Food Sci. Technol. Res., 14 (6), 525–532, 2008

Determination of the Viscoelastic Properties of Agar/Agar-Gelatin Gels Based on Finite

Element Method Optimization

Gyeong-Won Kim1*, Gab-Soo do

2, Yeonghwan Bae3 and Yasuyuki Sagara

4

1 National Institute of Agricultural Engineering, Rural Development Administration, 249 Seodun-dong, Gwonson-gu, Suwon, 441-707,

Korea2 Department of Bioenvironmental and Agricultural Engineering, College of Bioresource Sciences, Nihon University, 1866 Kameino,

Fujisawa-shi, Kanagawa, 252-8510, Japan3 Department of Industrial Machinery Engineering, Sunchon National University, 315 Maegok-dong, Sunchon, 540-742, Korea4 Department of Global Agricultural Sciences, Graduate School of Agricultural and Life Sciences, University of Tokyo, 1-1-1 Yayoi,

Bunkyo-ku, Tokyo, 113-8657, Japan

Received December 3, 2007; Accepted July 1, 2008

An FEM algorithm was developed to determine the viscoelastic properties of soft tissues of agar/agar-gelatin gels based on the curve-averaged data from stress relaxation experiment of parallel plate compres-sion and FEM optimization technique. This approach enabled more realistic and pertinent expression of the mechanical behavior of the gels than conventional methods, and allowed simultaneous and logi-cal characterization of all viscoelastic parameters, based on geometry, relating to both Prony series and Maxwell model such as elastic modulus, Poisson’s ratio, relative modulus, relaxation time, and dynamic viscosity, etc. Several assumptions were made in the FEM model such that the soft tissue materials were homogeneous in phase and isotropic, gravity effect was negligible, and the response was transient and controlled by displacement. To demonstrate the validity of the FEM model, the results of FEM optimiza-tion were compared with those of conventional method of nonlinear regression for agar/agar-gelatin gels, and also the predicted mechanical behavior of FEM on compressive creep as an interrelation with stress relaxation by the FEM model was compared with the experimental creep of 1% agar gel. The reliability of the FEM optimization method was confirmed by small stress deviation within 4.7% between experimental data and the FEM simulation using optimized parameters for stress-relaxation evaluation for agar/agar-gelatin gels and by strain deviation within 3.4% for creep prediction of 1% agar gel.

Keywords: agar, gelatin, gel, homogeneous structure, stress relaxation, creep, FEM optimization, viscoelastic properties

*To whom correspondence should be addressed.

Email: kim8415@rda.go.kr

IntroductionMechanical properties of solid/gel food materials are

considered as one of the most important factors in the de-

sign, manufacturing, and sensory evaluation of food. Agar/

agar-gelatin is widely used base material for food as well

as biomedical and industrial applications (Watase and Ara-

kawa,1968; Ishihara et al., 2000; Anand and Scanlon, 2002).

As food material, the agar/agar-gelatin gels possess the char-

acteristics of viscoelastic rheology properties (Kobayashi

and Nakahama, 1986) which are expressed by elastic modu-

lus, Poisson’s ratio, and relaxation time (Anand and Scan-

lon, 2002). To determine the viscoelastic properties of soft

food materials, indentation tests using flat-, cone-, and ball-

type indentor (Anand and Scanlon, 2002; Lu et al., 2006),

or compression tests using parallel plates (Kim et al., 1992)

were conducted, and then nonlinear regression analyses were

performed on viscoelastic models such as generalized Max-

well model for stress relaxation (Chen, 2000; Wang, 2003;

Telis et al., 2003), or Burgers model for compression creep

(Wang et al., 1992, Park, 1993). However, the viscoelastic

parameters obtained by these statistics-based analysis models

in general are insufficient to express the mechanical behavior

of original material since they do not reflect the geometry of

specimen (ANSYS, 2004; Kim et al., 2008). Furthermore,

separate specimens are needed to determine the mechanical

parameters such as elastic modulus, Poisson’s ratio, etc, and

thus the lack of uniformity among the specimens might re-

duce the reliability of the parameters especially for soft food

materials.

