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Appendix 9: Viscoelasticity 265
https://t.me/med1917
Fig. 4.22 A generalized
Maxwell viscoelastic model
(a linear spring in serial with
m Voigt elements)
εrtðÞ¼ε0þ ε
0
0
tðÞ¼
μ0, ε
σ
ε
m
X
ε0β
m
1 þ βðÞ
i¼1
μ1, ε
0
η1, ε
i1
1 þ βðÞ
μ2, ε
1
2
…
μm, ε
m
σ
ε
1
η2, ε
2
1i
ρ
t=τ
e
: ð4:140Þ
ηm, ε
m
Note that for m ¼ 1, the model in Fig. 4.22 is mathematically equivalent to a
spring in parallel with a Maxwell body (Fung, 1993; Orosz, Molnarka, & Monos,
1997). Therefore, the Kelvin model considered by Rehal et al. (2006) is a special
case of the generalized Maxwell model in Eq. (4.140).
For a vessel ring cut from no-load state, the analysis assumes the following three
steps: (1) Loaded artery is fully relaxed (stress is σ
obtained after elastomer removal (stress is σ
(3) Vessel ring is cut open at t ¼ t
and all stress is released. It is noted that the second
1
at t ¼ 0); (2) No-load state is
0
from t ¼ 0+to t ¼ t1¼ 1800 s);
1
step is an approximation (i.e., not strictly under constant stress).
According to the superposition principal (Findley et al., 1989), the strain recovery
in the artery after the radial cut can be written as:
ε
tðÞ¼σ0J 1ðÞþσ1 σ
n
ðÞJtðÞσ1Jt t
0
ðÞt > t
ðÞ, ð4:141Þ
1
1
where ε
(t) ! 0 when t !1(fully recovered zero-stress state). The substitution of
n
Eqs. (4.137)–(4.140) into Eq. (4.141) results in
1i
i1
ρ
t1=τ
e
m
1 þ βðÞ
1
1i
ρ
e
t=τ
t > t
ðÞ, ð4:142Þ
1
ε
n
where ε
P
m
ε1β
tðÞ¼εrtðÞþ
¼ σ1J(1) is the residual strain due to residual stress σ1.
1
1 þ βðÞ
i¼1
The artery is assumed to be incompressible. The loaded state is a tube with a
circular cross section. The circumferential stretch ratio at the inner surface is:
2πr
λ
θ
i
¼
, ð4:143Þ
C
i
and at the outer surface is:
2πr
o
¼
, ð4:144Þ
C
o
where r
λ
θ
and roare the inner and outer radii at the loaded state, Ciand Coare the inner
i
and outer circumferences at the fully relaxed zero-stress state, respectively.
266 4 Constitutive Models of Coronary Vasculature
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In the generalized Maxwell model, a linear stress–strain relation is expected.
Although arteries are known to exhibit nonlinear constitutive behavior, Zhang and
Kassab (2007) proposed to absorb the material nonlinearity with a new strain
measure in the two-dimensional case (Appendix 5). The same idea is then extended
to the three-dimensional Hooke’s law (Appendix 6) where the logarithmicexponential (log-exp) strain is defined as (no shear deformation):
D
¼ ln λiexp nJ1 3ðÞ½i ¼ θ; z; rðÞ, ð4:145Þ
ii
where λ
and J
are stretch ratios, n is a constant that characterizes the material nonlinearity,
i
is the first invariant of the right Cauchy-Green deformation tensor:
1
2
2
J
¼ λ
þ λ
1
θ
2
þ λ
: ð4:146Þ
z
r
The second Piola–Kirchhoff stress and the log-exp strain in the circumferential
direction can be written as (Zhang, Wang, et al., 2007):
S
¼ c11Dθθþ c12Dzzþ c13Drr, ð4:147Þ
θθ
where c’s are elastic moduli wi th respect to the log-exp strains.
As an estimate, it is assumed that D
axially relaxed. In such a case, D
rr
0(λz 1) which implies that the artery is
zz
¼Dθθ(λr¼ 1/λθ) according to Eq. (4.145). The
circumferential strain becomes:
ε ¼ D
¼ ln λθexp n λ
θθ
2
θ
It is noted that Eq. (4.147) can naturally reduce to a 1D linear model (S
þ 1=λ
2
2
: ð4:148Þ
θ
θθ
/ ε).
