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Appendix 6: 3D Linearization of Fung’s Exponential Strain... 245
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Table 4.18 Material constants of the constitutive equation obtained from experimental data of
LAD (left anterior descending) artery
(a) Generalized Hooke’s law. Units of linear material parameters, i.e., c
are kPa. Nonlinear
11–c23
parameter n is a nondimensional number
2
2
R
nc
11c22
c
33c12c13c23
R
(FT)
( pi)
RMS%
(FT)
RMS%
( pi)
Heart71.41 9.54 37.68 50.53 4.56 5.77 15.92 0.96 0.97 15.7 23.7
Heart81.45 18.33 59.11 18.00 18.16 13.71 29.40 0.99 0.99 7.6 9.1
Heart91.88 31.86 48.00 47.00 29.70 30.25 36.26 0.92 0.98 29.6 17.4
Heart101.44 37.45 55.86 61.34 15.00 24.01 26.64 0.98 0.99 13.0 13.4
Heart111.11 43.74 70.35 30.00 35.56 32.38 37.61 0.98 0.98 13.9 16.7
Heart121.49 37.43 139.92 50.00 56.25 39.42 80.89 0.95 0.98 22.0 11.1
Mean 1.46 29.73 68.49 42.81 26.54 24.26 37.79 0.96 0.98 17.0 15.2
SD 0.25 13.09 36.67 15.82 18.22 12.53 22.51 0.03 0.01 7.7 5.2
(b) Fung’s exponential model. Material parameter C has the units of stress (kPa), b
are dimensionless constants
and b
6
Cb
b2b3b
1
b5b
4
2
2
R
R
(FT)
6
( pi)
1
RMS%
(FT)
, b2, b3, b4, b5,
RMS%
( pi)
Heart711.64 1.03 2.09 0.38 0.37 0.02 0.06 0.98 0.97 26.3 21.1
Heart85.94 1.27 2.78 0.62 0.45 0.13 0.13 0.97 0.99 25.1 21.4
Heart94.16 1.58 2.66 0.73 0.30 0.03 0.08 0.96 0.98 34.5 18.4
Heart1011.39 1.39 2.62 0.51 0.32 0.06 0.06 0.99 0.99 28.0 18.5
Heart115.48 1.54 4.29 1.31 0.05 0.22 0.21 0.98 0.99 12.6 13.0
Heart124.43 1.48 5.15 0.24 0.25 0.46 0.01 0.88 0.97 33.2 22.1
Mean 7.17 1.38 3.27 0.63 0.29 0.15 0.09 0.96 0.98 26.6 19.1
SD 3.43 0.21 1.18 0.37 0.13 0.17 0.07 0.04 0.01 7.8 3.3
(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
k2φρR2(FT) R2( pi) RMS% (FT) RMS% ( pi)
1
Heart 7 10.87 0.47 0.65 37.89 0.26 0.99 0.98 10.6 19.5
Heart 8 11.62 0.83 0.79 89.71 0.47 0.97 0.98 15.7 19.8
Heart 9 10.14 0.01 1.39 6.97 0.51 0.98 0.99 9.3 19.3
Heart 10 13.80 0.27 1.18 39.13 0.56 0.99 0.99 10.4 14.0
(continued)
246 4 Constitutive Models of Coronary Vasculature
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(c) Bi-phasic model. Material parameter μ > 0 and k1> 0 have the units of stress, k2> 0 and
ρ 2 [0, 1] are dimensionless parameters, and the unit of φ is degree
μ k
k2φρR2(FT) R2( pi) RMS% (FT) RMS% ( pi)
1
Heart 11 9.72 0.51 0.96 1.15 0.41 0.98 0.98 12.7 18.5
Heart 12 17.97 0.60 2.17 90.00 0.91 0.98 0.91 24.3 35.6
Mean 12.35 0.45 1.19 44.14 0.52 0.98 0.97 13.8 21.1
SD 3.10 0.28 0.55 38.66 0.22 0.01 0.03 5.6 7.4
2
R
is correlation coefficient; RMS% is the percentage of the root mean square error of the fit
