A New Viscoelastic Model for Preconditioning in Ligaments and Tendons

Abstract

In this paper, a new viscoelastic model is presented to describe the elastic and viscoelastic behaviors of ligaments and tendons during the preconditioning process. The model is formulated by accounting for the mechanical contribution of the main structures of ligaments and tendons, i. e., the collagen fibers, the proteoglycan-rich matrix and the coupling between the collagen fibers and the matrix. The coupling between the collagen fibers and the matrix is assumed to behave as the links binding the collagen fibers and the matrix together. According to the previous studies, the friction loss in ligaments and tendons plays a significant role on the stress softening and the decreasing of the hysteresis during preconditioning. Therefore, in this work, the links are assumed to gradually break during preconditioning and there will be no links breaking when the preconditioning is completed. Finally, the viscoelastic model can easily describe the stress softening and the decreasing of the hysteresis during preconditioning process. We believe that our model can provide a simple way to interpret the physical mechanism during the preconditioning process in ligaments and tendons.

Full text

A New Viscoelastic Model for Preconditioning in
Ligaments and Tendons
Ratchada Sopakayang
Abstract—In this paper, a new viscoelastic model is presented
to describe the elastic and viscoelastic behaviors of ligaments
and tendons during the preconditioning process. The model is
formulated by accounting for the mechanical contribution of
the main structures of ligaments and tendons, i. e., the collagen
fibers, the proteoglycan-rich matrix and the coupling between
the collagen fibers and the matrix. The coupling between the
collagen fibers and the matrix is assumed to behave as the links
binding the collagen fibers and the matrix together. According
to the previous studies, the friction loss in ligaments and
tendons plays a significant role on the stress softening and the
decreasing of the hysteresis during preconditioning. Therefore,
in this work, the links are assumed to gradually break during
preconditioning and there will be no links breaking when the
preconditioning is completed. Finally, the viscoelastic model
can easily describe the stress softening and the decreasing of
the hysteresis during preconditioning process. We believe that
our model can provide a simple way to interpret the physical
mechanism during the preconditioning process in ligaments and
tendons.
Index Terms—viscoelastic models, stress softening, precondi-
tioning, ligaments, tendons.
I. INTRODUCTION
LIGAMENTS and tendons are viscoelastic material in
which the loading/stretching history has a great effect
on the mechanical properties of the tissues. Therefore, before
testing the soft biological specimens, the tissues need to
experience the preconditioning process for canceling the
loading/stretching history and setting up the same initial
properties for all specimens. Preconditioning is a process
that the successive cyclic loading/stretching is applied to the
specimens until the mechanical properties of the specimens
are unchanged. Preconditioning is usually be performed
in two types of experiments; the load controlled and the
displacement controlled. In the load controlled experiments,
the maximum load of each successive cycle is fixed as a
constant while in the displacement controlled experiments,
the maximum displacement of each successive cycle is kept
constant. As the previous studies, it is believed that the
stress softening and the decreasing of hysteresis observed
in the successive cycles during preconditioning are due to
the structural changes occurring in the tissues [1].
Ligaments and tendons are usually preconditioned before
performing mechanical testing, such as tensile, relaxation,
creep and hysteresis tests. This preconditioning significantly
influences their mechanical properties [2], [3], [4], [5], [6],
[7], [8], [9], [10], [11]. In previous studies, [3] performed
tensile tests of human quadriceps tendons and patellar lig-
aments before and after preconditioning. They found that
Manuscript received March 11, 2013; revised April 12, 2013.
R. Sopakayang is with the Department of Mechanical Engineering,
Faculty of Engineering, Ubon Ratchathani University, Warinchumrap, Ubon
Ratchathani, 34190 Thailand, e-mail: enratcso@ubu.ac.th.
the ultimate failure load and stiffness of these tissues were
higher after preconditioning. In [2], Graf et al. performed
relaxation experiments of primate patellar tendons before and
after preconditioning and showed that the relaxation times
were lower after preconditioning.
