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Appendix 5: 2D Linearization of Fung’s Exponential Strain... 235
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2
2
R
Animal#C(kPa) b
( pi)
RMS%
(FT)
b2b
1
b4b5b
3
(FT)
6
R
RMS%
( pi)
Heart15 1.23 4.18 4.77 0.95 0.51 0.02 0.15 0.98 0.99 20.0 10.1
Mean 5.11 2.47 3.09 0.95 0.45 0.06 0.10 0.98 0.98 22.4 18.0
SD 2.65 1.00 0.95 0.55 0.10 0.04 0.05 0.01 0.01 6.4 5.7
(a) Material constants of the five intact LADs; (b) material constants of the five corresponding
intima-media layers. R
error of the fit compared to the mean value. Both were computed for the total axial force (F
inner pressure ( p
2
is correlation coefficient; RMS% is the percentage of the root mean square
). Reproduced from Wang et al. (2006) with permission
i
) and
T
Table 4.16 Material constants of exponential strain energy function obtained from experimental
data of left anterior descending (LAD) artery
2
2
R
Animal#C(kPa) b
( pi)
RMS%
(FT)
b2b
1
b4b5b
3
(FT)
6
R
RMS%
( pi)
(a) Intact
Heart16 4.43 1.48 5.15 0.24 0.25 0.46 0.01 0.88 0.97 33.2 22.1
Heart17 13.87 0.66 4.90 0.12 0.48 0.07 0.01 0.91 0.99 31.4 18.2
Heart18 4.26 0.92 7.84 2.42 0.31 0.20 0.32 0.97 0.99 20.8 38.4
Heart19 13.86 1.26 1.89 0.52 0.27 0.05 0.14 0.98 0.98 20.9 25.6
Heart20 6.55 1.52 1.75 1.79 0.40 0.01 0.10 0.98 0.99 25.9 19.3
Mean 8.59 1.17 4.31 1.02 0.34 0.16 0.12 0.95 0.98 26.5 24.7
SD 4.90 0.37 2.55 1.03 0.10 0.18 0.13 0.05 0.01 5.8 8.2
(b) Adventitia
Heart16 9.11 0.29 2.94 1.81 0.38 0.06 0.04 0.80 0.91 44.5 34.0
Heart17 13.29 0.68 1.56 1.72 0.34 0.16 0.04 0.96 0.94 40.7 46.9
Heart18 5.06 0.84 2.08 0.81 0.28 0.02 0.03 0.95 0.92 46.0 54.9
Heart19 9.68 0.50 1.77 2.24 0.43 0.27 0.21 0.97 0.98 30.9 30.4
Heart20 8.09 0.77 3.01 1.79 0.26 0.04 0.04 0.90 0.99 38.0 10.0
Mean 9.05 0.62 2.27 1.67 0.34 0.11 0.07 0.92 0.95 40.0 35.2
SD 2.97 0.22 0.67 0.52 0.07 0.10 0.08 0.07 0.04 6.0 17.2
(a) Material constants of the five intact LADs; (b) material constants of the five corresponding
adventitia layers. R
error of the fit compared to the mean value. Both were computed for the total axial force (F
inner pressure ( p
2
is correlation coefficient; RMS% is the percentage of the root mean square
). Reproduced from Wang et al. (2006) with permission
i
) and
T
Appendix 5: 2D Linearization of Fung’s Exponential Strain
Energy Function (SEF) (Zhang & Kassab, 2007)
To linearize the constitutive relation, the major task is to find a strain measure that
can best approximate the nonlinear curvature of load versus stretch ratio plots
obtained in experiments. Since Fung’s model represents the experimental measurements well, the only task is to examine if Fung’s model can be linearized. In the 1D
236 4 Constitutive Models of Coronary Vasculature
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strain case where Ezz¼ 0(E
¼ E
θθ
¼ 0 can be assumed as well), the second Piola–
zz
Kirchhoff stress in circumferential direction is given by (Appendix 3):
∂W
S
¼
θθ
¼ Ca1Eθθexp a1E
∂E
θθ
where c
¼ Ca1and Dθθ¼ Eθθexp a1E
11
law can be obtained in the 1D case if D
2
¼ c
θθ
2
. Equation (4.47) shows that Hooke’s
θθ
is viewed as a strain measure. A similar
θθ
, ð4:47Þ
11Dθθ
relation exists in the axial direc tion.
