application of digital image correlation method to...

Application of Digital Image Correlation Methodto Biogel

H.J. Kwon,1 Allan D. Rogalsky,1 Christopher Kovalchick,2 Guruswami Ravichandran2

1 Department of Mechanical and Mechatronics Engineering, University of Waterloo, Waterloo,Ontario, Canada N2L 3G1

2 Division of Engineering and Applied Science, California Institute of Technology, Pasadena,California 91125

This study adopts the digital image correlation (DIC)method to measure the mechanical properties undertension in agarose gels. A second polynomial stress–strain equation based on a pore model is proposed inthis work. It shows excellent agreement with experi-mental data and was verified by finite element simula-tion. Evaluation of the planer strain field by DIC allowsmeasurement of strain localization and Poisson’s ratio.At high stresses, Poisson’s ratio is found to exceed thestandard assumption of 0.5 which is shown to be aresult of pore water leakage. Local failure strains arefound to be approximately twice those determined bycrosshead displacements. Viscous properties of aga-rose gels are investigated by performing the tensiletests at various loading rates. Increases in loading ratedo not cause much difference in the shape of stress–strain curves, but result in increases in ultimate stressand strain. POLYM. ENG. SCI., 50:1585–1593, 2010. ª 2010Society of Plastics Engineers

INTRODUCTION

Polymer gels can be classified by the types of cross-

links joining chains together [1]. Chemical gels are joined

by permanent covalent bonds, while physical gels are

linked by polymer–polymer interactions. In chemical gels,

long flexible chains make up much of the network allow-

ing rubber elastic theory to be applied. The network in

physical gels may consist of bundles of chains which are

much less flexible, and therefore do not exhibit classical

rubber elastic behavior [2].

Agarose forms a type of physical gel, with the junc-

tions arising from a phase transition. Such junction zones

can be created and removed by cooling and heating,

respectively. Agarose exhibits significant hysteresis in its

melting and setting temperatures. Setting of the gel occurs

when a hot solution is cooled down below the ordering

temperature, which is around 408C for normal agarose,

while complete melting requires heating above 908C. Inthe hot solution state, agarose chains exist in a stiff and

disordered configuration, while gels consist of bundles of

agarose chains and large pores of interstitial fluid [3]. The

existence of these bundles, consisting of 10–30 helices

per clump [4], has been well established, and the charac-

teristics of the gel structure have been studied by various

methods [4–9]. It has been noted that increasing the ionic

strength of the solution when casting an agrose gel

increases the pore size [4, 7], while increasing the agrose

concentration has the opposite effect [4]. This has impli-

cations, both for the diffusion of large molecules through

the gel, and for the gel’s mechanical properties.

Because of their porous structure, agarose gels are use-

ful for the size-separation of large molecules, such as pro-

teins and DNA fragments. They are also widely used in

tissue culture systems because they permit cell and tissue

growth in a three-dimensional suspension. Agarose gels

have numerous applications in biomedical engineering

and biotechnology as load-bearing structures, such as for

cartilage repair [10]. Part of there appeal is there visco-

elastic nature [11], making an understanding of the me-

chanical behavior of agarose gels particularly important.

Despite this there are relatively few studies on the rate-

dependency of the mechanical behavior of agarose [2].

Several studies have been performed on the mechanical

properties of agarose gels. These include static and

dynamic stiffness [12, 13], stress relaxation [14], fluid

pressurization [15], and hydraulic permeability [16].

Because of the softness of hydrogels contact type sensors

such as strain gages can not be used for strain measure-

ment. For this reason, in most cases where uniaxial tensile

tests were conducted [3, 12, 17–20], strain was estimated

using the overall elongation of the specimen based on the

assumption that the deformation occurs uniformly in the

gauge section. However, FEM simulation shows that

Correspondence to: H.J. Kwon; e-mail: [email protected]

Contract grant sponsors: National Science Foundation (NSF), Natural

Sciences and Engineering Research Council of Canada (NSERC).

DOI 10.1002/pen.21636

Published online in Wiley InterScience (www.interscience.wiley.com).