compared to the mean value. Both were computed for the total axial force (F
). Reproduced from Zhang, Wang, et al. (2007) with permission
( p
i
) and inner pressure
T
Table 4.19 Results of the sensitivity analysis for generalized Hooke’s law (a), Fung’s exponential
model (b), and bi-phasic model (c)
(a) Generalized Hooke’s law. Units of linear material parameters, i.e., c
are kPa. Nonlinear
11–c23
parameter n is a nondimensional number
Ratio nc
11
R
( pi)
RMS
2
%
(FT)
2
c
c
c
22
33
c
12
13
R
c
(FT)
23
RMS
%
( pi)
0.8 0.65 21.32 60.86 30.00 23.75 18.93 28.41 0.97 0.97 15.9 21.8
0.9 0.83 25.63 62.47 30.00 23.92 19.95 28.53 0.97 0.98 15.7 20.4
1.0 1.11 43.74 70.35 30.00 35.56 32.38 37.61 0.98 0.98 13.9 16.7
1.1 1.54 54.59 59.02 30.01 31.85 34.30 31.67 0.98 0.99 12.5 14.2
1.2 2.00 74.38 37.30 30.09 24.59 36.18 18.09 0.98 0.99 11.3 13.9
Mean 1.23 43.93 58.00 30.02 27.93 28.35 28.86 0.98 0.98 13.9 17.4
SD 0.55 21.72 12.35 0.04 5.44 8.25 7.09 0.01 0.007 2.0 3.6
Note: Ratio of inner and outer circumference C
constants, coefficient of determination (R
and Cois indicated in the first column. Material
i
2
), and percent root mean square error (percentage of the
root mean square error of the fit compared to the mean value, RMS%) of the different data sets are
shown. The values of mean and standard derivation for all of the parameters are given as well.
F and P represent the axial force and pressure, respectively
(b) Fung’s exponential model
Ratio Cb
0.8 6.91 0.43 3.32 0.34 0.07 0.15 0.01 0.98 0.98 13.8 47.0
0.9 3.76 0.94 4.43 1.70 0.02 0.15 0.08 0.99 0.99 12.8 27.7
1.0 5.48 1.54 4.29 1.31 0.05 0.22 0.21 0.98 0.99 12.6 13.0
1.1 6.91 0.43 3.32 0.34 0.07 0.15 0.01 0.98 0.98 13.8 47.0
1.2 4.74 4.69 5.08 0.32 0.21 0.07 0.02 0.99 0.99 12.3 46.0
Mean 5.56 1.61 4.09 0.80 0.09 0.15 0.06 0.98 0.99 13.1 36.1
SD 1.38 1.79 0.76 0.65 0.07 0.06 0.09 0.002 0.003 0.7 15.3
Note: Ratio of inner and outer circumference C
units of stress (kPa), b
determination (R
of the fit compared to the mean value, RMS%) of the different data sets are shown. The values of
mean and standard derivation for all of the parameters are given as well. F and P represent the
axial force and pressure, respectively
2
2
R
( pi)
RMS%
(FT)
b2b3b4b5b
1
and Cois indicated in the first column. C has the
, b2, b3, b4, b5, and b6are dimensionless constants. Coefficient of
1
2
), and percent root mean square error (percentage of the root mean square error
i
(FT)
6
R
RMS%
( pi)
(continued)
Appendix 7: Shear Modulus in Reference to New Strain Measure Zhang... 247
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Table 4.19 (continued)
(c) Bi-phasic model
Ratio μ k
0.8 3.57 5.51 0.38 90.00 0.83 0.98 0.98 15.1 18.6
