Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

49, 4, pp. 1203-1216, Warsaw 2011

A COMPARISON BETWEEN THE EXPONENTIAL AND

LIMITING FIBER EXTENSIBILITY PSEUDO-ELASTIC

MODEL FOR THE MULLINS EFFECT IN ARTERIAL TISSUE

Eva Gultova, Lukas Horny, Hynek Chlup
Czech Technical University in Prague, Faculty of Mechanical Engineering,

Department of Mechanics, Biomechanics and Mechatronics, Prague, Czech Republic

e-mail: eva.gultova@fs.cvut.cz; lukas.horny@fs.cvut.cz; hynek.chlup@fs.cvut.cz

Rudolf Zitny
Czech Technical University in Prague, Faculty of Mechanical Engineering,

Department of Process Engineering, Prague, Czech Republic

e-mail: rudolf.zitny@fs.cvut.cz

This study compares the capability of two differentmathematical forms
of the so-called softening variable to describe strain-induced stress so-
ftening observed within cyclic uniaxial tension of the human thoracic
aorta. Specifically, the softening variable, which serves as the stress re-
duction factor, was considered to be tangent hyperbolic-based and error
function-based.Themechanical response of the aortawas assumed to be
pseudo-hyperelastic, incompressible and anisotropic. The strain energy
density function was employed in a classical exponential form and in a
not well-known limiting fiber extensibility model. This study revealed
that both the limiting fiber extensibility and exponential models of the
strain energy describe mechanical the response of the material with si-
milar results. It was found that it is not a matter which kind of the
softening variable is employed. It was concluded that such an approach
can fit the Mullins effect in the human aorta, however the question of
the best fitting model still remains.

Key words: aorta, limiting fiber extensibility, Mullins effect

1. Introduction

Due to cardiac cycle, arteries are subjected to cyclic loading and unloading in
their physiological conditions. In vitro, the mechanical response of arteries is
mostly studiedwithin the cyclic inflation-extension and the tensile test.Oneof



1204 E. Gultova et al.

the irreversible effects observed during these experiments is theMullins effect.
This strain-induced softening phenomenon is well known in elastomer mecha-
nics. It is characterized by the following features: when a so-called virgin ma-
terial (previously undeformed) is loaded to a certain value of the deformation,
the stress-strain curve follows the so-called primary loading curve. Subsequ-
ent unloading exhibits the stress softening. Next reloading follows the former
unloading curve until the previous maximum strain is reached. At this mo-
ment, when the previous deformation maximum is exceeded, the stress-strain
path starts to trace the primary loading curve (Diani et al., 2009). This defi-
nition describes quite ideal material behavior.Within the real experiment the
unloading and the reloadingmay notmatch exactly due to the hysteresis. The
comparison between the ideal and true cyclic softening is depicted in Fig.1.

Fig. 1. (a) IdealizedMullines effect, (b) true experimental data

The elastic response of blood vessel walls is significantly nonlinear, ani-
sotropic and requires large strains to be incorporated (Holzapfel et al., 2000;
Humphrey, 2003). Such behavior is usually modeled within the framework of
hyperelasticity which presumes existence of an elastic potential (strain energy
density function, SEDF), and constitutive equations are obtained by differen-
tiation of the elastic potential with respect to the strain tensor.
Although existence of the stress softening within in vitro cyclic loading

of blood vessels has been known for long time only few attempts have be-
en made to develop new theories. Fung et al. (1979) proposed to model the
mechanical response of arteries as pseudo-elastic. In this concept, an artery
wall is considered to be elastic, however loading and unloading responses are
defined with different constitutive equations. Nowadays, models capture the
Mullins effect using either the Continuum Damage Mechanics (CDM) or the
pseudo-elasticity theory.
Damage models describe the Mullins effect incorporating a damage para-

meter which serves as a reduction factor in the strain energy density func-



A comparison between the exponential and limiting fiber... 1205

tion. The damage parameter is considered as an internal variable (Simo, 1987)
and can be applied in a more general manner for an arbitrary irreversible
process (Holzapfel, 2000, chap. 6.9-11). The damage parameter can be full-
strain-history dependent (continuous damage) or maximum-value dependent
(discontinuous damage).

