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Chun-Yuh Huang
VanC.Mow
Gerard A. Ateshian
Departments of Mechanical Engineering and
Biomedical Engineering,
Columbia University,
New York, NY 10027
The Role of Flow-Independent
Viscoelasticity in the Biphasic
Tensile and Compressive
Responses of Articular Cartilage
A long-standing challenge in the biomechanics of connective tissues (e.g., articular car-
tilage, ligament, tendon) has been the reported disparities between their tensile and com-
pressive properties. In general, the intrinsic tensile properties of the solid matrices of
these tissues are dictated by the collagen content and microstructural architecture, and
the intrinsic compressive properties are dictated by their proteoglycan content and mo-
lecular organization as well as water content. These distinct materials give rise to a
pronounced and experimentally well-documented nonlinear tension–compression stress–
strain responses, as well as biphasic or intrinsic extracellular matrix viscoelastic re-
sponses. While many constitutive models of articular cartilage have captured one or more
of these experimental responses, no single constitutive law has successfully described the
uniaxial tensile and compressive responses of cartilage within the same framework. The
objective of this study was to combine two previously proposed extensions of the biphasic
theory of Mow et al. [1980, ASME J. Biomech. Eng., 102, pp. 73–84] to incorporate
tension–compression nonlinearity as well as intrinsic viscoelasticity of the solid matrix of
cartilage. The biphasic-conewise linear elastic model proposed by Soltz and Ateshian
[2000, ASME J. Biomech. Eng., 122, pp. 576–586] and based on the bimodular stress-
strain constitutive law introduced by Curnier et al. [1995, J. Elasticity, 37, pp. 1–38], as
well as the biphasic poroviscoelastic model of Mak [1986, ASME J. Biomech. Eng., 108,
pp. 123–130], which employs the quasi-linear viscoelastic model of Fung [1981, Biome-
chanics: Mechanical Properties of Living Tissues, Springer-Verlag, New York], were com-
bined in a single model to analyze the response of cartilage to standard testing configu-
rations. Results were compared to experimental data from the literature and it was found
that a simultaneous prediction of compression and tension experiments of articular car-
tilage, under stress-relaxation and dynamic loading, can be achieved when properly tak-
ing into account both flow-dependent and flow-independent viscoelasticity effects, as well
as tension–compression nonlinearity. 关DOI: 10.1115/1.1392316兴
Introduction
Over the past two decades, several studies have established that
the viscous drag induced by interstitial fluid flowing within the
porous-permeable collagen–proteoglycan matrix of cartilage im-
parts viscoelasticity to the mechanical response of this tissue. This
flow-dependent viscoelastic phenomenon has been the basis of
successful porous media models 关1–7兴, which can describe the
response of articular cartilage under various compressive loading
conditions, including confined 关1,2,4兴 and unconfined 关7,8兴 com-
pression of cylindrical cartilage discs, as well as indentation of
cartilage layers with a flat or spherical indenter 关9,10兴. In addition
to this mechanism of flow-dependent viscoelasticity, some inves-
tigators have proposed that there also exists an intrinsic, flow-
independent viscoelasticity in the solid matrix 关11,12兴, leading to
the formulation and application of porous media models with a
viscoelastic solid phase 关13–15兴. Incorporation of intrinsic vis-
coelasticity into porous media models has often produced better
agreement between theory and experiments than in the absence of
modeling such effects 关16,17兴, though not always 关18兴.
However, it has been difficult to assess whether this improved
agreement has indeed resulted from the existence of intrinsic
solid–matrix viscoelasticity, or was caused by the increased math-
ematical flexibility of the governing equations and the number of
material parameters in the model. A long-standing argument in
favor of the former interpretation has been the experimental ob-
servation of a frequency-dependent response in the dynamic shear
loading of cartilage 关11,12,19,20兴; indeed, porous media models
of isotropic materials undergoing infinitesimal deformation pre-
dict an isochoric deformation under torsional shear, which would
preclude interstitial fluid pressurization and flow. Thus, the obser-
vation of a viscoelastic response in torsional shear should support
the premise of intrinsic viscoelasticity of the solid matrix 关11,19兴.
However, it is important to recognize the limiting assumptions of
this analysis, namely that cartilage is not necessarily isotropic and
that for certain classes of anisotropy torsional shear can produce
nonzero dilatation, and that the prediction of isochoric deforma-
tion under infinitesimal strain is a mathematical idealization that
neglects higher order deformation effects that remain present ex-
perimentally 共i.e., higher order changes in dilatation may produce
non-negligible interstitial fluid pressurization and flow兲.
Another confounding factor in the assessment of intrinsic solid
matrix viscoelasticity has been the observation that other model-
ing assumptions of cartilage can equally improve agreement be-
tween theoretical predictions and experimental data. For example,
in confined and unconfined compression and indentation, good
agreement with experiments has been found not only with linear
isotropic poroviscoelastic biphasic models 关14,17,21兴, but also
when using a linear transversely isotropic biphasic model 关8,9,10兴,
or nonlinear bimodular biphasic models 关7,22兴. Modeling the in-
Contributed by the Bioengineering Division for publication in the JOURNAL OF
BIOMECHANICAL ENGINEERING. Manuscript received by the Bioengineering Divi-
sion December 13, 2000; revised manuscript received May 16, 2001. Associate Edi-
tor: L. A. Setton.