The expression of Eq. (4.47) in a linearized form does not highlight the advantages in the 1D model, but it certainly requires less material constants than a
polynomial form (Sokolis, Boudoulas, & Karayannacos, 2002). In 2D, the linearization of the stress–strain relationship is preferred since it may be difficult to
experimentally determine material constants due to the nonlinearity of the constitutive law (Pandit et al., 2005).
A Generalized Strain Measure
Although Green strain is the best choice for Fung’s model since it directly yields
second Piola–Kirchhoff stress from the differentiation of strain energy function,
other strain measures can also be selected to establish a stress–strain relationship. In
the spirit of Seth-Hill strain family (Farahani & Naghdabadi, 2000), a new strain
measure is proposed that takes into account the exponential increase of stress with
Green strain to absorb the nonlinearity and thus result in a simpler stress–strain
relation. In the circumferential (θ) and axial (z) directions, the new strains are defined
as:
where n is a material constant characterizing the strain nonlinearity with respect to
Green strain E
Appendix 4). These generalized strains (Green-exponential strains) are products of
Green strains and their exponential functions.
A Bilinear Stress–Strain Relation
Given the coupling between circumferential and axial directions in a biaxial deformation, a simple bilinear constitutive relation is postulated between second Piola–
Kirchhoff stress and the new strain measure as:
D
¼ Eθθexp nE
θθ
and Ezz(for n ¼ 0, Eq. (4.48) reduces to Eqs. (4.28) and (4.29),
θθ
2
θθ
, D
¼ Ezzexp nE
zz
2
, ð4:48Þ
zz
Appendix 6: 3D Linearization of Fung’s Exponential Strain... 237
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Sθθ¼ c11Dθθþ c12Dzzþ b12DθθD
Szz¼ c21Dθθþ c22Dzzþ b21DθθD
zz
, ð4:49Þ
zz
in which there are six constants having the unit of stress and one constant n that
characterizes the nonlinearity of strain. The constant c
circumferential direction when the strain in the axial direction is zero, c
elastic modulus of S
direction, and b
versus Dzzwhen there is no deformation in the circumferential
θθ
is the bilinear coupling constant for circumferential stress. The
12
is the elastic stiffness in the
11
reflects the
12
other three parameters have similar meanings in the axial direction.
In general, the material constants in Eq. (4.49) do not have symmetric properties,
e.g., c
6¼ c21since the hyperelasticity assumption (the existence of strain energy
12
potential (Fung, 1993)) cannot be invoked which can reduce the number of
constants.
Evaluation of the Bilinear Model
The nonlinearity parameter n in the strain defined in Eq. (4.48) can be obtained with
experimental data using a least square s method. If the stretch ratio in the axial
direction is kept constant (e.g., E
to Eq. (4.47), where a
¼ n is a constant to be determined. Of course, if n is extracted
1
from data in the axial direction, it may differ from that obtained from the circumferential direction. Fortunately, n is not very sensitive to the direction chosen and
thus eith er circumferenti al or axial data (or an average value) can be used for the
determination of n. In practice, one set of “the most reliable” (or the most representative) experimental data can be used to determine n. Once n is determined, the
values of the elastic constants can be easily obtained using least squares method for
the bilinear model.
¼ 0), the first equation in Eq. (4.49) will reduce
zz
Appendix 6: 3D Linearization of Fung’s Exponential Strain
Energy Function (SEF) (Zhang, Wang, & Kassab, 2007)
This appendix extends the strain linearization notion to the 3D stress–strain relationship. By introducing a more complicated but more rigorous strain measure in
tensor form, the constitutive equation can be written in the form of a generalized
Hooke’s law which connects the second Piola–Kirchhoff stresses with the new
strains. In general, the deformation gradient tensor in a cylindrical coordinate system
can be written as:
238 4 Constitutive Models of Coronary Vasculature
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3
7
7
7
r∂θ
7
, ð4:50 Þ
7
∂Z
7
5
F ¼
2
∂r∂R∂r
6
6
6
r∂θ∂Rr∂θ
6
6
6
4
∂z∂R∂z
R∂Θ∂r∂Z
R∂Θ
R∂Θ∂z∂Z
where (R, Θ, Z) and (r, θ, z) are spatial coordi nates in the un deformed and deformed
configurations, respectively.
