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A Bimodular Theory for Finite Deformations: Comparison of Orthotropic Second-order and Exponential Stress Constitutive Equations for Articular Cartilage

06 Apr 2006-Biomechanics and Modeling in Mechanobiology (Springer-Verlag)-Vol. 5, Iss: 2, pp 90-101

TL;DR: An orthotropic stress constitutive equation that is second-order in terms of the Biot strain tensor is derived as an alternative to traditional exponential type equations and suggests that the bimodular second- order models may be appealing for some applications with cartilaginous tissues.

AbstractCartilaginous tissues, such as articular cartilage and the annulus fibrosus, exhibit orthotropic behavior with highly asymmetric tensile–compressive responses. Due to this complex behavior, it is difficult to develop accurate stress constitutive equations that are valid for finite deformations. Therefore, we have developed a bimodular theory for finite deformations of elastic materials that allows the mechanical properties of the tissue to differ in tension and compression. In this paper, we derive an orthotropic stress constitutive equation that is second-order in terms of the Biot strain tensor as an alternative to traditional exponential type equations. Several reduced forms of the bimodular second-order equation, with six to nine parameters, and a bimodular exponential equation, with seven parameters, were fit to an experimental dataset that captures the highly asymmetric and orthotropic mechanical response of cartilage. The results suggest that the bimodular second-order models may be appealing for some applications with cartilaginous tissues.

Summary (2 min read)

Introduction

  • The work in this paper is motivated by the difficulty of, and the need for, developing accurate stress constitutive equa tions for fiber-reinforced cartilaginous tissues.
  • For articular cartilage, there has been no finite deformation model presented that accurately describes its orthotropic response for multiple experimental protocols including tension and compression.
  • Those models were based on a bimodular theory for infinitesimal strains (Curnier et al. 1995) in which the material constants may be discon tinuous (or jump) across a surface of discontinuity in strain space, provided that the stress continuity conditions are sat isfied at the surface.

2 Methods

  • The authors outline the derivation of a second-order stress constitutive equation for orthotropic materials.
  • Then, the authors propose a general theory for bimodular elastic materials and, consequently, derive a bimodular second-order stress constitutive equation.
  • Finally, the authors study the abilities of bi modular second-order and exponential models to describe the mechanical response of articular cartilage in uniaxial tension (UT) and confined compression (CC).
  • The deformation gradient tensor F is uniquely decomposed by the polar decomposition theorem as F = RU, (1) where R is a proper-orthogonal tensor and the right stretch tensor U is a symmetric positive-definite tensor.

T = RT̂(U)RT = RT̃(E)RT , P = RP̂(U) = RP̃(E), (6)

  • For isotropic elastic materials, various second-order theo ries for Green-elastic materials have been proposed using different strain tensors (Hoger 1999; Murnaghan 1937, 1951; Rivlin 1953) which differ depending on which strain tensor is used (Ogden 1984).
  • First, the authors require that the first-order constants {λ11, λ22, λ33} be continuous across the surfaces of discontinuity.
  • The bimodular stress constitutive equation corresponding to the general second-order orthotropic material may have a total of 61 mate rial constants.
  • The Poisson’s ratio at 16% strain was specified using the data reported in Huang et al. (1999).
  • Four second-order models were studied, with 6, 7, 8, and 9 parameters (Table 2), and one exponential model was studied, with 7 parameters.

3 Results

  • Qualitatively, the theoretical predictions of the UT re sponses were the same for all models, while the theoretical predictions of the CC response and Poisson’s ratios were different (Figs. 2–6 and Table 4).
  • For the exponential model, the CC responses were nonlinear and equal in all 3 directions while the Poisson’s functions were nonlinear and equal in all 3 directions.

4 Discussion

  • A bimodular theory for finite deformations was developed with the aim of accurately modeling the orthotrop ic and asymmetric mechanical response of cartilage.
  • For articular cartilage, a stress constit utive equation for finite deformations that can accurately de scribe the orthotropic and asymmetric mechanical response for multiple experimental protocols has not been proposed.
  • As mentioned before, Ateshian and colleagues (Soltz and Ateshian 2000; Wang et al. 2003) used a bimodular model for infinitesi­ mal strains.
  • The bimodular second-order stress constitutive equation has several features that may make it desirable for some appli rial constants.
  • The results of the present study revealed that the numerical regression algorithm did not initially con verge to a positive-definite stiffness matrix for the secondorder models.