VVC 2010 Society of Plastics Engineers

POLYMER ENGINEERING AND SCIENCE—-2010

the deformation is neither uniform within the gauge sec-

tion, nor is it ignorable outside, particularly when the stress

is large enough to initiate failure [21]. This makes stress–

strain relationships based on a uniform strain assumption

inaccurate, and casts doubt on the validity of the overall

strain at failure as a measure of material ductility.

This study adopts digital image correlation (DIC)

method to measure in-plane strain fields [22]. DIC works

by comparing images of the specimen acquired at differ-

ent stages of deformation, allowing the in-plane displace-

ment and strain fields to be evaluated in a large domain

without resorting to contact sensors that might influence

the test result. For the recognition of patterns, the surface

is traditionally decorated with a black and white paint

mist. However, since the agarose specimen is soaked in a

liquid, this method will not work here. Fluid is also

squeezed out onto the surface when the agarose gel is

deformed, making it more challenging to generate the

speckle pattern by this conventional way. In this study,

hygroscopic particles are sprinkled on the surface of aga-

rose gel to generate random patterns. The particles

adhered to the gel surface and were not separated during

the deformation. An internally developed DIC algorithm

successfully recognized the patterns formed by sprinkled

particles and followed the movement of subimages accu-

rately. By applying the DIC method to continuous images

of the agarose specimen during the tensile test, variations

of displacement, and in-plane strain field were evaluated.

Knowledge of the full planar strain field also allows cal-

culation of Poisson’s ratio, which for most hydrogels has

been assumed without verification to be 0.5 (i.e., volume is

conserved) [18]. This is despite the porous nature of hydro-

gels and the observation that at large strains water is

squeezed out onto the surface indicating a possible volume

decrease, and hence a Poisson’s ratio greater than 0.5.

The ultimate strain, defined as the overall strain at fail-

ure, is frequently regarded as the failure strain [18], and is

used to express the ductility of a material. For materials

where marked strain localization occurs, such as hydrogels,

a better estimate of ductility is provided by the local strain

at the spot where failure is about to initiate. In this study,

this has been obtained by adaptation of the DIC method.

DEVELOPMENT OF A STRESS–STRAINEQUATION

The majority of previous studies on the mechanical

properties of agarose gel regarded it as an isotropic

continuum, and employed a linear elastic model to

describe its response to loading [17, 18]. However, this

ignores porous and biphasic nature of agarose gel, which

develops complicated mechanical response to loading

conditions.

It is known that the volumetric response of a porous

medium is nonlinear with respect to both changes in total

pressure P and pore pressure p [23, 24]. To accommodate

this, the volumetric response of a porous material can be

expressed in incremental form for an infinitesimal

increase of the loading from {P, p, V} to {P þ dP, p þdp, V þ dV} where V is the pore volume. As in most

studies tests are performed at ambient pressure, P can be

ignored. Using simple continuum mechanics, the follow-

ing relations can be derived:

dV

V¼ dp

p¼ dekk (1)

where ekk is dilatational strain tensor. Eq. 1 can be inte-

grated to yield the instantaneous pore pressure as:

p� p0 ¼ p0ðeekk � 1Þ � p0 ekk þ 1

2e2kk þ � � �

� �(2)

where p0 is the initial pore pressure.

On the other hand, the increment of deviatoric stress

exerted on network can be expressed as:

dsij ¼ 2mdeij

eij ¼ eij � ekk3dij

(3)

where m is Lame’s constant and eij the deviatoric strain

tensor. As pore pressure is supported by the surrounding

network and the pores and network are geometrically

compatible, Eq. 3 is integrated on the assumption of con-

stant m, and combined with Eq. 2 to yield the network

stress as:

sij ¼ 2meij þ p0 ekk þ 1

2e2kk

� �dij (4)

If the loading condition is uniaxial plane-stress, the stress

in the loading direction can be expressed as a function of

strain in the same direction as:

s11 ¼ Kne11 þ Kve211 (5)

where Kn is mainly a function of network modulus while

Kv is dependent on the pore pressure and drainage condi-

tion. Zile et al. [19] and Barrangou et al. [3] proposed

similar relationships between nominal stress and nominal

strain from stretch test results. However, their approaches

were empirical without any theoretical consideration. Both

constants in Eq. 5 can be evaluated through uniaxial ten-

sile (UT) tests.