410 Õ Vol. 123, OCTOBER 2001 Copyright © 2001 by ASME Transactions of the ASME
homogeneity of cartilage observed in compression 关23–25兴 simi-
larly appears to have a potential for improving agreement between
the isotropic biphasic theory and experimental data in confined
and unconfined compression 关26,27兴.
In our own studies, we have been able to demonstrate good
agreement between the infinitesimal or finite deformation isotro-
pic biphasic models 关1,28兴 and experiments in confined compres-
sion creep and stress-relaxation 关4,29兴, and dynamic loading 关18兴,
not only by curve-fitting the corresponding experimental load or
deformation response but also by predicting the measured inter-
stitial fluid pressurization. For unconfined compression of cylin-
drical samples of cartilage, where the tissue is simultaneously
subjected to axial compression and radial and circumferential ten-
sion 关30兴, we have recently proposed to incorporate the Conewise
Linear Elasticity 共CLE兲 model of Curnier et al. 关31兴 into the bi-
phasic theory of Mow et al. 关1兴 to account for the disparity in the
tensile and compressive moduli of articular cartilage observed in
various studies 共e.g., 关1,4,32–39兴兲. This model was shown to pro-
duce good agreement between theory and experiments in uncon-
fined compression, including in its ability to predict the cartilage
interstitial fluid response 关22兴. However, an interesting prediction
of this biphasic-CLE model, which is presented below, is that the
response of cartilage under uniaxial tension 共an experimental con-
figuration often investigated in the literature 关32,33,35–38,40兴 but
not yet predicted successfully with a porous media model兲 exhib-
its almost no transient response, unlike experimental observations
关35,37,40兴. This finding suggests that uniaxial tension of articular
cartilage may indeed be a discriminating testing configuration for
investigating the intrinsic viscoelasticity of the solid matrix of
articular cartilage, in the context of a porous media model that
also describes the tension-compression nonlinearity of articular
cartilage.
Therefore, the short-term objective of this study is to under-
stand the role of flow-independent viscoelasticity in the context of
a porous media model of cartilage which accounts for its tension–
compression nonlinearity. The long-term objective is to develop a
more comprehensive framework for understanding the mechanical
behavior of cartilage than the currently available theoretical mod-
els, which can better interpret the diverse experimental outcomes
reported in the literature; this framework should further help ex-
plain how cartilage is able to sustain the high compressive stresses
typical of in vivo loading conditions that far exceed its equilibrium
compressive modulus 关41,42兴. The specific aims are to combine
existing theories of cartilage that incorporate intrinsic viscoelas-
ticity and tension–compression nonlinearity of the solid matrix
within a biphasic model; and to investigate the response of such a
model to standard testing configurations and compare the out-
comes to experimental data reported in the literature.
Model Formulation
Mak 关13兴 developed a formulation for an isotropic biphasic
model of cartilage whose solid phase is described by the quasi-
linear viscoelasticity 共QLV兲 theory of Fung 关43兴, with the tissue
modeled as a binary mixture of an intrinsically incompressible
solid phase, representing primarily the collagen fibers, proteogly-
cans, and chondrocytes, and an intrinsically incompressible fluid
phase representing the interstitial water 关1兴. The governing equa-
tions for this model, known as the biphasic poroviscoelastic
共BPVE兲 theory, are the momentum equation for the mixture 共ne-
glecting inertia and in the absence of body forces兲,
ⵜ•
⫽⫺ⵜp⫹ ⵜ•
v
e
⫽ 0, (1)
where
represents the total stress tensor, which is the sum of the
interstitial fluid pressure p and the viscoelastic or effective stress
v
e
resulting from deformation of the solid matrix (
⫽⫺pI
⫹
v
e
), ⵜ• denotes the divergence operator, and ⵜ is the gradient
operator; and the continuity equation for the mixture,
ⵜ•
共
v
s
⫹ w
兲
⫽ 0, (2)
where w⫽
f
(v
f
⫺ v
s
) is the flux of fluid relative to the solid,
f
is
the fluid volume fraction 共tissue porosity兲 and v
s
, v
f
are the solid
and fluid phase velocities, respectively. In one of the simpler em-
bodiments of the QLV theory,
1
the viscoelastic stress tensor
v
e
can be related to the stress tensor under equilibrium conditions,
e
, through
v
e
共
t
兲
⫽ g
共
t
兲
e
关
E
共
0
兲
兴
⫹
冕
0
t
g
共
t⫺
兲
e
关
E
共
兲
兴
d
. (3)
The infinitesimal strain tensor E, which appears above, is related
to the solid displacement u through E⫽ (1/2)(ⵜu⫹ ⵜu
T
), and the
displacement to the solid velocity through v
s
⫽ D
s
u/Dt 共material
derivative following the solid phase兲. The reduced relaxation
function, g(t), is given by
g
共
t
兲
⫽ 1⫹ c
冋
E
i
冉
t
2
冊
⫺ E
i
冉
t
1
冊
册
(4)
where E
i
(•) represents the exponential integral function, and c,
1
,
2
are material properties of the QLV theory. Physically,
关
1/
2
,1/
1
兴
represents the frequency range over which most of the
intrinsic viscoelastic energy dissipation occurs under dynamic
loading, whereas (1⫹ c ln
2
/
1
) is the ratio of instantaneous to
equilibrium moduli resulting from the intrinsic viscoelasticity
alone. 关Note that unlike the classical formulation of Fung 关43兴,we
adopt the trivial modification of the function g(t) such that
g(0)⫽ 1⫹ c ln(
2
/
1
) and g(⬁)⫽ 1, so that the viscoelastic stress
v
e
reduces to the elastic stress
e
at equilibrium.兴 The remain-
ing constitutive relations adopted in the current formulation are
Darcy’s law,
w⫽⫺kⵜp, (5)
which relates the fluid flux to the pressure gradient, with k repre-
senting the hydraulic permeability 共assumed isotropic and con-
stant here, though it is generally recognized to be strain-dependent
关44兴兲, and the Conewise Linear Elasticity model of Curnier et al.