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A bimodular theory for finite deformations: comparison
of orthotropic second-order and exponential stress constitutive
equations for articular cartilage
Stephen M. Klisch - California Polytechnic State University, San Luis Obispo
Abstract Cartilaginous tissues, such as articular cartilage and the annulus fibrosus, exhibit orthotropic behavior with
highly asymmetric tensile–compressive responses. Due to this complex behavior, it is difficult to develop accurate stress
constitutive equations that are valid for finite deformations. Therefore, we have developed a bimodular theory for finite
deformations of elastic materials that allows the mechanical properties of the tissue to differ in tension and compres-
sion. In this paper, we derive an orthotropic stress constitutive equation that is second-order in terms of the Biot strain ten-
sor as an alternative to traditional exponential type equations. Several reduced forms of the bimodular second-order equa-
tion, with six to nine parameters, and a bimodular exponential equation, with seven parameters, were fit to an experimental
dataset that captures the highly asymmetric and orthotropic mechanical response of cartilage. The results suggest that the
bimodular second-order models may be appealing for some applications with cartilaginous tissues.
1 Introduction
The work in this paper is motivated by the difficulty of, and the need for, developing accurate stress constitutive equa-
tions for fiber-reinforced cartilaginous tissues. Cartilage is composed of chondrocytes embedded in an extracellular ma-
trix consisting primarily of proteoglycans, a crosslinked collagen network, and water. The proteoglycans are negatively
charged molecules that mainly resist compressive loads (Basser et al. 1998; Lai et al. 1991) while the collagen net-
work primarily resists tensile and shear loads (Mow and Ratcliffe 1997; Venn and Maroudas 1977). Due to this molecular
structure, articular cartilage typically exhibits a mechanical response with marked anisotropy and tension–compression
asymmetry (Akizuki et al. 1986; Laasanen et al. 2003; Soltz and Ateshian 2000; Wang et al. 2003; Woo et al. 1976, 1979),
and likely experiences finite, multi-dimensional strains due to typical in vitro and in vivo loads. Although MRI mea-
surements of in situ and in vivo joints have predicted that cartilage is subject to average strains of less than 10% under
physiologic loading conditions (Eckstein et al. 2000; Herberhold et al. 1999), local strains may be much higher due to
nonhomogeneous mechanical properties that depend on both anatomic location (Laasanen et al. 2003) and depth from the
articular surface (Schinagl et al. 1997; Wang et al. 2001). For physiologic loading conditions, FEM contact analyses
(Donzelli et al. 1999; Krishnan et al. 2003) suggest that in situ cartilage may experience local strains up to 26%, suggest-
ing that the tissue is in the nonlinear range of its stress–strain relationship (Huang et al. 1999).
Due to the complex mechanical behavior of cartilaginous tissues, the development of accurate finite deformation models
of the equilibrium elastic response has been difficult. Lotz and colleagues developed an orthotropic finite deformation
model for the annulus fibrosus using an exponential strain energy function; however, maximum errors between the the-
oretical and experimental stresses in uniaxial tension (UT) were approximately 50% (Klisch and Lotz 1999; Wagner
and Lotz 2004). For articular cartilage, there has been no finite deformation model presented that accurately describes
its orthotropic response for multiple experimental protocols including tension and compression. However, an elastic stress
constitutive equation for finite deformations has been used in more complex models, including multiphasic models with
isotropic (Ateshian et al. 1997; Holmes and Mow 1990; Kwan et al. 1990) and transversely isotropic (Almeida and Spilker
1997) material symmetry.
For infinitesimal strains, Ateshian and colleagues (Soltz and Ateshian 2000; Wang et al. 2003) have employed
elastic and biphasic models with a bimodular stress constitutive equation that allows for different mechanical proper-
ties in tension and compression. Their model can describe the mechanical response in unconfined compression in three

orthogonal directions while providing reasonable predictions
for the other protocols (Wang et al. 2003). Those models were
based on a bimodular theory for infinitesimal strains (Curnier
et al. 1995) in which the material constants may be discon-
tinuous (or jump) across a surface of discontinuity in strain
space, provided that the stress continuity conditions are sat-
isfied at the surface. Recently, several exponential models
for finite deformations that allow for different mechanical
properties in tension and compression have been used for the
arterial wall (Holzapfel et al. 2004) and the annulus fibro-
sus (Baer et al. 2004). One reason that an exponential strain
energy function is often used may be due to its ability to
model the highly asymmetric tension–compression response
without invoking the bimodular feature (Almeida and Spilker
1997; Klisch and Lotz 1999; Wagner and Lotz 2004).
Our long-term goal is to develop an accurate stress–strain
equation that can simultaneously describe the equilibrium
elastic response in tension, confined and unconfined com-
pression, and torsional shear. Accurate stress constitutive
equations for cartilaginous tissues have practical applica-
tions. They may be used in FEMs of in vivo joints; the re-
sults of Donzelli et al. (1999) and Krishnan et al. (2003) sug-
gest that more accurate stress constitutive equations for large
deformations may lead to an improved understanding of car-
tilage degeneration and failure. They may be used in micro-
structural finite element models to estimate the mechanical
microenvironment of the cell in order to improve our under-
standing of the mechanotransduction process (Baer et al.
2004; Guilak and Mow 2000). Also, accurate stress constit-
utive equations are needed for conducting robust validation
tests of the cartilage growth mixture models that we have
developed (Klisch et al. 2000, 2001, 2003, 2005; Klisch and
Hoger 2003) and to estimate how the mechanical properties
of cartilage evolve during growth using these models.
In this study, we hypothesize that a bimodular second-
order stress constitutive equation can be used to accurately
model the anisotropic and asymmetric tensile–compressive
response of articular cartilage. The specific objectives are to:
(1) derive a general bimodular theory for finite deformations;
(2) derive a bimodular second-order stress constitutive equa-
tion for orthotropic materials; and
(3) compare the predictive capability of several bimodular
second-order models and a bimodular exponential model
using experimental data gathered from the literature.
In the Discussion, we present ongoing aims that relate to
the integration of the derived phenomenological model with
microstructurally based models.
2 Methods
In this section, we outline the derivation of a second-order
stress constitutive equation for orthotropic materials. Then,
we propose a general theory for bimodular elastic materials
and, consequently, derive a bimodular second-order stress
constitutive equation. Finally, we study the abilities of bi-
modular second-order and exponential models to describe the
mechanical response of articular cartilage in uniaxial tension
(UT) and confined compression (CC).
2.1 Background
The deformation gradient tensor F is uniquely decomposed
by the polar decomposition theorem as
F = RU, (1)
where R is a proper-orthogonal tensor and the right stretch
tensor U is a symmetric positive-definite tensor. The Biot
strain tensor E and the right Cauchy–Green deformation ten-
sor C are
E = U I, C = F
T
F, (2)
where the superscript T signifies the transpose operator. The
Cauchy and first Piola–Kirchhoff stress tensors (denoted as
T and P, respectively) are related by
J T = PF
T
, (3)
where J is the determinant of F. The stress constitutive equa-
tions for a Green-elastic material may be expressed as
n
W
W I
i
P = 2F = 2F , (4)
C I
i
C
i=1
where W is a scalar strain energy function that depends on a
set of invariants of C, I
C
={I
i
(C); i = 1, n}, corresponding
to the material symmetry group:
W = W
ˆ
(C) = W
˜
(I
1
(C), I
2
(C), ..., I
n
(C)) W
˜
(I
C
). (5)
In the second-order theory, an alternative form of the
stress constitutive equation for a Green-elastic material is
used:
T = RT
ˆ
(U)R
T
= RT
˜
(E)R
T
, P = RP
ˆ
(U) = RP
˜
(E), (6)
where the functions T
ˆ
(U) and P
ˆ
(U) are derived from W :
W = W
ˆ
(U) = W
˜
(I
1
(U), I
2
(U), ..., I
n
(U)) W
˜
(I
U
) (7)
and I
U
={I
i
(U); i = 1, n} is the set of basic polynomial in-
variants of U corresponding to the material symmetry group.
2.2 Second-order orthotropic materials
For isotropic elastic materials, various second-order theo-
ries for Green-elastic materials have been proposed using
different strain tensors (Hoger 1999; Murnaghan 1937, 1951;
Rivlin 1953) which differ depending on which strain tensor
is used (Ogden 1984). In this paper, the Biot strain tensor
is used in the second-order equations for two reasons. First,
it has a clear physical interpretation; the eigenvalues of the
Biot strain tensor represent the principal extensions. Second,
the results of Van Dyke and Hoger (2000) suggested that the
second-order stress equations using the Biot strain tensor, as
compared to other strain measures, provided a better approx-
imation of the exact solutions to a group of boundary-value

