On the Measurement of Fracture Toughness ofSoft Biogel

H.J. Kwon, Allan D. Rogalsky, Dong-Woo KimDepartment of Mechanical and Mechatronics Engineering, University of Waterloo,200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1

In this study, the rate dependent energy dissipationprocess and the fracture toughness of physical gelswere investigated using agarose as a sample material.Both the J-integral and Essential work of Fracture(EWF) methods were examined. To assess the quasi-static fracture toughness of gels, linear regression wasperformed on critical J (Jc) values at different loadingrates resulting in a quasi-static Jc value of 6.5 J/m2.This is close to the quasi-static EWF value of 5.3 J/m2

obtained by performing EWF tests at a quasi-staticloading rate (crosshead speed of less than 2 mm/min).Nearly constant crack propagation rates at low loadingrates, regardless of crack length, suggest viscoplasticchain pull-out is the fracture mechanism. At high load-ing rates failure was highly brittle, which is attributedto sufficient elastic energy accumulation to precipitatefailure by chain scission. We conclude that in physicalgels quasi-static fracture toughness can be evaluatedby both the J-integral and EWF methods provided theeffects of loading rate are investigated and accountedfor. POLYM. ENG. SCI., 51:1078–1086, 2011. ª 2011 Society ofPlastics Engineers

INTRODUCTION

Hydrogels can be classified by the types of crosslinks

joining chains together [1]. In chemical gels, the cross-

links between polymer chains are made of covalent

bonds, while physical gels are linked by secondary bonds

that are much weaker and more prone to break down

under applied mechanical stress. In most of them the net-

work is formed by biopolymers [2], e.g., proteins (gelatin,

elastin) or polysaccharides (agarose, alginates). Biopoly-

mer-based gels have been attracting substantial interest

motivated by their wide use in biomedical applications

such as drug delivery and tissue engineering [3, 4]. All

these implementations call for the accurate assessment

and control of their mechanical properties such as elastic

modulus and fracture behavior.

Elastic properties of gels have been extensively stud-

ied, both in the small- [2, 5] and large-deformation

regimes [6, 7]; while fracture studies have been mainly

concerned with crack nucleation [8] and ultimate strength

measurements [6, 7]. There are relatively few studies on

the fracture toughness (the material resistance to the prop-

agation of cracks and associated rate dependent energy

dissipation), partly due to the difficulty of applying con-

ventional fracture mechanics schemes to soft hydrogels.

Despite this, there is a fair degree of consensus regarding

the differences in fracture behavior between physical and

chemical gels.

In chemical gels, stiffness and toughness have been

found to be negatively correlated, i.e., the stiffer the gel

is, the less tough it becomes by both Tanaka et al. [9, 10]

using polyacrylamide and Kong et al. [11] using alginate.

On the other hand, Kong et al. showed in physical gels

the elastic moduli and toughness both increase with cross-

linking density. A reason for this is suggested by the

work of Baumberger et al. [12] who investigated the frac-

ture mechanisms of physical gels using gelatin. They

proposed that, unlike chemical hydrogels, physical gels

do not fracture by chain scission but by viscous pull-out

of whole chains from the network via plastic yielding of

the cross-links. This accounts for the dependence of

toughness on crack tip velocity.

Though the above studies qualitatively agree, the

measured energy dissipation values are not necessarily

comparable due to the limitations of the testing methods

employed. Complex phenomena inherent in hydrogel frac-

ture such as viscous deformation, stress relaxation and

large plastic deformation prevent the adoption of tough-

ness measures such as the stress intensity factor [13] or

energy release rate which are based upon Linear Elastic

Fracture Mechanics (LEFM) theory. Also the softness of

the gels prevents the use of the common contact based

strain measurement schemes. To cope with gel softness,

Tanaka et al. [9, 10] measured the fracture energy of gels

using a modified version of the tearing energy concept

proposed for elastomers by Rivlin and Thomas [14],

based on the similarity between elastomers and chemical

gels. This scheme is an extension of Griffith’s approach

Correspondence to: H.J. Kwon; e-mail: hjkwon@uwaterloo.ca

Contract grant sponsor: Natural Sciences and Engineering Research

Council of Canada (NSERC).

DOI 10.1002/pen.21923

Published online in Wiley Online Library (wileyonlinelibrary.com).