There is little known about the micro-structural changes
occurring during preconditioning. In order to understand the
mechanism of preconditioning, [12] compared the energy
absorption (area in a hysteresis cycle) of the first and the
last hysteresis cycle during preconditioning of normal canine
anterior cruciate ligaments (ACLs) and treated canine ACLs
in which the hyaluronic acid was enzymatically digested.
They found that the energy absorption of the first hysteresis
cycle of treated ACLs was much smaller than the energy
absorption of the first hysteresis cycle of normal ACLs.
The energy absorption was not significantly different for
the last hysteresis cycle for both groups. Yahia and Drouin
[12] suggested that the interfibrillar matrix containing water
and other material such as proteoglycans and hyaluronic
acid might be responsible for the observed hysteresis during
preconditioning .
There have been several investigations which have ex-
plained the stress softening and energy dissipation during
preconditioning [13], [14], [15], [16], [17]. In [13], Carew et
al. tried to establish a protocol for preconditioning of porcine
aortic valve cusps by using quasilinear viscoelastic (QLV)
theory. Their simulation did not predict well the maximum
stress of each cyclic loading due to the fact that the authors
did not incorporate any structural mechanism into the model.
Later, Nava et al. [15] introduced a phenomenological soften-
ing mechanism into a QLV model to describe preconditioning
of bovine liver. Softening was assumed to be a function of
the deformation history and a softening variable that changed
over time. The model showed good prediction although the
values of the model parameters were not presented by the
authors. Both these phenomenological models did not help in
understanding the mechanisms that govern preconditioning.
Currently, there is still no standard protocol used for pre-
conditioning by experimental biomechanicians [18]. There-
fore, developing mechanical models which can elucidate the
role of the structural components of biological tissue during
preconditioning could help in defining important standard
parameters that can guide the design of the experiments. The
main goal of this study is to present a viscoelastic model for
parallel-fibered collageneous tissues which describes the vis-
coelastic behavior exhibited during preconditioning process.
II. MO DE L FORMULATION
A. Basic Assumptions
The model is formulated by accounting for the mechanical
contribution of the main structures of ligaments and tendons:
Proceedings of the World Congress on Engineering 2013 Vol III,
WCE 2013, July 3 - 5, 2013, London, U.K.
ISBN: 978-988-19252-9-9
ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
WCE 2013

the collagen fibers, the proteoglycan-rich matrix and the
coupling between the collagen fibers and the matrix. The
collagen fibers is modeled as multiple linear elastic springs
arranged in parallel. Each spring which represents a collagen
fiber has different wavinesses and therefore, is activated at
different strains. The elastic modulus of each straight fiber
is defined as Ef. The proteogycan-rich matrix surrounded
collagen fibers is modeled as a Maxwell element which is
a series of an elastic spring with an elastic modulus Em
and a viscous dashpot with a coefficient of viscosity ηm.
The multiple elastic springs which represent the collagen
fibers and the Maxwell element which represents the ma-
trix are then connected in parallel. The links binding the
collagen fibers and the matrix together are modeled as two
identical groups of the parallel arrangement of numerous
elastic springs with an elastic modulus Ec. The strain of
the first group of the elastic springs is assumed to be the
same with the strain of the elastic springs representing the
collagen fibers while the strain of the second group of the
elastic springs is assumed to be the same with the strain of
the elastic spring representing the elastic component of the
matrix. These two groups of elastic springs are assumed to
gradually break simultaneously during preconditioning which
contributes the stress softening and the decreasing of the
hysteresis in preconditioning process. The schematic of the
proposed model described is shown in Fig. 1.
ε(t)%
εEm(t)%
ηm%
Ef%
σ(t)%σ(t)%
Em%
εηm(t)%
Ec%
Ec%
Fig. 1. Schematic of Viscoelastic Model.
B. Modeling Framework
The total stress of the tissue, σ(t), where tdenotes the
time, is given by
σ(t) = σf(t) + σc,f (t) + σc,m(t) + σm(t),(1)
where σf(t)is the stress of the collagen fibers. σc,f (t)and
σc,m(t)are the stresses of the links between the collagen
fibers and the matrix which associate to the collagen fibers
and the matrix, respectively. σm(t)is the stress of the matrix.