problems using specific nonlinear elastic materials. For an
arbitrary material symmetry group, general stress constitu-
tive equations that are second-order in terms of the Biot strain
tensor have been presented (Hoger 1999). Those equations
are presented in Appendix A [see (27) and (28)]. Explicit
stress constitutive equations were obtained only for isotropic
and transversely isotropic materials in Hoger (1999); conse-
quently, the derivation for orthotropic materials was one aim
of the present work.
For orthotropic materials, we assume that the material
symmetry group includes reflections about three planes de-
fined by a set of three basis vectors (E
1
, E
2
, E
3
). Structural
tensors (M
1
, M
2
, M
3
) are defined as
M
1
= E
1
E
1
, M
2
= E
2
E
2
, M
3
= E
3
E
3
, (8)
where is the tensor dyadic product. We use the set of in-
variants
{I
i
(U)}={M
1
·U, M
1
·U
2
, M
2
·U, M
2
·U
2
,
M
3
· U, M
3
·U
2
, I · U
3
}
≡{I
1
, I
2
, I
3
, I
4
, I
5
, I
6
, I
7
}. (9)
In Appendix B, we outline the derivation of the general
orthotropic second-order stress constitutive equation. How-
ever, that general equation has 46 material constants and is
not practical for use. Although we initially derived the nec-
essary conditions for stress continuity postulated in the bi-
modular theory for that general equation, here we first obtain
a reduced model for the second-order orthotropic stress con-
stitutive equation.
To obtain a reduced model, we make the following assump-
tions regarding the dependence of the strain energy function
W (7) with respect to the invariants I
i
(U) (9):
(i) W is independent of {I
2
, I
4
, I
6
};
(ii) W is a polynomial function of terms that are uncoupled
with respect to the invariants {I
1
, I
3
, I
5
, I
7
};and
(iii) W is at most a quadratic function of the invariant I
7
.
1
The resulting equation is:
P = RP
˜
(E) =R{λ
11
E
11
M
1
+ λ
22
E
22
M
2
+λ
33
E
33
M
3
+λ[(E
22
+ E
33
)M
1
+ (E
11
+ E
33
)M
2
+(E
11
+ E
22
)M
3
]+2μE
+(1/2
11
E
11
(EM
1
M
1
E)+ (1/2
22
×E
22
(EM
2
M
2
E)
+(1/2
33
E
33
(EM
3
M
3
E)+ (1/2
×[(E
22
+ E
33
)(EM
1
M
1
E)
+(E
11
+ E
33
)(EM
2
M
2
E)
+(E
11
+ E
22
)(EM
3
M
3
E)]
+μE
2
+ λ(I · E
2
)I+ 2λ(I · E)E
+γ
1
(E
11
)
2
M
1
+ γ
2
(E
22
)
2
M
2
+ γ
3
(E
33
)
2
M
3
}. (10)
1
These restrictions can be relaxed somewhat because only the cor-
responding derivatives of W as evaluated in the reference configuration
must vanish in the second-order theory.
This model has eight material constants {λ
11
22
33
,
μ, γ
1
2
3
}, which are defined in terms of the strain energy
function W in Appendix B.
2.3 Bimodular elastic materials
Curnier et al. (1995) developed a bimodular theory for lin-
ear elastic materials in terms of the second Piola–Kirchhoff
stress and Lagrange strain tensors. Here, a bimodular theory
for finite deformations is posed in terms of the right stretch
tensor U or, equivalently, the Biot strain tensor E. The elas-
ticity tensors associated with P
ˆ
(U) and P
˜
(E) [defined in (6)
2
]
are defined as
P
ˆ
(U) P
˜
(E)
P
U
= , P
E
= , (11)
U E
where it can be shown that P
U
(E + I) = P
E
(E).Werequire
the existence of a stress-free reference configuration; i.e.,
P
ˆ
(I) = P
˜
(0) = 0. A scalar valued function of U or E that
identifies a surface of discontinuity in the six-dimensional
strain space is defined as
g
U
(U) = 0, g
E
(E) = g
U
(E + I) = 0, (12)
where it can be shown that
g
U
g
E
(E + I) = (E). (13)
U E
Due to the surface of discontinuity, the stress constitu-
tive equation and, consequently, the elasticity tensor may be
different on either side of the surface of discontinuity; here
we define
P
˜
+
(E) if g
E
(E)> 0 P
E+
if g
E
(E)> 0
˜
P(E) = , P
E
=
if g
E
(E)< 0
. (14)
P
˜
(E) if g
E
(E)< 0
P
E
In Curnier et al. (1995), a theorem that established necessary
and sufficient conditions for the stress–strain equation to be
continuous across the surface of discontinuity was proved.
Here, that theorem is slightly modified, because the major
symmetry of the elasticity tensor for linear elastic materi-
als was invoked in Curnier et al. (1995) whereas the elas-
ticity tensor for finitely elastic materials need not possess
major symmetry. By not invoking that symmetry assumption,
a minor modification of the proof presented in Curnier et al.
(1995) leads to the following necessary and sufficient con-
ditions for the stress–strain equation to be continuous across
the surface of discontinuity:
˜
P(E) = P
˜
+
(E) = P
˜
(E), [P
E
]] = P
E+
P
E
g
E
= s(E)M(E) , (15)
E
for all E such that g
E
(E) = 0, where s(E) is a scalar valued
function and M(E) is a second-order tensor.
2
Due to material
symmetry, the surface of discontinuity must satisfy
g
E
(E) g
U
(I
U
)
|
U=EI
. (16)
2
See Lemma 3.2 in Curnier et al. (1995).

