J. Biomedical Science and Engineering, 2008, 1, 195-199
Published Online November 2008 in SciRes. http://www.srpublishing.org/journal/jbise JBiSE
Application of modified superposition model to
viscoelastic behavior of periodontal ligament
1Javad Hazrati Marangalou, 1Farzan Ghalichi & 2Behnam Mirzakouchaki
1Division of Biomechanics, Mechanical Engineering Department, Sahand University of Technology, Tabriz, Iran. 2Division of Orthodontics, School Of Dentistry,
Tabriz University of Medical Sciences, Tabriz, Iran. Correspondence should be addressed to J. Hazrati (hazrati_biomech@yahoo.com).
Received July 17, 2008; revised October 17, 2008; accepted October 17, 2008
The periodontal ligament (PDL) is a soft bio-
logical tissue which shows a strongly nonlinear
and time dependent mechanical behavior. Re-
cent experiments on rabbit PDL revealed that
the rate of stress relaxation is strain dependent.
This nonlinear behavior of PDL cannot be de-
scribed well by the separable quasi linear vis-
coelasticity theory which is usually used in tis-
sue biomechanics. Therefore, PDL requires a
more general description which considers this
nonlinearity and time dependency. The purpose
of this study was to model strain dependent
stress relaxation behavior of PDL using modi-
fied superposition method. It is shown herein
that modified superposition method describes
viscoelastic nonlinearties well and shows a
good compatibility with available experimental
PDL data. Hence, the modified superposition
model is suggested to describe periodontal
ligament data, because it can suitably demon-
strate both elastic nonlinearity and
strain-dependent stress relaxation behavior of
Keywords: periodontal ligament, viscoelasticity,
modified superposition method, stress relaxa-
A tooth is secured to the alveolar bone by fibrous con-
nective tissue that is called the periodontal ligament
(PDL) and schematically is showed in Figure 1. The
human PDL stabilizes the tooth in bone and provides
nutritive, proprioceptive and reparative functions [1]. It
is composed of collagenous fibers and a gelatinous
ground substance including cells and neurovascular tis-
sue [2]. The PDL not only strongly binds the tooth root
to the supporting alveolar bone but also absorbs occlusal
loads and distributes the resulting stress over the alveolar
bone [3]. This causes PDL to play a major role in tooth
mobility which is very important in prosthodontic and
orthodontic treatment and selection of an optimal force
system for orthodontic treatment [4]. The PDL has a de-
terminant influence on tooth instantaneous mobility be-
cause of lower stiffness in comparison with surrounding
tissues [5], and also long term movement because of
bone remodeling [6].
Ligaments display time dependent behavior which is
typical of viscoelastic materials [7]. Viscoelastic behav-
ior has been observed and studied in articular cartilage
[8, 9], periodontal ligament [10-12], ligament [7]. For-
mulation for modeling viscoelasticity in nonlinear mate-
rial was first used for modeling the time dependent be-
havior in soft tissue by Fung [13] and called it “quasi
linear viscoelasticity” (QLV). The QLV theory has been
applied for PDL by Natali et al. [14]; Toms et al. [2].
These models are composed of two parts: an instantane-
ous elastic response (often a hyperelastic strain energy
function) and a relaxation function. Relaxation function
governs the fading memory of current constitutive state
on deformation history [15].
It is known from literature, periodontal ligament is a
nonlinear viscoelastic material [16, 17], and stress re-
laxation behavior is a nonlinear function of strain. The
rate of stress relaxation decreases with increasing strain
[7]. The behavior in these data cannot be described well
using Fung quasi linear method [13], because in this
formulation stress relaxation is independent of strain.
Therefore, two previously presented QLV model for PDL
is discussed:
1) Komatsu et al. [12] studied stress relaxation on
PDL at different deformations. Experiments were done
on seventeen 4-month-old rabbits. The stress relaxation
process was well described by a function (relaxation
function) with three exponential decay terms and a con-
stant. On the other hand the experiment showed that
stress relaxation is a function of strain too, and G (t) (re-
laxation function) differs at different strains.
Komatsu's model for relaxation function is as follows:
0.390.09 exp(/0.35)
0.13exp(/ 4.12)0.39exp(/ 403)
() t
Gt =+− +
−+ −
In the present study, experimental data obtained by
Komatsu et al. [12] has been used to describe stress re-
laxation process for PDL.