FEM analysis for food, fruits, and biological organisms

has been applied to predict or compare the mechanical be-

havior of the material of interest by utilizing the viscoelastic

parameters obtained from mechanical tests (Chen and De

Baerdemaeker, 1993; Wu and Pitts, 1999; Rong et al., 2004;

Lu et al., 2006). Kim et al. (2008) suggested a two-stage

FEM algorithm to determine the viscoelastic properties of

apple flesh based on homogeneous structure model. In the

first stage, they determined the elastic modulus and Pois-

son’s ratio of apple flesh specimen by using Prony series

model (Chen, 2000; Wang, 2003; ANSYS, 2004) and FEM

optimization method. In the second stage, they obtained

elastic modulus, relative elastic modulus, and relaxation time

based on stress-relaxation model. However, the difference in

textural properties between surface and core of apple flesh

makes the assumption of homogeneous phase questionable,

and also the process of determining viscoelastic properties

was a bit too complicated. It is considered to be very practi-

cal to develop a method of determining all pertinent visco-

elastic properties of soft and homogeneous food materials

simultaneously without performing multiple and independent

tests.

The objective of this research is to develop an FEM algo-

rithm of simultaneously determining viscoelastic properties

of agar/agar-gelatin gels based on the results of stress-relax-

ation tests. This methodology is expected to allow more re-

alistic expression of the mechanical behavior of soft textures

following generalized Maxwell model based on the results

of penetration, indentation, as well as parallel plate uniaxial

compression tests.

Materials and MethodsSpecimens for compression tests Agar gel of 1% con-

centration (weight per volume) was used as a food model

having homogeneous and isotropic structure. Pure agar re-

agent (Lot No. Mta6118, Nacalai Tesque, Inc., Tokyo, Japan)

was dissolved in distilled water solvent of 90 to 95℃ and

the obtained solutions were held at the same temperature for

5 minutes with magnetic hot stirrer. And then the solutions

were cooled down to 50℃ in room temperature condition

with continuous stirring. Finally, the solutions were poured

into acrylic cylindrical molds of 26 mm inner diameter and

40 mm height before they were transferred into an incuba-

tor to make gels and to get stabilized at 25℃ for 20 h. Also,

1.0% agar-0.5% gelatin, 1.0% agar-0.75% gelatin, and 1.0%

agar-1.0% gelatin gel specimens were prepared as additional

food models. To make agar-gelatin gels, the gelatin solu-

tions of pertinent concentrations were prepared by dissolving

gelatin reagent 061020 (Taiyokagaku Co., Ltd., Shinmatchi,

Yokkaichi, Mie, Japan) in distilled water at room tempera-

ture with magnetic stirring for five minutes, and they were

heated to 70℃. The two solutions of agar and gelatin were

mixed at 70℃ and cooled down to 50℃. The agar/agar-gel-

atin gel specimens were cut into pieces of 25 mm height at

one end of the cylindrical molds. The average heights of the

agar/agargelatin gel specimens are listed in Table 1. To avoid

spontaneous dehydration, all specimens were kept covered

with PARAFILM (ALCAN Packing Inc. Chicago, USA) un-

til they were tested.

Experiments and stress-relaxation model Three dif-

ferent types of compression experiments were conducted

with RHEONER Ⅱ texture analyzer (Yamaden Co., Tokyo,

Japan). The pre-contact force and trigger force were 0.04 N

and 0.01 N, respectively, and a constant cross-head speed of

0.1 mm/s was used. Prior to each compression experiment,

a piece of friction paper (Kimtowel wiper, KURESYA Inc.,

Tokyo, Japan) of 0.03 mm thickness was put on the lower

compression plate and the sectioned plane of each specimen

was placed on the friction paper. To determine the amount of

initial deformation in stress relaxation, compression rupture

experiments were performed with three agar gel specimens.