The development in (11)–(15) can be easily modified if
one prefers to work with the Cauchy stress. The functions
T
ˆ
(U) and T
˜
(E) replace P
ˆ
(U) and P
˜
(E) and the elasticity
tensors are defined as in (11); e.g. T
E
= T
˜
(E) E.
2.4 Bimodular second-order orthotropic materials
For a second-order elastic material, not all surfaces that sat-
isfy the material symmetry restriction (16) will satisfy the
continuity conditions (15). In order to model tension-com-
pression asymmetry along the three directions defining or-
thotropy, we use three surfaces of discontinuity:
g
1
= M
1
· E = E
11
= 0,
g
2
= M
2
· E = E
22
= 0, (17)
g
3
= M
3
· E = E
33
= 0.
Consider the surface g
1
= 0. Inspection of the stress
constitutive equation for P, i.e. Eq. (10), reveals that the first
continuity condition (15)
1
(i.e., P
˜
+
(E) = P
˜
(E) for all E
such that E
11
= 0) requires that the only material constants
that may jump across this surface are {λ
11
1
}. We adopt the
notation
λ
11+
if E
11
> 0
[[λ
11
]]=λ
11+
λ
11
11
[E
11
]= . (18)
λ
11
if E
11
< 0
The terms in P
˜
(E) that involve the jump constants are
highlighted as follows:
P
˜
(E) = ···+ λ
11
E
11
M
1
+
1
λ
11
E
11
(EM
1
M
1
E)
2
+γ
1
(E
11
)
2
M
1
+··· (19)
Using (19) and calculating the jump in the elasticity ten-
sor P
E
using (11)
2
we obtain, switching to indicial notation,

˜

[[P
E
]] =
P
AB
E
KL
1
=[[λ
11
]] δ
1A
δ
1B
δ
1K
δ
1L
+
[
E
12
(
δ
2A
δ
1B
δ
1A
δ
2B
)
2
+E
13
3A
δ
1B
δ
1A
δ
3B
)
]
δ
1K
δ
1L
(20)
where the condition E
11
= 0 at the interface was used. Fur-
thermore,
g
1
E
11
= = δ
1K
δ
1L
, (21)
E
KL
E
KL
so that the second continuity condition (15)
2
becomes
[[P
E
]] = s(E)M
AB
δ
1K
δ
1L
. (22)
Comparison of (20) and (22) reveals that the second conti-
nuity condition may be satisfied. Therefore, a bimodular sec-
ond-order material with a surface of discontinuity defined by
g
1
= E
11
= 0 may be represented by replacing the material
constants {λ
11
1
} with {λ
11
[E
11
]
1
[E
11
]} where we have
used the notation of (18)
2
. Using a similar analysis, or by
interchanging the indices appropriately, a bimodular mate-
rial with additional surfaces of discontinuity g
2
= E
22
= 0
and g
3
= E
33
= 0 may have discontinuous material con-
stants {λ
22
2
} and {λ
33
3
}, respectively. Consequently,
the bimodular stress constitutive equation corresponding to
the reduced second-order orthotropic material (10) may have
a total of 14 material constants.
3
A similar analysis using
the Cauchy stress reveals that the same material constants
can jump across these surfaces of discontinuity.
Finally, we make two additional simplifying assumptions
for the analyses of the present study. First, we require that the
first-order constants {λ
11
22
33
} be continuous across the
surfaces of discontinuity. The rationale for this requirement
is that previous analyses suggested that material stability is
difficult to ensure if the first-order constants jump, because
eight stiffness matrices must be positive-definite (see, Klisch
et al. 2004; Wang et al. 2003). Second, we neglect the second-
order terms associated with λ and μ [see Eqs. (10) or (37)].
4
The rationale for this reduction is that preliminary results for
models that included these second-order terms exhibited a
non-convex mechanical response in CC. Thus, we consider
the reduced model
P = RP
˜
(E) = R
{
λ
11
E
11
M
1
+ λ
22
E
22
M
2
+ λ
33
E
33
M
3
+λ[(E
22
+ E
33
)M
1
+ (E
11
+ E
33
)M
2
+(E
11
+ E
22
)M
3
]+ 2μE
+(1/2
11
E
11
(EM
1
M
1
E)
+(1/2
22
E
22
(EM
2
M
2
E)
+(1/2
33
E
33
(EM
3
M
3
E)
+(1/2[(E
22
+ E
33
)(EM
1
M
1
E)
+(E
11
+ E
33
)(EM
2
M
2
E) + (E
11
+ E
22
)
×(EM
3
M
3
E)]
+γ
1
[E
11
](E
11
)
2
M
1
+ γ
2
[E
22
](E
22
)
2
M
2
+ γ
3
[E
33
](E
33
)
2
M
3
, (23)
which results in a maximum of 11 parameters. It is important
to emphasize that in (23), only the second-order material con-
stants may jump. Consequently, the elasticity tensor (in addi-
tion to the strain energy function and stress–strain equation)
is continuous across the surfaces of discontinuity whereas the
gradient of the elasticity tensor may jump. Finally, for pure
stretch deformations, (23) simplifies to
P = P
˜
(E) = λ
11
E
11
M
1
+ λ
22
E
22
M
2
+ λ
33
E
33
M
3
+λ[(E
22
+ E
33
)M
1
+ (E
11
+ E
33
)M
2
+(E
11
+ E
22
)M
3
]+ 2μE
+γ
1
[E
11
](E
11
)
2
M
1
+ γ
2
[E
22
](E
22
)
2
M
2
+γ
3
[E
33
](E
33
)
2
M
3
. (24)
3
The bimodular stress constitutive equation corresponding to the
general second-order orthotropic material may have a total of 61 mate-
rial constants.
4
Consequently, the equation studied is no longer an exact second-
order approximation.