Another concept describing theMullins effect is the theory of the pseudo-
elasticity developed by Ogden and Roxburgh (1999). They suggested the in-
troduction of the a variable (softening variable) into the strain energy function
which is thereafter called the pseudo-strain energy density function.The softe-
ning variable then governs the energy density and switch on and off between
the primary and softened response of a material. The particular model of the
pseudo-energy function suggested byOgden andRoxburgh (1999) results in si-
milar symbolic formas inCDM.Successful application of the pseudo-elasticity
in rubbermechanics were reported byDorfmann andOgden (2003, 2004) and
Eĺıas-Zúñiga (2005). Peña andDoblaré (2009) proposed the application of this
theory for anisotropic biologicalmaterials consideringdifferent softeningvaria-
bles for an extracellular matrix and fibers. This model successfully described
the softening behavior of sheep vena cava.

The aim of this paper is to compare the pseudo-elasticmodels for theMul-
lins effect using different forms of the SEDF. In the first case, the mechanical
response of the artery wall is described with the strain energy function adop-
ted in the exponential form (Holzapfel et al., 2000). In the second case, not
well-known the limiting fiber extensibility form of the SEDF is applied. The
comparison is shownbyfitting the experimental data recordedwithin uniaxial
tension of human thoracic aorta. The artery is considered to be nonlinear,
incompressible and anisotropic continuum. Here we focus only on the strain-
induced softening. Temperature, heat and strain-rate effects are not concerned
as well as the active response (smooth muscle fibers) of arterial tissue.

2. Methods

2.1. Experiment – uniaxial tension

In order to illustrate the Mullins effect in human aorta, cyclic uniaxial
tension testswereperformedonMTSMiniBionix testingmachine (MTS,Eden
Prairie, USA). Samples of healthy human thoracic aorta were resected from
cadaveric donor (male, 47 years old)with the approval of theEthicCommittee
of theUniversityHospitalNaKralovskychVinohradech inPrague.Respecting



1206 E. Gultova et al.

the anisotropy of an aorta, samples were resected in the circumferential and
longitudinal direction. The total number of samples was eight (five oriented
longitudinally, three oriented circumferentially).
Five levels of maximum stretch were performed during the tests: λm =

= 1.1, 1.2, 1.3, 1.4 and λm = 1.5, where λm is the maximum ratio between
the current length l and the referential length L. The representative of the
recorded data is shown in Fig.1b. Each λm level was preformed as four-cycle
of the loading and unloading. Considering the incompressibility of the tissue,
the loading stress was obtained according to the following relation

σ=
F

s
=
Fl

LBH
(2.1)

Here F denotes the applied force and s the current cross-section. B and H
denote thewidth and the thickness of the sample in the reference (zero-stress)
configuration. Dimensions of the samples in the reference configurations were
determined within the analysis of digital photographs (thickness) and by a
caliper (length and width). Strain rate was 120mm/min.

2.2. Constitutive modeling – pseudo-elasticity

A deformation is considered as the homeomorphic mapping ϕ between
material particles embedded in the reference Cartesian coordinate system
{O;X1X2X3} and the same particles embedded in the spatial Cartesian coor-
dinate system {O;x1x2x3}. The reference position vector X is mapped onto
x according to x = ϕ(X). The deformation is then described with the de-
formation gradient F = ∂ϕ/∂X which generates right Cauchy-Green strain
tensor C=F⊤F. Within the modeling of the uniaxial tension we restrict the
attention only to pure homogenous strains xi = λiXi (i = 1,2,3). Thus the
deformation gradient is of the form F= diag[λ1,λ2,λ3].
The response of the artery during the primary loading by uniaxial tension

is modeled as incompressible, hyperelastic and anisotropic. The anisotropy
is generated with two preferred directions in continuum which are perfectly
aligned with β and −β angles. These angles lay in the X1X2-plane of the
sample. It is assumed that both preferred directions are mechanically equiva-
lent andhence the resulting degree of anisotropy is called local orthotropy (for
details see p.272 in Holzapfel, 2000; or in Holzapfel et al., 2000). The hype-
relastic behavior of the material is determined by the strain energy density
function W0. Here index 0 denotes the primary loading response. The stored
energy is additively decoupled into isotropic and anisotropic part

W0 =W
ISO
0 (I1)+W

ANISO
0 (I4,I6) (2.2)



A comparison between the exponential and limiting fiber... 1207

I1, I4 and I6 denote the invariants of the Right Cauchy-Green deformation
tensor C. The first invariant of C can be expressed as I1 = λ

2
1 +λ

2
2 +λ

2
3.