The force-deformation (F-D) curves of the compression

rupture experiments indicated that elastic region of initial

deformation was up to 8% strain. Therefore, 5% strain was

selected as the initial strain of agar/agar-gelatin gel speci-

mens for stress-relaxation experiments. The agar/agar-gelatin

specimens were deformed at a constant rate of 0.1 mm/s up

to the initial strain (ε0 = 0.05) and allowed to relax for 240 s.

Three nominal stress-time curves obtained from the stress-

relaxation experiments for each type of agar/agar-gelatin

g-W. Kim et al.

1% agar a1.0-g 0.5 a1.0-g 0.75 a1.0-g 1.0

24.80 24.03 24.41 24.59

a1.0-g 0.5 = 1.0% agar-0.5% gelatin, a1.0-g 0.75 = 1.0% agar-0.75% gelatin, a1.0-g

1.0 = 1.0% agar-1.0% gelatin.

Table 1. Average height of agar/agar-gelatin specimens (mm).

526

gels were averaged in time (curve-average) by using Micro-

cal Origin (Version 6, Microcal Software Inc., Northampton,

MA, USA) and this was utilized as the representing stress-

time curve for each agar/agar-gelatin gel type. The results of

nominal stress obtained by curve-average and FEM analysis

were compared at five points in time, t0 /1.5, t0 , 30 s, 60 s

and final relaxation time.

By utilizing Maxwell model, the viscoelastic parameters

for each agar/agargelatin gel type were obtained as the ini-

tial values of viscoelastic parameters of FEM optimization

and the results were compared with those of conventional

method based on nonlinear regression analysis. The param-

eters were estimated by Marquardt-Levenberg nonlinear

regression method (Chen, 2000) of Microcal Origin. The

generalized Maxwell model for each agar/agar-gelatin gel

with two parallel Maxwell elements can be expressed as Eq.

1 using equilibrium and decay moduli and relaxation times

of Prony series, i.e., a series of the form itn

ii e τα /

1

−

=∑ ˙ where αi

is the exponential coefficients with two terms (Chen, 2000;

Wang, 2003; ANSYS, 2004; Kim et al., 2008).

(1)

)]/)(exp()/)(exp([)( 2021010 ττεσ ttEttEEt e −−+−−+=

)( 210 EEEE e ++=

iii Eτη = ( 2,1=i )

where σ(t) and ε0 are nominal relaxation stress and nominal

initial strain, respectively. E0 and Ee are the instantaneous

and equilibrium elastic moduli, respectively. E1 and E2 are

the first and second decay elastic moduli, and τ1 and τ2 are

the first and second relaxation times of the Prony series, re-

spectively. ηi is the specific viscosity (dynamic viscosity) of

i-th term. t ( t ≥ t0 ) and t0 are current time and the past time

until the beginning of the relaxation, respectively.

FEM model and optimization algorithm A hexahedral

FEM element model was created by using ANSYS 8.1

(ANSYS Co., Canonsburg, PA, USA), a commercial FEM

program, to express the homogeneous and isotropic structure

for agar/agar-gelatin gel specimens. The elements used for

homogeneous model had three degrees of freedom at each

node: translations in the nodal x, y, and z directions. The ge-

ometry of this model was a three-dimensional cylinder and

made symmetrical to the three axes to save computer run

time. Every element in this model was meshed to a length of

about 1.0 mm with the exception of the height being about

2.0mm by a quadrilateral meshing key in ANSYS, 2004 (Fig.

1).

The procedure of FEM optimization is illustrated in Fig.