Note that we have again used the notation introduced in (18)
2
for the discontinuous material constants {γ
1
2
3
}.
2.5 A bimodular exponential orthotropic material
To compare the predictive capability of (24) with current
exponential models, we considered a strain energy function
of the form (5):
3
W = W
˜
(I
C
) =
a
1
[I
7
(I
7
)
1
]
2
+
b
i
e
c
i
(I
i
1)
2
1 ,
2 2c
i
i=1
(25)
with seven parameters (a
1
, b
1
, c
1
, b
2
, c
2
, b
3
, c
3
) where, for
convenience, we have ordered the invariants of C as
{I
i
(C)}={M
1
· C, M
2
· C, M
3
· C, M
1
· C
2
, M
2
·C
2
, M
3
· C
2
, detC}
≡{I
1
, I
2
, I
3
, I
4
, I
5
, I
6
, I
7
}. (26)
In (25), the isotropic part (i.e., the term dependent on
I
7
) was based on the orthotropic model of Wagner and Lotz
(2004) while the anisotropic part was based on the study of
Holzapfel et al. (2004). Recently, exponential strain energy
functions of the type (25) have been employed that are “bi-
modular” (Baer et al. 2004; Holzapfel et al. 2004). In those
studies the anisotropy is attributed to the presence of collagen
fibers that are hypothesized to support only tensile stresses.
To be consistent with those studies, it suffices to investigate
whether the constants b
1
, b
2
,and b
3
can jump across the
surfaces of discontinuity defined by I
1
= 1, I
2
= 1, and
I
3
= 1, respectively. First, (4) is used to calculate P,which
can then be expressed as P = RP
˜
(E). Then, using the meth-
ods outlined above, it can be confirmed that the continuity
conditions (15) can be satisfied while allowing the constants
b
1
, b
2
,and b
3
to jump across the surfaces I
1
= 1, I
2
= 1,
and I
3
= 1, respectively. Thus, to employ the exponential
model (25) and to be consistent with Baer et al. (2004) and
Holzapfel et al. (2004), we consider b
1
= 0, b
2
= 0, and
b
3
= 0when I
1
< 1, I
2
< 1, and I
3
< 1, respectively,
resulting in a 7-parameter model.
2.6 Material stability conditions
For analyses using the reduced orthotropic bimodular model
(23), necessary and sufficient conditions for material stabil-
ity are straightforward to derive and apply. Here, we follow
the definition and interpretation of incremental stability for
the conjugate pair of Biot stress and Biot strain as presented
in Ogden (1984). Appendix C outlines the derivation of the
following two necessary and sufficient conditions for mate-
rial stability: (1) the first-order material constants λ
11
, λ
22
,
λ
33
, λ,and μ must correspond to a positive-definite stiffness
matrix; and (2) the second-order constants, if non-zero, must
satisfy γ
1+
> 0
2+
> 0
3+
> 0
1
< 0
2
< 0, and
γ
3
< 0. Stability restrictions are difficult to obtain and ver-
ify when using an exponential strain energy function such as
(25). In Wagner et al. (2002), it was noted that a
1
> 0 is nec-
essary for having a positive definite strain energy function.
In Holzapfel et al. (2004), it was argued that b
i
> 0isa suf-
cient condition for strong ellipticity. Thus, these relations are
sufficient for incremental stability.
2.7 Experimental data
We constructed a hypothetical experimental dataset (Table 1)
that may approximate the mechanical response of adult hu-
man cartilage in the surface region, which typically exhibits
the strongest anisotropy. The data corresponded to UT and
CC experiments along three directions: (1) parallel to the
split line, (2) perpendicular to the split line and parallel to
the surface, and (3) perpendicular to the surface (Fig. 1). To
develop the UT datasets, a 2-parameter exponential func-
tion was used to generate data from 0 to 20% strain in 2%
increments. For UT in the 1 and 2 directions, the parameters
were adopted from Huang et al. (1999, 2005). For UT in the
3 direction, the parameters from the UT in the 1-direction
were scaled down using a ratio of the infinitesimal Young’s
moduli reported in Chahine et al. (2004) for bovine cartilage.
Poisson’s ratios were assumed to be the same in all three UT
experiments, and to be linearly increasing functions of ax-
ial strain. The Poisson’s ratio at 0% UT strain was assumed
to correspond to the Poisson’s ratio for infinitesimal defor-
mations in unconfined compression since the stress–strain
equation is continuous through the origin, and was chosen
using theoretical predictions with bovine tissue (Wang et al.
2003). The Poisson’s ratio at 16% strain was specified using
the data reported in Huang et al. (1999). To develop the CC
datasets, a 2-parameter exponential function (Ateshian et al.
1997) was used to generate data from 0 to 20% strain in 2%
increments; the parameters were based on results of Huang
et al. (1999, 2005). It was assumed that the CC response was
the same in the 1 and 2 directions.
2.8 Regression analysis
A simultaneous nonlinear regression algorithm was performed
in Mathematica (Wolfram, V5.0) based on an approach devel-
Table 1 Values of Young’s modulus E, Poisson’s ratio ν, and aggregate
modulus H
A
at 0% and 16% strain levels in the 1,2, and 3 directions
for the experimental dataset used (E and H
A
in MPa)
Parameter Direction
1 2 3
E
0
E
0.16
ν
0
ν
0.16
H
A0
H
A0.16
7.8
42.8
0.05
1.33
0.18
0.26
5.9
26.3
0.05
1.33
0.18
0.26
1.2
9.0
0.05
1.33
0.10
0.15