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196 J. Hazrati / J. Biomedical Science and Engineering 1 (2008) 195-199
SciRes Copyright © 2008 JBiSE
Figure 1. Transversal section of a double rooted tooth sur-
rounded by PDL and alveolar bone.
2) Toms et al. [2] studied the quasi linear viscoelastic
model and applied to mechanical tests of the human
PDL. Transverse sections of cadavric premolars were
subjected to relaxation tests. In their study the relaxation
function described by using an exponential equation.
This model was similar to Komatsu's model.
Many reasonably general constitutive models have
been proposed to describe nonlinearly viscoelastic mate-
rials. A brief description of these models has been stud-
ied by Provenzano et al.[7]. In this paper, we have used
the modified superposition (or nonlinear superposition)
method to describe nonlinear viscoelasticity of PDL to
determine whether this method can model strain de-
pendent stress relaxation behavior of PDL.
2.1. Modified Superposition Method
The single integral formulation of the modified super-
position method [18, 19] allows the relaxation function
to depend on strain level:
(,)(, ())
tEt d
= (2)
The form of the relaxation modulus will be chosen as
a non separable strain-dependent power law:
(,)() (,)Et AGt
= (3)
(,) B
= (4)
The function()A
represents the initial modulus0
which can be obtained from a stress–strain curve or iso-
chronal curve describing the nonlinear elastic behavior.
is the relaxation function which itself is function
of strain and time. The function()B
describes the
strain-dependent rate of stress relaxation. Substituting a
Heaviside function into Eq.(2) results in:
() ()
(,) BB
tEt t
σεε σ
where 0
Eand 0
represent isochronal values of the tan-
gent modulus and stress, respectively, and can be func-
tions of strain to account for nonlinearities in the elastic
Figure 2. Histological section through an anterior tooth is taken
from a pig mandible. The periodontal ligament is enclosed by the
alveolar bone (B) and the tooth roots (T) and is pervaded by a
dense network of collagen fibers. (staining: hematoxylin and
eosin, original magnification ×400).
response. In addition, Eq.(5) can take on a more predic-
tive form once relaxation rates over a range of strain
values are obtained, the dependence of the rate function
as a function of strain is known. Stress–strain or
isochronal curves can be used to obtain the initial
modulus or stress terms,()A
and a polynomial can be
fit to the rate range to obtain the function()B
; so me-
chanical behavior of the tissue can be predicted. Hence,
the non separable form of modified superposition is able
to represent both the elastic and strain-dependent rate
nonlinearities that are experimentally observed.
2.2. Application of Modified Superposition
Method to PDL
Modified superposition method Eqs.(2-5) was applied to
experimental stress relaxation data from rabbit periodon-
tal ligament [12]. As mentioned above, these data dem-
onstrate that the rate of stress relaxation decreases sig-
nificantly with increasing tissue strain, this behavior
have shown in Figure 10. For rabbit PDL modified su-
perposition method fits the experimental data well for all
strain levels:0.078
=, 0.215
values are 0.265, 0.53, 1.19, 1.85, MPa for 0.078,
0.124, 0.17, 0.215 strain respectively.0
is the stress at
the start of stress relaxation test (1t= sec), this is rea-
sonable if assumed that the stress relaxation have not
started up to 1 sec, so it is supposed that 1t= sec is ini-
tial time for this method. Strain dependent rate of stress
relaxation term or ()Be can be seen to decrease in mag-
nitude as strain increase (-0.172, -0.127, -0.102, -0.096
for 0.078, 0.124, 0.17, 0.215 strain, respectively) which
shows a nonlinearity in the strain dependent rate of stress
The rate function or ()e
and ()e
is a polyno-
mial which is obtained by curve fitting using commercial
software MATLAB 7.0 (The Mathwork Inc., Natick,
MA). Curve fitting rate function (Fig 4), and initial
modulus (Fig 5) versus strain were done using experi-
mental data from Komatsu et al. [12].
J. Hazrati / J. Biomedical Science and Engineering 1 (2008) 195-199 197
SciRes Copyright © 2008 JBiSE
Figure 3. Relaxation function with three exponential decay
terms and a constant presented by Komatsu [12] under the
deformation of 78m
Figure 4. Fitting of polynomial function (curve) to experimental
stress relaxation rate of Komatsu et al. [12][12] for multiple
rabbit periodontal ligament at multiple strain levels.