Due to the arrangement of the elastic springs of the links
which associate to the matrix, the elastic spring of the matrix
and the viscous dashpot of the matrix, one has that
σc,m(t) + σm(t) = ση(t),(2)
where ση(t)is the stress of the viscous component of the
matrix.
Moreover, the total strain of the tissue, ε(t), is
ε(t) = εf(t) = εc,f (t) = εc,m(t) + εη(t),(3)
where εc,m(t) = εm(t).εf(t)is the strain of the fibers.
εc,f (t)and εc,m(t)are the strains of the links between
the fibers and the matrix which associate to the fibers and
the matrix, respectively. Moreover, εm(t)and εη(t)are the
strains of the elastic and viscous components of the matrix,
respectively.
The stress of the fibrous component of the tissue, σf(t),
is defined by using a structural approach as previously done
by other investigators [19], [20]. The collagen fibers are
assumed to become straight at different strains, εs≥0,
defined by the following exponential probability density
function
p(εs) = αe−αεswith Z∞
0
p(εs)dεs= 1 ,(4)
where α > 0denotes the so-called rate parameter and, for
εs≥0,P(ε)denotes the exponential cumulative distribution
function defined as
P(ε) = Zε
0
p(εs)dεs= 1 −e−αε .(5)
The stress of collagen fibers is given for ε≥εsby
σf(t) = Zε(t)
0
Ef(ε(t)−εs)p(εs)dεs,(6)
where Efdenotes the elastic modulus of each straight
collagen fiber.
The total stress of links between the collagen fibers and the
proteoglycan-rich matrix is defined as σc(t)and is assumed
to be a contribution of two structural components. The first
structural component is the contact surface between the
links and the fibers which contributes the stress of the links
associating to the fibers, σc,f (t)while the second structural
component is the contact surface between the links and the
matrix which contributes the stress of the links associating
to the matrix, σc,m(t). Therefore, the total stress of the links
between the fibers and the matrix is given by
σc(t) = σc,f (t) + σm,f (t),(7)
The stress determined by the links between the collagen
fiber and the proteoglycan-rich matrix is defined by using an
approach similar to the one presented by Raischel et al. [21]
and De Tommasi et al. [22] for different materials. The links
are assumed to break when their strains, εc,f and εc,m, reach
some values εb≥0that are defined by an exponential
probability density function. Specifically, the stress of the
links associating to the fibers is given by
σc,f (t) = Ecεc,f (t)(1 −P(εc,f (t)))
+EcZεc,f (t)
0
εbp(εb)dεb,(8)
Proceedings of the World Congress on Engineering 2013 Vol III,
WCE 2013, July 3 - 5, 2013, London, U.K.
ISBN: 978-988-19252-9-9
ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
WCE 2013

and the stress of the links associating to the matrix is given
by
σc,m(t) = Ecεc,m (t)(1 −P(εc,m(t)))
+EcZεc,m(t)
0
εbp(εb)dεb,(9)
where Ecis the elastic constant of the links.
In Eq. 8 and Eq. 9, p(εb)is the probability density function
of an exponential distribution defined, for εb≥0, as
p(εb) = βe−β εb,(10)
where β > 0denotes the so-called rate parameter. The
exponential cumulative distribution function of the links
associating to the fibers, P(εc,f ), and the matrix, P(εc,m)
can be written as follows.
P(εc,f ) = Zεc,f
0
p(εb)dεb= 1 −e−β εc,f .(11)
P(εc,m) = Zεc,m
0
p(εb)dεb= 1 −e−β εc,m .(12)
The first term on the right-hand side of Eq. 8 and Eq. 9
are the stresses of all the unbroken links while the second
term are the stresses of all the broken links. These stresses
of the broken links are not transferred to the unbroken links.
The stress of the elastic component of the matrix is defined
as
σm(t) = Emεm(t),(13)
where Emdenotes the elastic modulus of the matrix.
The stress of the viscous component of the matrix is
defined as
ση(t) = ηmε0
η(t),(14)
where ηmdenotes the viscous modulus of the matrix and a
prime denotes the differentiation with respect to t.