Citations
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Journal ArticleDOI
01 Dec 2008-Spine
TL;DR: A hyperelastic fiber-reinforced constitutive model characterized the functional evolution of engineered AF constructs, and was used to simulate physiologically relevant loading configurations and demonstrated that fibers resist shear even when the shearing direction does not coincide with the fiber direction.
Abstract: STUDY DESIGN Integrating theoretical and experimental approaches for annulus fibrosus (AF) functional tissue engineering. OBJECTIVE Apply a hyperelastic constitutive model to characterize the evolution of engineered AF via scalar model parameters. Validate the model and predict the response of engineered constructs to physiologic loading scenarios. SUMMARY OF BACKGROUND DATA There is need for a tissue engineered replacement for degenerate AF. When evaluating engineered replacements for load-bearing tissues, it is necessary to evaluate mechanical function with respect to the native tissue, including nonlinearity and anisotropy. METHODS Aligned nanofibrous poly-epsilon-caprolactone scaffolds with prescribed fiber angles were seeded with bovine AF cells and analyzed over 8 weeks, using experimental (mechanical testing, biochemistry, histology) and theoretical methods (a hyperelastic fiber-reinforced constitutive model). RESULTS The linear region modulus for phi = 0 degrees constructs increased by approximately 25 MPa, and for phi = 90 degrees by approximately 2 MPa from 1 day to 8 weeks in culture. Infiltration and proliferation of AF cells into the scaffold and abundant deposition of s-GAG and aligned collagen was observed. The constitutive model had excellent fits to experimental data to yield matrix and fiber parameters that increased with time in culture. Correlations were observed between biochemical measures and model parameters. The model was successfully validated and used to simulate time-varying responses of engineered AF under shear and biaxial loading. CONCLUSION AF cells seeded on nanofibrous scaffolds elaborated an organized, anisotropic AF-like extracellular matrix, resulting in improved mechanical properties. A hyperelastic fiber-reinforced constitutive model characterized the functional evolution of engineered AF constructs, and was used to simulate physiologically relevant loading configurations. Model predictions demonstrated that fibers resist shear even when the shearing direction does not coincide with the fiber direction. Further, the model suggested that the native AF fiber architecture is uniquely designed to support shear stresses encountered under multiple loading configurations.

78 citations


Journal ArticleDOI
01 Aug 2011-Spine
TL;DR: Model predictions indicate that the properly sized implant retains the mobility, load sharing, and contact force magnitude and distribution of the intact case, and support the continued use of motion sparing implants in the lower cervical spine.
Abstract: Study design: A parametric finite element investigation of the cervical spine. Objective: To determine what effect, if any, cervical disc replacement has on kinematics, facet contact parameters, and anterior column loading. Summary of Background Data: Anterior cervical discectomy and fusion (ACDF) has been a standard treatment for certain spinal degenerative disorders, but evidence suggests that fusion contributes to adjacent-segment degeneration. Motion-sparing disc replacement implants are believed to reduce adjacent-segment degeneration by preserving kinematics at the treated level. Such implants have been shown to maintain the mobility of the intact spine, but the effects on load transfer between the anterior and posterior elements remain poorly understood. Methods: In order to investigate the effects of disc replacement on load transfer in the lower cervical spine, a finite element model was generated using cadaver-based Computed Tomography (CT) imagery. Mesh resolution was varied in order to establish model convergence, and cadaveric testing was undertaken to validate model predictions. The validated model was altered to include a disc replacement prosthesis at the C4/C5 level. The effect of disc-replacement on range of motion, antero-posterior load distribution, contact forces in the facets, as well as the distribution of contact pressure on the facets were examined. Three sizes of implants were examined. Results: Model predictions indicate that the properly-sized implant retains the mobility, load sharing, and contact force magnitude and distribution of the intact case. Mobility, load sharing, nuclear pressures, and contact pressures at the adjacent motion segments were not strongly affected by the presence of the properly sized implant, indicating that disc replacement may not be a significant cause of post-operative adjacent-level degeneration. Implant size affected certain mechanical parameters, such as antero-posterior load sharing, and did not affect compliance or range of motion. Conclusions: The results of this work support the continued use of motion sparing implants in the lower cervical spine. Load sharing data indicate that implant size may be an important factor that merits further study, although the deleterious effects of improper size selection may be less significant than those of fusion.

58 citations


Journal ArticleDOI
TL;DR: The long range objective of this work is to develop a cartilage growth finite element model (CGFEM) based on the theories of growing mixtures that has the capability to depict the evolution of the anisotropic and inhomogeneous mechanical properties, residual stresses, and nonhomogeneities that are attained by native adult cartilage.
Abstract: The long range objective of this work is to develop a cartilage growth finite element model (CGFEM), based on the theories of growing mixtures that has the capability to depict the evolution of the anisotropic and inhomogeneous mechanical properties, residual stresses, and nonhomogeneities that are attained by native adult cartilage. The CGFEM developed here simulates isotropic in vitro growth of cartilage with and without mechanical stimulation. To accomplish this analysis a commercial finite element code (ABAQUS) is combined with an external program (MATLAB) to solve an incremental equilibrium boundary value problem representing one increment of growth. This procedure is repeated for as many increments as needed to simulate the desired growth protocol. A case study is presented utilizing a growth law dependent on the magnitude of the diffusive fluid velocity to simulate an in vitro dynamic confined compression loading protocol run for 2 weeks. The results include changes in tissue size and shape, nonhomogeneities that develop in the tissue, as well as the variation that occurs in the tissue constitutive behavior from growth.