Other models using Green strain measure (e.g., Fung model) can be used approximately since the strain is small (thus stress–strain relation can be linearized) during
the creep process.
The open sector is characterized by an opening angle Φ. Considering the volumetric incompressibility condition λ
¼ 1, the cross-sectional wall area A0can be
θλzλr
calculated by Chuong and Fung (1986):
where L
and Loare inner and outer circu mferences of the sector (not fully relaxed).
i
Equation (4.149) yields the opening angle expressed by:
A
¼ πλzr
0
Φ tðÞ¼π
2
r
o
2
L
2
o
¼
i
4 π Φ tðÞðÞ
2
L
tðÞL
o
4A
2
tðÞL
0
tðÞ
i
, ð4:149Þ
2
tðÞ
i
: ð4:150Þ
Appendix 10: Active Mechanical Properties (Huo et al., 2012) 267
https://t.me/med1917
For a given A0, Li, and Lo, Φ can be computed from Eq. (4.150). It is found that
the change of these measures is negli gible 2 h after the radial cut. Thereafter, the
artery is assumed to be fully relaxed, i.e., L
The stretch ratio of the open sector λ
(t) ¼ L(t)/C is equal to the circumference at a
θ
, Lo¼ Co, and Φ ¼ Φ0at t ¼ 7200 s.
i¼Ci
given time over that at the fully relaxed state, which differs from that at the loaded
state (Eqs. 4.143 and 4.144). The strain is computed with Eq. (4.148).
Appendix 10: Active Mechanical Properties (Huo et al., 2012)
The blood vessel is assumed to be a thin-walled elastic tube deformed in the
circumferential (θ) and axial (z) directions. Green strains are defined as:
D
where λ
¼
θ
D
0
tively; D and D
L and L
are axial lengths in the loaded and no-load states, respectively. If the
0
1
2
E
θθ
and λz¼
are diameters in the loaded and zero-stress states, respectively;
0
λ
¼
1
θ
2
L
are the circumferential and axial stretch ratios, respec-
L
0
and Ezz¼
1
2
λ
1
z
2
ð4:151Þ
densities of vessel wall are identical in the loaded and no-load state, first Piola–
Kirchhoff and Cauchy stresses can be written as:
where S
T
¼ λθS
θθ
Tzz¼ λzS
and SZZare the circumferential and axial second Piola–Kirchhoff stresses,
θθ
θθ
zz
and
σ
σzz¼ λ
2
¼ λ
S
θθ
θθ
θ
2
S
zz
z
ð4:152Þ
respectively.
The total strain energy function consists of passive and active components as:
W
where W
passive
caused by the K
energy of the K
¼ W
total
is the strain energy of passive vessel; W
+
-induced smooth muscle contraction; and W
+
-induced active vessel.
passive
þ W
active
is the active strain energy
active
is the total strain
total
ð4:153Þ
Passive Strain Energy Function
In Fung’s 2D model without shear deformation, the strain energy per unit volume
(W
) is given by:
passive
268 4 Constitutive Models of Coronary Vasculature
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C
1
W
passive
exp QðÞ1½ ð4:154aÞ
¼
2
and
where C
1,a1,a2
Q ¼ a
2
E
1
θθ
, and a4are constants. The passive second Piola–Kirchhoff stresses
þ a2E
2
þ 2a4EθθE
zz
zz
ð4:154bÞ
are obtained by differentiating the strain energy with respect to the corresponding
Green strains as:
8
>
>
<
>
>
:
S
θθ
S
zz
passive
passive
∂W
passive
¼
∂E
∂W
¼
∂E
¼ C1a1Eθθþ a4E
θθ
passive
¼ C1a2Ezzþ a4E
zz
ðÞexp QðÞ
ðÞexp QðÞ
zz
θθ
ð4:155Þ
The passive first Piola–Kirchhoff stresses can be written as:
where C
1,a1,a2
(
T
¼ C1a1Eθθþ a4E
θθ
passive
T
¼ C1a2Ezzþ a4E
zz
passive
ðÞ
ðÞ
, and a4for coronary arteries are determined by experimental
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2Eθθþ 1pexp QðÞ
zz
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2Ezzþ 1pexp QðÞ
θθ
ð4:156Þ
measurements.