VVC 2011 Society of Plastics Engineers

POLYMER ENGINEERING AND SCIENCE—-2011

and is equivalent to the energy rate interpretation of the

J-integral [15]. However, under this scheme tearing

energy is coupled with the energy for plastic deformation

occurred at the sharply bent corner adjacent to the crack

tip. Therefore, this scheme is not suitable to measure the

toughness of physical gels where plastic deformation

energy is large compared with the fracture energy. Also

this scheme is not regarded as a standard procedure to

assess the fracture toughness.

Baumberger et al. [12] overcome the challenges posed

by hydrogel softness through the use of optical techni-

ques to measure crack position and velocity. They were

able to assess the quasi-static toughness using a linear

regression technique based on the observation that the

energy release rate G is linearly proportional to crack ve-

locity. A constant crack speed was required for them to

extrapolate the average fracture energy release rate (frac-

ture energy divided by fracture area) to zero crack speed.

For this reason, all measurements of crack growth were

taken well after crack initiation in the center of rather

large specimens. However, in most of the fracture pro-

cess, the crack speed as well as energy release rate varies

with the crack growth; therefore, their approach is limited

to special cases where crack speed can be assumed

constant.

The essential work of fracture (EWF) method [16–18]

employed by Kong et al. [11] for gel study is more gener-

ally applicable. It overcomes the difficulty posed by plas-

tic deformation by separating the essential work (that con-

sumed for crack growth within process zone) from the

nonessential work (that dissipated outside the process

zone) through empirical dimensional analysis. It also has

the advantage that it can use crosshead force and dis-

placement data from a standard tensile tester. Unfortu-

nately this method is rather cumbersome requiring several

tests to be performed at differing initial ligament widths

before enough data are available for the mathematical

manipulation. Its accuracy is also dependent on the

underlying assumptions of the derivation which does

not consider the effects of loading rate in its formulation,

a potential problem in physical gels, most of which

demonstrate high rate-dependency in both stiffness and

toughness.

The traditional method for studying the fracture tough-

ness of ductile materials is the J-integral method which

can handle the relatively large plastic zone developed

around the crack tip. It does not seem to be widely

adopted for hydrogels, likely due to the difficulty in

applying the conventional measurement scheme to soft

and fragile hydrogels. This can be overcome by using op-

tical techniques to their full potential. The digital image

correlation (DIC) method not only overcomes the issues

involving specimen contact but also allows the complete

deformation field in the vicinity of the crack tip to be

directly observed. With this data, the critical strain energy

release rate (Jc) can be obtained in a very generally appli-

cable manner.

EXPERIMENTS

Agarose Preparation

Agarose sols were prepared by dispersing the agarose

powder (Product codes: A426-05, CAS No: 9012-36-6,

J. T. Baker) in standard 0.5 TBE buffer (Tris/Borate/EDTA,

ph 8.0) at 2% concentration (% w/v, assuming powder to be

100% agarose). The agarose solution was heated to 958Cfor 30 min and poured into a Teflon mold. It was then

cooled quickly to 108C. Preliminary tests indicated no

measurable change in mechanical properties after 30 min

casting time. Based on this, a conservative casting time of

�1 h was adopted for the remainder of the study. After

casting, specimens were immersed in buffer for 30 min to

allow swelling and tested immediately thereafter.

Test Scheme

Identical rectangular gel plates with the dimensions of

75 mm wide (W), 100 mm long (H), and 2 mm thickness

(t0) were prepared for both types of fracture tests, EWF

and J-integral. Specimens were held on the top and the

bottom by custom built grips machined from Delrin

(Dupont) faced with sandpaper to create a coarse gripping

surface.

The following gripping procedure was employed to

prevent damage to specimens prior to testing: (1) Upper

grip was removed from the load frame and attached to

the top part of the specimen in a jig that allowed the

specimen to be supported in a horizontal orientation. (2)

Upper grip holding the specimen was assembled into the

load frame and the bottom part of the specimen was

loosely held by the lower grip. (3) Load zeroed and the

lower grip tightened. (4) Compressive force developed by

gripping was released by moving the crosshead upwards

at 0.2 mm/min until zero load. This restored the specimen

to be vertically straight from the slightly bent shape

obtained during gripping.

Due to the fragility of the material notches were only

made after gripping using a fresh razor blade and a cus-

tom-built jig. Notch length (a in Fig. 1a) was approxi-

mately a half of the specimen width in single-edge-

notched tension (SENT) specimens for J-integral tests.