Figure 5. Fitting of power function (curve) to experimental initial
modulus of Komatsu et al. [12] for multiple rabbit periodontal liga-
ment at multiple strain levels.
Figure 6. Comparison of experimental data (points) and pre-
dicted (line) relaxation function behavior, for strain of 0.215
and initial stress of 1.85 MPa.
Figure 7. Comparison of experimental data (points) and pre-
dicted (line) relaxation function behavior, for strain of 0.17 and
initial stress of 1.19 MPa.
Figure 8. Comparison of experimental data (points) and pre-
dicted (line) relaxation function behavior, for strain of 0.124 and
initial stress of 0.53 MPa.
Figure 9. Comparison of experimental data (points) and pre-
dicted (line) relaxation function behavior, for strain of 0.078 and
initial stress of 0.265 MPa.
Figure 10. Relaxation function at different strain levels, the rate
of stress relaxation decreases significantly with increasing tissue
198 J. Hazrati / J. Biomedical Science and Engineering 1 (2008) 195-199
SciRes Copyright © 2008 JBiSE
Figure 11. Stress relaxation curve for rabbit PDL at the strain
of 21.5% and initial stress of 1.86 MPa.
In order to obtain a set of constitutive parameters for
PDL, invitro experimental data were analyzed, and the fit
of the rate function was done with2
value of 0.999 for
tension relaxation using nonlinear least square method.
The results of curve fitting have been depicted in Figure
4. The rate function obtained as follows:
5.9377.2992.286 0.3094
The function ()A
was determined by fitting the tan-
gential modulus as a function of strain using nonlinear
least square method from typical stress–strain data for
the rabbit periodontal ligament with 2
Rvalue of 0.967.
The results of curve fitting have been depicted in Figure
5. The function ()A
obtained as follows:
= (7)
So, the relaxation modulus will take the following
42.66 1.039
5.9377.2992.286 0.3094
Four separate stress relaxation tests at four different
strains were then fitted using the predicted rate function.
The results for each strain and comparison between the
experimental data and the model curve have been dem-
onstrated in Figures 6-9. Figure 10 shows that the rate
of stress relaxation decreases significantly with increas-
ing tissue strain which shows the dependence of relaxa-
tion function to tissue dilatation. Stress relaxation proc-
ess for the initial stress of 1.86 MPa and strain of 0.215
which is followed by a gradual reduction up to 600s has
been depicted in Figure 10, after 600s the stress de-
creases approximately to 1.0 MPa and the relaxation
function reaches 0.54.
In this study the ability of modified superposition
(nonlinear superposition) method as a nonlinear viscoe-
lastic model has been investigated to describe periodon-
tal ligament behavior which has been experimentally
observed. In previous studies, QLV model had been ap-
plied for PDL, with QLV theory, the time dependent
portion of model G(t) (relaxation function) is independ-
ent of strain, however, as experimental data shows the
relaxation behavior differs at different strain levels and
depends on tissue dilatation too. In this study the time
dependent portion of model depends on strain to consider
strain history dependence of tissue, too.
The stress relaxation property of PDL was expressed
by a non separable strain dependent power law, which
showed a good fitting with experimental data under dif-
ferent deformations, but in previous studies stress relaxa-
tion property of PDL had been often expressed by expo-
nential decay terms which were functions of time alone,
as in the human [2]; rabbit [12]; bovine [20]; pig [14],
and strain history dependence of tissue had not been
taken into account.
The mechanisms driving viscoelastic behavior in
ligament are not yet completely defined. It has been
speculated that ‘‘the decrease in relaxation rate with in-
creasing strain could be the result of larger strains caus-
ing greater water loss (wringing out effect) which causes
the tissue to be more elastic (less viscous) than tissues
subjected to lower strains’’ [21]. In other words, at
greater deformations because of water loss, the viscous
components (matrix) have less influence in tissue me-
chanics than elastic components, and this causes the tis-
sue to show more elastic properties than viscous.
There are several limitations which must be consid-
ered while examining the presented model in this study.
In the presented model other effects such as diseased
state, effects of age, biochemical changes, temperature,
hydration and others have not been considered. The other
limitation is related to experimental data, in which col-
lagen fiber bundles in PDL of the specimens are assumed
to run almost parallel to the direction of testing.
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