MATERIALS AND METHODS

Sample Preparation

Commercially available agarose gel (Product codes:

A426-05, CAS No: 9012-36-6, J.T. Baker) were used in

1586 POLYMER ENGINEERING AND SCIENCE—-2010 DOI 10.1002/pen

this study. Agarose powder was dispersed in standard 0.5

Tris/Borate/EDTA buffer (TBE, pH 8.0), to form sols

with concentrations ranging from 1 to 5%w/v, assuming

the powder to be 100% agarose. These solutions were

heated to 958C for 1 h, poured into a Teflon mold and

cooled quickly to 108C. Samples were held at this temper-

ature for 1 h for complete gelation. In preliminary tests,

significant differences in mechanical properties were not

detectable after 30 min. After casting, to simulate their

response to a physiological environment, specimens were

swollen in buffer for 30 min and tested immediately

thereafter. For ideal simulation of a physiological environ-

ment it is sometimes suggested that samples be tested

while immersed [12]; however, preliminary tests found

little difference in the load-displacement curves between

gels tested while immersed and those tested immediately

after removal from the buffer, provided they had been

swollen for more than 10 min.

Specimens

Two types of specimens were used in this study (see

Fig. 1). For evaluating the stress–strain relationship, a

dog-bone type specimen with geometry given in Fig. 1a

was used. For evaluating failure strain, notches were

added on both sides of the width in the middle of the

gage length, Fig. 1b. These notches were molded into the

specimen and are within the limit of width reduction rec-

ommended by ASTM standard (ASTM D 638-01) for uni-

axial tensile tests.

Mechanical Testing

The in-house built uniaxial tensile tester shown in

Fig. 2 was used for this study. The main components of

the setup were a motorized translation stage (Physik

Instrumente, minimum displacement step size 0.1 lm),

and a load cell (AL Design, load range 5 N, sensitivity

0.03 N). During testing, a displacement transducer

(LVDT, Omega) was used for crosshead control. The

ideal set-up should be equipped with a closed-loop control

to keep the strain rate constant [25]; however, using DIC

to measure strain makes real time closed-loop control of

strain rate difficult to implement [19]. For this reason con-

stant crosshead speed was used in this study. Cross-head

speed was 10 mm/min if not otherwise indicated. Instru-

ment control software was implemented in LabVIEW,

(National Instruments). An acrylic hydration chamber sur-

rounded the grip region to allow testing in a hydrated

environment. To prevent slipping in the grips, grip size

was customized and sandpaper was applied to create a

coarse gripping surface.

A CCD camera captured images at between 2 fps for 1

mm/min cross-head speed and 25 fps for 100 mm/min for

the post-processing by DIC. To generate the stochastic pat-

tern required for DIC, chalk powder was uniformly applied

to the specimen surface immediately before each test.

To capture images of stress induced water flow, fine carbon

powder collected from laser toner was substituted for chalk

powder. The carbon powder is so fine that it is easily washed

off by the water creating good imaged contrast. To ensure de-

formation and failure occurred within the camera field of view

notched specimens, Fig. 1b, were used for these tests.

Digital Image Correlation

Various image processing techniques [26–29] have

been used to compare images in DIC algorithm. Due to

FIG. 1. Tensile specimens used in this study: (a) specimen geometry

for both notched and un-notched specimens, and (b) the detailed view of

the notch.

FIG. 2. Test set-up for uniaxial tensile test.

DOI 10.1002/pen POLYMER ENGINEERING AND SCIENCE—-2010 1587

its computational efficiency and robustness to lighting

condition the fast normalized cross-correlation (FNCC)

proposed by Lewis [26] was employed in this study.

Figure 3a is an image of an undeformed specimen, on

which the speckle patterns are generated with chalk pow-

der. Base grids defined by the DIC algorithm are visible

(dark cross marks). To measure the strain field in the

specimen, a subimage around each grid point in the first

image is compared to the second image Fig. 3b, to track

the movement of each grid (white cross marks). The gap

between grids points and the subimage size were adjusted

to optimize processing speed and tracking accuracy. For

each test, a series of images were processed and the strain

field at each image was evaluated from the displacements

of the grids using Lagrangian finite strain as

eij ¼ 1

2

quiqxj

þ qujqxi

þX2k¼1

qukqxi

qukqxj

" #(6)

where ui is the displacement vector of each grid point and

xi initial grid spacing.