For EWF tests, the ligament length (L0 in Fig. 1b) of dou-

ble-edge-notched tension (DENT) specimens was varied

from 15t0 (30 mm) to 3t0 (6 mm). The final dimensions

were measured by counting the number of pixels in the

image captured by high resolution CCD camera (1028 31008 pixels, STC-CL202A, SENTECH).

As shown in Fig. 2a, fine chalk powders were

sprinkled on the surface of the specimens before the tests

to produce the patterning required for DIC. Tests were

conducted at ambient environment using a TA.XT Plus

load frame (Stable Micro Systems), with the force and

displacement being recorded. Crosshead speed was varied

from 1 mm/min to 100 mm/min. For each fracture

DOI 10.1002/pen POLYMER ENGINEERING AND SCIENCE—-2011 1079

experiment, a uniaxial specimen was also prepared from

the same batch and tested following the protocol proposed

in our previous study [6].

METHODS

Digital Image Correlation

DIC [19] is a robust experimental method enabling the

measurement of displacement field by tracking multiple

points on the surface. The surface is usually decorated with

speckle patterns, and the tracking is performed by comparing

the patterns in the digital images before and after deforma-

tion. This method has many advantages over conventional

methods using contacting sensors, such as being capable of

evaluating the strain field in a large domain without contact-

ing material. It can also be applied to various length scales

as long as proper speckle pattern can be generated on the sur-

face, which is a very useful feature when the preparation of

large specimens is untenable or very expensive.

Among the various image tracking algorithms, such as

cross-correlation [19], gradient descent search [20], snake

method [21], and sum of squared differences [22], the fast

normalized cross-correlation algorithm [23] was adopted

for this study because of its computational efficiency and

robustness to lighting conditions [24].

In this practice, the reference template is chosen from unde-

formed image. Its location in the deformed image is found by

performing pixel-by-pixel comparison of intensity values

using the following normalized cross-correlation formula

gðu; vÞ ¼P

x;y½f ðx; yÞ � f u;v�½tðx� u; y� vÞ � t�fPx;y½f ðx; yÞ � f u;v�2

Px;y½tðx� u; y� vÞ � t�2g0:5

(1)

where f is the intensity of deformed image, t the intensity of

reference template, t mean of the reference template, fu,vmean of the image overlapped by the reference template,

(u,v) current location of the reference image, and g(u,v) cor-

FIG. 1. Specimens used in this study: (a) Single-edge-notched tensile (SENT) specimen for J-integral test,and (b) double-edge-notched tensile (DENT) specimen for EWF test.

FIG. 2. (a) Undeformed SENT specimen with the initial grids (þ) gen-

erated around the J-integral path, and (b) integration paths around the

crack tip for the calculation of J-integral.

1080 POLYMER ENGINEERING AND SCIENCE—-2011 DOI 10.1002/pen

relation coefficient at (u,v). The new location in the deformed

image is where the maximum g(u,v) occurs. By tracking

the movement of multiple reference templates, the displace-

ment fields in a large domain can be estimated. This algo-

rithm was coded into internally developed DIC Matlab code.

J-integral Evaluation

For J-integral testing, the DIC method was adopted to

evaluate the displacement and strain field on the images

photographed by CCD camera during the test. As a first

step, the initial notch and the J-integral path around the

notch tip were defined on the image of an undeformed

SENT specimen, as shown in Fig. 2a. Initial grids (þ)

were generated within the rectangular regions of interest

enclosing the path. Then, the sub-images around the grid

points in the deformed image (not shown) were correlated

to sub-images in the reference (undeformed) image using

DIC algorithm to estimate the displacements u and v at

the grid points. By smoothing the in-plane displacement

components (u, v) from DIC [25], the strain components

along the path were determined using infinitesimal strain

tensor in accordance with the assumption of J-integralderivation as:

eij ¼ 1

2

quiqXj

þ qujqXi

� �(2)

where ui is the displacement at each grid and Xi material

coordinate. Since the J-integral path is relatively distant

from the crack-tip, the differences between finite and in-

finitesimal strain tensors along the path are less than 3%.

Conventionally, Hooke’s law based on linear elasticity has

been used to determine the stresses and the energy from strains.