After noting that σc,m(t) + σm(t) = ση(t)from Eq. 2,
ση(t) = ηmε0
η(t)from Eq. 14 and that σ(t) = σf(t) +
σc,f (t) + σc,m(t) + ση(t)from Eq. 1, Eq. 1 can be rewritten
as
σ(t) = σf(t) + σc,f (t) + ηmε0
η(t).(15)
Moreover, since εη(t) = ε(t)−εc,m(t)from Eq. 3, Eq. 15
becomes
σ(t) = σf(t) + σc,f (t) + ηm(ε0(t)−ε0
c,m(t)) .(16)
By recalling that σc,m(t) = Ecεc,m (t)(1 −P(εc,m(t))) +
EcRεc,m(t)
0εbp(εb)dεbfrom Eq. 9, σm(t) = Emεm(t)from
Eq. 13, ση(t) = ηmε0
η(t)from Eq. 14 and that σc,m(t) +
σm(t) = ση(t)from Eq. 2, Eq. 2 can be rewritten as
Ecεc,m(t)(1 −P(εc,m (t)))
+EcZεc,m(t)
0
εbp(εb)dεb+Emεm(t)
=ηmε0
η(t).(17)
Moreover, since εc,m(t) = εm(t)and εη(t) = ε(t)−
εc,m(t)from Eq. 3, Eq. 17 can be rearranged and rewritten
as
ε0
c,m(t) = ε0(t)−Ec
ηm
εc,m(t)(1 −P(εc,m (t)))
−Ec
ηmZεc,m(t)
0
εbp(εb)dεb
−Em
ηm
εc,m(t).(18)
By recalling p(εb) = βe−β εband P(εc,m)=1−e−βεc,m
from Eqs. 10 and 12, respectively, Eq. 18 becomes
ε0
c,m(t) = ε0(t) + Ec
βηm
(e−βεc,m (t)−1) −Em
ηm
εc,m(t).(19)
By recalling that σf(t) = Rε(t)
0Ef(ε(t)−εs)p(εs)dεs
from Eq. 6 and σc,f (t) = Ecεc,f (t)(1 −P(εc,f (t))) +
EcRεc,f (t)
0εbp(εb)dεbfrom Eq. 8 and ε(t) = εc,f (t)from
Eq. 3, Eq. 16 becomes
σ(t) = Zε(t)
0
Ef(ε(t)−εs)p(εs)dεs
+Ecε(t)(1 −P(εc,f (t))) + EcZε(t)
0
εbp(εb)dεb
+ηm(ε0(t)−ε0
c,m(t)) .(20)
By substituting Eq. 19 into Eq. 20, Eq. 20 can be rewritten
as
σ(t) = Zε(t)
0
Ef(ε(t)−εs)p(εs)dεs
+Ecε(t)(1 −P(εc,f (t))) + EcZε(t)
0
εbp(εb)dεb
+ (Emεc,m(t)−Ec
β(e−βεc,m −1)) .(21)
After differentiating both sides of Eq. 21 with respect to
t, one obtains that
σ0(t) = d
dt[Zε(t)
0
Ef(ε(t)−εs)p(εs)dεs]
+Ec
d
dt[ε(t)(1 −P(εc,f (t)))]
+Ec
d
dt[Zε(t)
0
εbp(εb)dεb))]
+d
dt[Emεc,m(t)−Ec
β(e−βεc,m −1)] .(22)
Firstly, by applying Leibniz’s rule for differentiation of an
integral to Eq. 22 one obtains that
d
dt[Zε(t)
0
Ef(ε(t)−εs)p(εs)dεs]
=Efε0(t)Zε
0
p(εs)dεs.(23)
It must be noted that for p(εs)defined by Eq. 4,
Rε
0p(εs)dεs= 1−e−αε , which is the exponential cumulative
density function. Thus, Eq. 23 can be written as
d
dt[Zε(t)
0
Ef(ε(t)−εs)p(εs)dεs]
=Efε0(t)1−e−αε.(24)
Proceedings of the World Congress on Engineering 2013 Vol III,
WCE 2013, July 3 - 5, 2013, London, U.K.