Finite Element Simulation

The stress–strain equation evaluated by DIC was veri-

fied using finite element method (FEM). The load-dis-

placement curve from tensile test was simulated using

ABAQUS 6.5 Standard, and compared to the experimen-

tal curve. For the simulation, the stress–strain equationwas coded using UMAT, a user-defined module for mate-

rial properties in ABAQUS.

RESULTS AND DISCUSSION

Stress–Strain Curve

Typical strains of 2% agarose in the axial (loading)

and lateral (width) directions are plotted with respect to

time are shown in Fig. 4a, with the sign of strain in lat-

eral direction reversed. The variation of nominal (engi-

neering) stress against time is also shown in Fig. 4a. By

combining true strain and the nominal stress in Fig. 4a,

true stress was calculated using a plane stress assumption.

Image time stamps were used to mach calculated strain to

the measured stress to plot true stress-strain curve as pre-

sented in Fig. 4b. Least squares method was used to fit

the proposed relationship, Eq. 5, to the data and estimate

Kn and Kv. The proposed equation showed good agree-

ment with the experimental curve when Kn ¼ 150 and Kv

¼ 2200 kPa. As described earlier, most of the previous

studies assume that the mechanical behavior of agarose is

linear elastic and attempt to evaluate Young’s modulus.

However, the true stress–strain curve in Fig. 4b illustrates

FIG. 3. Agarose specimen: (a) undeformed with base grid, (b)

deformed with base (black þ) and displaced (white þ) grids. Back-

ground speckles are chalk powder.

FIG. 4. Typical uniaxial tensile test results for 2% agarose: (a) true

strain–time curves in the axial (——), and lateral (��) direction, and nom-

inal stress-time curve (l), and (b) true stress–strain curves from: DIC (*),

conventional scheme using overall elongation (——), and Eq. 5 (��).


that the stress–strain relationship is nonlinear, especially

above 0.05 strain.

Previous researchers also determined the stress–strain

curve using overall elongation and length of the gauge

section [17–20]. The same scheme was adopted to evalu-

ate nominal stress–strain curve, and then converted into

true stress–strain curve. The result is presented in Fig. 4b

(dotted line) which is considerably higher than the curve

determined by DIC. This difference increases with strain.

Therefore, the conventional scheme to determine the

stress–strain curve should not be suitable for soft hydro-

gels such as agarose. The accuracies of both stress–strain

curves are considered using FEM simulation in the later

section.

Poisson’s Ratio

The variation of Poisson’s ratio (m) was examined

using the true strains in the loading and width directions

shown in Fig. 4a and presented in Fig. 5. m has been

assumed to be 0.5 in previous studies of agarose gel’s

[18]. However, Fig. 5 demonstrates that m increases with

the elongation of the specimen. When strain is small, m is

around 0.4. As the specimen is elongated, m exceeds 0.5

at 0.89 strain and reaches 0.65 at around 0.15 strain. This

implies that the volume decreases with the elongation. In

conventional solid mechanics m can never be larger than

0.5 based on the assumption of mass conservation. How-

ever, in the porous hydrogels such as agarose, the pore as-

pect ratio changes with elongation, reducing the available

pore volume and increasing the pore pressure. This causes

the leak of pore water, and a loss of volume. The leak of

pore water will be discussed further under the heading of

flow stress.

FEM Simulation Results

A FEM model was implemented in ABAQUS 6.5

Standard, using 2565 20-node brick elements as shown in

Fig. 6a. Geometrical symmetry allowed half of a speci-

men with a quarter of the cross section to be used. On the

basis of the experimental results in Fig. 5, m was assumed

to be constant at 0.45 until 0.05 strain, and to increase

from 0.45 to 0.65 between 0.05 and 0.15 strain. Eq. 5was used as a stress–strain equation.