Active Strain Energy Function
The active strain energy function caused by the K+-induced SMC contraction is
proposed as:
W
active
where C
b
0
b
¼
b
, b2, b3, and b4are constants and Erf(X) is the Gauss error function. If
2,b1
b
3
4
þ
, Eq. (4.157) can be simplified as:
b
1
2
¼ C2Erf
¼ C
2
2E
þ 1p b
θθ
b
λθ b
Erf
W
active
1
λz b
3
þ
b
1
b
¼ C2Erf
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
þ 1p b
2E
3
4
2
λ
θ
b
1
zz
þ
b
4
2
1
ð4:157Þ
1
λ
z
þ
0
b
b
2
1
ð4:158Þ
Appendix 10: Active Mechanical Properties (Huo et al., 2012) 269
https://t.me/med1917
Equation (4.159) has 4 unknown parameters instead of 5 in Eq. (4.157). The
active second Piola–Kirchhoff stresses are obtained by differentiating the strain
energy with respect to the corres ponding Green strains as:
8
>
>
>
>
>
<
>
>
>
>
>
:
∂W
S
θθ
active
S
zz
active
active
¼
∂E
θθ
∂W
active
¼
∂E
2C
¼
b
1
2C
¼
b
zz
2
2
ffiffiffiπp
p
2
p
ffiffiffiπp
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2E
θθ
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
þ 1
2E
zz
!
exp
þ 1
!
exp
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
2E
θθ
b
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
2E
θθ
b
1
þ 1
þ 1
þ
þ
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2E
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2E
b
þ 1
zz
b
2
þ 1
zz
2
b
b
0
2
0
ð4:159Þ
The active first Piola–Kirchhoff stresses are obtained by differentiating the strain
energy with respect to the corres ponding stretch ratios as:
2
8
>
>
T
<
>
>
:
with
λ
λ
0
Q
¼
z
θ
þ
b
b
b
1
2
θθ
T
zz
2
0
active
active
¼
2C
¼
b
1
2C
¼
b
2
λ
θ
b
1
2
2
2
þ
ffiffiffiπp
exp Q
ffiffiffiπp
exp Q
λ
z
b
2
0
ðÞ
0
ðÞ
2
λ
λ
θ
þ 2
b
b
1
λ
z
2
2
θ
þ
b
1
λ
z
b
b
2
0
þ b0ðÞ
2
ð4:160Þ
The material constants in Eq. (4.160) are determined by biaxial experimental
measurements for intact and intima-media layers of coronary artery wall. The
material parameters for six RCA coronary arteries are listed below in Tables 4.21
and 4.22 for passive and active states of intact wall and in Tables 4.23 and 4.24 for
the intima-media layer, respectively. Reproduced from Huo et al. (2012) with
permission.
Table 4.21 Material constants of Fung’s passive strain energy function of intact coronary artery
Animal no. C
Heart 1 1.27 2.61 1.59 0.6 0.94 0.73
Heart 2 0.94 2.86 2.09 1.14 0.90 0.94
Heart 3 0.87 3.39 3.91 0.42 0.94 0.94
Heart 4 1.51 2.89 3.29 0.43 0.74 0.90
Heart 5 0.89 3.18 2.82 0.75 0.72 0.74
Heart 6 1.25 2.59 1.55 0.59 0.96 0.94
Fit of all experi-
mental data of
hearts 1–6
Mean SE for
hearts 1–6
(kPa) a
1
1.33 3.08 2.48 0.49 0.96 0.90
1.12 0.11 2.92 0.13 2.54 0.39 0.66 0.11
1
a
2
a
4
R2for
T
θθ
passive
R2for
T
zz
passive
(continued)
270 4 Constitutive Models of Coronary Vasculature
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Animal no. C1(kPa) a
CV ¼
SD
Mean
100%
23.0% 10.8% 37.7% 40.8%
1
a
2
a
4
T
R2for
R2for
T
θθ
passive
zz
Reproduce from Huo et al. (2012) with permission
Note: Material constants, C
coronary artery (RCA). R
2
Table 4.22 Material constants of K
, and a4, are obtained from experimental measurements of right
1,a1,a2
represents the goodness-of-curve-fit
+
-induced active strain energy function (Eq. 4.160) in the
pressure range of 60–200 mmHg for intact coronary artery wall as corresponding to Table 4.21
Animal no. C
(kPa) b
2
1
b
2
0
b
R2for
T
R2for
T
θθ
active
zz
Heart 1 4.29 0.109 0.212 21.2 0.78 0.83
Heart 2 3.37 0.105 0.185 21.9 0.55 0.79