0.9 6.46 2.31 0.57 90.00 0.68 0.99 0.98 12.5 19.5
1.0 9.72 0.51 0.96 65.72 0.41 0.98 0.98 12.7 18.5
1.1 10.07 2.54 0.68 0.07 0.10 0.97 0.98 17.8 16.8
1.2 9.54 10.94 0.79 0.11 0.29 0.92 0.96 36.9 31.1
Mean 7.87 4.36 0.68 49.18 0.46 0.97 0.98 19.0 20.9
SD 2.81 4.09 0.22 45.90 0.29 0.03 0.009 10.2 5.8
Note: Ratio of inner and outer circumference C
parameters μ > 0 and k
parameters, and the unit of φ is degree. Coefficient of determination (R
square error (percentage of the root mean square error of the fit compared to the mean value, RMS
%) of the different data sets are shown. The values of mean and standard derivation for all of the
parameters are given as well. F and P represent the axial force and pressure, respectively
Reproduced from Zhang, Wang, et al. (2007) with permission
k
1
1
φρR2(FT) R2( pi) RMS% (FT) RMS% ( pi)
2
and Cois indicated in the first column. Material
> 0 have the units of stress, k2> 0 and ρ 2 [0, 1] are dimensionless
i
2
) and percent root mean
Appendix 7: Shear Modulus in Reference to New Strain
Measure Zhang, Wang, et al. (2007)
The coronary arteries are assumed to be homogeneously elastic, cylindrically
orthotropic, and volumetrically incompressible solids. The arterial wall is also
assumed to remain as an axisymmetric tube with a uniform cross section at various
combinations of inflation pressure P
Fig. 4.20. Therefore, the cylindrical coordinates of a vessel segment in the deformed
state (r, θ, z) are related to those in the zero-stress state (R, Θ, Z ) by:
, axial stretch λz, and torque T, as illustrated in
i
where χ ¼ π/(π Φ) with Φ being the opening angle that characterizes the residual
strain, α is the twist angle, L
ratio with L being the length at deformed state. The origins of the coordinate systems
are assumed to be at one end of the vessel segment. The change of axial stretch from
zero-stress to no-load state is neglected.
The deformation gradient matrix F with respect to the zero-stress state is given
by:
where j, k ¼ r, θ, z and
αZ
r ¼ rRðÞ, θ ¼ χΘ þ
the length at zero-stress state, λz¼L/L0the axial stretch
0
2
λ
F
jk
r
4
¼
0 λ
00λ
, z ¼ λzZ, ð4:74Þ
L
0
3
00
5
, ð4:75Þ
ξ
θ
θ
z
248 4 Constitutive Models of Coronary Vasculature
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dr
dR
, λ
λr¼
The incompressibility condition, det[F
R
λ
¼
r
χλ
θ
, r ¼
r
z
χr
, ξ
¼
R
] ¼ 1 in Eq. (4.75), yields
jk
s
αr
¼
¼
θ
L
0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R2 R
2
r
þ
i
χλ
z
λzαr
L
2
i
ð4:76Þ
ð4:77Þ
Therefore, the right Cauchy-Green deformation tensor C (Ogden, 1984) is given by:
C ¼ F
T
F ¼
2
2
λ
r
4
0 λ
0 λθξθλ
00
2
θ
λθξ
2
z
þ ξ
3
5
θ
2
θ
ð4:78Þ
Lagrangian strains can be defined by the symmetric tensor C.