The invariants I4 and I6 arise from material anisotropy. Due to the mecha-
nical equivalency between the preferred directions I4 = I6. Therefore W0 is
considered to be W0 =W0(I1)+2W0(I4).

Two specific forms of strain energy (2.2) were incorporated. The first one
corresponds to the exponential function proposed by Holzapfel et al. (2000)

WHGO0 =
c

2
(I1−3)+

k1
k2

[

exp
(

k2(I4−1)
2
)

−1
]

(2.3)

where cand k1 denote stress-likematerial parameters and k2 is thedimension-
less parameter. Such a model is invariant-based modification of the classical
Fung-type exponential model which was many times successfully applied in
soft tissue biomechanics. The second form of the SEDFwas incorporated via
the limiting fiber extensibility model proposed by Horgan and Saccomandi
(2005), see equation (2.4)

WHS0 =
c

2
(I1−3)−µJf ln

(

1−
(I4−1)

2

J2
f

)

(2.4)

Here c and µ are stress-like material parameters and Jf is the dimensionless,
so-called limiting extensibility parameter. Both WHGO0 and W

HS
0 include the

isotropic Neo-Hookean term linked with the matrix of biological composite.
The material nonlinearity, related to the presence of wavy collagen fibers in
the soft tissue is, however in (2.3) and (2.4), captured in significantly different
manners. Exponential function (2.3) reflects the famous result of Y.C. Fung
that the stiffness is proportional to the applied stress. In contrast to (2.3)
the limiting extensibility model restricts admissible deformation to a certa-
in maximum value under which the stored energy approaches infinity. The
deformation admissible in (2.4) has to satisfy condition

(I4−1)
2

J2
f

< 1 (2.5)

which implies that I4 < Jf +1. When the unit vector aligned with the pre-
ferred direction M = cos(β)E1 +sin(β)E2 is considered then the additio-
nal invariant I4 can be expressed as I4 = M(CM). Combining equations
F= diag[λ1,λ2,λ3]C=F

⊤
F, and I4 =M(CM) we arrive at

I4 =λ
2
1cos

2β+λ22 sin
2β (2.6)



1208 E. Gultova et al.

Now it is clear that I4 can be considered as the square of the stretch in
the preferred direction that must be invariant under a change of the frame of
reference. Here introduced the limiting fiber extensibility model was proposed
by Horgan and Saccomandi (2005) and is adopted with minor modification
in the square of Jf. It may be regarded as the extension of the simple phe-
nomenological model originally proposed by Gent (1996) which is suitable to
capture large-strain stiffening behavior of isotropic materials. The applicabili-
ty of such a class ofmodels in soft tissuemechanicswas pointed out byHorgan
and Saccomandi (2003) andOgden and Saccomandi (2007). It is worth noting
that limiting fiber extensibility model (2.4) offers some advantages. Incorpo-
rating SEDF in form (2.4) one can obtain closed analytical solutions of some
boundary-value problems important in blood vessel biomechanics like e.g. an
inflation-extension of a thick-walled tube (Horny et al., 2008). This is in con-
trast to classical (Fung-type) exponential models. Constitutive equations for
the primary response of the hyperelastic incompressible material are now ob-
tained as expressed in (2.7). Here the principal stresses are denoted by σ0i,
and p0 denotes a Lagrange multipiler associated with the incompressibility
constraint λ1λ2λ3 =1

σ0i =λi
∂W0
∂λi
−p0 i=1,2,3 (2.7)

In order to reproduce softened behavior during unloading and reloading,
we introduce the softening factors η into constitutive equations (2.7). Thus
the same form of the strain energy W0 still takes place here

σi = ηλi
∂W0
∂λi
−p i=1,2,3 (2.8)

Thestress is reducedby the factor η∈ [0;1]. In this studyweonly concernwith
the idealizedMullins effect, thus unloading and reloading pathsmatch exactly.
Now the softening factors must govern the constitutive equations into expres-
sion (2.7) for the primary loading and into (2.8) for the unloading/reloading.
Within uniaxial tension of the sample in the direction j, it is satisfied by
definition

η=

{

1 for λj =λjmax

η(λ1,λ2,λ3) for λj <λjmax
(2.9)

Definition (2.9) says that if the sustained stretch in thedirection of the loading
is maximal then there is no softening. And when the sustained stretch in the
direction of the loading is smaller than the maximum value in the history of
the loading then the softening occurs.