Heart 3 7.04 0.156 0.297 14.0 0.57 0.61
Heart 4 2.81 0.112 0.127 24.2 0.66 0.74
Heart 5 7.65 0.110 0.209 20.1 0.64 0.74
Heart 6 3.97 0.108 0.21 21.3 0.73 0.85
Fit of all experi-
4.72 0.120 0.182 20.3 0.71 0.78
mental data of
hearts 1–6
Mean SE for
4.86 0.82 0.117 0.008 0.207 0.022 20.5 1.4
hearts 1–6
CV ¼
SD
Mean
100%
41.3% 16.6% 26.5% 16.8%
Reproduced from Huo et al. (2012) with permission
Note: Material constants, C
coronary artery (RCA). R
, b1, b2, and b0, are obtained from experimental measurements of right
2
2
represents the goodness-of-curve-fit
passive
active
Table 4.23 Material constants of Fung’s passive strain energy function for intima-media layer of
right coronary artery (RCA)
Animal no. C
Heart 1 4.16 4.20 3.02 0.60 0.99 0.99
Heart 2 6.09 1.72 2.93 0.43 0.91 0.98
Heart 3 7.94 3.26 2.19 0.45 0.99 0.99
Heart 4 4.07 3.96 1.64 1.61 0.94 0.93
Heart 5 5.13 3.18 2.14 0.70 0.98 0.99
Heart 6 5.30 1.93 0.51 1.92 0.97 0.96
Fit of all
experimental
data of hearts
1–6
Mean SE
for hearts 1–6
Reproduced from Huo et al. (2013) with permission
Note: Material constants, C
intima-media layer of RCA using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method. R
represents the goodness-of-curve-fit
(kPa) a
1
1
a
2
a
4
T
θθ
passive
4.55 3.05 2.32 1.15 0.98 0.99
5.45 0.59 3.04 0.42 2.07 0.38 0.95 0.26
, a1, a2, and a4, are obtained from experimental measurements of
1
R2for
R2for
T
zz
passive
2
Appendix 11: Micromechanics of Heterogeneous Materials (Chen, Zhao... 271
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Table 4.24 Material constants of K+-induced active strain energy function for intima-media layer
in the pressure range of 30–110 mmHg
Animal no. C
Heart 1 9.36 0.22 0.44 9.08 0.81 0.75
Heart 2 5.19 0.29 0.73 6.65 0.76 0.66
Heart 3 4.58 0.20 0.39 10.0 0.82 0.52
Heart 4 14.8 0.20 0.32 11.5 0.67 0.73
Heart 5 16.8 0.29 0.46 7.29 0.78 0.58
Heart 6 4.21 0.18 0.34 11.4 0.81 0.62
Fit of all experi-
mental data of
hearts 1–6
Mean SE for
hearts 1–6
Reproduced from Huo et al. (2013) with permission
Note: Material constants, C
intima-media layer of right coronary artery (RCA). R
(kPa) b
2
7.88 0.24 0.43 8.60 0.93 0.73
9.17 2.25 0.23 0.02 0.45 0.06 9.33 0.83
1
, b1, b2, and b0, are obtained from experimental measurements of
2
b
2
2
represents the goodness-of-curve-fit
0
b
R2for
T
θθ
active
R2for
T
zz
active
Appendix 11: Micromechanics of Heterogen eous Materials
(Chen, Zhao, Lu, & Kassab, 2013)
The idea of homogenization of heterogeneous nonbiological materials has been used
to predict the macroscopic or effective mechanical properties of composites (Milton,
2002). Rigorous and reliable methods are well established and widely applied to
linear elastic composites, including the classical Voigt and Reuss bounds (Hill,
1952), the variational principles of Hashin-Shtrikman (Hashin & Shtrikman, 1962,
1963), and the general self-consistent approximations (Hershey, 1954; Hutchinson,
1976). For nonlinear composites, however, rigorous methods have become available
only more recently because of difficulties in addressing both strong material
nonlinearity and heterogeneity. Earlier efforts are made to predict the effective
constitutive behaviors of composites by extending linear methods to nonlinear
materials, such as the extensions of the self-consistent procedures (Hill, 1965), or
the Hashin-Shtrikman variational principles for linear composites (Talbot & Willis,
1985; Willis, 1983).