However, our previous study [6] indicates that the stress-strain

behavior of agarose follows the nonlinear relationship as

s ¼ c1 eþ c2 e2 (3)

where c1 and c2 are fitting constants, s and e are the stress

and the strain in the loading direction, respectively. Since

this relationship was estimated from uni-axial tension test,

the stress and strain in Eq. 3 can be regarded as effective

(or equivalent) stress r and effective strain e.It can be assumed that agarose gel is almost incom-

pressible (Poisson ratio m � 1/2) [12] and that a plane-

stress condition, and monotonic loading exist in the J-in-tegral test [26]; thus, the deformation theory of plasticity

was invoked as [27]:

sxx ¼ 4

3

�s�eðexx þ 1

2eyyÞ

syy ¼ 4

3

�s�eðeyy þ 1

2exxÞ

sxy ¼ 2

3

�s�eexy

(4)

where e in Eq. 4 is

�e ¼ffiffiffi2

p

3

�ðexx � eyyÞ2 þ ðeyy � ezzÞ2 þ ðezz � exxÞ2

þ 6 ðe2xy þ e2yz þ e2zxÞ�1=2

ð5Þ

and r is from Eq. 3.Consider the J-integral along the contours around the

crack tip, composed of horizontal and vertical paths as

shown in Fig. 2b. The J integral can be evaluated along

these paths as [28]:

J ¼ JV1 þ JH2 � JV2 � JH1

JV1 ¼ZV1

½w� ðsxxexx þ sxyqvqxÞ�dy

JV2 ¼ZV2

½w� ðsxxexx þ sxyqvqxÞ�dy

JH1 ¼ �ZH1

ðsyy qvqxþ sxyexxÞdx

JH2 ¼ �ZH2

ðsyy qvqxþ sxyexxÞdx

(6)

where w is the strain energy density, and u and v the dis-

placement in x and y direction, respectively. The strain

energy density can be approximated as

w ¼Z �e

0

�s d�e (7)

Essential Work of Fracture

The EWF concept is the scheme to estimate the

fracture toughness of ductile materials that involve large

plastic deformation prior to fracture [16–18]. In this

concept, the total work provided to fracture, Wf, is clas-

sified into two parts. The first part is the work for

deforming the region around the ligament, Wp, which is

proportional to the volume of deformation zone. Since

the width of the deformation zone is proportional to the

initial ligament length [16], Wp is proportional to L20 3t0 where L0 is initial ligament length in Fig. 1b and t0initial thickness. The second part is the essential work

required for the formation of fracture surface, We,

which is proportional to the cross sectional area of the

ligament (L0 3 t0).From energy equilibrium, Wf is the sum of the two

parts of energy consumption as:

Wf ¼ We þWp ¼ weL0t0 þ b wpL20t0 (8)

where we is the specific EWF, wp the average deformation

work, and b the shape factor for the deformation zone.

By measuring the total work of fracture for specimens

of different ligament lengths, and dividing Wf by the

DOI 10.1002/pen POLYMER ENGINEERING AND SCIENCE—-2011 1081

ligament cross sectional area (L0 � t0), the specific work

of fracture, wf, can be expressed as:

wf ¼ we þ b wp L0 (9)

The specific EWF, we, which represents the fracture

toughness can then be determined by extrapolating wf to

zero ligament length.

RESULTS AND DISCUSSION

J Integral Evaluation

The constants c1 and c2 in Eq. 3 as estimated from uni-

axial tension tests were 150 and 2200 kPa, respectively. In

our previous study [6], it was observed that this constitutive

relation is relatively insensitive to the crosshead speed

range used in this study (1 mm/min �100 mm/min).

J-integral values computed along the path in Fig. 2a

are plotted in Fig. 3. This test was conducted at a cross-

head speed of 10 mm/min. Close investigation of the

images revealed that the crack was initiated at the

moment indicated by the arrow (image no. 54 in Fig. 3).

The small discontinuity between this and the subsequent

point (image no. 55 in Fig. 3) is attributed to sudden

crack development. The J-integral value on this image is

14.95 J/m2, which is taken as the critical J-integral, Jc,for crack initiation at this crosshead speed.

According to the definition of J-integral [28], the

J-integral should be path-independent, i.e., constant

along any path around the notch. To investigate the path

dependence of the Jc estimated by DIC method, multiple

J paths were generated on the image at which the crack

initiated (image no. 54), as shown in Fig. 4a, and J-inte-grals were calculated along each path. As can be seen in

Fig. 4b, the J-integral values for all paths are reasonably

consistent with the average value of 14.62 6 0.68 J/m2.

Therefore, J-integral value evaluated by the DIC method

is path-independent.

