100, 500, 1000, and 104. The arrows illustrate the direction of solvent
migration in the upper half specimen.
This journal is ª The Royal Society of Chemistry 2012
Fig. 4 The relative volumetric swelling ratios, at material points of
different distances away from the crack tip, plotted as functions of the
dimensionless time, tD/L2. The numbers in parentheses are the dimensionless coordinates (X/L,Y/L) of the material particles.
of the crack, and plot the results in Fig. 6a. Due to the normalization of the time using tp, the size-independent viscoelastic
deformation appears to be different in Fig. 6a. With the viscoelastic stage being unimportant here, the readers should direct
their focuses to the poroelastic stage. Despite the visco-poroelastic coupling, the shape of G(t) in the poroelastic stage is almost
unchanged while plotted in a dimensionless manner. It seems
that the evolution of instantaneous energy release rate is
captured by a master curve, independent of the coupling with
viscoelastic state. The only difference between gels with longer
and shorter cracks is the starting point of the poroelastic stage.
Furthermore, we have studied the effect of the dimensionless
remote load, s ¼ s/E, and plotted the results in Fig. 6a and b. As
shown in Fig. 6b, the instantaneous energy release rate scales
with the applied stress approximately as s2 in both the rubbery
limit and the long-term limit, just as the scaling relation of
a linear elastic material. Following these observations, we may
write the expression of instantaneous energy release rate, eqn
(10), approximately into
GE
t EU
m0
;
c;
¼
;
G
;
s2 L
tp kT
kT
(12)
in the poroelasticity-dominant stage, t [ tv. The right hand side
of eqn (12) is the dimensionless energy-release-rate function, G ¼
GE/s2L. More specifically, when constant values of parameters
EU/kT, c, and m0/kT are chosen, we find the following functional
Fig. 5 Time evolution of instantaneous energy release rate and global
creep strain after a step load of remote stress. Due to the separation in
time scales, the energy-release-rate curve shows a viscoelastic regime and
a poroelastic regime, and monotonously increases in both regimes.
in G(t) in both stages can be understood as follows. Upon the
application of the step load, the viscosity of the gel prevents it from
deforming instantaneously. At t ¼ 0, the gel is rigid and thus G(0)
¼ 0. At 0 < t < tv, the gel undergoes a viscoelastic creep. With the
concentrated creep stretch at the crack tip, the gel stores locally
higher elastic energy which contributes to the instantaneous
energy release rate. The contribution from poroelasticity is similar,
although the corresponding creep is due to long range solvent
migration and thus takes a longer time. It is well known that some
dry amorphous elastomers also experience delayed rupture
beyond critical elongations,28 although the main cause there is
viscoelastic creep. To this point, we can easily differentiate the
poroelastic and viscoelastic effects by examining the global creep
stretch, lY(t), as plotted in Fig. 5. During viscoelastic creep, the
entire specimen deforms. In contrast, the solvent migration in the
poroelastic stage does not induce significant global deformation
on the gel. For ease of description, let us denote the two limits of
the instantaneous energy lease rate as Gr and GN, namely the
rubbery and the long-term limit, respectively.
Although the assumption on the separation of time scales is of
theoretical interest, it may not always be the case for an actual
material. To explore more general cases, we vary the relative size
This journal is ª The Royal Society of Chemistry 2012
Fig. 6 (a) Time evolution of instantaneous energy release rate as
a function of dimensionless time for various crack sizes and applied
stress. FEM results (squares and circles) are compared with the fitting
curves using eqn (13). (b) Instantaneous energy release rates at the
rubbery and long-term limits plotted as functions of the remote load.
FEM results are compared with fitting curves of a quadratic relation.
Soft Matter, 2012, 8, 8171–8178 | 8175
that accurately captures the transition from the
form of G(t)
rubbery limit to the long-term limit:
GðtÞ ¼ Gr þ
GN Gr
;
1 þ ðaL2 =tDÞ
(13)
where Gr and GN are the values of G in the rubbery and longterm limits, respectively. The dimensionless parameter a captures
the onset of the transition, and is determined by geometry and
material properties. By fitting eqn (13) to the numerical data on
Fig. 6a, we obtain that a ¼ 700.
It was observed in experiments12,13 that the fracture delay time
is often much larger than the characteristic time of viscoelastic
deformation, tf [ tv. We therefore hypothesize that the delay is
mostly due to the poroelastic deformation of the gels, and in the
following discussion we will thus utilize eqn (13), which captures
the poroelasticity-dominant stage. We insist the criterion for
crack propagation: G $ G0. If the remote load is relatively high,
Gr > G0, the crack will propagate shortly after loading during the
viscoelastic stage, when G(t) ¼ G0. In the other extreme, when the
applied load is so low that even the long-term energy release rate
is lower than the intrinsic fracture energy, GN < G0, the crack will
never propagate. Only when an intermediate load is applied, such
that Gr < G0 < GN, a delay due to poroelastic deformation may
be observed. The approximate delay time for a pre-existing edge
crack of length L is, from eqn (13),
tf D
G0 Gr
¼a
:
L2
GN G0
(14)
Containing crack size L, the dimensionless time tD/L2 may not
be an objective measure for the fracture delay time. Alternatively, we may normalize the fracture time by G02/DE2, which
contains only material parameters, and write eqn (14) into
tf DE 2
a G 0 G r s2
¼
;
2
2
G0 2
G0 G N s G0
(15)
0 ¼ G0/EL is the dimensionless intrinsic fracture energy.