2. The FEM optimization techniques applied in this research

were to simulate the same experimental conditions as the

compression experiments (Kim et al., 2008), and to obtain

optimum viscoelastic parameters such as elastic modulus (E),

Poisson’s ratio (ν), relative moduli (β1E , β2

E ) and relaxation

time (τ1 , τ2) in the Prony series which minimized the sum of

nominal stress errors at the five points in time by Sequential

Unconstrained Minimization Technique (SUMT) algorithm

which is used to solve an unconstrained problem using pen-

alty functions (Kim et al., 2008; ANSYS, 2004).

The viscoelastic parameters obtained from the Maxwell

model were utilized as initial estimates for FEM simula-

tion, and Poisson’s ratio of 0.4 was adapted from literature

(Anand and Scanlon, 2002). The difference in nominal stress

between the experiments and FEM was used as a state vari-

able and its value was maintained below 1% at initial strain

(ε0) and below 3% at the other four points in time to enforce

convergence during FEM optimization.

The bounds for the design variables are: E0 ± 50% for

elastic modulus, 0.3 to 0.5 for ν , 0.01 to 1.0 for β1E and β2

E,

and 0.1s to 1000 s for τ1 and τ2 . And all FEM optimization

analyses were transient without gravity effect and controlled

by displacement.

In the kernel function of FEM analysis, it was assumed

Viscoelastic Properties of Agar/Agar-Gelatin Gels by FEM

Fig. 1. Original experiment geometry (left) and the simplified symmetric FEM geometry model of agar/agar-gelatin gel.

527

that the quotient of decay modulus and instantaneous modu-

lus for each term of Prony series was held constant for elas-

tic, shear, and bulk modulus components as follows:

0

i

0

i

0

i

KK

GG

EE ===== K

iGi

Ei βββ ( 2,1=i )

(2)

where β1E , β1

G , and β1K indicate relative decay moduli for

Young’s, shear, and bulk in relaxation, respectively, for the

i-th term of the Prony series. Also, G i and K i are the i-th

term of decay shear modulus and decay bulk modulus, re-

spectively. Eq. 3 is the constitutional equation of stress which

is composed of shear and bulk kernel functions (ANSYS,

2004; Kim et al., 2008) based on hereditary integration for

FEM Cauchy stress or true stress.

∫ ∫ Δ−+−=tt

ddd

tKIddde

tG00

)()(2 ττ

τττ

τσ (3)

where G(t) is the kernel function for shear relaxation and K(t)

is for bulk relaxation. Also, σ and I are the Cauchy stress and

unit tensor, respectively; τ is past time; and e and Δ indicate

deviatory strain and volumetric strain, respectively. Objec-

tive function for the FEM optimization algorithm is given by

the following equation:

Stress_diff. = ∑ ∑==

−5

1

5

1Re )(

iiFEMiExperiigi iiii

wworw σσσ (4)

where σReg and σFEM are nominal stresses of nonlinear regres-

sion model and FEM, respectively.

Evaluation of FEM optimization Initial validations

were made on the FEM simulation results of stress relax-

ation for agar/agar-gelatin gel specimens with reference to

the experimental data. Two sets of viscoelastic parameters,

one by FEM optimization (FEM using optimization) and

the other by nonlinear regression (FEM using regression),

were fed into each FEM model for the simulation of valida-

tion, and the nominal stress values at the initiation and end

of relaxation for each case were compared with those of

experimental averages. The second validations were made

with the set of viscoelastic parameters obtained by FEM

optimization on the compressive creep behavior of 1% agar

specimens. Therefore, separate creep experiments of three

specimens (average height of 24.51 mm) for the validation

g-W. Kim et al.

Fig. 2. Flow diagram of FEM optimization procedure to determine the viscoelastic properties of agar/agar-gelatin gels.