关31兴, in its cubic symmetry embodiment 关22兴,
e
共
E
兲
⫽
兺
a⫽1
3
再
1
关
A
a
:E
兴
tr
共
A
a
E
兲
A
a
⫹
兺
b⫽1
b⫽a
3
2
tr
共
A
a
E
兲
A
b
冎
⫹ 2
E, (6)
which describes a bimodular response, or tension–compression
nonlinearity, of the solid matrix. Here, tr(•) is the trace operator
that yields the first invariant of its tensorial argument, and A
a
:E
⫽ tr(A
a
T
E). A
a
is a texture tensor corresponding to each of three
preferred material directions defined by the unit vectors a
a
共a
a
•a
a
⫽ 1, no sum over a, • denoting the dot product of vectors兲,
with A
a
⫽ a
a
丢 a
a
共丢 denoting the dyadic product of vectors, no
sum over a兲. For cubic material symmetry, a
a
•a
b
⫽ 0 when b
⫽a, and the three directions are generally taken to be: a
1
parallel
to the split line direction,
2
a
2
perpendicular to the split line direc-
tion, and a
3
normal to the articular cartilage surface. The term
A
a
:E represents the component of normal strain along the pre-
ferred direction a
a
. Tension–compression nonlinearity stems
from the conditional statement,
1
The function g(t) in Eq. 共3兲 is taken to be a scalar function for simplicity in this
analysis. A most general formulation could employ a fourth-order tensor instead; for
cartilage, it has sometimes been proposed to separate the viscoelastic response in
bulk deformation from that in shear deformation when using an isotropic model
关13,15兴, though this is not done here.
2
Split lines have been used as indicators of predominant collagen fibril directions
on the articular surface 共Hulkrantz, W., 1898, ‘‘Ueber die Spaltrichtungen der Gelen-
kknorpel,’’ Verh. D. Anat. Ges., 12, pp. 248–256兲.
Journal of Biomechanical Engineering OCTOBER 2001, Vol. 123 Õ 411
1
关
A
a
:E
兴
⫽
再
⫺ 1
, A
a
:E⬍ 0
⫹ 1
, A
a
:E⬎ 0
. (7)
This signifies that the material properties
1
differ whether the
normal strain component along the direction a
a
is compressive or
tensile. The physical meaning of these elastic constants is as fol-
lows: H
⫺ A
⫽
⫺ 1
⫹ 2
is the equilibrium confined compression
modulus of the tissue 共the ‘‘aggregate’’ modulus兲 and H
⫹ A
⫽
⫹ 1
⫹ 2
is the equivalent modulus in tension;
2
is the ‘‘off-
diagonal’’ modulus, which could be determined from the equilib-
rium ratio of radial stress to axial strain in confined compression
共the radial stress being measurable on the side wall, e.g., see the
experiments of Khalsa and Eisenberg 关45兴兲. Note that the choice
of an isotropic permeability is consistent with the choice of cubic
symmetry for the stress–strain law, since isotropic and cubic sym-
metry are identical in second-order tensors such as the permeabil-
ity tensor; however, it is also possible to adopt a more general
orthotropic model for permeability, as in our earlier study 关22兴.
In summary, the model presented above, which is valid for
infinitesimal strains and can describe tension–compression non-
linearity 关Eqs. 共6兲 and 共7兲兴 as well as intrinsic viscoelasticity 关Eqs.
共3兲 and 共4兲兴 of the solid matrix of a solid-fluid biphasic mixture,
has eight material constants:
⫺ 1
,
⫹ 1
,
2
,
,c,
1
,
2
,k. This
model can be reduced to the isotropic biphasic poroviscoelastic
model of Mak 关13兴 by letting
⫺ 1
⫽
⫹ 1
⫽
2
⬅ 共and noting that
A
1
⫹ A
2
⫹ A
3
⫽ I兲. It can be reduced to our recently proposed
biphasic-CLE model 关22兴 by letting c⫽ 0. It can also be reduced
to the linear isotropic biphasic theory of Mow et al.关1兴 by imple-
menting both of the reductions described above.