Quasi-Static Jc

J-integral was originally formulated based on the

assumption of quasi-static fracture [28] where thermody-

namic energy equilibrium holds. The general form of

energy equilibrium in cracked body is given by:

DW ¼ DEþ DK þ DG (10)

where DW is the increment of external work supplied to

the body, DE and DK are the changes of the internal

energy and kinetic energy, and DG is the change of irre-

versibly dissipated energy. The internal energy E com-

prises elastic strain energy Ue and plastic deformation

energy Up.

Assuming energy dissipation occurs only through

quasi-static fracture process (DK � 0 in Eq. 10), energyrelease rate G or J can be defined as:

J ¼ dW

dA� dUe

dA¼ dG

dAþ dUp

dA(11)

where dA is crack area increment. Note that G is for

linear elastic materials (Up � 0), while J for nonlinearFIG. 3. Variation of J-integral with crack growth.

FIG. 4. (a) Deformed SENT specimen with multiple J-integral paths

generated, and (b) the critical J values (Jc) for different paths in (a).

1082 POLYMER ENGINEERING AND SCIENCE—-2011 DOI 10.1002/pen

elastic materials. Since nonlinear elastic and plastic

stress-strain curves are not discernible in monotonically

increasing load, J can be also applied to elasto-plastic

materials where a part of the energy is dissipated by plas-

tic deformation (Up . 0). Under mode I loading, the J-in-tegral fracture criterion for crack initiation takes the form

J ¼ Jc where Jc is the quasi-static fracture toughness. The

implicit restrictions of this criterion are:

i. Energy dissipation rate with respect to time ( _C) is higherthan or equal to energy supply rate ( _W) so that the meas-

ured J value represents the instantaneous energy release

rate of a cracked body at the moment, and

ii. Jc is rate-independent. Otherwise, a separate scheme to

decouple K from J in Eq. 11 needs to be arranged.

If the fracture process or estimated J value does not

conform to these restrictions, the aforementioned J-inte-gral method may not be a suitable testing scheme and the

estimated Jc is not true fracture toughness.

The fracture process in agarose was examined by

observing the crack growth behavior in a separate SENT

test. In this test, SENT specimens with an initial notch

(Fig. 5a) were loaded in tension by moving the crosshead

upward. As soon as crack initiation was observed at the

notch tip, the crosshead was arrested (Fig. 5b). If it is a

quasi-static fracture conforming to the restriction (i), thecrack should stop growing immediately or shortly aftercrosshead arrestment, since the external energy supply,dW, in Eq. 11 is zero so that J falls below Jc, as strainenergy is dissipated by crack growth. However, crackgrowth continued at a slow but almost constant speed(Fig. 5c), to the end of the specimen (Fig. 5d). Repeatedtests revealed that once crack was initiated, the crackalways propagated through the remaining body of thespecimen regardless of initial notch length or notch shape.This suggests that _C is much less than _W in the crack

nucleation and development process; thus, much larger

strain energy is always accumulated before crack initia-

tion than that for crack development across the entire

specimen width. Therefore, the fracture process in agarose

is not quasi-static and the Jc estimated by the above J-in-tegral scheme should not be the quasi-static fracture

toughness.

The rate effect on Jc was investigated by performing

the fracture tests at various crosshead speeds ranging from

1 to 100 mm/min, and the results are presented in Fig. 6.

The variation of Jc against loading rate clearly demon-

strates that fracture toughness is not constant, but a func-

tion of loading rate. A similar trend was also found in the

literature [29] in a study on metals. Our previous study

on agarose [6] elucidated that ultimate stress and strain of

FIG. 5. The crack growth behavior: initial notch (a) was opened and crack started growing as the crosshead

moved upward. Crosshead was arrested at (b), but the crack kept growing (c) to the end of the specimen (d).

DOI 10.1002/pen POLYMER ENGINEERING AND SCIENCE—-2011 1083

agarose specimen under uniaxial tension have power law

relationships with crosshead speed, i.e. linear relationship

on log-log scale. Since Jc is the critical strain energy

release rate corresponding to the material resistance to

crack initiation, where strain energy is given as stress

times strain, Jc is expected to follow a similar power law

relationship. As this is the case in the high speed fracture

(HSF) region of Fig. 6, the second restriction regarding

rate independence of fracture toughness is also not met,

and a scheme for stripping rate dependency needs to be

arranged.