Generalized Hooke’s Law
To simplify the constitutive relation of blood vessels, a logarithmic-exponential
(log-exp) strain tensor is defined as (Appendix 6):
1
ln Cexp nJ
D ¼
2
where n is called the nonlinearity parameter and J
3ðÞ½ ð4:79Þ
1
denotes the first invariant of the
1
right Cauchy-Green deformation tensor C. The D reduces to the Hencky (logarithmic) strain (ln C)/2 when n ¼ 0. In the specific case of Eq. (4.78), one finds:
2
2
2
J
¼ tr CðÞ¼λ
1
θ
þ λ
z
þ λ
2
þ ξ
r
θ
ð4:80Þ
It is postulated that the second Piola–Kirchhoff stress and the log-exp strain are
connected by a generalized Hooke’s law (Appendix 6), namely,
0
B
B
B
B
B
B
@
1
2
S
θθ
C
S
zz
C
C
S
rr
C
C
S
zr
C
A
S
θr
S
θz
c
11c12c13
6
c
12c22c23
6
6
c
13c23c33
6
¼
6
000c
6
4
0000c
00000c
000
000
000
00
44
55
3
0
7
B
7
B
7
B
7
B
7
B
7
B
5
@
0
66
D
D
D
2D
2D
2D
1
θθ
C
zz
C
C
rr
C
C
zr
C
A
θr
θz
ð4:81Þ
Φ
Θ
R
R
o
R
i
θ
r
o
r
i
r
α
r
T
T
αro/L
r
o
r
i
L
λ
z
z
P
i
Appendix 7: Shear Modulus in Reference to New Strain Measure Zhang... 249
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Fig. 4.20 (a) The zerostress state, (b) the deformed
cross section, and (c) the
deformed coronary artery
under transmural pressure
, axial stretch λz, and twist
P
i
torque T. R
and Ro, riand ro,
i
are the inner and outer radii
in the (R, Θ, Z ) and (r, θ, z)
(a) (b)
coordinate systems,
angle, L is vessel segment
length, α is twist angle.
Reproduced from Zhang,
Wang, et al. (2007) with
permission
λ
(c)
where c’s (with unit of stress) can be interpreted as the elastic moduli with respect to
the log-exp strains. In particular, c
Fig. 4.20.
To compute the log-exp strains, the logarithm of tensor C (Eqs. 4.78 and 4.79)
must be calculated. In terms of spectral decomposition (Ogden, 1984), if u
2, 3) are the unit vectors (eigenvectors) along the Lagrangian principal axes of C,
then:
is the shear modulus for the torsion test in
66
(i)
(i ¼ 1,
z
The eigenvalues and eigenvectors of tensor C (the first equation) must be
determined, and subsequently the logarithm of those eigenvalues must be calculated
to project the result back to the original coordinates.
Shear Modulus
In the experiments of Lu et al. (2003), a typical angle of twist is α ¼ 0.4 radian (23).
The representat ive radius and length of porcine left anterior descending artery are
r ¼ 2.5 mm and L ¼ 20 mm, respectively, and the typical value of ξ
L ¼ 0.05λ
first-order terms of ξ
which is much smaller than λθand λz(on the order of unity). If only the
z
C ¼
3
X
2
iðÞ
u
iðÞ
,lnC ¼
λ
u
i
i¼1
is retained, the logarithm of tensor C in Eq. (4.78) can be
θ
3
X
i¼1
ln λ
2
iðÞ
u
iðÞ
is ξθ¼ λzαr/
θ
u
i
ð4:82Þ
250 4 Constitutive Models of Coronary Vasculature
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approximated by the following Taylor expansion (which is obtained from Eq. (4.82)
and then expanded with respect to ξ
ln C
):
θ
2
2
ln λ
r
4
0lnλ
02eθzln λ
2e
3
5
θz
2
z
00
2
θ
ð4:83Þ
where
λθξθln λθ=λ
ðÞ
e
¼
θz
2
λ
θ
λ
z
2
z
ð4:84Þ
Note that e
ξθ/(2λz) ¼ ar/(2L ) when λθ λz, which is the small shear strain
θz
used in Lu et al. (2003).