ISBN: 978-988-19252-9-9
ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
WCE 2013

Secondly, by recalling P(εc,f)=1−e−βεc,f and ε(t) =
εc,f (t)from Eq. 11 and Eq. 3, respectively, the second term
of Eq. 22 can be rewritten as
Ec
d
dt[ε(t)(1 −P(ε(t)))] = Ec
d
dt[εe−βε ]
=Ecε0(t)e−βε (1 −βε).(25)
Thirdly, by applying Leibniz’s rule for differentiation of
an integral to Eq. 22 one obtains that
d
dt[Zε(t)
0
εbp(εb)dεb] = ε(t)p(ε)ε0(t).(26)
It must be noted that for p(εb) = βe−β εbdefined by
Eq. 10, p(ε(t)) = βe−β ε(t), which is the exponential proba-
bility distribution function. Thus, Eq. 26 can be written as
d
dt[Zε(t)
0
εbp(εb)dεb] = βε(t)e−β εε0(t).(27)
By using Eq. 24, Eq. 25 and Eq. 27, Eq. 22 can be
rewritten as
σ0(t) = Efε0(t)(1 −e−αε) + Ecε0(t)e−β ε(t)
+ε0
c,m(t)(Em+βEce−βεc,m (t)).(28)
Eqs. 19 and 28 form a system of ordinary differential
equations that can be solved numerically to find εc,m(t)and
σ(t)after assigning the initial conditions.
In order to describe the stress-strain relationship of each
cyclic loading in preconditioning by solving numerically the
system of ordinary differential equations, Eqs. 19 and 28, the
cyclic strain history of the tissue, ε(t), is assumed to have
the form
ε(t) = 


bt −(2i−2)bt0for (2i−2)t0≤t < (2i−1)t0,
−bt + (2i)bt0for (2i−1)t0≤t < (2i)t0,
(29)
where band t0are constants, iis a positive integer that
denotes the number of loading cycles.
C. Softening Model
The stress-strain relationships for all the cycles that fol-
low the 1st cycle can be computed by accounting for the
decreasing of the elastic modulus of the links between the
collagen fibers and the matrix, Ec, by using the softening
model, Eq. 30, which can be written as
Ei+1
c=E1
ce−(i−1) .(30)
where E1
cand Ei+1
cdenote the Ecfor the 1st cycle and the
(i+ 1)th cycle, respectively.
III. RES ULTS
The preconditioning process is simulated by applying the
strain history, Eq. 29, and the softening model, Eq. 29, to
the ordinary differential equation system, Eq. 19 and Eq. 28,
therefore the stress strain relationships for all cycles can
be presented. In this study, the preconditioning is simulated
by assuming the parameters that ε0= 0.2/s,εM= 0.04,
t0= 0.2s,Ef= 500 MPa, Em= 1 MPa, η = 1MP a.s,
α= 10,β= 0.01 and E1
c= 10 MPa. The stress-strain
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
ï1
0
1
2
3
4
Strain
Stress (MPa)
Fig. 2. Preconditioning simulation for the 1st cycle.
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
ï1
0
1
2
3
4
Strain
Stress (MPa)
Fig. 3. Preconditioning simulation for the 1st and 2nd cycles.
curves for the 1st cycle are shown in Fig. 2 while the stress-
strain curves of the 1st and the 2nd cycles are shown in Fig.
3.
In this study, it has been calculated that the stress-strain
curves of preconditioning remain constant since the 6th
cycle, therefore the preconditioning is simulated and shown
in Fig. 4. for 7 cycles. The decreasing of the maximum stress
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
ï1
0
1
2
3
4
Strain
Stress (MPa)
Fig. 4. Preconditioning simulation for 7 cycles.
in each cycle during the preconditioning is shown in Fig.
5. In this simulation case, it is observed that the maximum
stress of the first hysteresis cycle is 3.96 MPa and it remains
constant as 3.55 MPa since the 6th cycle.