528

of the parameters were carried out with 1% agar gels using

a constant load speed of 0.072 N/s up to 11.9 s and holding

the load of 0.86 N until it being removed at 311.9 s. It was

assumed that the generalized Maxwell model and hereditary

integration for FEM Cauchy stress (Eq. 3) were not sufficient

to predict the compressive creep behavior of viscoelastic

materials since there are two different deformation natures

on Newtonian dash pots in Burgers model which consists of

a Maxwell element joined in series with a Voigt (or Kelvin)

element (Park, 1993). Therefore, recovery behavior was ig-

nored and only unidirectional loading was considered in the

prediction of creep behavior in the FEM creep analysis. Two

simulation tests were performed on the analysis of creep be-

havior: quasi-static loading as in the case of stress-relaxation

analysis and dynamic loading considering gravity effect in

transient analysis. The strains at the times of creep initiation

and unloading were evaluated by the two simulation tests

and compared with experimental data.

Results and DiscussionNominal stress The stress-relaxation experiments for

1% agar gel specimen provided nominal stress values of 1.061

± 0.029 kPa,1.626 ± 0.058 kPa, 1.382 ± 0.040 kPa, 1.221 ±

0.033 kPa, and 0.901 ± 0.015 kPa at 8.1 s (t0 /1.5), 12.4 s (t0),

30 s, 60 s, and 240 s, respectively. The representing curve

through curve averaging in time provided nominal stress val-

ues of 1.070 kPa, 1.620 kPa, 1.381 kPa, 1.218 kPa, and 0.898

kPa at the five time points as above, which were within ±

0.84% of the average value at each time point. Similar curves

were obtained by curve averaging for the three types of agar-

gelatin gels and they were utilized for non-linear regression

analysis and to compare directly with the results of FEM op-

timization.

Table 2 shows the values of the objective function (Eq. 4,

sum of stress errors) ranging from 0.039 to 0.103 kPa for the

four types of agar/agar-gelatin gel specimens as a result of

FEM optimization. Also the final values of the state variables

(nominal stresses at t0 /1.5, t0 , t1 , t2 , and t3 ) resulted in 1.073

to 0.893 kPa for 1% agar gel. At t0 , which determines instan-

taneous elastic modulus, FEM nominal stress decreased to

1.491 kPa as the gelatin contents increased to 1.0% in 1.0 %

agar specimen. The difference in nominal stress values be-

tween FEM analysis and experimental data remained within

5.1% for the four types of agar/agar-gelatin gels. Especially

at the initial stage of stress-relaxation, t0 , the difference in

stress values ranged from -2.4 to -1.0%.

Viscoelastic properties The viscoelastic parameters of

1% agar solution obtained by FEM optimization were: E =

31.787 kPa, ν = 0.437, β1E = 0.128, β2

E = 0.45, τ1 = 14.41 s,

and τ2 = 159.62 s. As in the case of the elastic modulus by

nonlinear regression analysis, the modulus obtained by FEM

analysis decreased as the gelatin contents increased, and the

differences in the values obtained by the two methods were

within 2.7%.

As shown in Fig. 3(a), the FEM elastic modulus of 1%

agar-1% gelatin was reduced by 6.6% than that of 1% agar

gel. In Fig. 3(b) the FEM Poisson’s ratio for all gels con-

verged well within the boundary condition values of 0.3 to

0.5 which were adapted from literature (Anand and Scanlon,

2002). Adding 0.5% gelatin to 1% agar gel reduced Pois-

son’s ratio to 0.362 (-17%), while adding 0.75% gelatin and

1% gelatin increased Poisson’s ratio to 0.390 and to 0.403,

Viscoelastic Properties of Agar/Agar-Gelatin Gels by FEM

Samples

0t /1.5 0tt = Relaxed time ( 0tt − ) Objective

function

(kPa)

Eq. (4)

1tt = 2tt = 3tt =

t (s) kPa t (s) kPa kPa kPa kPa

1.0 % agar 8.301.073

12.41.604 1.422 1.267 0.893

0.039(0.3) (-1.0) (3.0) (4.0) (-0.6)

a1.0-g 0. 5 8.001.015

12.021.529 1.409 1.303 0.987

0.076(5.1) (-2.4) (1.1) (1.2) (-4.2)

a1.0-g 0.75 8.130.996

12.201.507 1.373 1.269 1.024

0.103(4.4) (-1.6) (-0.1) (-1.4 ) (-4.7)

a1.0-g 1.0 8.200.994

12.301.491 1.361 1.292 1.105

0.070(4.2) (-2.3) (-0.6) (0.4) (0.0)

Table 2. FEM optimization results of state variables (nominal stresses (kPa)) at five points in time and the objective function values in stress-relaxation for agar/agargelatin gel specimens.