where G
The normalized delay time as in eqn (15) is plotted as a function
of the remote stress in Fig. 7. Just as that observed in experiments,12–14 the delayed fracture only happens in a very narrow
range of subcritical stresses, outside which fracture either
happens immediately (after a short time comparable to tv) or
never happens. The dependence of the fracture delay time on the
crack size L is also shown in Fig. 7, through the dimensionless
0. With a longer pre-existing crack, the
intrinsic fracture energy G
failure stress takes a lower value, and the delay time is also
longer, since solvent will migrate through a larger region during
poroelastic deformation. Taking an estimate of the intrinsic
fracture energy G0 ¼ 0.1 Nm1 and the modulus G0 ¼ 12 kPa, for
a crack of size L ¼ 100 mm, the delay time increases by two orders
of magnitude from 10 s to 1000 s, when stress increases slightly
from 2 kPa to 2.4 kPa. Moreover, the special scaling law for
poroelasticity-dominant processes dictates that the delay time of
a specimen with a pre-existing macroscopic crack is much longer,
although no experimental result is yet available, perhaps due to
the very narrow stress window for delayed fracture of a macroscopic crack. For example, the delay time for a millimetre-size
crack may be as long as several days.
The reduced functional form of energy release rate in eqn (12)
and (13) allows us to study the effect of material parameters
simply by looking at their contributions to the two limits, Gr and
GN. In general, Gr and GN are constants depending on material
parameters as well as the shape of the specimen. In the case when
the viscoelastic and poroelastic processes have separated time
scales, there is hardly any solvent migration during the viscoelasticity-dominant stage. As a result, the energy release rate at
the rubbery limit, Gr, would only be dependent on the stiffness of
the gel, instead of the Flory–Huggins parameter c or chemical
potential m0. In contrast, the long-term limit, GN, would be
influenced more by the swelling behavior. Here we numerically
test the effects of material parameters c and initial swelling
stretch l and plot the results in Fig. 8. As shown in Fig. 8a,
when we gradually change the polymer to be more hydrophobic
(c ¼ 0–0.2) while keeping the initial swelling ratio constant, GN
shows an increase. If the same type of gel is more swollen, the
value of GN is also higher, as shown in Fig. 8b. The more
hydrophobic polymers and more swollen gels with higher GN
give rise to a more significant delay effect, with a wider stress
window and a longer delay time.
The dependences on both parameters could be easily verified
through experiments, and may be understood as follows. While
making a polymer more hydrophobic or adding more solvent has
the same effect to a gel by increasing its chemical potential (or
osmotic pressure), the regions where the gel is more swollen
(e.g. near a crack tip) will have a relatively smaller change. The
non-uniform increase in chemical potential results in a convergent flux of solvent towards the crack tip, causing the increase of
the instantaneous energy release rate. More numerical results
show that in the case when gels of different polymers are all
brought to equilibrium with the same solvent (same m0/kT), the
difference in GN turns out to be smaller than the two cases shown
in Fig. 8. This is because a more hydrophobic gel would swell
less, so that the two contributions compete against each other
and diminish the effect.
4.2 Statistical lifetime theory of gels
Fig. 7 Dimensionless fracture delay time as a function of the applied
load for various values of intrinsic fracture energy.
8176 | Soft Matter, 2012, 8, 8171–8178
While the discussion in previous sections relies on the knowledge
of the size of a pre-existing crack, such information may not be
available in practice. On the other hand, polymeric gels are
known to be highly heterogeneous and contain a large amount of
defects. The uncertainty in defect size in gels may contribute to
the scattering in the experimental data of delayed fracture. In this
This journal is ª The Royal Society of Chemistry 2012
Fig. 10 Characteristic lifetime t^f as a function of the remote stress, for
gels made of the same polymer at different initial swelling ratios.
Fig. 8 Normalized instantaneous energy release rate as a function of
dimensionless time during poroelastic deformation regime for gels with
(a) the different type of polymer network with fixed initial swelling ratio
and (b) the same type of polymer network with a different initial swelling
ratio.
Fig. 9 Probability of survival of a poroelastic gel as a function of the
dimensionless time, tfDE2/G02, under different values of the remote load s/E.
section, we propose a lifetime theory to predict the statistics of
gel fracture. For simplicity, we regard the pre-existing defects as
non-interacting microcracks of various sizes. The results of
previous discussion direct our focus to the largest crack in a gel,
which is the most vulnerable too. The delay in fracture is limited
by the largest crack in a gel, and the statistics of its lifetime could
This journal is ª The Royal Society of Chemistry 2012
be determined from the probability of finding the largest crack of
a certain size.