24 citations


Cites background or methods from "A Bimodular Theory for Finite Defor..."

  • ...In Klisch (2006b), we derived an equation for determining δFg from the growth law using a first-order Taylor series expansion....

    [...]

  • ...Future work will extend the bimodular stress– strain equation for the collagen constituent to include anisotropic behavior (Klisch 2006a)....

    [...]

  • ...(A3) For orthotropic, transversely isotropic, and isotropic materials, if F2e and T̂κ1(δFe) satisfy F2e = δFeF1e, T̂κ1(δFe) = T̂κ0(δFeF1e), (A4) then these are sufficient conditions for (A3) to hold Klisch (2006b)....

    [...]

  • ...These two tools interactively solve the equilibrium growth boundary value problem (BVP) as proposed in (Klisch et al. 2001; Klisch 2006b), subject to the growth theories presented in the CGMM (Klisch et al. 2000, 2003; Klisch and Hoger 2003)....

    [...]

  • ...This result was first presented in Klisch (2006b)....

    [...]


Journal ArticleDOI
25 Aug 2008-Wear
Abstract: The creep of a polymer sphere in contact with a rigid flat under a constant normal load is investigated numerically by a finite element method. A simplified form of the modified time hardening is used for the constitutive material model. A new dimensionless form of the creep displacement is suggested, which enables a universal solution of the problem for any contact time based on a conveniently selected reference time. The model may be applicable to other materials which can be described by the modified time hardening model including biological materials.

22 citations


Journal ArticleDOI
TL;DR: The results suggest that the main advantage of a model employing the strong interaction terms is to provide the capability for modeling anisotropic and asymmetric Poisson's ratios, as well as axial stress-axial strain responses, in tension and compression for finite deformations.
Abstract: A strain energy function for finite deformations is developed that has the capability to describe the nonlinear, anisotropic, and asymmetric mechanical response that is typical of articular cartilage. In particular, the bimodular feature is employed by including strain energy terms that are only mechanically active when the corresponding fiber directions are in tension. Furthermore, the strain energy function is a polyconvex function of the deformation gradient tensor so that it meets material stability criteria. A novel feature of the model is the use of bimodular and polyconvex "strong interaction terms" for the strain invariants of orthotropic materials. Several regression analyses are performed using a hypothetical experimental dataset that captures the anisotropic and asymmetric behavior of articular cartilage. The results suggest that the main advantage of a model employing the strong interaction terms is to provide the capability for modeling anisotropic and asymmetric Poisson's ratios, as well as axial stress-axial strain responses, in tension and compression for finite deformations.

16 citations


References
More filters

Book
01 Jan 1984
Abstract: non linear elastic deformations iwsun non linear elastic deformations erpd non linear elastic deformations hneun non-linear elastic deformations (dover civil and non-linear elastic deformations of multi-phase fluid systems non linear elastic deformations dover civil and mechanical ogden nonlinear elastic deformations pdf wordpress non-linear, elastic researchgate chapter 6 non linear material models international journal of nonlinear mechanics nonlinear elastic deformations ogden pdfslibforme international journal of non-linear mechanics 1 rubber elasticity: basic concepts and behavior non linear elastic deformations dover civil and mechanical on a non-linear wave equation in elasticity non linear elastic deformations (pdf) by r. w. ogden (ebook) exact formulations of non-linear planar and spatial euler introduction to nonlinear analysis mit opencourseware manual for the calculation of elastic-plastic materials non linear elastic axisymmetric deformation of membranes types of analysis: linear static, linear dynamic and non fracture mechanics, damage and fatigue non linear fracture chapter 2 linear elasticity freie universität the influence of non-linear elastic systems on the a simple geometric model for elastic deformations

3,706 citations


"A Bimodular Theory for Finite Defor..." refers background or methods in this paper

  • ...2.2 Second-order orthotropic materials For isotropic elastic materials, various second-order theo ries for Green-elastic materials have been proposed using different strain tensors (Hoger 1999; Murnaghan 1937, 1951; Rivlin 1953) which differ depending on which strain tensor is used (Ogden 1984)....

    [...]

  • ...(40) In Ogden (1984) 7, this inequality is interpreted as stabil ity “under tractions which follow the material, i.e. rotate with the local rotation R . . . for an isotropic material.”...

    [...]

  • ...For isotropic elastic materials, various second-order theo­ ries for Green-elastic materials have been proposed using different strain tensors (Hoger 1999; Murnaghan 1937, 1951; Rivlin 1953) which differ depending on which strain tensor is used (Ogden 1984)....

    [...]

  • ...Here, we follow the defini­ tion and interpretation of incremental stability for the conju gate pair of Biot stress and Biot strain as presented in Ogden (1984) and derive necessary and sufficient material stabil ity conditions for the reduced orthotropic bimodular secondorder model (23)....

    [...]

  • ...…are the symmetric and skew parts of a tensor, and ∂Ŵ ∂2W ∂ I j ∂ Ik �1[E] = D (I)[E] = (II ) (I) (I) · E ∂U ∂ Ik ∂ I j ∂U ∂U ∂W ∂2 I j+ (II ) (I)[E], ∂ I j ∂U2 5 For example, see the discussion on molecular and phenomenologi cal models of rubber elasticity presented in Chapter 7 of Ogden (1984)....

    [...]


Book
01 Jan 1981
Abstract: Preface. Acknowledgments. Tensor Algebra. Tensor Analysis. Kinematics. Mass. Momentum. Force. Constitutive Assumptions. Inviscid Fluids. Change in Observer. Invariance of Material Response. Newtonian Fluids. The NavierStokes Equations. Finite Elasticity. Linear Elasticity. Appendix. References. Hints for Selected Exercises. Index.