1 2 3 4 5 6 7
3
3.2
3.4
3.6
3.8
4
Number of cycles
Maximum stress (MPa)
Fig. 5. The decreasing of the maximum stress in each cycle during the
preconditioning process.
The stress softening and the decreasing of hysteresis is
influenced from the breaking of the links between the fibers
and the matrix during preconditioning. Therefore, the elastic
Proceedings of the World Congress on Engineering 2013 Vol III,
WCE 2013, July 3 - 5, 2013, London, U.K.
ISBN: 978-988-19252-9-9
ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
WCE 2013

modulus of the links is assumed to be exponentially de-
creased until it goes to zero as shown in Fig. 6.
1 2 3 4 5 6 7
0
2
4
6
8
10
Number of cycles
Ec (MPa)
Fig. 6. The decreasing of Ecin each cycle during the preconditioning
process.
IV. DISCUSSION AND CONCLUSIONS
A new viscoelastic model, for which the schematic is
shown in Fig. 1, is presented to describe the viscoelastic
behavior exhibited during preconditioning process. This vis-
coelastic model is formulated by accounting for the me-
chanical contributions of the collagen fibers, the interven-
ing proteoglycan-rich matrix and the coupling between the
collagen fibers and the matrix. The coupling between the
collagen fibers and the matrix is assumed to behave as the
links binding the collagen fibers and the matrix together. It is
believed that the stress softening and the decreasing of hys-
teresis (energy dissipation) during preconditioning occur due
to the friction loss and the changes of the internal structure of
the tissues [1]. It has been experimentally observed that the
maximum stresses and hysteresis of successive stress-strain
cycles gradually decrease during preconditioning and they
finally remain unchanged when the tissues are completely
preconditioned [1], [7]. Therefore, in this study, the links
between the collagen fibers and the matrix are assumed to
progressively fail while the collagen fibers are assumed to
gradually become active under the loading paths of all cycles.
For the unloading paths, the collagen fibers and the unbroken
links are assumed to gradually move back to their original
positions while the broken links occurred during the loading
paths are not transferred to the unbroken links. Because of
the decreasing of the amount of the broken links occurred
in successive cycles, in the softening model (Eq. 30), the
elastic modulus of the links which associates to amounts of
the broken links between the collagen fibers and the matrix
is assumed to exponentially decreases when the number of
cycle increases.
Because of the lack of some information from the ex-
periment in the paper [7], the exact quantitative simulation
can be done by requiring a specific experimental data of
preconditioning. The parameters in the proposed model are
directly related to the physiology of the internal structure of
ligaments and tendons during preconditioning. In general,
they can be found by validating the model with some
experimental data which can be carried out in future works.
For this work, in order to describe the viscoelastic behavior
during preconditioning process, the parameters in the model
are assumed as Ef= 500 MPa, Em= 1 MPa, η = 1M P a.s,
α= 10,β= 0.01 and E1
c= 10 MPa while the constants
in the cyclic strain history are assumed as ε0= 0.2/s,
εM= 0.04 and t0= 0.2s. The values of these parameters are
estimated based on the values of the parameters in a similar
model presented in a previous work [23].
The results of the model simulations are shown in Fig. 2
- Fig. 4. The preconditioning simulation for the 1st cycle
is presented in Fig. 2. which is observed that the model
could capture the nonlinearities of the loading and unloading
stress-strain curves, and the hysteresis (energy dissipation)
during the cyclic loading. According to the model, the
nonlinearities are due to the parameters, αand β, which
indicate the rate of the amount of the active fibers and the
rate of the amount of broken links between the fibers and the
matrix, respectively. The hysteresis is due to the broken links
occurred in the loading path of the cyclic loading which is
indicated by the parameter, β, and the viscoelastic property
of the matrix which is specified by the parameters, Emand
η, which represent the elastic modulus and the viscosity of
the matrix, respectively. The preconditioning simulation for
the 1st and 2nd cycles is shown in Fig. 3. It is observed that
the maximum stress and the hysteresis of the 1st cycle are
greater than the maximum stress and the hysteresis of the 2nd
cycle, respectively. The model parameter that captures these
behaviors is the Ecwhich exponentially decreases when the
number of cyclic loading increases. Ecindicates the value
of the modulus stiffness of the links between the collagen
fibers and the matrix. Because of the breakage of the links in
every loading cycle, Ecis assumed to decrease in successive
cycles as the softening model presented in Eq. 30. Therefore,
the softening model could capture both the stress softening
and the decreasing of the hysteresis in successive cycles
during preconditioning. Fig. 4. shows the preconditioning
simulation for 7 stress-strain cycles. It could be noticed that
the stress-strain curves remain unchanged since the 6th cycle
which could be interpreted that the coupling between the
collagen fibers and the matrix would not play the role on the
structural changes during preconditioning since the 6th cycle
of the cyclic loading. Therefore, the maximum stress of each
cycle would exponentially decrease until the 6th cycle and
remain unchanged in successive cycles as shown in Fig. 5.