Since the Lagrangian strain tensors are of interest, the following unique polar
decomposition of the deformation gradient is considered (Ogden, 1984):
F ¼ RU, ð4:51Þ
where R is an orthogonal second-order tensor which reflects the rigid rotation and
U is a positive definite symmetric second-order tensor (the right stretch tensor).
The general form of Seth-Hill (Doyle-Ericksen) strain tensor family is defined as
(Mahnken, 2005; Miehe & Lambrecht, 2001; Ogden, 1984):
1
mðÞ
E
¼
m
U
IðÞ, ð4:52Þ
m
where I is the identity matrix (second-order tensor) and m is a real number. An
important example is the Green strain tensor (classically denoted by E) which
corresponds to m ¼ 2, namely:
It is noted that U
(or matrix) transpose.
Hencky (natural, logarithmic, true) strain tensor is a special case of the Seth-Hill
strain family when m ! 0:
The Hencky strain has many good features such as additively separable dilatation
and distortion (Criscione, Humphrey, Douglas, & Hunter, 2000). To take advantage
of these properties, a generalized Hencky (logarithmic-exponential) strain tensor is
defined as:
1
2ðÞ
E ¼ E
2¼FT
F because RTR ¼ I, where superscript T denotes the tensor
E
2
U
¼
0ðÞ
I
2
1
ln U
¼
2
D ¼ ln Uexp nJ
1
T
F
¼
2
¼ ln U: ð4:54Þ
1
F I
2
: ð4:53Þ
3ðÞ½, ð4:55Þ
Appendix 6: 3D Linearization of Fung’s Exponential Strain... 239
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where J1is the first invariant of the right Cauchy-Green deformation tensor FTF ¼ U
(Ogden, 1984) and the nondimensional n characterizes the nonlinear variation of
strain D with respect to the Hencky strain measure and is thus called the nonlinearity
parameter. The objectivity (i.e., independence on observer (Ogden, 1984) of strain
measure D is the same as the Hencky strain since J
is an invariant in an observer
1
transformation.
For simplicity, it is assumed that the blood vessel is a cylindrically orthotropic
material (Fung, 1993). In an inflation–stretch test without shear deformation, the
principal stretches (λ
, λz, λr) are aligned in the circumferential (θ), axial (z), and
θ
radial (r) directions. In such cases, the principal Green strains (Eq. 4.53) can be
written as:
1
2
E
λ
¼
ii
1
i
2
i ¼ θ; z; rðÞ, ð4:56Þ
and the principal logarithmic-exponential (log-exp) strains in Eq. (4.55) becomes:
D
¼ ln λiexp nJ1 3ðÞ½i ¼ θ; z; rðÞ, ð4:57Þ
ii
where
2
2
J
¼ λ
þ λ
1
θ
2
þ λ
: ð 4 :58Þ
z
r
No summation is assumed in Eqs. (4.56) and (4.57 ). Similar to the Hencky strain,
a spectral decomposition method (Ogden, 1984) can be used to compute the general
log-exp strain tensor when the principal stretches are known.
To gain a better understanding of the log-exp strain measure, it will be compared
with the Green and Hencky strains (note that D reduces to Hencky strain when
n ¼ 0). n is found to typically vary in a range of 0 < n < 2 for coronary vessels.
2
Strain Potential
For a hyperelastic material, it is assumed that a strain energy density function exists
and the stress components can be deriv ed by differentiating the strain potential with
respect to the corresponding strains. For example, the second Piola–Kirchhoff stress
can be obtained by Fung (1993) and Ogden (1984):
where W(E
) is the strain potential, q is an arbitrary scalar, V(Eij) ¼ 0 is a possible
ij
internal constraint (e.g., incompressibility), and S
constraint.
^
S
¼ Sijþ q
ij
∂V
∂E
∂W
¼
∂E
ij
ij
þ q
∂V
∂E
i; j ¼ θ; z; rðÞ, ð4:59Þ
ij
¼ ∂W/∂Eijis the stress with no
ij
240 4 Constitutive Models of Coronary Vasculature
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The Fung’s 3D strain energy per unit volume is given by:
C
W ¼
exp QðÞ1½, ð4:60Þ
2
where
Q ¼ b
2
E
1
θθ
þ b
7
Note that the shear strains E
þ b2E
2
E
θz
þ E
2
zz
2
zθ
2
þ b3E
þ 2 b4EθθEzzþ b5EzzErrþ b6EθθE
ðÞ
rr
2
zr
þ E
2
rz
E
þ b
8
¼ Eji(i, j ¼ θ, z, r) are written separately for the
ij
2
þ b
E
9
2
þ E
θr
rθ
: ð4:61Þ
rr
convenience of deriving stress.