The nominal stresses determined from the tensile test

(�) and those generated by FEM simulation (�) are plotted

with respect to cross-head displacement in Fig. 6b. The

FEM results were very similar to the experimental data

up to 5.5 mm. Above this the FEM result became slightly

higher than the experimental data due to the deviation of

the experimental curve from Eq. 5 as the strain

approached its maximum. Another simulation was per-

formed employing the true stress-strain curve determined

through the conventional scheme based on overall elonga-

tion (dotted line in Fig. 4b). This simulation yielded nom-

inal stress-displacement plots significantly higher than the

experimental results (~) in Fig. 6b. Thus, this it can be

concluded that the conventional scheme using overall

elongation did not produce an accurate stress–strain curve,

and had a tendency to overestimate the stresses. The FEM

result also implied that the effective gauge length should

be shorter than the actual one, which could be attributed

to the nonuniform distribution of deformation and a tri-

axial stress state around the interface with the nongauge

section, as shown in Fig. 6a. Further information on the

stress state within a uniaxial tensile specimen can be

found in reference [30].

Effect of Concentration

The stress–strain curves for 1 and 4% agarose are pre-

sented in Fig. 7a and b. As can be seen in Figs. 7a and

4b, the stress–strain relationship defined by Eq. 5 is in

good agreement with the experimental results for 1 and

2% agarose to failure. However, Fig. 7b shows that it

deviates from the results for 4% agarose above 0.1 strain.

It is speculated that the deviation is attributed to the

expulsion of pore water under high stress. Note that Eq. 5was derived based on undrained condition. As higher

stresses are supported by more concentrated specimens,

the deviation was observed in the stress-strain curve of

4% specimen. This is discussed further under the heading

FIG. 5. Dependence of Poisson’s ratio on the true strain.

FIG. 6. FEM Simulation results: (a) deformed shape of the model at 6

mm elongation, and (b) the plots of the normalized load versus displace-

ment from: experiment (l), and FEM simulations using the stress–strain

curve from DIC (l), and from the conventional scheme (~).


of flow stress. The best fit values of Kn and Kv for all

concentrations tested are listed Table 1.

Kn and Kv for various concentrations are normalized

by to the values obtained for 1% agarose and plotted in

Fig. 8. Kn increases almost linearly with concentration.

This is consistent with the derivation of Eq. 5 where the

main contribution to Kn is from the polymer network

modulus. As the concentration of the agarose increased,

the crosslinking density and the degree of entanglement

of polymer chains increased in proportion to the concen-

tration, resulting in the steep rise of the network modulus.

On the other hand, Kv, which mainly accounts for the

effect of pore pressure in the derivation of Eq. 5, is much

less strongly influenced by concentration. This is because

pore pressure is generated by the geometry change which

is relatively independent of concentration. The weak de-

pendence of Kv on the concentration, observed above 1%

agarose, is likely because highly entangled networks

formed the pore walls that were less permeable, i.e.,

drainage condition tended toward undrained condition.

The large difference in Kv between 1 and 2% agarose

may indicate the existence of a critical threshold in the

pore network permeability.

As the dependence of Kn on concentration is much

higher than that of Kv, the significance of the first order

term relative to that of the second order term in Eq. 5increases with concentration. In Fig. 7b, it can be seen

that at a concentration of 4% the curve is almost linear.

Concentrations above 4% followed the linear trend which

is typical in network polymers [30] rather than nonlinear

quadratic polynomial.

Effect of Strain Rate

Typical stress–strain curves for 2% agarose at cross-

head speeds of 1, 10, and 100 mm/min are shown in

Fig. 9. The curves for 1 and 10 mm/min are almost over-

lapped, and showed good agreement with Eq. 5 until the

initiation of failure. The curve for 100 mm/min shows

good agreement up to 0.18 strain, above which Eq. 5overestimates the stress. This is very similar to the behav-

ior of 4% agarose at 10 mm/min cross-head speed and

should also be caused by the leakage of pore water at the

higher stresses reached. The increase of modulus against

loading rate is not significant when compared to the

experimental error. Zile et al. observed the same trend

TABLE 1. Kn and Kv fit values (kPa) versus agarose concentration (w/

v%).

1% 2% 3% 4% 5%

Kn 50 150 400 800 1,000

Kv 800 2,200 2,450 2,650 2,750

FIG. 8. Normalized Kn (*) and Kv (^) with respect to agarose con-

centration.

FIG. 7. True stress–strain curves for (a) 1%, and (b) 4% agarose: test

data (*) and the curve from Eq. 5 (��).