Since it is found that the measured torque (T) and the angle of twist per unit
length (α/L) are linearly proportional in the range of interest, Cauchy shear stress has
been used to compute shear modulus with respect to shear strain (αr/(2L )) (Lu et al.,
2003). According to the theory of continuum mechanics (Humphrey & Na, 2002),
the Cauchy stress of incompressible materials can be derived from the second Piola–
Kirchhoff stress as:
σ
¼ FjlFkmSlmþ Hδjkj; k; l; m ¼ r; θ; zðÞð4:85Þ
jk
where δ
is the Kronecker delta and H is an arbitrary scalar that needs to be
jk
determined from boundary conditions. Einstein summation has been assumed in
Eq. (4.85). Equations (4.75) and (4.85) yield the Cauchy shear stress in the inflation–
stretch–torsion test as:
σ
¼ λθSθzþ ξθS
ðÞλ
θz
zz
z
ð4:86Þ
Equation (4.86) shows that Cauchy shear stress depends on both shear and axial
second Piola–Kirchhoff stresses. Substituting the incompressibility condition
D
+ Dzz+ Drr¼ 0 (i.e., λθλzλr¼ 1, also see Eqs. (4.79) and (4.83)) into
θθ
Eq. (4.81), the following holds:
S
¼ 2c66Dθz, Szz¼ d21Dθθþ d22D
θz
zz
ð4:87Þ
where d
¼ c12 c23and d22¼ c22 c23. Equation (4.83) can be substituted into
21
Eq. (4.79) to obtain the following log-exp strains needed in Eq. (4.87):
D
¼ eθzexp nJ1 3ðÞ½ ð4:88Þ
θz
D
¼ ln λθexp nJ1 3ðÞ½ ð4:89Þ
θθ
Appendix 8: Incompressibility in the Generalized Hooke’s Law (Li ... 251
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Dzz¼ ln λzexp nJ1 3ðÞ½ ð4:90Þ
If Eqs. (4.87)–(4.90) are substituted into Eq. (4.86), the result is:
2c66λθe
σ
¼
θz
θz
þ d21ln λθþ d22ln λ
ξ
θ
λzexp nJ1 3ðÞ½: ð4:91Þ
ξ
z
θ
The shear modulus of Cauchy stress with respec t to Cauchy small strain can be
evaluated as (Lu et al., 2003):
G ¼
σ
αr=LðÞ
λzσ
θz
θz
¼
ξ
θ
ð4:92Þ
The substitution of Eqs. (4 .84) and (4.91) into Eq. (4.92) yields
"#
2
2c66λ
ln λθ=λ
ðÞ
Since λ
J
λ
1
G ¼
2
þ λ
θ
θ
2
λ
θ
¼ 1/(λθλz) and ξθis a small value in Eq. (4.80), i.e.,
r
2
þ 1= λ
z
z
þ d21ln λθþ d22ln λ
2
λ
z
2
2
, one can readily conclude from Eq. (4.93) that the incre-
λ
θ
z
2
exp nJ1 3ðÞ½ð4:93Þ
λ
z
z
mental shear modulus G depends on the circumferential and axial stretches
nonlinearly. Providing that c
for given λ
and λz, independent of Cauchy small shear strain ξθ/(2λz) ¼ αr/(2L ),
θ
, d21and d22are material constants, G will be constant
66
consistent with the findings in (Lu et al., 2003). The small shear strain assumption is
still valid in the above analysis although the strains in the circumferential and axial
directions are large in the inflation–stretch–torsion test.
Appendix 8: Incompressibility in the Generalized Hooke’s
Law (Liu, Zhang, et al., 2011)
A cylindrically orthotropic elastic model that accounts for large deformation of
arteries is necessary to capture the mechanical response of the vessel. For simplicity,
a linearized stress–strain relation (constitutive model) is used that makes the analysis
easier and more amenable to interpretation as outlined Appendix 6.
As per standard notations in continuum mechanics, the deformation gradient
F can be decomposed as tensor product of a rigid rotation tensor R (R
right stretch tensor U; i.e., F ¼ RU. The right Cauchy-Green deformation tensor C ¼
T
F
F ¼ U2. Following the approach in Appendix 6, a logarithmic-exponential
(log-exp) strain tensor D that absorbs the material nonlinearity is defined as:
T
R ¼ I) and
252 4 Constitutive Models of Coronary Vasculature
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1
Q ln C ¼ exp nJ
D ¼
2
3ðÞ½ln U ð4:94Þ
1
with
Q ¼ exp nJ
3ðÞ½, J1¼ tr CðÞ¼tr U
1
2
, ð4:95Þ
where n is a nonlinearity parameter, i.e., the tensor D is the product of the logarithmic (natural, Hencky) strain and a scalar Q which is proposed to account for the
strong exponential strain– stress relation found in vascular tissues (Fung, 1993). It
can be shown, under the principal-axis representation, that D satisfies all the
mathematical requirements for a strain tensor (Ogden, 1997). It can also be proven
that tr(D)¼ ln(J )Q, where J ¼ det(F) ¼ det(U), i.e., tr(D) ¼ 0 for incompressible
deformation where J 1. In fact, this property arises from the classical Hencky
strain. Furthermore, D is a Lagrangian strain similar to the Green strain E.