In a special case without shear deformation, the second Piola–Kirchhoff stresses
obtained from the Fung model are:
0@1
S
θθ
S
zz
S
rr
It is noted that only stresses resulting from the strain energy function W(E
considered, while the contribution of the internal constraint V(E
A
¼ C
2
b
1b4b6
4
b4b2b
b6b5b
3
0@1
5
5
3
E
θθ
A
exp QðÞ: ð4:62Þ
E
zz
E
rr
) can be addressed
ij
) are
ij
separately.
Analogous to the classical theory of elasticity, a nominal strain energy W
n(Dij
)is
proposed from which the second Piola–Kirchhoff stress can be symbolically derived
as:
^
S
¼ Sijþ p
ij
∂V
∂D
where p is an arbitrary scalar, and V
∂W
n
¼
∂D
ij
n(Dij
∂V
n
þ p
ij
∂D
n
i; j ¼ θ; z; rðÞ, ð4:63Þ
ij
) ¼ 0 is a possible internal constraint.
The nominal strain potential defined in Eq. (4.63) merely serves the purpose of
mathematical derivation of stresses where the physical meaning is different from the
conventional definition where strain energy function connects the conjugate stress
and strain pairs (Ogden, 1984). Since the strain tensor in Eq. (4.55) is designed to
absorb the nonlinearity of the stress–strain relationship and the nonlinear coupling
between different directions, a quadratic nominal strain potential is proposed as:
W
1
c
¼
n
2
þc13DθθDrrþ c23DzzDrrþ c44D
þc
66
2
D
11
θθ
2
D
θz
þ c22D
2
þ D
zθ
2
þ c33D
zz
:
2
þ c
12DθθDzz
rr
2
þ D
zr
2
rz
2
D
þ c
þ D
55
θr
ð4:64Þ
2
rθ
Appendix 6: 3D Linearization of Fung’s Exponential Strain... 241
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According to Eq. (4.63), the resulting stress–strain relation is linear. Note the
contribution of constraint V
) are discarded for brevi ty.
n(Dij
Generalized Hooke’s Law
With the definitions of Eqs. (4.63) and (4.64), the second Piola–Kirchhoff stresses
and the log-exp strains (without shear components) are connected by:
0@1
S
θθ
S
zz
S
rr
which, including n in Eq. (4.55), requires 7 model parameters (same as in the Fung
model).
Equation (4.65) is in the form of the generalized Hooke’s law, where the
constants c’s (with the unit of stress) can be interpreted as the elastic moduli with
respect to the log-exp strains. Fo r the convenience of illustration, it will be further
assumed that the vessel wall is volumetrically incompressible. As a result, the third
invariant of the right Cauchy-Green deformation tensor U
A
2
c
4
¼
c12c22c
c13c23c
11c12c13
3
5
23
33
0@1
D
θθ
A
D
zz
D
rr
2
, ð4:65Þ
must be unity:
2
2
J
¼ λ
3
2
λ
λ
¼ 1, ð4:66Þ
θ
z
r
which implies that the principal log-exp strains in Eq. (4.57) are linearly related as:
D
þ Dzzþ Drr¼ 0: ð4:67Þ
θθ
This result can be used to reduce the 3D Hooke’s law (Eq. 4.65) to a 2D form.
Substitution of the D
solved from Eq. (4.67)andλr¼ 1/(λθλz) from Eq. (4.66)
rr
into Eqs. ( 4.65), (4.57) and (4.58 ) yields the following equation:
0@1
S
θθ
S
zz
S
rr
A
¼
2
d
4
d21d
d31d
11d12
3
D
5
22
D
32
θθ
, ð4:68Þ
zz
where
d
¼ c11 c13, d12¼ c12 c13,
11
d
¼ c12 c23, d22¼ c22 c23,
21
d
¼ c13 c33, d32¼ c23 c33:
31
Equation (4.68) is actual ly a 2D formulation in which stresses depend on D
D
.
zz
ð4:69Þ
and
θθ
242 4 Constitutive Models of Coronary Vasculature
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Under the assumption of incompressibility, the 3D Fung model (Eq. 4.62) also
reduces to a 2D form. Based on Eqs. (4.56) and (4.66), one can obtain:
E
rr
2EθθEzzþ Eθθþ E
¼
2Eθθþ 1ðÞ2Ezzþ 1ðÞ
zz
: ð4:70Þ
Substitution of Eq. (4.70) into Eq. (4.62), the second Piola–Kirchhoff stresses in
the Fung model can be expressed as functions of the Green strains E
and Ezz. In the
θθ
following sections, the material constants of the generalized Hooke’s law can be
determined from the Fung model for blood vessels.