Uniaxial Tension and Unconfined Compression
Typical experiments of unconfined compression of articular car-
tilage are performed on cylindrical samples 共e.g., 1 mm thick, 6
mm in diameter兲, which can be easily harvested with a circular
core punch cutting perpendicularly to the articular surface 共e.g.,
关1兴兲. In contrast, uniaxial tensile tests are performed on long strips
of cartilage, either dumbbell shaped or prismatic 共e.g, 10 mm
long兲, with a rectangular cross section 共e.g., 0.2⫻1.5 mm兲, which
are typically harvested with a pair of blades perpendicularly to the
surface 共e.g., 关33,35兴兲. The solution of Armstrong et al. 关30兴 for
unconfined compression of a linear isotropic biphasic material, as
well as many subsequent solutions 共e.g., 关7,8兴兲 assumed friction-
less conditions at the loading platens, which is reasonable in view
of the typically low friction coefficient of articular cartilage. This
simplification, along with the assumption of axisymmetric condi-
tions facilitated by the specimen geometry, leads to equations
amenable to a closed-form analytical solution. In contrast, the
biphasic or biphasic poroviscoelastic analysis of the uniaxial ten-
sile response of cartilage has generally been performed on pris-
matic bars with rectangular cross sections, requiring either a sim-
plification of the boundary conditions 关46兴 or a numerical scheme
such as finite element analysis 关47兴; the reason is that the analysis
of the biphasic response of a prismatic bar with rectangular cross
section, whose lateral boundaries are free draining, is fully three
dimensional and does not lend itself to an analytical closed-form
solution because of the complexity of the transient interstitial fluid
flow fields that would result from loading. Therefore, in order to
achieve a closed-form solution for the biphasic-CLE-QLV analy-
sis of uniaxial tension for the purpose of examining the different
responses of cartilage in tension and compression, we assume in
this study that the cartilage specimen is a prismatic bar with a
circular cross section, while recognizing that such a specimen
geometry cannot be easily obtained in practice by typical speci-
men preparation. It will be demonstrated below that this assump-
tion is not as restrictive as it first may seem. By St. Venant’s
principle, the effects of the clamping conditions at the two ends of
the prismatic bar are also neglected, since the specimen length is
typically much greater than its characteristic cross-sectional
width, so that the analyses of uniaxial tension and unconfined
compression are treated in a similar fashion, with the only differ-
ence in those two configurations arising from the bimodular con-
stitutive assumptions of the CLE theory.
The reduction of the general biphasic equations to the configu-
ration of unconfined compression with frictionless platens and
axisymmetric conditions has been described previously for the
linear isotropic biphasic model 关30兴, the poroviscoelastic biphasic
model 关21兴, and the biphasic-CLE model 关22兴. In these analyses,
the shear traction at the interface between cartilage and the load-
ing platens is set to zero and the axial normal strain is homoge-
neous; the interstitial fluid pressure and normal traction are also
set to zero on the lateral boundary. The same approach can be
followed with the constitutive equations of Eqs. 共3兲–共6兲, hence
only a summary of the results is presented here. As is typical for
this type of problems, the governing equations and closed-form
solution is given in Laplace transform space. The differential
equation for the radial displacement is given by
2
u
¯
r
r
2
⫹
1
r
u
¯
r
r
⫺
u
¯
r
r
2
⫺ f
⫾
u
¯
r
⫽
rf
⫾
2
¯
共
s
兲
, (8)
whereas the boundary conditions reduce to
u
¯
r
兩
r⫽0
⫽ 0, H
⫿ A
u
¯
r
r
⫹
2
冉
u
¯
r
r
⫹
¯
共
s
兲
冊
冏
r⫽r
0
⫽ 0. (9)
The fluid pressure can be determined from the radial displacement
using
p
¯
共
r,s
兲
⫽⫺
1
k
冕
r
r
0
关
su
¯
r
共
,s
兲
⫹ s
¯
共
s
兲
/2
兴
d
. (10)
The solution then reduces to
u
¯
r
⫾
共
r,s
兲
⫽
r
0
2
冋
冉
1⫺
2
H
⫿ A
冊
I
1
冉
冑
f
⫾
r
r
0
冊
冑
f
⫾
I
0
共
冑
f
⫾
兲
⫺
冉
1⫺
2
H
⫿ A
冊
I
1
共
冑
f
⫾
兲
⫺
r
r
0
册
¯
⫾
共
s
兲
,
(11)
p
¯
⫾
共
r,s
兲
⫽
H
⫿ A
2
冉
1⫹ c ln
1⫹
2
s
1⫹
1
s
冊
⫻
再
冑
f
⫾
冉
1⫺
2
H
⫿ A
冊
冋
I
0
冉
冑
f
⫾
r
r
0
冊
⫺ I
0
共
冑
f
⫾
兲
册
冑
f
⫾
I
0
共
冑
f
⫾
兲
⫺
冉
1⫺
2
H
⫿ A
冊
I
1
共
冑
f
⫾
兲
冎
¯
⫾
共
s
兲
,
(12)
F
¯
⫾
共
s
兲
r
0
2
⫽ H
⫿ A
冉
1⫹ c ln
1⫹
2
s
1⫹
1
s
冊
再
冉
2H
⫾ A
⫺ 3
2
⫹ H
⫿ A
2H
⫿ A
冊
冑
f
⫾
I
0
共
冑
f
⫾
兲
⫹
冉
1⫺
2
H
⫿ A
冊冉
2
2
⫺ H
⫾ A
⫺ H
⫿ A
H
⫿ A
冊
I
1
共
冑
f
⫾
兲
冑
f
⫾
I
0
共
冑
f
⫾
兲
⫺
冉
1⫺
2
H
⫿ A
冊
I
1
共
冑
f
⫾
兲
冎
¯
⫾
共
s
兲
,
(13)
412 Õ Vol. 123, OCTOBER 2001 Transactions of the ASME
F
¯
p
⫾
共
s
兲
r
0
2
⫽ H
⫿ A
冉
1⫹ c ln
1⫹
2
s
1⫹
1
s
冊
⫻
再
冉
1⫺
2
H
⫿ A
冊
冋
I
1
共
冑
f
⫾
兲
⫺
1
2
冑
f
⫾
I
0
共
冑
f
⫾
兲
册
冑
f
⫾
I
0
共
冑
f
⫾
兲
⫺
冉
1⫺
2
H
⫿ A
冊
I
1
共
冑
f
⫾
兲
冎
¯
⫾
共
s
兲
,
(14)
where
f
⫾
⫽
r
0
2
s
H
⫿ A
k
冉
1⫹ c ln
1⫹
2
s
1⫹
1
s
冊
. (15)
In these equations, r is the radial coordinate, u
r
refers to the radial
displacement, p is the interstitial fluid pressure, is the axial
strain, F is the total axial force across the specimen, and F
p
is that
component of the force supported by interstitial fluid pressure,
i.e., F
p
⫽ 2
兰
0
r
0
rpdr, where r
0
is the specimen radius. I
0
(•) and
I
1
(•) are modified Bessel functions of the first kind, of order 0
and 1, respectively. Overbars indicate Laplace transformation
from the time domain and s is the Laplace transform variable.