FIG. 9. Stress–strain curves for 2% agarose gel at cross-head speeds of

1 mm/min (x), 10 mm/min (o), and 100 mm/min (þ). Ultimate strain at

each cross-head speed is indicated by an arrow.


and concluded that agarose gel has no viscous properties

[19]. However, different types of viscous properties are

observed in Fig. 9. Ultimate strains increased substantially

with increase of strain rate as indicated by the vertical

dashed lines. When ultimate strain was plotted against

cross-head speed on log-log scale, Fig. 10, a strong linear

relationship was found. Each point is the mean value of

at least five replicates. A similar trend was found between

ultimate stress and cross-head speed, which was expected

based on Eq. 5. This type of relationship was observed at

all agarose concentrations.

McEvoy et al. [2] reported a similar relationship based

on tests using 5% gelatin. Gelatin can also be classified

as a physical gel, suggesting that a power law dependency

between strain rate and ultimate properties may be a char-

acteristic of gel structure. The crosslink’s in agarose gel

are junction zones that are formed by cooperative sequen-

ces of ordered chains [2, 9, 11], which can be broken and

reformed perturbations allowing polymer chains to reor-

ient and rearrange into new equilibrium positions under

an applied load. This can result in disentanglement and

dissociation at the bonds, but these processes are not in-

stantaneous and require a relatively long period of time,

as long polymer chains are involved in. At low strain rate

there is sufficient time for these dissociation processes to

occur at the junction zones. However, as the strain rate

increases, it becomes harder for polymer chains to be dis-

entangled allowing the junction zones carry higher

stresses. This allows agarose gels to be elongated much

farther before failure at a high strain rate, explaining the

increase of ultimate strain with increasing strain rate.

Flow Stress

It is generally known that hydrogels contain two types

of water [31]. Strongly bound water is in direct interac-

tion with hydrophilic chains, while free water is filling

the pores between polymer networks. In the undeformed

state, the polymer network is well aligned to contain the

free water without leak. However, if the specimen is elon-

gated significantly, misalignment occurs in the polymer

networks, which results in the leakage of free water.

The images showing the progress of specimen defor-

mation and the development of water flow are presented

in Fig. 11. In Fig. 11a, a specimen tested at low strain

rate fails without any visible flow, while in Fig. 11b at

high strain rate, water begins to flood the specimen sur-

face prior to visible crack initiation. This set of flow tests

suggested that at large deformations and high strain rates

the pore pressure should exceed the networks’ ability to

contain it, resulting in a partial breakdown of the network

and a large leak of pore water. This is consistent with the

speculation that deviations of experimental stress–strain

curves from Eq. 5 shown in Figs. 7b and 9 are attributed

to the leakage of pore water. On the contrary, at moderate

deformations and low strain rates Eq. 5 provides a good

fit to the data, suggesting that the leak of pore water from

the specimen is not so great as to violate the undrained

assumption used in its derivation.

The nominal stress at which visible water flow started

is plotted against cross-head speed in Fig. 12 on a semi-

log scale. No data is available for 1 mm/min because fail-

ure initiated before flow was observed. Fig. 12 illustrates

that the flow stress exhibits a weaker dependence on

strain rate. It was previously shown in Fig. 10 that ulti-

mate strain, and hence stress, increased very rapidly with

the cross-head speed. As the increasing rate of flow stress

is much less than that of ultimate stress, the difference

between them became more significant as the cross-head

speed increased. As a result, the network deformation and

misalignment continued after inception of leakage until

the occurrence of failure at high cross-head speed, which

FIG. 10. Dependence of ultimate strain on the cross-head speed.

FIG. 11. Specimens images taken during tensile test at cross-head speeds

of: (a) 1 mm/min, and (b) 100 mm/min. Cross-head displacements are indi-

cated on images. Arrows point to water flow on the surface.


explains the substantial water flow at 100 mm/min cross-

head speed, as shown in Fig. 11b.

It was previously proposed that gel volume decreased

with elongation, based on the variation of Poisson’s ratio.

Good agreement is found between the true strain at which

flow initiates in 2% agarose gels and the point in Fig. 5,

where Poisson’s ratio first exceeds 0.5. Therefore, the pro-

posal based on the DIC results is supported by direct ob-

servation of water leakage.