A comparison between the exponential and limiting fiber... 1209

A particular form of the mathematical expression for η must be defined.
We adopt forms for softening factors as was originally introduced by Ogden
andRoxburgh (1999) andDorfmann andOgden (2003). Hence, let η be of the
form

η=1−
1

r
f
(Wm−W0(λ1,λ2,λ3)

m

)

(2.10)

where f(t) is erf (t) or tanh(t). Wm denotes the maximum value of W0
reachedwithin the loading history. r and m are real positive parameters. The
resultingmodel, regardless if erf (t) or tanh(t) is operative in (2.10), contains
sixmaterial parameters. It is explicitly c,µ, Jf,β, r and mwhen the limiting
fiber extensibility model WHS0 is applied, and c, k1, k2, β, r and m in the
case of the exponential model WHGO0 .

3. Results – fitting the model

The capability of the introducedmodels was tested within the regression ana-
lysis of uniaxial tension experimental data.The total numberof tested samples
was eight and they exhibited similar results.Only onepair of sample (one strip
in the circumferential and one in the longitudinal direction of the artery) was
considered for the regression. The selected samples were obtained from one
donor and resected in the same region of the thoracic aorta.

With respect to anisotropy exhibited byhumanarteries, W0 has to be con-
sidered as a function of λ1, λ2 and λ3. Incorporating the incompressibility as-
sumption, the out-of-plane stretch λ3 is eliminated. However, our experimen-
tal equipment does not allow one tomeasure transversal stretches. In order to
overcome this drawback we employed the boundary condition σtransversal =0
which is used to calculate transversal stretches upon the uniaxial state of
stress. The data from the circumferential and longitudinal experiment were
optimized simultaneously to find the minimum of objective function

Q=
1

Mean2(σEXPi,j )

2
∑

i,j=1

∑

k

(

σEXPi,j −σ
MOD
i,j

)2

k
(3.1)

Here upper indices EXP and MOD indicate the experimental observation
and the model prediction, respectively. The observed stresses were calculated
according to (2.1). Model predictions were based on equation (2.8) incorpo-
rating the definition of stress reduction factors (2.9) and (2.10). The lower
indices i and j are operative representing the direction of demanding stress



1210 E. Gultova et al.

and the direction of the uniaxial tensile test, respectively. It means that σi,j
denotes the stress in the direction i during the loading in the direction j.
Only the data from the first two cycle-levels were included in the regression
(λm =1.1 and 1.2). The regression was performedwith the optimization pac-
kage inMaple 13 (Maplesoft, Waterloo, Canada).
Both WHGO0 and W

HS
0 were used in order to compare their suita-

bility. The softening factor η was employed in both mathematical forms;
η=1−r−1 tanh(t) and η=1−r−1erf (t). The estimatedmaterial parameters
are listed inTable 1.Themodel predictions are comparedwith the experiment
in Fig.2. The regression results were also checked on the condition I4 > 1.
Because I4 models reinforcement with collagen fibers, theymay contribute to
the stored energy only in tensile strains. It was found that this condition was
satisfied in all data points. Figure 3 shows the stress ratio σ0/σ computed
from the experimental data which can be considered as the observation of η.

Fig. 2. (a), (b) TheMullins effect – circumferential strip, (c), (d) theMullins
effect – longitudinal strip

Fig. 3. The softening variable



A comparison between the exponential and limiting fiber... 1211

4. Discussion

This study presents the comparison between two (primary) strain energy den-
sity functions, WHS0 and W

HGO
0 , used in the pseudo-elastic model for the

Mullins effect observed within the periodic uniaxial tension of arterial tis-
sue. The strain-induced stress softening has been described by means of the
stress-reduction factors η which can be simply considered as the stress ra-
tio σ0i/σi, where σ0i takes place during the primary loading and σi corre-
sponds to the softened behavior. A particular mathematical form of η has
been adopted from the pseudo-elasticity theory introduced byOgden andRo-
xburgh (1999) and Dorfmann and Ogden (2003, 2004). These reduction fac-
tors, η=1−r−1 tanh(t) and η=1−r−1erf (t) were reported to be successful
in the description of theMullins effect observed in particle-reinforced rubber,
and herein were used for healthy human thoracic aorta. Based on the com-
parison in Fig.2, one can conclude that the primary response of the aorta
can be successfully modeled by both WHS0 and W

HGO
0 strain energy func-

tions. But the results of predictions obtained for the softening behavior are
not quite satisfactory. The data suggests that it is not a matter if tanh(t) or
erf (t) is employed in the model for the softening variable (reduction factor).
Almost the same predictions are also obtained by incorporating WHS0 and
WHGO0 into the softening model. Nevertheless, the quality of the models is
controversial. They can mimic the softening behavior in principle, however
the character seems to be almost piecewise linear which is in contrast to the
experimental data. It was tried to find parameters with better approxima-
tion ability but the optimization procedure always converged to the presented
parameters.