Superscripted ⫾ on these parameters refer to the solution for ten-
sion 共⫹兲 or compression 共⫺兲, as it can be observed that these
solutions differ by the interchange of the material constants H
⫹ A
⫽
⫹ 1
⫹ 2
and H
⫺ A
⫽
⫺ 1
⫹ 2
. Note that the solution does not
depend on the axial dimension 共thickness or length兲 of the cylin-
drical specimen.
These solutions are valid for a variety of loading conditions.
For example, for a step application of strain with a magnitude
0
,
use
¯
⫾
(s)⫽⫾
0
/s; if the strain is ramped over a ramp time of t
0
and subsequently kept constant at a magnitude of
0
, use
¯
⫾
(s)
⫽⫾(1⫺ e
⫺ st
0
)
0
/s
2
t
0
. Alternatively, the total axial load F
¯
⫾
(s)
may be prescribed in a similar way, and a solution for
¯
⫾
(s)
obtained from Eq. 共13兲 then substituted into the remaining
expressions.
For step or ramp strain application, inverse Laplace transforma-
tion into the time domain can be performed numerically, e.g.,
using the INLAP routine from the IMSL library 共Visual Numerics,
Inc., Houston, TX兲.
3
For the steady-state solution to a sinusoidal
axial load or strain, it suffices to substitute s⫽ i
into the solu-
tions of Eqs. 共11兲–共15兲, where
is the angular frequency of the
applied load or strain and i⫽
冑
⫺ 1, to derive the frequency-
dependent amplitude and phase response of the corresponding
parameter.
The dynamic modulus G
¯
⫾
(s) of this biphasic-CLE-QLV mate-
rial can be derived from the ratio of F
¯
⫾
(s)/
r
0
2
and
¯
⫾
(s) in Eq.
共13兲. To determine the material’s ‘‘instantaneous’’ modulus 共or the
modulus in the limit of loading at high frequency兲, denoted by
E
⫾ Y
0
⫹
, it suffices to take the limit of the resulting expression as s
→ ⬁, which corresponds to the real time limit of t→ 0
⫹
:
E
⫾ Y
0
⫹
⫽
冉
1⫹ c ln
2
1
冊冉
H
⫾ A
⫺
3
2
2
⫹
H
⫿ A
2
冊
. (16)
Similarly, the modulus at equilibrium 共or in the limit of loading at
very low frequency兲, E
⫾ Y
共Young’s modulus in classical elastic-
ity兲, can be obtained by taking the limit of the dynamic modulus
as s→ 0,
E
⫾ Y
⫽ H
⫾ A
⫺
2
2
2
H
⫿ A
⫹
2
. (17)
It can be observed that, as expected, the equilibrium modulus is
independent of the QLV parameters. Finally, the instantaneous 共or
high-frequency兲 fluid load support can be obtained by taking the
limit, as s→ ⬁, of the ratio of the expressions in Eqs. 共14兲 and
共13兲,
⫺
F
¯
p
⫾
F
¯
⫾
⫽
1
1⫹ 2
H
⫾ A
⫺
2
H
⫿ A
⫺
2
. (18)
It is noteworthy that this expression is independent of the QLV
parameters, even though it represents an instantaneous response.
Results
A variety of testing configurations in tension and compression
can be simulated from the solutions described above, a subset of
which are presented here to provide sufficient insight into the
behavior of this biphasic-CLE-QLV model. Since this compound
model has not been employed previously in the literature, repre-
sentative material properties used in the simulations here derive
from two separate sources: For the biphasic-CLE properties, we
employ the results of our recent analysis 关22兴, derived from con-
fined and unconfined compression stress-relaxation and torsional
shear experiments on bovine articular cartilage; H
⫹ A
⫽ 13.2 MPa, H
⫺ A
⫽ 0.64 MPa,
2
⫽ 0.48 MPa,
⫽0.17 MPa, k
⫽ 6.1⫻ 10
⫺ 16
m
4
/N.s; these properties were obtained under a total
strain of approximately 18 percent 共including tare loading兲. For
the QLV parameters, we use the results of Setton et al. 关14兴 who
curve-fit the biphasic poroviscoelastic theory of Mak 关13兴 to con-
fined compression creep data, also on bovine articular cartilage:
c⫽ 0.16,
1
⫽ 0.06 s,
2
⫽ 201 s.