In the same way, the stiffness of the links, Ec, of each cycle
exponentially decreases and becomes zero since the 6th cycle
as shown in Fig. 6. which means the mechanical contribution
of the links or the coupling between the collagen fibers and
the matrix does not influence the viscoelastic properties of
the tissues after the 6th cycle. At this stage, it could be stated
that the tissues are completely preconditioned.
In conclusion, a new modeling framework is presented
for describing the mechanical response of ligaments and
tendons exhibited during preconditioning process. The model
simulation seems to have excellent qualitative agreement
with the experimental data of Dorow [7]. The proposed
model here could be extended to describe other viscoelastic
phenomena in ligaments and tendons, such as creep and
relaxation. Moreover, it could be also applied to describe
the viscoelasticity of other collagenous tissues. More im-
portantly, it could be employed to illustrate the role of the
mechanical response of the coupling between the collagen
fibers and the matrix in collagenous tissues. According to the
model simulations in this study, it could be suggested that the
energy dissipation during preconditioning might due to the
effect of the structural changes in the coupling between the
collagen fibers and the matrix which associate to the friction
Proceedings of the World Congress on Engineering 2013 Vol III,
WCE 2013, July 3 - 5, 2013, London, U.K.
ISBN: 978-988-19252-9-9
ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
WCE 2013

loss or the disconnection of some contact surfaces between
the collagen fibers and the matrix. However, based on the
excellent qualitative agreement and the adjustable parameters
in the model, the quantitative agreement could be done in the
future experimental and theoretical studies.
REFERENCES
[1] Y. C. Fung, Biomechanics: Mechanical properties of living tissues,
2nd ed. New York: Springer-Verlag, 1993.
[2] K. Graf, B., J. Vanderby, R., J. Ulm, M., P. Rogalski, R., and J. Thielke,
R., “Effect of preconditioning on the viscoelastic response of primate
patellar tendon,” The Journal of Arthroscopic and Related Surgery,
vol. 10, pp. 90–96, 1994.
[3] L. Schatzmann, P. Brunner, and H. U. Staubli, “Effect of cyclic precon-
ditioning on the tensile properties of human quadriceps tendons and
patellar ligaments,” Knee Surgery Sports Traumatology Arthroscopy,
vol. 6, pp. 56–61, 1998.
[4] U. Staubli, H., L. Schatzmann, P. Brunner, L. Rincon, and L. P. Nolte,
“Mechnical tensile properties of the quadriceps tendon and patellar
ligament in young adults,” The American Journal of Sports Medicine,
vol. 27, pp. 27–34, 1999.
[5] J. R. Funk, G. W. Hall, J. R. Crandall, and W. D. Pilkey, “Linear and
quasi-linear viscoelastic characterization of ankle ligaments,” Journal
of Biomechanical Engineering, vol. 122, pp. 15–21, 2000.
[6] J. A. Weiss and J. C. Gardiner, “Computational modeling of ligament
mechanics,” Critical Reviews in Biomedical Engineering, vol. 29, pp.
1–70, 2001.