Model parameters obtained within the minimization of objective function
(3.1) are listed inTable1.Numericvalues ofparameters obtained for WHS0 and
WHGO0 are similar, which confirms the graphically displayed results in Fig.2.
The shearmodulus relatedwith theNeo-Hookean term,which is usually linked
to the response of isotropic matrix, was found to be almost the same in every
model (∼ 110kPa).This is slightly higher thanusually reported values around
tens of kPa. Stress-like parameters in thenonlinear termsof WHS0 and W

HGO
0 ,

µ and k1, were obtained in hundreds of kPa, which is in accordancewith some
values summarized in Holzapfel (2009).

There are only few papers reporting the limiting extensibility parame-
ter Jf. In our previous studies, values of Jf ranging from 0.1 up to 1.044
were found (Jf =0.1202 for thoracic aorta inHorny et al. (2010); Jf =1.044
for abdominal aorta in Horny et al. (2008); Jf = 0.7498 for saphenous vein



1212 E. Gultova et al.

coronary artery bypass graft after 35 months of remodeling in Horny et al.
(2009); and Jf ≈ 0.3 for human vena cava Horny et al. (2011)). Ogden and
Saccomandi used Jf = 0.422 in their simulation (Ogden and Saccomandi,
2007).

The parameter β is interpreted as the orientation of reinforcement fibers.
The estimated value is around 52◦. However, the artery wall is significan-
tly heterogeneous and fibers show dispersed alignment (Gasser et al., 2006).
Thus, this parameter without histological observation is rather phenomeno-
logical.

The softening parameter r (dimensionless) was reported to be 1.05 for
soft-bodied arthropod (Dorfmann et al., 2007) and 1.105 for vaginal tissue
and sheep vena cava (Peña et al., 2009). Our values are also of the order of
unity (approx. r = 2.6). The parameter m was obtained as m=0.00725 for
WHGO0 and η=1−r

−1 tanh(t); m=0.0082 for WHGO0 and η=1−r
−1erf (t);

m= 0.735 for WHS0 and η = 1− r
−1 tanh(t); and m= 0.937 for WHS0 and

η=1−r−1erf (t). It can be comparedwithOgden andDorfmann (2003) who
found it to be 0.3 (for particle reinforced rubber), andDorfmann et al. (2007)
whok reported 0.0038 (for muscle of soft-bodied arthropod).

For the sake of completeness, we have to note that the primary response
of thematerial was fitted at first (parameters c, k1, k2, β in W

HGO
0 ; and c,µ,

Jf, β in W
HS
0 ). Subsequently, the regression of r and mwas performedwith

fixed values of the parameters in W0. Such away of the fitting procedurewas
established due to still remaining lack of clear (physical) interpretation of the
softening parameters.

Finally, the employed model for the softening variable η was isotropic. It
means that the stress ratio σ0i/σi is independent of the direction inwhich the
tension was applied. There is no explicit dependence of r and m on the di-
rection of the stress. There is only implicit anisotropy generated with relation
η = η(W0) because W0 is naturally anisotropic. It was justified by the obse-
rvation presented in Fig.3.

Weconclude that the exponential and the limiting extensibility strain ener-
gy functions are both suitable for the description of the primary response
within uniaxial tension of the thoracic aorta. They can be coupled with the
pseudo-elastic softening variable η in order to capture the idealized Mullins
effect. Nevertheless, themodel predictions suggested that specific forms of the
softening variable may not be quite appropriate.



A comparison between the exponential and limiting fiber... 1213

Conflict of interest

None of the authors have a conflict of interest related to the research described
in this manuscript.

Acknowledgements

This work has been supported by Czech Ministry of Education pro-

ject MSM 6840770012, and Czech Science Foundation GACR 106/08/0557,

and Grant Agency of the Czech Technical University in Prague, grant No.