Deformation of blood vessels is considered incompressible under physiological
loading. The Cauchy stress (σ) for incompressible material is decomposed as
e
σ ¼ σ
pressure and σ
interpreted as the deviatoric stress, i.e., tr(σ
pI, where p is a Lagrange multiplier that accounts for the hydrostatic
e
is the “extra stress” due to deformation of the material. σeis also
e
) ¼ 0. It is noted for incompressible
materials that hydrostatic stress σ ¼pI does not lead to any deformation. In other
words, the hydrostatic pressure p must be determined from the boundary condition
rather than the stress–strain relation. The second Piola–Kirchhoff stress (S), however, cannot be physically decomposed into hydrostatic stress and “extra stress”
since S ¼ F
1σFT
¼ F1σeFT pC1(note J ¼ 1). This presents some
complexity for modeling anisotropic incompressible materials in Lagrangian
frame, e.g., the linear relationship between D and S becomes problematic since it
is difficult to ensure tr( D) ¼ 0 under arbitrary stress S.
A modified formulation is presented to directly address the hydrostatic pressure
and material incompressibility, with the introduction of a co-rotational Cauchy stress
T (Ogden, 1984) as:
Although T is in Lagrangian frame (because both U and S are in Lagrangian
frame), it preserves the decomposition of σ, i.e., T
generated by material deformation:
and the trace tr(T) is exclusively contributed by the hydrostatic stress p I which is
independent of deformation.
T ¼ USU ¼ R
e
tr T
ðÞ¼tr RTσeR
T
σR ¼ RTσeR pI ¼ Te pI ð4:96Þ
e
is a deviatoric “extra stress”
¼ tr σeðÞ¼0 ð4:97Þ
Appendix 8: Incompressibility in the Generalized Hooke’s Law (Li ... 253
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Incompressibility Condition
It is proposed that the stress T and the log-exp strain D are conn ected by a constant
fourth-order compliance tensor M as:
D ¼ M : T ¼ M : T
e
pIðÞ: ð4:98Þ
The tensor M has major and minor symmetries, and must satisfy certain conditions such as M : I ¼ 0 (Mehrabadi & Cowin, 1999), i.e., to ensure zero deformation
(D ¼ 0) when the stress is hydrostatic (T ¼pI). If the orthotropic symmetry of
vessel tissue is considered, Eq. (4.98) can be expressed in a matrix form (generalized
Hooke’s law):
0
B
B
B
B
B
B
B
B
@
1
2
6
6
6
6
6
6
6
6
4
0
B
B
B
B
B
B
B
B
B
@
ν
ν
1=E
θz=Eθ
θr=Eθ
T
T
T
ffiffiffi2p
ffiffiffi2p
ffiffiffi2p
νθz=Eθνθr=E
θ
1=E
νzr=E
νzr=E
z
1=E
z
r
0001=2G
000
θ
000
z
000
zr
00001=2G
000001=2G
1
e
θθ
C
e
C
zz
C
e
C
rr
C
,
C
e
T
C
zr
C
C
e
T
A
θr
e
T
θz
00
θr
D
θθ
C
D
zz
C
C
C
D
rr
C
ffiffiffi2p
ffiffiffi2p
ffiffiffi2p
¼
C
D
zr
C
C
A
D
θr
D
θz
3
7
7
7
7
7
7
7
7
5
0
θz
ð4:99aÞ
or
where θ, z, and r denote circumferential, axial, and radial directions. The parameters
E
, Ez, and Ercan be interpreted as Young’s moduli in reference to the log-exp
θ
strains, G
, Gθr, and Gzrare shear moduli, and vθz, vθr, and vzrare Poisson ’ s ratios.