529

respectively. The maximum difference in the relative moduli

(β1E , β2

E ) between the two methods, FEM and nonlinear

regression, was recognized as up to 42.8%. The variation

in FEM β1E among the agar/agar-gelatin gels was relatively

small as compared with FEM β2E , which was reduced mark-

edly by 45.1% ( β2E of nonlinear regression was 37.9%) when

1% gelatin was added to 1% agar gel. The results discussed

above illustrated that the relative moduli were related to the

reduction in the equilibrium modulus (Eq. 1) and the final re-

duction in the magnitude of stress-relaxation. The maximum

differences in relaxation time (τ1 , τ2 ) obtained by the two

methods were 22.5% for τ1 and 6.1% for τ2 .

As indicated in Table 3, Maxwell model (Eq.1) analysis

of 1% agar gel for viscoelastic parameters obtained by FEM

optimization produced Ee =13.414 kPa ( E0 =31.787 kPa),

E1 =4.069 kPa, E2 =14.303 kPa, η1 =58.63 kPa·s, and η2 =2,283.23 kPa·s. The values of Ee increased greatly with

the increase in the contents of gelatin, addition of 1% gelatin

to 1% agar of gel mixtures resulted in Ee of 18.791 kPa, an

increase by 40.1%. The Ee values by FEM optimization were

lower than those by nonlinear regression by 6.2 to 14.0%.

The values of dynamic viscosity for the four types of agar/

agar-gelatin gels by FEM optimization were in the range of

25.10 to 58.63 kPa·s for η1 and 1270.71 to 2359.46 kPa·s for η2 , the maximum differences were 77.3% for η1 and 14.8%

for η2 when comparing the results obtained by the two meth-

ods.

Each of the viscoelastic properties (Prony series param-

eters and Maxwell parameters) of agar/agar-gelatin gels

obtained by FEM optimization at an initial strain of 5%

exhibited similar trend to the nonlinear regression param-

eters in variation according to the contents of gelatin in the

gels. However, the maximum differences in the viscoelastic

parameters between FEM optimization and nonlinear regres-

sion for agar/agar-gelatin gels were as small as 2.7% for

instantaneous elastic modulus, 14% for equilibrium elastic

modulus, 42.8% for relative elastic modulus, and as large as

g-W. Kim et al.

(a) Elastic modulus (b) Poisson’s ratio

(c) Relative moduli (d) Relaxation time

Fig. 3. Effect of added gelatin contents (wt.%) on the viscoelastic parameters of 1.0 wt.% agar-gelatin gel solution using FEM optimization.

530

77.2% for specific viscosity. These discrepancies might have

been induced by the facts that the nonlinear regression, un-

like FEM optimization, has no provision for geometry, Pois-

son’s ratio, and the incipient-mechanical behavior up to the

initial strain of stress relaxation (Kim et al., 1992; Cheng,

2000); and is based only on statistical manipulation of the

coefficients of the exponential decay function.

Evaluation of stress relaxation FEM stress-relaxation

simulation for the four types of agar/agar-gelatin gel speci-

mens was carried out by using two sets of viscoelastic pa-

rameters to compare the performance of FEM optimization

method (FEM using optimization) and conventional method

of nonlinear regression (FEM using regression), and the

resulting nominal stresses at t0 and t3 were evaluated with

reference to experimental values. For four types of gels, the

errors in the resulting nominal stress values for FEM using

optimization ranged from -1.6 to -2.4% (1% agar- 0.5% gela-

tin) of experimental value at t0 and from -4.7 (1% agar- 0.75%

gelatin) to 0.0% at t3 . However, at t0 and t3 the errors in the

nominal stress values for FEM using regression ranged from

-7.5 (1% agar- 0.75% gelatin) to 1.0% and from 6.2 to 7.0%,

respectively (Fig. 4). This indicated that the results of FEM

optimization method explained the stress-relaxation behavior

of agar/agar-gelatin gels better than those of nonlinear re-

gression method.