In uniaxial tension, a representative radius of r
0
⫽ 0.345 mm is
employed, which produces a surface area equivalent to a rectan-
gular cross section of dimensions 1.5 mm ⫻ 0.25 mm. In the first
analysis, a step tensile strain of magnitude
0
⫽ 0.10 共10 percent兲
is applied to the sample and the time-dependent stress-relaxation
response of the biphasic-CLE-QLV model is presented in Fig. 1,
together with the specialized case of the biphasic-CLE model
共with c⫽ 0兲. It can be noted that, unlike the biphasic-CLE-QLV
response, the biphasic-CLE response exhibits almost no transient
relaxation under this testing configuration. Substitution of these
material constants into Eq. 共16兲 confirms that the instantaneous
tensile modulus of the biphasic-CLE-QLV material, E
⫹ Y
0
⫹
3
Numerical inverse Laplace transformation is best achieved by nondimensional-
izing the expressions in Eqs. 共6兲–共12兲 to avoid numerical overflow or underflow.
Though IMSL has a built-in routine to evaluate Bessel functions with a complex
argument, a custom-written routine implementing asymptotic expansions for large
arguments was employed instead.
Fig. 1 Uniaxial tensile response of the biphasic-CLE-QLV and
biphasic-CLE models, to a step strain of
0
Ä0.10 „10 percent…,
as derived from Eq. „13… by numerical inverse Laplace transfor-
mation
Journal of Biomechanical Engineering OCTOBER 2001, Vol. 123 Õ 413
⫽29.4 MPa, is considerably greater than that of the biphasic-CLE
material, E
⫹ Y
0
⫹
⫽ 12.8 MPa. In contrast, for both models, the equi-
librium modulus in tension is E
⫹ Y
⫽ 12.3 MPa. The dynamic
modulus in tension is displayed for both models in Fig. 2, which
displays the amplitude and phase angle as a function of frequency
over the range f⫽
/2
⫽ 10
⫺ 6
⫺ 10
2
Hz. For reference, the three
characteristic frequencies for the material are f
⫹
⫽ H
⫺ A
k/r
0
2
⫽ 0.0033 Hz, 1/
2
⫽ 0.005 Hz, 1/
1
⫽ 16.7 Hz. As already evi-
denced by the values of E
⫹ Y
0
⫹
and E
⫹ Y
, there is virtually no fre-
quency dependence of the dynamic tensile modulus in the
biphasic-CLE model, whereas the inclusion of QLV produces a
characteristic flow-independent viscoelastic response.
In unconfined compression, a radius of r
0
⫽ 2.39 mm is
assumed.
4
In the first of these analyses, a ramped-strain stress-
relaxation test is employed, with
0
⫽ 0.10 and t
0
⫽ 3 s, 150 s, and
300 s. The resulting stress-relaxation responses are presented in
Fig. 3, with and without QLV effects. Unlike the case for tension,
a very significant transient response is observed in unconfined
compression in both models. Differences in the two models are
most evident only in the fastest of the ramp rates employed (t
0
⫽ 3 s), with the peak stress at the end of the ramp achieving a
larger value for the biphasic-CLE-QLV model. The amplitude and
phase angle of the unconfined compression dynamic modulus is
presented in Fig. 4, for both models; for this testing configuration,
the three characteristic frequencies for the material are f
⫺
⫽ H
⫹ A
k/r
0
2
⫽ 0.0014 Hz, 1/
2
⫽ 0.005 Hz, 1/
1
⫽ 16.7 Hz. From
this figure, as from Eq. 共16兲, it can be observed that E
⫺ Y
0
⫹
⫽15.0 MPa for the biphasic-CLE-QLV model, and E
⫺ Y
0
⫹
⫽ 6.5 MPa for the biphasic-CLE model, whereas the equilibrium
compressive modulus is only E
⫺ Y
⫽ 0.57 MPa for both cases.
Discussion
The short-term objective of this study was to investigate the
role of intrinsic viscoelasticity of the solid matrix of cartilage
4
Representative radii are not required when performing these analyses with the
nondimensional form of the equations; they are used here for illustration.
Fig. 2 Dynamic modulus versus frequency under uniaxial ten-
sile loading, for the biphasic-CLE-QLV and biphasic-CLE mod-
els: „
a
… magnitude; „
b
… phase angle. In the limit of high frequen-
cies, the magnitude of the dynamic modulus is given by
E
¿
Y
0
¿
in
Eq. „16…, whereas at low frequencies it is given by
E
¿
Y
in Eq.
„17…
Fig. 3 Unconfined compression stress-relaxation response of
the biphasic-CLE-QLV and biphasic-CLE models, to a ramp
strain of magnitude
0
Ä0.10 „10 percent…, and for three differ-
ent ramp times „
t
0
Ä3 s, 150 s, and 300 s…, as derived from Eq.