SGS10/247/OHK2/3T/12.

References

1. ChagnonG.,VerronE.,MarckmannG.,GornetL., 2006,Development
of new constitutive equations for theMullins effect in rubber using the network
alteration theory, International Journal of Solids and Structures,43, 6817-6831

2. Diani J., Fayolle B., Gilormini P., 2009, A review on the Mullins effect,
European Polymer Journal, 45, 601-612

3. Dorfmann A., Ogden R.W., 2003, A pseudo-elasticmodel for loading, par-
tial unloadingand reloadingofparticle-reinforcedrubber, International Journal
of Solids and Structures, 40, 2699-2714

4. Dorfmann A., Ogden R.W., 2004, A constitutive model for the Mullins
effect with permanent set in particle-reinforced rubber, International Journal
of Solids and Structures, 41, 1855-1878

5. DorfmannA.,TrimmerB.A.,Woods J.W.A., 2007,A constitutivemodel
for muscle properties in a soft-bodied arthropod, Journal of the Royal Society
Interface, 4, 257-269

6. Eĺıas-ZúñigaA., 2005,A phenomenological energy-basedmodel to characte-
rize stress-softening effect in elastomers,Polymer, 46, 10, 3496-3506

7. FungY.C., FronekK., Patitucci P., 1979,Pseudoelasticity of arteries and
the choice of itsmathematical expression,TheAmerican Journal of Physiology,
237, H620-H631

8. Gasser, T.C., Ogden R.W., Holzapfel G.A., 2006, Hyperelastic model-
ling of arterial layerswith distributed collagenfibre orientations, Journal of the
Royal Society Interface, 3, 6, 15-35

9. Gent A.N., 1996, A new constitutive relation for rubber, Rubber Chemistry
and Technology, 69, 819-832



1214 E. Gultova et al.

10. GraciaL.A.,PeñaE.,RoyoJ.M.,PelegayJ.L.,CalvoB., 2009,Acom-
parison between pseudo-elastic and damage models for modelling the Mullins
effect in industrial rubber components, Mechanics Research Communications,
36, 769-776

11. Gregersen H., Emery J.L., McCulloch A.D., 1998, History-dependent
mechanical behavior of guinea-pig small intestine, Annals of Biomedical Engi-
neering, 26, 850-858

12. Holzapfel G.A., 2000, Nonlinear Solid Mechanics: A Continuum Approach
for Engineering, JohnWiley & Sons, Chichester

13. Holzapfel G.A., 2009, Arterial tissue in health and diseases: experimental
data, collagen-basedmodeling and simulation, including aortic dissection,Bio-
mechanical Modelling at the Molecular, Cellular and Tissue Levels, 259-344

14. Holzapfel G.A., Gasser T.C., Ogden R.W., 2000, A new constitutive
framework for arterial wall mechanics and a comparative study of material
models, Journal of Elasticity, 61, 1-48

15. HolzapfelG.A.,OgdenR.W., 2009,Onplanar biaxial tests for anisotropic
nonlinearly elastic solids. A continuum mechanical framework, Mathematics
and Mechanics of Solids, 14, 474-489

16. Horgan C.O., Ogden R.W., Saccomandi G., 2004, A theory of stress
softening of elastomers based on finite chain extensibility, Mathematics and
Mechanics of Solids A, 460, 1737-1754

17. Horgan C.O., Saccomandi G., 2002, A molecular-statistical basis for the
Gent constitutive model of rubber elasticity, Journal of Elasticity, 68, 167-176

18. Horgan C.O., Saccomandi G., 2003, A description of arterial wall mecha-
nics using limiting chain extensibility constitutive models, Biomechanics and
Modeling in Mechanobiology, 1, 251-266

19. Horgan C.O., Saccomandi G., 2005, A new constitutive theory for fiber-
reinforced incompressible nonlinearly elastic solids, Journal of the Mechanics
and Physics of Solids, 53, 1985-2015

20. Horny L., Chlup H., Zitny R., Konvickova S., Adamek T., 2009, Con-
stitutive behavior of coronary artery bypass graft, IFMBE Proceedings, 25, 4,
181-184

21. HornyL., GultovaE., ChlupH., SedlacekR., Kronek J., Vesely J.,
Zitny R., 2010,Mullins effect in an aorta and limiting extensibility evolution,
Bulletin of Applied Mechanics, 6, 1-5