The experimental data are fitted with the Hooke’s law in Eq. (4.68) to determine
the nonlinearity parameter n which defines the log-exp strains (Eq. 4.57) that best
linearize the Fung model. In other wor ds, an optimal n that makes the linear model
best represent the “experimental data” must be found. Specifically, it is required that
n results in a minimum relative least squares error (RLSE) defined as:
where S
LSEθþ LSEzþ wLSE
RLSE ¼
e
e
, S
θθ
zz
, and S
e
are the experimental data, N is the total number of sampled
rr
P
N
e
S
θθ
þ S
e
zz
þ wS
r
, ð4:71Þ
e
rr
points, w is the weight of least squares error in the radial direction (w ¼ 1 for the 3D
and w ¼ 0 for the 2D model) and
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s
LSE
X
¼
i
e
S
ii
N
S
2
f
i ¼ θ; z; rðÞ ð4:72Þ
ii
denotes the error contr ibution from θ, z, and r directions, respectively. The superscript f in Eq. (4.72) means “fitted” value.
For comparison purpose, please see Appendix 4 for the Fung’s exponential model
while the bi-phasic model is introduced below. The strain energy function of the
bi-phasic model is proposed by Hol zapfel et al. (2005) which can be written as:
W ¼ μ I
where I
stress, and k
3ðÞþ
1
2
¼ λ
þ λ
1
θ
> 0 and ρ 2 [0, 1] are dimensionless parameters. μ is associated with
2
the non-collagenous matrix of the material, which describes the isotropic part of the
overall response of the tissue (Holzapfel et al., 2000). The constants k
associated with the aniso tropic contribution of collagen to the overall response
(Holzapfel et al., 2000). Since there are no shear loadings in the experiments
(Wang et al., 2006) and the assumption that all the fibers are embedded in the
k
1
k
2
2
2
þ λ
z
r
hino
exp k21 ρðÞI1 3ðÞ2þ ρ I4 1ðÞ
2
1
, ð4:73aÞ
and I4> 1 are invariants, μ > 0 and k1> 0 have the units of
and k2are
1
Appendix 6: 3D Linearization of Fung’s Exponential Strain... 243
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tangential surface of the tissue (no components in the radial direction), I4in
Eq. (4.73a) can be expressed as:
2
I
¼ λ
cos2φ þ λ
4
θ
2
sin2φ, ð4:73bÞ
z
The collagen fibers in this model are assumed to be arranged in helical structures,
and φ is the angle of the fibers with respect to circumferential direction. The second
term in Eq. (4.73a) contributes to W only when I
> 1 (Holzapfel et al., 2005;
4
Holzapfel, Gasser, & Ogden, 2004).
Determination of Material Constants
Experimental data are provided by a previous study on the passive mechanical
properties of porcine coronary arteries (Chap. 3). Briefly, a series of inflation tests
are done on cannulated vessels under different axial stretch ratios (Wang, Zhang, &
Kassab, 2008). Outer radius r
Vessel rings are taken from the specimen and a radial cut is made to the vessel ring to
reveal the zero-stress state. Inner circumference C
area A are recorded. There are two steps in determining the material constants for
Hooke’s law. The first step is to derive the equations that express the external
loadings (internal pressure p
constants. The second step is to use a separable nonlinear least squares method to
determine the material constants by minimizing the differences between theoretical
and measured values of external loadings. Material constants for the bi-phasic model
are determined by the standard nonlinear Levenberg-Marquardt method. The results
are summarized in Tables 4.17 and 4.18 for the RCA and LAD arteries, respectively.