„13… by numerical inverse Laplace transformation
Fig. 4 Dynamic modulus versus frequency under unconfined
compression loading, for the biphasic-CLE-QLV and biphasic-
CLE models: „
a
… magnitude; „
b
… phase angle. In the limit of high
frequencies the magnitude of the dynamic modulus is given by
E
À
Y
0
¿
in Eq. „16…, whereas at low frequencies it is given by
E
À
Y
in
Eq. „17…
414 Õ Vol. 123, OCTOBER 2001 Transactions of the ASME
within the context of a biphasic mixture theory that accounts for
tension–compression nonlinearity. The long-term objectives are to
help elucidate the mechanism by which articular cartilage can
sustain typical physiological stresses in the range of 2–6 MPa
共e.g., 关41,42兴兲, with occasional peak values as high as 18 MPa
关48兴, while exhibiting a compressive equilibrium modulus on the
order of 0.5 MPa only; and to provide a theoretical framework
that can simultaneously predict the tensile and compressive re-
sponses of articular cartilage, under transient and dynamic condi-
tions. As discussed below, the results of this study provide a more
comprehensive agreement between theory and experiments in the
literature, than achieved from the linear biphasic, the QLV, the
biphasic-QLV 共biphasic poroviscoelastic兲, or the biphasic-CLE
theories alone.
First, because the biphasic-CLE-QLV subsumes the linear bi-
phasic theory, it can provide the same successful agreement with
confined compression experiments as demonstrated previously
共e.g., 关1,2,18,29,49兴兲. Second, because the biphasic-QLV and the
biphasic-CLE are subsumed by the more general model, good
agreement can also be expected in the curvefitting of unconfined
compression stress-relaxation results 关17,22兴. However, under dy-
namic unconfined compression, cartilage has been shown to ex-
hibit a dynamic compressive modulus on the order of 12–20 MPa
关50,51兴 at the highest tested frequencies. Looking at Eq. 共16兲 with
c⫽ 0, the highest achievable dynamic modulus with the biphasic-
CLE model is E
⫺ Y
0
⫹
⬇H
⫹ A
/2 共given that H
⫺ A
,
2
Ⰶ H
⫹ A
兲, which
puts it on the order of 6.5 MPa when using the representative
value of H
⫹ A
employed in the simulations given above 共see the
high-frequency range in Fig. 4兲. With the biphasic-QLV 共i.e., let-
ting H
⫹ A
⫽ H
⫺ A
in Eq. 共16兲兲, the highest predicted dynamic
modulus in compression would be E
⫺ Y
0
⫹
⫽ (3/2)(1⫹ c ln
2
/
1
)(H
⫺A
⫺
2
), which is on the order of 0.5 MPa for the typical
material constants used above. Clearly, while the biphasic-CLE
model is a better predictor of the dynamic unconfined compres-
sion modulus than the biphasic-QLV, it still falls somewhat short
of the values observed experimentally. However, when using the
combined biphasic-CLE-QLV model, Eq. 共16兲 produces a dy-
namic unconfined compression modulus E
⫺ Y
0
⫹
on the order of 15
MPa 共Fig. 4兲, in better agreement with experiments in the
literature.
Looking at Eq. 共17兲, it can be observed that the biphasic-CLE
theory can account for the differences observed in the literature
between the compressive and tensile equilibrium moduli of carti-
lage 共e.g., E
⫺ Y
⫽ 0.55 MPa and E
⫹ Y
⫽ 12.3 MPa, using the above-
given typical material constants兲, represented for typical samples
of humeral head cartilage in Fig. 5; clearly, neither the linear
biphasic nor the biphasic-QLV models account for tension–
compression nonlinearity, so they are unable to model both ten-
sion and compression using consistent material constants. How-
ever, as observed in Fig. 1, the biphasic-CLE is unable to produce
a substantial transient response under a step strain in tension, even
though such responses have always been observed in the literature
共e.g., 关35,37,40兴, also see the typical experimental response in Fig.
6兲, whereas the biphasic-CLE-QLV can account for this effect. It
is important to appreciate that in tension, the flow-dependent vis-
coelasticity is curtailed by the large difference between the tensile
and compressive moduli in the biphasic-CLE and biphasic-CLE-
QLV theories, whereas in compression, the fluid pressurization is
enhanced by the tension–compression nonlinearity. This is evi-
dent from the fluid load support as given in Eq. 共18兲; whereas in
unconfined compression the fluid load support for the
biphasic-CLE or biphasic-CLE-QLV models is very high
(
兩
⫺ F
¯
p
⫺
/F
¯
⫺
兩
⫽ 97.6 percent), it becomes virtually negligible in
tension (
兩
⫺ F
¯
p
⫹
/F
¯
⫹
兩
⫽ 0.6 percent). For the linear biphasic or the
biphasic-QLV, the fluid load support in tension and compression
remains the same,
兩
⫺ F
¯
p
⫾
/F
¯
⫾
兩
⫽ 33.3 percent; indeed, the latter
theories would predict the same transient relaxation in tension as
in unconfined compression, though they otherwise appear to be
unsuitable for predicting the tension–compression nonlinearity as
discussed above.
Since flow-dependent viscoelasticity is negligible when
H
⫺ A
/H
⫹ A
Ⰶ 1, uniaxial tension experiments may conceivably be
used to extract only the QLV parameters, c,
1
,
2
, and the equi-
librium tensile modulus, E
⫹ Y
, even when using the framework of
the biphasic-CLE-QLV theory. This simplification in the analysis
of experimental data in uniaxial tension also implies that the pre-
cise geometry of the specimen cross section would not impact the
experimental outcome since fluid flow effects, which otherwise
depend on geometry, are negligible. The results of this analysis
also suggest that for biological hydrated soft tissues, which gen-
erally sustain very little compressive loads and thus exhibit very
significant tension–compression nonlinearity, such as tendons and
ligaments, flow-dependent viscoelasticity may not be a significant
effect.