From the above observations, it can be hypothesized

that if the loading rate becomes extremely small, _Cbecomes comparable with _W and quasi-static energy

equilibrium can be achieved to meet the first restriction.

Then, the Jc value at this equilibrium condition may rep-

resent material resistance at quasi-static fracture, i.e.,

true fracture toughness. Figure 6 illustrates that the trend

of Jc versus crosshead speed changes as the crosshead

speed decreases. In HSF region, Jc decreases almost lin-

early in the log-log plot with the decrease of crosshead

speed, as indicated by the inclined solid trend line in

Fig. 6. If the crosshead speed is less than 10 mm/min,

Jc decreases very rapidly until it reaches 2 mm/min, as

presented in the transition region in Fig. 6. A linear

trend is still maintained in this region. Below 2 mm/

min, Jc is almost constant, as illustrated by the horizon-

tal solid trend line in the low speed fracture (LSF)

region. Extrapolating to the zero abscissa using linear

regression the y-intercept is 6.5 J/m2. Since the fracture

process and estimated Jc satisfy both restrictions, this

value can be regarded as true Jc, a quasi-static fracture

toughness of 2% agarose.

Essential Work of Fracture

EWF method has much less stringent restrictions than

LEFM based methods or J-integral. Plastic energy dissi-

pation is allowed in a much larger scale, since it is to

be decoupled from the specific EWF, we, representing

fracture toughness, by linear regression method.

Therefore, this approach can be applied to the fracture of

various materials with diverse ductility. However, EWF

method also does not consider the rate-dependence in its

formulation.

The rate dependent fracture behavior of agarose gel

was investigated first by comparing load-displacement

curves of DENT specimens with 30 mm ligament length

tested at various crosshead speeds: 1 mm/min, 10 mm/

min, and 100 mm/min. The initial parts of load-displace-

ment curves from different loading rates are almost over-

lapped one another up to 2 mm displacement, as pre-

sented in Fig. 7. This may be attributed to the relatively

insensitive stress-strain relationship to the loading rate, as

observed in our previous study [6]. However, crack initia-

tion, crack growth and fracture behaviors were signifi-

cantly varied by the loading rate. At 100 mm/min cross-

head speed, crack initiated at around 3 mm displacement.

Interestingly, the load kept increasing almost linearly even

after crack initiation until the last moment of fracture at

which highly brittle failure occurred with a loud noise. In

this fracture process, a large amount of strain energy was

accumulated during crack growth, and was released in the

form of rapid failure, retraction, vibration, and sound at

the final fracture. As described above, this was caused by

the unbalance of energy flow in crack growth with energy

dissipation rate ( _C) being much lower than energy supply

rate ( _W). Since these energies are not scalable as other

energies in Eq. 8, the EWF method is not valid for this

type of fracture behavior.

On the other hand, the fracture test at 1 mm/min cross-

head speed demonstrated significantly different fracture

behavior. At this loading rate, crack initiation happened

much earlier at around 2 mm crosshead displacement.

This is consistent with the decreasing trend of Jc with the

decrease of loading rate in Fig. 6. As soon as a crack was

initiated, the load started decreasing, suggesting that the

FIG. 6. The variation of Jc value with respect to crosshead speed.

FIG. 7. Load-displacement curves from the fracture tests using DENT

specimens with 30 mm ligament length at the crosshead speeds of

1 mm/min (^), 10 mm/min (&) and 100 mm/min (~).

1084 POLYMER ENGINEERING AND SCIENCE—-2011 DOI 10.1002/pen

energy dissipation rate ( _C) and energy supply rate ( _W)

were relatively balanced and energy equilibrium was

maintained. Note that the shape of the load-displacement

is typical of quasi-static fracture of ductile materials [30]

to which EWF method has been successfully applied.

Based on this fracture behavior, EWF tests were

performed employing 1 mm/min crosshead speed to yield

the specific EWF value (we) as the quasi-static fracture

toughness.

The specific work of fracture wf was calculated using

the area under the load-displacement curve, and plotted in

Fig. 8 with respect to ligament length L0. Figure 8 clearly

shows a linear relationship between wf and L0 for the liga-

ment length used for this study (6 mm � L0 � 30 mm).

The y-intercept at zero ligament length corresponds to we

that is 5.3 J/m2 with the correlation coefficients (R) for

the curve fitting being 0.9991. This EWF value is reason-

ably close to the quasi-static Jc value, 6.5 J/m2, evaluated

above.