, internal pressure piand axial force F are measured.
o
, outer circumference Co, and wall
i
and axial force F) as functions of strains and material
i
Table 4.17 Material constants of the constitutive equation obtained from experimental data of
right coronary artery (RCA)
(a) Generalized Hooke’s law. Units of linear material parameters, i.e., c
parameter n is a nondimensional number
nc
Heart11.31 19.97 18.39 15.28 3.54 8.95 5.27 0.97 0.99 16.6 14.8
Heart21.28 11.54 33.37 15.41 3.92 2.95 8.74 0.98 0.97 15.6 22.4
Heart31.06 19.79 38.44 14.80 0.22 7.21 4.71 0.98 0.98 14.0 20.0
Heart41.33 11.94 17.18 25.92 2.97 6.92 6.50 0.98 0.99 15.2 14.1
Heart51.76 20.11 37.28 20.00 9.37 11.98 16.45 0.98 0.99 11.7 11.6
are kPa. Nonlinear
11–c23
2
R
( pi)
RMS
%
(FT)
2
c
c
11
22
c
33
12c13
R
c
(FT)
23
RMS
%
( pi)
(continued)
244 4 Constitutive Models of Coronary Vasculature
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(a) Generalized Hooke’s law. Units of linear material parameters, i.e., c11–c23are kPa. Nonlinear
parameter n is a nondimensional number
nc
11
2
R
( pi)
RMS
%
(FT)
2
c
c
22
c
33
12c13
R
c
(FT)
23
RMS
%
( pi)
Heart62.20 10.35 15.32 16.00 4.01 7.40 8.03 0.99 0.99 9.4 8.3
Mean 1.49 15.62 26.66 17.90 4.01 7.57 8.28 0.98 0.99 13.8 15.2
SD 0.42 4.78 10.80 4.36 2.98 2.94 4.29 0.005 0.01 2.7 5.2
(b) Fung’s exponential model. Material parameter C has the units of stress (kPa), b
are dimensionless constants
and b
6
Cb
b
1
b3b
2
b
4
5
2
2
R
R
b
(FT)
6
( pi)
1
RMS%
(FT)
, b2, b3, b4, b5,
RMS%
( pi)
Heart17.73 1.29 2.04 0.64 0.26 0.06 0.07 0.98 0.98 13.6 16.1
Heart27.78 1.32 2.73 0.16 0.38 0.08 0.02 0.99 0.98 14.9 20.5
Heart310.13 0.95 2.69 0.82 0.30 0.05 0.15 0.99 0.97 12.5 23.5
Heart44.37 1.21 2.68 0.75 0.28 0.03 0.03 0.99 0.99 18.9 16.1
Heart57.78 1.64 2.54 0.25 0.31 0.01 0.01 0.99 0.99 16.7 20.9
Heart64.35 1.30 3.57 0.90 0.26 0.09 0.03 0.99 0.96 10.9 17.4
Mean 7.02 1.29 2.71 0.59 0.30 0.05 0.05 0.99 0.98 14.6 19.1
SD 2.26 0.22 0.49 0.31 0.05 0.03 0.05 0.003 0.01 2.9 3.0
(c) Bi-Phasic model. Material parameter μ > 0 and k
> 0 have the units of stress, k2> 0 and
1
ρ 2 [0, 1] are dimensionless parameters, and the unit of φ is degree
μ k
k
1
φρR2(FT) R2( pi) RMS% (FT) RMS% ( pi)
2
Heart 1 8.57 0.40 0.98 43.60 0.69 0.99 0.98 10.8 21.3
Heart 2 9.36 0.10 1.46 46.27 0.35 0.99 0.97 12.4 22.6
Heart 3 6.11 2.64 0.64 89.98 0.55 0.99 0.97 16.4 25.6
Heart 4 12.02 0.49 0.40 39.92 0.32 0.94 0.96 27.5 31.2
Heart 5 10.58 0.06 1.40 8.73 0.32 0.99 0.99 13.3 10.2
Heart 6 10.90 1.52 0.88 74.17 0.72 0.99 0.93 7.2 27.7
Mean 9.59 0.87 0.96 50.44 0.49 0.98 0.97 14.6 23.1
SD 2.09 1.02 0.42 28.43 0.19 0.02 0.02 7.0 7.3
2
R
is correlation coefficient; RMS% is the percentage of the root mean square error of the fit
compared to the mean value. Both are computed for the total axial force (F
). Reproduced from Zhang, Wang, et al. (2007) with permission
( p
i
) and inner pressure
T
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