A general variational procedure for estimating the effective behavior of nonlinear
composites is proposed by Ponte Castañeda (1991). He introduced a “linear elastic
comparison composite” (LCC) with the same microstructure of the nonlinear composite, which allows for the use of numerous bounds and estimates for linear
composites. Ponte Castañeda further proposed an alternative approach with more
sophisticated LCC to generate estimates that are exact to the second order in the
phase contrast (Ponte Castañeda, 1996; Ponte Castañeda & Willis, 1999). This is the
so-called second-order estimate (SOE) homogenization, which yields significant
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improvements over previous micromechanics models. During the past decade,
successive developments have been made to the SOE approach by Ponte Castañeda
and his colleagues to better predict the nonlinear macroscopic properties of heterogeneous materials (Agoras, Lopez-Pamies, & Ponte Castañeda, 2009; Chen, Liu,
Zhao, et al., 2011; Kailasam, Ponte Castañeda, & Willis, 1997; Liu, Gilormini, &
Ponte Castañeda, 2003; Liu & Ponte Castañeda, 2004; Lopez-Pamies & Ponte
Castañeda, 2004b; Ponte Castañeda, 2002). Applications have been extended to
various nonlinear materials with randomly or periodically distributed microstructure,
including viscoplastic polycrystal, porous, or reinforced rubbers and fiber-reinforced
elastomers.
Framework of Nonlinear Micromechanics
Finite strain micromechanics can be used to determine the macroscopic constitutive
response of biological soft tissues based on structural and mechanical properties of
the microstructure. Based on the principle of minimum strain energy, the framework
can provide multiple approximate solutions for the macroscopic strain energy
function (SEF) of soft tissues. The “upper bound” and a “second-order” estimate
models are discussed below.
Hyperelastic Heterogeneous Material
A heterogeneous material, such as soft tissue, is made up of N +1(N 1) different
phases, which are distributed (randomly or with a certain distribution of orientation)
in a specimen with a volume Ω and boundary ∂Ω in the reference configuration. The
constitutive behavior of each inclusion is characterized by a respective SEF W
so that the local SEF W(X, F)of composites is written as:
N
X
r¼0
rðÞ
χ
W X; FðÞ¼
where F is defined as the deformation gradient tensor, and χ
XðÞW
rðÞ
FðÞ ð4:161Þ
(r)
¼ 1 when X 2 Ω
(the volume occupied by phase r), or 0 otherwise to describe the distribution of the
microstructure. The local stress field (i.e., the microscopic constitutive behavior) of
the inhomogeneous material is expressed, based on thermodynamics as:
SXðÞ¼
∂W X; FðÞ
∂F
pF
T
(r)
(F),
(r)
ð4:162Þ
Appendix 11: Micromechanics of Heterogeneous Materials (Chen, Zhao... 273
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S(X) is the first Piola–Kirchhoff stress tensor, which is related to the Cauchy stress
tensor σ by S ¼ σ F
T
, and the hydrostatic stress p is due to incompressibility (det
(F) ¼ 1) of the soft tissue.
According to the principle of minimum strain energy, the effective SEF of a
microscopically inhomogeneous hyperelastic material is defined as:
N
DE
F2κ
X
rðÞWrðÞ
c
F
r¼0
(r)
¼hχ
(r)
F I X in ∂Ω
F¼ min
W
F2κ
where the brackets hiand hi
over the r-th phase Ω
fraction of the r-th phase.