Dynamic Fracture Behavior of a Physical Gel

In most materials, crack tip velocity is very low at the

crack initiation stage, but accelerates with crack growth,

reaching the maximum at the final stage of the fracture.

In strong contrast to this, we observed that in the fracture

of physical gels, crack speed is almost constant or

increases at a much slower rate than that predicted by

LEFM [12]. This may be closely related to the positive

rate-dependence of fracture toughness in these materials.

Conventional fracture mechanics suggests that cracks

begin to move when the potential energy released by a

unit extension of a crack (middle term in Eq. 11) becomes

equal to critical fracture toughness Jc. Even if this condi-

tion of energy balance predicts the onset of motion, once

a crack starts to move the kinetic energy due to material

motion around the moving crack has to be taken into

account. Accounting for this energy led to the equation of

motion for a crack [31–33] as

V ¼ffiffiffiffiffiffi2pk

rc0 1� a0

a

� �� 0:38c0 1� a0

a

� �(12)

where c0 ¼ffiffiffiffiffiffiffiffiffiE=q

p, is the speed of sound for one-dimen-

sional wave propagation, k is a function of m [34], and a0the initial current crack length given by

a0 ¼ 2EJcps2f

(13)

Equation 12 predicts that a crack should continuously

accelerate as a function of its instantaneous length to a

limiting, but finite, asymptotic velocity. In 2% agarose

gel, c0 is around 14 m/s so that the asymptotic velocity is

around 5.3 m/s. This asymptotic limit can be reached by

either increasing the amount of energy driving the crack

or by reducing the fracture toughness to zero. Note that

Eq. 12 was derived based on rate-independent fracture

toughness Jc in the framework of LEFM.

Baumberger et al. [12] proposed that the physical gels

are fractured by viscous pull-out of whole polymer chains

from the network via plastic yielding of the cross-links. The

viscous nature of this fracture mechanism leads to the

increase of fracture energy with increased crack tip velocity

in their experiments. According to Eq. 12, a crack is initi-

ated at an infinitesimally low velocity, as a � a0. If Jc is

constant, V will increase as a0/a decreases with the crack

growth. However, in the physical gels that are fractured by

chain pull-out mechanism, Jc increases with V, and hence

a0 does. This in turn lowers the crack speed and energy dis-

sipation rate ( _C) in the fracture of physical gels compared

with the materials having rate-independent fracture tough-

ness. As a result, the energy supply rate ( _W) usually

exceeds the energy dissipation rate ( _C) and much larger

strain energy is accumulated within the gel than that

required for fracture, unless _W is fairly low as in the test

at 2 mm/min crosshead speed for 2% agarose gel as

described above. Previous studies [9, 12, 16] employed ap-

proximate scheme to compute fracture toughness using

Jc ¼ W/(A0) where A0 is the cracked area; however, it may

yield erroneous result overestimating the fracture tough-

ness, since W is much larger than fracture energy.

It needs to be mentioned that the fracture at 100 mm/min

in Fig. 7 showed quite brittle and unstable behavior with

extremely high fracture energy, which is significantly differ-

ent from those at lower loading rates. At this high loading

rate, _W is much larger than _C, thus most of the supplied

energy is piled up as an elastic strain energy. We speculate

that this allows the stored strain energy to reach the critical

energy level for the failure by chain scission, resulting in

rapid crack growth and highly brittle behavior, as opposed

to the slow and stable failure by chain pull-out mechanism

at low strain rate.

FIG. 8. Specific work of fracture (wf) from the fracture tests on DENT

specimens as a function of ligament length.

DOI 10.1002/pen POLYMER ENGINEERING AND SCIENCE—-2011 1085

CONCLUSIONS

The evaluation of fracture toughness of hydrogels is

often very challenging due to the difficulty of testing as

well as lack of standard testing methods. We evaluated

fracture toughness of a physical hydrogel using 2% aga-

rose, and adopting the J-integral test based on DIC

method. The quasi-static fracture toughness was deter-

mined by linear regression on critical Jc values with

respect to loading rate. When this was compared with

EWF results obtained using a quasi-static equilibrium

condition, the quasi-static we was found to be reasonably

close to the quasi-static Jc value. Rate dependent fracture

behavior of physical gels was considered. At low speeds,

results were consistent with the chain pull-out mechanism,

while at high speeds, elastic energy accumulated faster

than could be dissipated by crack growth resulting in brit-

tle fracture likely due to chain scission.

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