With the lack of detailed information on the statistics of
microcracks in a gel, we assume that the size of each crack is
a random variable, and the size distributions of individual cracks
are identical and independent. Let q(L) be the size
Ð distribution of
an arbitrary crack in the gel, and QðLÞ ¼ qðLÞdL be the
cumulative distribution, i.e. the probability of the crack size to be
smaller than L. Denote the total number of cracks in a gel as M.
Since the crack sizes are independent and identically distributed,
the probability of having all cracks in the gel to be smaller than L
is Q(L)M. For a crack of known size L, eqn (15) provides
a functional relation between the crack size and the expected
delay time before propagation, L(tf), for a specific type of gel
under known load. Utilizing this relation, we could further
write the probability for a gel to survive after time t as Qm(t) ¼
Q(L(t))M. With the knowledge of the size distribution of defects
in a gel, q(L), its lifetime could be predicted.
Here as an example, we assume the exponential distribution of
crack sizes:
1
L
qðLÞ ¼
;
(16)
exp La
La
where La is the mean crack length. Utilizing this specific size
distribution and the same material parameters as in Section 4.1,
we calculate the survival probability function, Qm(t), and plot the
results under various levels of loads in Fig. 9. The survival
probability decreases with the normalized applied stress s. Under
a large stress, almost all samples are expected to fail after a short
time, while almost no sample will fail even after very long time if
the load is below some critical value. Only when the level of the
remote stress falls within a narrow window, the survival probability Qm(t) starts from a higher value and decreases to a lower
value after a delay. If we take the threshold probability to be e1,
a characteristic lifetime t^f could be obtained from Qm(t^f) ¼ e1.
The resulting dependence of the dimensionless lifetime t^fDE2/G02
on the applied load is plotted in Fig. 10, and agrees qualitatively
with the experimentally measured lifetime of a gel.12 The
dependence of the lifetime on the initial swelling ratio is also
shown in Fig. 10, for gel samples made from the same type of
polymer (with the same crosslink density and hydrophobicity). A
highly swollen gel is expected to have a more significant delay
Soft Matter, 2012, 8, 8171–8178 | 8177
effect in fracture, so that a larger window in terms of the remote
stress exists. At the same time, a more swollen gel would have
a shorter lifetime, under the same level of remote stress.
It should be noted that if the mechano-chemical reaction on
polymer chains which causes subcritical crack growth is the
dominating mechanism, the fracture delay should not be strongly
affected by the swelling ratio, as the solvent molecules usually do
not participate in these reactions. The different dependences of
delayed fracture on the initial swelling of gels can easily be tested
through future experiments, and could be used to identify the
possible dominating mechanism and verify the theory.
homogenous gel, which contains a large amount of non-interacting microcracks with independently and identically distributed sizes, is used as a model system. Using a Weibull type
distribution for crack size as an example, our result qualitatively
recovers the experimental findings on the statistical feature and
stress sensitivity of the average delay time on similar gels. The
model also provides insights towards the dependence of the
lifetime of a gel on its material properties as well as the initial
swelling ratio. These predictions may be compared with future
experiments for verification of the current theory.
5 Conclusions
References
From a fracture-mechanics point of view, this paper suggests an
alternative mechanism for the delayed fracture phenomena of
polymeric gels. The coupled viscoelastic deformation and solvent
transportation processes in polymeric gels are modeled using
continuum visco-poroelasticity. The local driving force for crack
propagation is characterized by an instantaneous energy release
rate, to separate the intrinsic fracture energy from the dissipation
through viscous deformation and mass transportation. The
instantaneous energy release rate is then calculated numerically
using a cohesive-zone model. While both the viscoelastic creep
and the poroelastic deformation could result in an increase in the
instantaneous energy release rate of a crack, the delay in fracture
is usually limited by the transportation process, especially in
a system with separate time scales of viscoelasticity and poroelasticity (e.g. a system with relatively large cracks and low
viscosity). For the poroelasticity-dominant delay in fracture, the
delay time is expected to be size dependent – a larger crack will
have a longer delay before fracture. In this paper, we have limited
our discussion to the cases of impermeable crack faces, and
excluded liquid-filled cracks. If a crack tip is allowed to exchange
solvent directly through the crack faces, the diffusion length and
thus the delay time will be greatly reduced. This is consistent with
recent observations that crack propagation in gels is facilitated
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It should be noted that a large body of literature is available on
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deformation associated with solvent migration may redistribute
the stress field and increase the local driving force for crack
propagation.
Based on the scaling relation developed for the delayed fracture of a single crack, we further develop a statistical theory to
study the fracture delay time, i.e. the lifetime of a gel without
a macroscopic crack. A simple case of a macroscopically
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