The results of this study are primarily based on a theoretical
analysis at this time, though favorable qualitative comparisons are
observed relative to literature findings. Experiments that directly
Fig. 5 Typical experimental equilibrium tensile and compres-
sive responses of human humeral head articular cartilage from
the superficial and middle zones „55 y.o. male…. The tensile re-
sponse is measured from long prismatic samples with rectan-
gular cross section, harvested parallel to the surface, along the
split line direction; the compressive response is measured
from cylindrical plugs harvested normal to the articular surface
†34‡.
Fig. 6 Experimental uniaxial tensile response of the superfi-
cial zone specimen of Fig. 5 to a ramp strain of
0
Ä0.02 „2
percent… and ramp time of
t
0
Ä1 s, as a function of time
Journal of Biomechanical Engineering OCTOBER 2001, Vol. 123 Õ 415
test the predictions of the biphasic-CLE-QLV model represent the
next step in this effort to establish a more comprehensive theoret-
ical framework for articular cartilage. Recently, models that have
accounted for the tension–compression nonlinearity of cartilage
关7,22兴, such as the biphasic-CLE model reviewed above, have
demonstrated that interstitial fluid load support is considerably
enhanced in compression by the disparity in tensile and compres-
sive moduli of cartilage, producing a greater dynamic modulus.
The current study demonstrates that the addition of intrinsic vis-
coelasticity provides the necessary boost to the theoretical predic-
tion of the dynamic modulus to match experimental findings at
higher frequencies. In the biphasic-CLE-QLV model adopted in
this study, the dynamic modulus is greater than that of the
biphasic-CLE model by a factor of (1⫹ c ln
2
/
1
) according to
Eq. 共16兲. Interestingly, in this model, the incorporation of solid
matrix intrinsic viscoelasticity neither enhances nor defeats the
fluid load support, as indicated by Eq. 共18兲. As noted by others
关17兴, and evidenced by comparing the biphasic-CLE and biphasic-
CLE-QLV results of Fig. 3 or 4, it appears that the effect of
intrinsic solid matrix viscoelasticity is most evident at relatively
fast strain rates 共e.g., with a ramp time of t
0
⫽ 3 s in Fig. 3 in
stress-relaxation, or at dynamic frequencies above 10
⫺ 3
Hz ac-
cording to Fig. 4兲; the stress-relaxation results of Fig. 3 suggest
that a combination of stress-relaxation experiments with a slow
ramp time and a fast ramp time could also be used to discriminate
between flow-dependent and flow-independent effects 共see a typi-
cal experimental response in Fig. 7兲, though not as dramatically as
uniaxial tensile tests 共Fig. 1 or Fig. 6兲.
There are a number of limitations to the current study. For
example, the assumption of a linear response in tension is known
to be a simplification, since cartilage has been shown to behave
nonlinearly even at small tensile strains; furthermore, the current
model does not adequately describe the anisotropy of cartilage,
such as the known disparity in tensile moduli measured parallel
and perpendicular to the split lines, as well as the observation of
tensile Poisson’s ratios in excess of 0.5, though these could be
potentially addressed using an orthotropic model. The known in-
homogeneity of cartilage properties from the surface to the deep
zone would need to be taken into account to produce more accu-
rate results. A finite strain formulation of the constitutive equa-
tions would also be more appropriate to investigate higher physi-
ological loads. These issues, which typically increase the
complexity of cartilage modeling, can be addressed in future
analyses.
Conclusion
This study attempts to explain various observed experimental
findings in the testing of articular cartilage, by combining several
constitutive models previously described in the literature, each of
which had demonstrated success in one or more testing configu-
rations. By comparing theoretical predictions with experimental
findings in the literature, it is found that a simultaneous prediction
of compression and tension experiments, under stress-relaxation
and dynamic loading, can be achieved when taking into account
flow-dependent and flow-independent viscoelasticity effects, as
well as tension–compression nonlinearity. While a guiding prin-
ciple of constitutive modeling of biological tissues is to adopt the
simplest possible formulation that can describe experimental data,
it is becoming increasingly clear that the complexity of articular
cartilage mechanics requires more elaborate models than those
already presented in the literature. The biphasic-CLE-QLV model
adopted in this study has the advantage that it employs as building
blocks theories that have proved popular and are thus familiar to
the concerned research community. In the embodiment adopted
here, the model has eight material constants; clearly, this signifies
that an experimental characterization of the material properties of
a particular tissue sample could not come from a single test but
would rather require multiple experiments, either on the same
sample or on adjoining samples from the same tissue source. The
combination of tests that might provide sufficient data to deter-
mine these constants uniquely is the subject of ongoing studies.
Acknowledgments
This study was supported in part by the National Institute for
Arthritis, and Musculoskeletal and Skin Diseases of the National
Institutes of Health 共AR46532, AR43628, AR42850, and
AR41913兲.
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Ä0.126 s and
t
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Ä400 s…
416 Õ Vol. 123, OCTOBER 2001 Transactions of the ASME
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