θz
Except for the definition of strain and stress, this constitutive relation is identical to
the classical generalized Hooke’s law for orthotropic elast ic materials. Note that the
use of factor
presentation (Mehrabadi & Cowin, 1999).
Unlike the general orthotropic materials where the six parameters E’s and ν’sin
Eqs. (4.99a, 4.99b) are independent, the incompressibility condition M : I ¼ 0
imposes restrictions (Itskov & Aksel, 2002; Loredo & Klocker, 1997), namely:
D
fg
¼ M½T
e
fg
ð4:99bÞ
ffiffiffi2p
in Eqs. (4.99a , 4.99b) preserves the tensorial properties in the matrix
254 4 Constitutive Models of Coronary Vasculature
https://t.me/med1917
E
1
1
vij¼
i
þ
E
2
i
1
E
E
j
ij ¼ θz; θr; zr; k 6¼ i; jðÞð4:100Þ
k
such that Eqs. (4.99a, 4.99b) is rewritten for vessel tissue as:
ffiffiffi3p
1
z
T
e
θθ
2
1
E
θ
1
e
zz
C
C
C
C
C
C
C
C
C
C
C
C
C
A
1
1
E
E
θ
z
1
1
þ
E
2E
z
r
3
000
000
00
zr
θr
7
7
7
7
7
7
7
7
7
7
7
7
0
5
θz
0
2Drr Dθθ D
B
B
B
B
Dzz D
B
B
B
B
B
B
B
B
@
ffiffiffi2p
ffiffiffi2p
ffiffiffi2p
1
2
zz
ffiffiffi6p
C
C
C
C
θθ
C
ffiffiffi2p
C
C
¼
C
D
D
D
C
zr
C
C
C
θr
A
θz
3
6
2E
6
6
6
6
6
6
6
6
6
6
6
4
0
B
B
B
B
B
B
B
B
B
B
B
B
B
@
r
ffiffiffi3p
1
E
2
2T
E
θ
001=2G
0001=2G
00001=2G
e
e
T
rr
θθ
ffiffiffi6p
e
T
T
zz
ffiffiffi2p
ffiffiffi2p
e
T
zr
ffiffiffi2p
e
T
θr
ffiffiffi2p
e
T
θz
ð4:101aÞ
or
¼
^
D
which involves six, instead of nine, material parameters: {E
Together with tr(D) ¼ D
+ Dzz+ Drr¼ 0 and tr TeðÞ¼T
θθ
M
^
e
^
T
ð4:101bÞ
, Ez, Er, Gθz, Gθr, Gzr}.
θ
e
e
þ T
θθ
e
þ T
rr
¼ 0,
zz
Eqs. (4.101a, 4.101b) is identical to Eqs. (4.99a, 4.99b) with the advantage that the
relation is invertible (note: matrix [M] in Eqs. (4.99a, 4.99b) is singular), i.e.,
¼
e
^
T
M
1
^
^
D
ð4:102Þ
which is convenient for strain-based computations.
Equilibrium equati ons and boundary conditions for finite strain deformations are
conventionally formulated in terms of second Piola–Kirchhoff stress S or Cauchy
stress σ. Given a deformation gradient F, the log-exp strain D is calculated by
Eqs. (4.94) and (4.95), then the deviatoric co-rotational Cauchy stress T
lated by Eq. (4.102) and condition tr(T
e
) ¼ 0, and finally S or σ is obtained by
e
is calcu-
Eq. (4.96). For some deformation modes such as the inflation–stretch experiment on
vessels, the shear stress vanish, an d the stress–strain relation (Eqs. 4.101a, 4.101b
and 4.102) can be further simplified as in Eq. (4.104).
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