Prediction evaluation of FEM optimization Prior to the

evaluation of FEM simulation for compressive creep be-

havior, the applied force was determined by dynamic FEM

simulation based on the experimental reaction force. The ap-

plied force thus obtained was 0.99 N, and the reaction force

maintained the experimental value of 0.86 N. There were

-3.5% and -6.4% discrepancies from experimental values

at creep beginning time (11.9 s, 5.2% strain by experiment)

and at unloading time (311.9 s, 9.1% strain by experiment),

respectively, when quasi-static FEM simulation (FEM Opt.)

was applied for creep analysis. On the other hand, FEM us-

ing dynamic (with gravity effect) resulted in 3.4% and 0.0%

errors at creep beginning time and unloading time, respec-

tively (Fig. 5).

Therefore, the effect of dynamic mechanical behavior

was not negligible in the creep behavior of 1% agar gel un-

der constant compression loading. The creep behavior of

Viscoelastic Properties of Agar/Agar-Gelatin Gels by FEM

Items

)]/)(exp()/)(exp([)( 2021010 ττεσ ttEttEEt e −−+−−+=

eE

(kPa)1E

(kPa)1τ

(s)1η

(kPa·s)2E

(kPa)2τ

(s)2η

(kPa·s)

1.0% agarFEM 13.414 4.069 14.41 58.63 14.304 159.62 2283.23

Reg. 14.896 4.330 12.79 58.53 12.791 161.08 2060.34

a1.0-g 0.5FEM 14.934 2.309 10.87 25.10 12.745 185.13 2359.46

Reg. 17.367 2.881 10.58 39.23 10.584 197.13 2086.39

a1.0-g 0.75FEM 16.598 3.468 14.97 51.91 9.574 180.05 1723.75

Reg. 19.196 2.559 8.40 29.28 8.402 178.64 1500.92

a1.0-g 1.0FEM 18.791 3.562 9.58 34.13 7.332 173.3 1270.71

Reg. 20.038 2.521 7.45 31.15 7.447 174.58 1300.06

Table 3. Comparison of Maxwell model parameters of FEM optimization with those of nonlinear regression for stress relaxation of agar/agar-gelatin gel specimens ( ε0 = 0.05).

Fig. 4. Comparison of nominal stress of two FEM simulations using FEM optimization viscoelastic parameters and using parameters ob-tained from nonlinear regression with that of experiment of 1%-agar gel at initial strain of 0.05 relaxed for 240 seconds.

531

1% agar gel could be well predicted by the parameters from

stressrelaxation analysis once gravity effect was taken into

account. Considering the validity of FEM optimization meth-

od in analyzing stress-relaxation and creep behavior, it was

expected that this approach could be applied to determine

mechanical properties of various viscoelastic materials with

relatively high accuracy. Also, the geometry based FEM opti-

mization method is expected to be applied for frozen-thawed

and other biomaterials which are often difficult to analyze by

conventional methods.

Acknowledgement This work was supported in part by the Japan

Society for the Promotion of Science (JSPS). The JSPS RON-

PAKU Program is gratefully acknowledged by Gyeong-Won Kim

(KOSEF-10603).

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Fig. 5. Comparison of strain (%)-time values of two FEM creep simulations (dynamic behavior with gravity (FEM Opt. using dy-namic) and quasi-static without gravity (FEM Opt.)), using the same FEM optimization parameters-values, to those of experiment of 1%- agar gel.

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