22. Horny L., Zitny R., Chlup H., 2008, Strain energy function for arterial
walls based on limiting fiber extensibility, IFMBE Proceedings, 22, 1910-1913



A comparison between the exponential and limiting fiber... 1215

23. HornyL., ZitnyR., ChlupH.,AdamekT., SaraM., 2011,Limiting fiber
extensibility as parameter for damage in venous wall, International Journal of
Biological and Life Sciences, 7, 196-199

24. Humphrey J.D., 2003, Continuum biomechanics of soft biological tissues,
Proc. R. Soc. Lond.,A 459, 3-46

25. Liao D., Zhao J., Kunwald P., Gregersen H., 2009, Tissue softening of
guinea pig oesophagus tested by the tri-axial test machine, Journal of Biome-
chanics, 42, 804-810

26. Marckmann G., Verron E., Gornet L., Chagnon G., Charrier P.,
Fort P., 2002, A theory of network alteration for the Mullins effect, Journal
of the Mechanics and Physics of Solids, 50, 2011-2028

27. Miehe C., 1995, Discontinuous and continuous damage evolution in Ogden-
type large-strain elastic materials,European Journal of Mechanics – A/Solids,
14, 697-720

28. MuñozM.J., Bea J.A.,Rodŕıguez J.F.,Ochoa I.,Grasa J., Pérez del
PalomarA., ZaragozaP., OstaR., DoblaréM., 2008,An experimental
study of the mouse skin behaviour: Damage and inelastic aspects, Journal of
Biomechanics, 41, 93-99

29. OgdenR.W.,RoxburghD.G., 1999,A pseudo-elasticmodel for theMullins
effect in filled rubber,Proceedings of the Royal Society A, 455, 2861-2877

30. Ogden R.W., Saccomandi G., 2007, Introducing mesoscopic information
into constitutive equations for arterial walls, Biomechanics and Modeling in
Mechanobiology, 6, 333-344

31. PeñaE.,AlasturéV.,LabordaA.,Mart́ınezM.A.,DoblaréM., 2010,
A constitutive formulationof vascular tissuemechanics including viscoelasticity
and softening behavior, Journal of Biomechanics, 43, 984-989

32. Peña E., DoblareM., 2009,An anisotropic pseudo-elastic approach formo-
dellingMullins effect in fibrous biologicalmaterials,Mechanics Research Com-
munications, 36, 784-790

33. Peña E., Pena J.A., Doblare M., 2009, On the Mullins effect and hy-
steresis of fibered biological materials: A comparison between continuous and
discontinuous damage models, International Journal of Solids and Structures,
46, 1727-1735

34. Simo J.C., 1987,On a fully three-dimensional finite-strain viscoelastic damage
model: Formulation and computational aspects,Computer Methods in Applied
Mechanics and Engineering, 60, 153-173



1216 E. Gultova et al.

Porównanie modelu wykładniczego z pseudo-sprężystym modelem

o ograniczonej rozszerzalności włókien przy opisie efektu Mullinsa

w tkance tętniczej

Streszczenie

Praca zawiera analizę porównawcząpoprawności dwóch różnychmatematycznych
sformułowań tzw. zmiennej osłabienia przy opisie zjawiska osłabienia naprężeń in-
dukowanych odkształceniem obserwowanym podczas cyklicznego jednoosiowego roz-
ciągania aorty piersiowej. W szczególności, zmienną osłabienia jako czynnika redu-
kującego poziom naprężeń opisano funkcją typu tangens hiperboliczny oraz funkcją
błędu. Założono, że mechaniczne właściwości aorty odpowiadają modelowi pseudo-
hipersprężystemu, nieściśliwemu i anizotropowemu.Funkcję gęstości energii odkształ-
cenia przyjęto w klasycznej formie wykładniczej i mało rozpoznanej postaci, któ-
ra ogranicza zakres rozszerzalności włókien. Badania wykazały, że obydwa podejścia
opisują właściwości mechaniczne tkanki z podobnym skutkiem. Pokazano, że rodzaj
przyjętej zmiennej osłabienia nie ma wpływu na rezultaty badań. W konkluzji pod-
kreślono, że obydwamodele nadają się do analizy efektuMullinsawaorcie, jakkolwiek
nadal otwartą kwestią pozostaje problem znalezienia najlepiej dopasowanegomodelu
do opisu tego zjawiska.

Manuscript received January 3, 2011; accepted for print February 25, 2011