Evidence of water loss raises the suspicion that

increased specimen toughness (greater elongation and ulti-

mate stress) with increased strain rate might be caused by

the loss of water by effectively increasing the agarose

concentration. However, Fig. 12 illustrates that water flow

started at a lower stress level when the cross-head speed

was lower, presumably allowing more time for water flow

prior to failure at lower speeds and resulting in greater

loss of water as a percentage of total specimen mass.

Therefore, it can be concluded that the loss of water is

not the main cause of the increase in toughness.

Failure Strain

Previous studies determined the failure strain from a

measurement of the overall elongation at failure [2, 18],

so that the strain presented is a maximum average strain

in the gauge section, not a true local failure strain. If con-

siderable strain localization occurs before failure, the av-

erage strain is varied by the gauge length and the location

of strain localization. For those materials, failure strain is

the maximum local strain at the onset of failure. As aga-

rose gel showed significant strain localization, local fail-

ure strain needs to be measured.

One of the difficulties in measuring the local failure

strain lies in the prediction of the spot where strain is

localized. In this study, notched specimens shown in Fig.

1b were used to ensure that the strain localization

occurred at the notch. Failure strain was measured by

applying reverse DIC method to the images of the

notched specimen. Under this procedure, the base grid is

defined on the last image before crack initiation and grid

points are tracked backwards to their initial location in

the undeformed specimen. This allows precise determina-

tion of the failure strain with a fine grid at the location of

failure without having to analyze the entire gage section.

Figure 13 shows the strain distribution in the width

direction just before failure, which clearly shows significant

strain localization. In Fig. 13, the outermost grid point was

the one closest to the point of crack initiation and corre-

sponds to distance 0. The middle of specimen width corre-

sponds to a distance of 4 mm. Because of the strain concen-

tration induced by the notches, the local strain was the high-

est at the grid points closest to the notch. The local strain

decreased to a constant value of approximately 0.12 at a

distance of 1.5 mm and varied minimally from there to the

specimen center. As the highest local strain before failure

initiation was recorded at distance 0, this was taken as the

failure strain, as summarized in Table 2.Table 2 indicates that the local failure strain is approx-

imately 0.32 regardless of agarose concentration. There is

a weak trend of increasing strain with concentration, but

the differences are of the same order as the error, casting

doubt on its significance. The overall failure strains were

also constant with respect to concentration but were

roughly half as large as the maximum local strain. Nor-

mand et al. [18] reported a nominal strain at failure under

tension of about 0.14 for a similar type of agarose to that

used in this study which is similar to what was found

here. They also proposed that the nominal strains at fail-

ure are invariant regardless of the concentration, which is

also consistent with this study.

CONCLUSIONS

The stress–strain relationship of agarose gel was eval-

uated using the DIC method. A stress-strain equation for

FIG. 12. Dependence of flow stress on the cross-head speed.

FIG. 13. Strain distribution in sample width just before failure is initi-

ated. Agarose concentrations are indicated in image legend.

TABLE 2. Failure strain versus agarose concentration (w/v%).

1% 2% 4%

Local strain at

crack initiation

0.315 6 0.012 0.322 6 0.010 0.324 6 0.008

Overall strain at

crack initiation

0.162 6 0.018 0.158 6 0.015 0.160 6 0.022


the porous gel was proposed considering the presence of

water within the pores. The proposed equation exhibited

excellent agreement with experimental results for different

concentrations of agarose. Its validity was verified by a

FEM simulation where the generated load-displacement

curve is in good agreement with the experimental curve.

Deviation of the proposed equation from the experimental

results at high stresses is explained by the leakage of pore

water under these conditions. This leakage was predicted

by Poisson’s ratio values above 0.5 and confirmed by

direct observation.

The viscous behavior of agarose gel was investigated

through the tensile tests at different loading rates.

Increases in loading rate did not cause much difference in

the shape of stress–strain curve, but resulted in an

increase in ultimate stress and strain. To measure the

local failure strain, a reverse DIC method was proposed

and applied to notched agarose tensile specimens. Meas-

ured local failure strains were roughly twice as large as

the maximum overall strains which have conventionally

been used as a measure of agarose ductility.

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