[7] C. Dorow, N. Krstin, and F. G. Sander, “Experiments to determine the
material properties of the periodontal ligaments,” Journal of Orofacial
Orthopedics, vol. 63, pp. 94–104, 2002.
[8] F. Genna, L. Annovazzi, C. Bonesi, P. Fogazzi, and C. Paganelli,
“On the experimental determination of some mechanical properties of
porcine periodontal ligament,” Meccanica, vol. 43, pp. 55–73, 2008.
[9] T. Cheng and R. Z. Gan, “Mechanical properties of anterior malleo-
lar ligament from experimental measurement and material modeling
analysis,” Biomechanics and Modeling in Mechanobiology, vol. 7, pp.
387–394, 2008.
[10] A. Teramoto and Z. P. Luo, “Temporary tendon strengthening by
preconditioning,” Clinical Biomechanics, vol. 23, pp. 619–622, 2008.
[11] A. Teramoto, W. R. Su, and Z. P. Luo, “Effect of cyclic preconditioning
on the failure load, strength and stiffness of multiple rat tendons and
ligaments,” Journal of Musculoskeletal Research, vol. 11, pp. 89–95,
2008.
[12] L. H. Yahia and G. Drouin, “Study of the hysteresis phenomenon on
canine anterior cruciate ligaments,” Journal of Biomedical Engineer-
ing, vol. 12, pp. 57–61, 1990.
[13] E. O. Carew, J. E. Barber, and I. Vesely, “Role of preconditioning and
recovery time in repeated testing of aortic valve tissues: Validation
through quasilinear viscoelastic theory,” Annals of Biomedical Engi-
neering, vol. 28, pp. 1093–1100, 2000.
[14] A. Sverdlik and Y. Lanir, “Time-dependent mechanical behavior
of sheep digital tendons, including the effects of preconditioning,”
Transaction of the American Society of Mechanical Engineers: Journal
of Biomechanical Engineering, vol. 124, pp. 78–84, 2002.
[15] A. Nava, E. Mazza, O. Haefner, and M. Bajka, “Experimental ob-
servation and modeling of preconditioning in soft biological tissues,”
Lecture Notes in Computer Science, vol. 3078, pp. 1–8, 2004.
[16] P. Ciarletta, P. Dario, and S. Micera, “Pseudo - hyperelastic model of
tendon hysteresis from adaptive recruitment of collagen type i fibrils,”
Biomaterials, vol. 29, pp. 764–770, 2008.
[17] O. Lokshin and Y. Lanir, “Viscoelasticity and preconditioning of rat
skin under uniaxial stretch: Microstructural constitutive characteriza-
tion,” Transaction of the American Society of Mechanical Engineers:
Journal of Biomechanical Engineering, vol. 131, pp. 1–10, 2009.
[18] N. Conza, “Engineering issues in experimental biomedicine series. part
3: Tissue preconditioning,” Experimental Techniques, vol. 29, pp. 43–
46, 2005.
[19] Y. Lanir, “A structural theory for the homogeneous biaxial stress-strain
relationship in flat collageneous tissues,” Journal of Biomechanics,
vol. 12, pp. 423–436, 1979.
[20] R. De Vita and W. S. Slaughter, “A structural constitutive model
for the strain rate-dependent behavior of anterior cruciate ligaments,”
International Journal of Solids and Structures, vol. 43, pp. 1561–1570,
2006.
[21] F. Raischel, F. Kun, and H. Herrmann, “Failure process of a bundle
of plastic fibers,” Physical Review E, vol. 73, p. 66101, 2006.
[22] D. De Tommasi, G. Puglisi, and G. Saccomandi, “A micromechanics-
based model for the mullins effect,” Journal of Rheology, vol. 50, p.
495, 2006.
[23] R. Sopakayang and R. De Vita, “A mathematical model for creep,
relaxation and strain stiffening in parallel-fibered collagenous tissues,”
Medical Engineering Physics, vol. 33, pp. 1056–1063, 2011.
Proceedings of the World Congress on Engineering 2013 Vol III,
WCE 2013, July 3 - 5, 2013, London, U.K.
ISBN: 978-988-19252-9-9
ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)
WCE 2013