κ
F¼ FjFXðÞ¼I þ ∇
W X; FðÞ
hi
F
(r)
, respectively, so that c
uXðÞin Ω; and uXðÞ¼
X
¼ min
(r)
denote volume averages over the composite Ω and
rðÞ
FðÞ
, ð4:163Þ
i represents the volume
denotes
all the admissible deformation gradient field in the representative volume element
(RVE) of the composite with an affine displacement boundary condition u(X). In
analogy with the local expression of Eq. (4.162), assuming sufficient smoothness for
W
F, the macroscopic stress of the composite is defined by:
∂W
S ¼
F
∂F
pF
T
ð4:164Þ
whereS ¼ ShiandF ¼ Fhi, which are also called the average stress and average
deformation gradient, respectively. The effective SEFW physically represents the
average elastic energy stored in the composite. In general, the rigorous minimization
solution for Eq. (4.163) is difficult to obtain for a heterogeneous material with
complex microstructure since a set of highly nonlinear partial differential equations
must be solved. Consequently, approximate solutions or estimates have been pursued (Hill, 1965; Lopez-Pamies & Ponte Castañeda, 2006; Ponte Castañeda, 1991,
2002; Ponte Castañeda & Willis, 1999; Talbot & Willis, 1985). The efforts to seek
an approximate minimizing field F(X) have resulted in several classes of finite strain
micromechanics models including the uniform-field upper bound model and the
second-order estimate model.
Uniform-Field Upper Bound Model
The simplest trial field for Eq. (4.163)is FXðÞ¼F, which assumes a uniform
deformation field in the composite. Hence, the macroscopic SEFW
the volumetric sum of the constitu ents,
Fis simply
274 4 Constitutive Models of Coronary Vasculature
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N
X
W
FW
U
F¼
c
r¼0
rðÞWrðÞ
F
ð4:165Þ
This solution leads to an upper bound of the exact SEFW asF is an admissible
deformation field that does not minimize the total strain energy. It is usually referred
to as the Voigt upper bound in linear micromechanics. Consequently, the effective
stressS
¼ ∂WU=∂F pFTis an upper bound of the exact macroscopic stress for
U
a givenF. This approximation requires that all phases must deform with the same
deformation regardless of their different material properties (e.g., stiffness). This
assumption is not true for composites that have widely different phases. In addition,
the affine deformation assumption only includes information on the phase volume
fraction c
(r)
(as indicated by Eq. (4.165)) but neglects the inte ractions between
phases that may affect the overall mechanical response of a composite.
Second-Order Estimate Approach
A second-order estimate (SOE) homogenization approach, utilizing the concept of
an LCC, is proposed by Ponte Castañeda (Ponte Castañeda, 1996; Ponte Castañeda
& Willis, 1999). This method has been widely applied to nonlinear elastomeric
composites, including reinforced and porous elastomers, as well as other heterogeneous elastomeric systems (Agoras et al., 2009;Lopez-Pamies & Ponte Castañeda,
2004a; Ponte Castañeda, 2002). This approach accounts for the statistical micro-
structure beyond the volume fraction of the phases and incorporates interactions
among different phases. Hence, it provides a more accurate prediction of the
constitutive behavior of nonlinear composites than the uniform-field upper bound.
Following previous derivations (Lopez-Pamies & Ponte Castañeda, 2004a; Ponte
Castañeda, 2002), an LCC is introduced with effective SEFW
rðÞ
W
FðÞto denote that it has the same microstructure as the nonlinear composites, but
T
each of the phases is linearly elastic with the SEF W
approximation to the nonlinear SEF W
(r)
). According to generalized Legendre
rðÞ
(i.e., the second-order Taylor
T
transform of the strain energy (Lopez-Pamies & Ponte Castañeda, 2004a ; Ponte
Castañeda, 2002), the exact macroscopic SEF expressed in Eq. (4.163), can be
approximated by:
F min
W
()
sðÞ;LsðÞ
F
fg
W
no
sðÞ
T
F; F
; L
sðÞ
N
X
þ
r¼0
rðÞVrðÞ
c
P
N
F¼
T
rðÞ
F
; L
rðÞ
rðÞ
χ
r¼0
XðÞ
, ð4:166Þ
where L
(r)
is the unknown reference modulus and F
deformation gradient of the r-th phase of LCC, both of which will be determined
by the minimization procedure of Eq. (4.166).
rðÞ
; L
rðÞ
rðÞ
V
F
no
¼ sup
^
F
W
rðÞ
^
F W
rðÞ
^
F
T
is a “corrector” function, i.e., the
(r)
is the virtual residual
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