1,751 citations


Book
01 Jan 1967

1,439 citations


"A Bimodular Theory for Finite Defor..." refers background in this paper

  • ...2.2 Second-order orthotropic materials For isotropic elastic materials, various second-order theo ries for Green-elastic materials have been proposed using different strain tensors (Hoger 1999; Murnaghan 1937, 1951; Rivlin 1953) which differ depending on which strain tensor is used (Ogden 1984)....

    [...]


Book
01 Jan 2010
TL;DR: This new edition offers expanded coverage of the subject matter both in terms of details and contents, providing greater flexibility for either a one or two-semester course in either continuum mechanics or elasticity, and contains expanded and improved problem sets providing both intellectual challenges and engineering applications.
Abstract: Continuum Mechanics is a branch of physical mechanics that describes the macroscopic mechanical behavior of solid or fluid materials considered to be continuously distributed. It is fundamental to the fields of civil, mechanical, chemical and bioengineering. This time-tested text has been used for over 35 years to introduce junior and senior-level undergraduate engineering students, as well as graduate students, to the basic principles of continuum mechanics and their applications to real engineering problems. The text begins with a detailed presentation of the coordinate invariant quantity, the tensor, introduced as a linear transformation. This is then followed by the formulation of the kinematics of deformation, large as well as very small, the description of stresses and the basic laws of continuum mechanics. As applications of these laws, the behaviors of certain material idealizations (models) including the elastic, viscous and viscoelastic materials, are presented. This new edition offers expanded coverage of the subject matter both in terms of details and contents, providing greater flexibility for either a one or two-semester course in either continuum mechanics or elasticity. Although this current edition has expanded the coverage of the subject matter, it nevertheless uses the same approach as that in the earlier editions - that one can cover advanced topics in an elementary way that go from simple to complex, using a wealth of illustrative examples and problems. It is, and will remain, one of the most accessible textbooks on this challenging engineering subject. It offers significantly expanded coverage of elasticity in Chapter 5, including solutions of some 3-D problems based on the fundamental potential functions approach. It includes a new section at the end of Chapter 4 devoted to the integral formulation of the field equations. Seven new appendices appear at the end of the relevant chapters to help make each chapter more self-contained. This book contains expanded and improved problem sets providing both intellectual challenges and engineering applications.

1,165 citations


Journal ArticleDOI
TL;DR: The results show that all three mechanisms are important in determining the overall compressive stiffness of cartilage.
Abstract: Swelling of articular cartilage depends on its fixed charge density and distribution, the stiffness of its collagen-proteoglycan matrix, and the ion concentrations in the interstitium. A theory for a tertiary mixture has been developed, including the two fluid-solid phases (biphasic), and an ion phase, representing cation and anion of a single salt, to describe the deformation and stress fields for cartilage under chemical and/or mechanical loads. This triphasic theory combines the physico-chemical theory for ionic and polyionic (proteoglycan) solutions with the biphasic theory for cartilage. The present model assumes the fixed charge groups to remain unchanged, and that the counter-ions are the cations of a single-salt of the bathing solution. The momentum equation for the neutral salt and for the intersitial water are expressed in terms of their chemical potentials whose gradients are the driving forces for their movements. These chemical potentials depend on fluid pressure p, salt concentration c, solid matrix dilatation e and fixed charge density cF. For a uni-uni valent salt such as NaCl, they are given by mu i = mu io + (RT/Mi)ln[gamma 2 +/- c(c + cF)] and mu w = mu wo + [p-RT phi (2c + cF) + Bwe]/pwT, where R, T, Mi, gamma +/-, phi, pwT and Bw are universal gas constant, absolute temperature, molecular weight, mean activity coefficient of salt, osmotic coefficient, true density of water, and a coupling material coefficient, respectively. For infinitesimal strains and material isotropy, the stress-strain relationship for the total mixture stress is sigma = - pI-TcI + lambda s(trE)I + 2 musE, where E is the strain tensor and (lambda s, mu s) are the Lame constants of the elastic solid matrix. The chemical-expansion stress (-Tc) derives from the charge-to-charge repulsive forces within the solid matrix. This theory can be applied to both equilibrium and non-equilibrium problems. For equilibrium free swelling problems, the theory yields the well known Donnan equilibrium ion distribution and osmotic pressure equations, along with an analytical expression for the "pre-stress" in the solid matrix. For the confined-compression swelling problem, it predicts that the applied compressive stress is shared by three load support mechanisms: 1) the Donnan osmotic pressure; 2) the chemical-expansion stress; and 3) the solid matrix elastic stress. Numerical calculations have been made, based on a set of equilibrium free-swelling and confined-compression data, to assess the relative contribution of each mechanism to load support. Our results show that all three mechanisms are important in determining the overall compressive stiffness of cartilage.

1,050 citations


"A Bimodular Theory for Finite Defor..." refers background in this paper

  • ...The proteoglycans are negatively charged molecules that mainly resist compressive loads (Basser et al. 1998; Lai et al. 1991) while the collagen net­ work primarily resists tensile and shear loads (Mow and Ratcliffe 1997; Venn and Maroudas 1977)....

    [...]

  • ...The proteoglycans are negatively charged molecules that mainly resist compressive loads (Basser et al. 1998; Lai et al. 1991) while the collagen net work primarily resists tensile and shear loads (Mow and Ratcliffe 1997; Venn and Maroudas 1977)....

    [...]


Frequently Asked Questions (1)
Q1. What are the contributions mentioned in the paper "A bimodular theory for finite deformations: comparison of orthotropic second-order and exponential stress constitutive equations for articular cartilage" ?

In this paper, the authors derive an orthotropic stress constitutive equation that is second-order in terms of the Biot strain ten­ sor as an alternative to traditional exponential type equations. The results suggest that the bimodular second-order models may be appealing for some applications with cartilaginous tissues.