Добавил:
Sekretar
kiopkiopkiop18@yandex.ru
t.me/Prokururor I Вовсе не секретарь, но почту проверяю
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_3787_Библиотеки_им_академика_М_И_Перельмана
.pdf
Appendix 8: Incompressibility in the Generalized Hooke’s Law (Li ... 255
https://t.me/med1917
As formulated in (Appendix 7), the right Cauchy-Green deformation tensor C has
the following non-zero components:
C
where λ
radial directions, respectively, computed from the applied axial stretch (λ
deformed radius r
The torsion term (ξ
L
is the initial (no-load) length of the specimen. In the log-exp strain (Eqs (4.94) and
0
(4.95) J
2
¼ λ
, Cθθ¼ λ
rr
r
, λz, and λrare the local material stretches in the circumferential, axial, and
θ
e
o
) is calculated by ξθ(r) ¼ αr/L0, where r is the current radius and
θ
¼ tr CðÞ¼Crrþ Cθθþ Czz¼ λ
1
2
, Czz¼ λ
θ
2
2
þ ξ
, Cθz¼ Czθ¼ λθξθ, ð4:103Þ
z
θ
, and the zero-stress data (R
2
2
þ λ
θ
z
, Ro, and Φ) (Wang et al., 2006).
i
2
þ λ
2
þ ξ
r
for axial torsion defor-
θ
mation according to Eq. (4.103). The stress–strain relation is given by:
0
2Drr Dθθ D
B
B
B
Dzz D
B
B
@
ffiffiffi2p
2
1
zz
ffiffiffi6p
θθ
ffiffiffi2p
D
θz
C
C
C
C
C
A
6
6
6
6
¼
6
6
4
0
B
B
B
B
B
@
3
2E
r
ffiffiffi3p
1
1
2
E
E
θ
001=2G
e
2T
e
T
rr
T
T
θθ
ffiffiffi6p
e
T
zz
θθ
ffiffiffi2p
ffiffiffi2p
e
T
θz
ffiffiffi3p
1
1
E
2
1
þ
E
z
θ
1
e
zz
C
C
e
C
:
C
C
A
E
θ
z
1
1
E
2E
z
0
0
r
θz
), the
z
3
7
7
7
7
7
7
5
ð4:104Þ
Identification of Material Parameters
The material parameters involved in the stress–strain relation Eq. (4.104), {n, Eθ, Ez,
E
}, are first fitted from stretch–inflation data where the axial twist angle is α ¼ 0. As
r
formulated in Appendix 6, the complete axisymmetric Cauchy stress field σ
e
(¼T
pI noting that R ¼ I here) is solved by integrating the Eulerian equilibrium
equation in radial direction and taking into account the stress-free condition on the
outer surface of the vessel. The calculated (denoted by superscript c) internal
pressure P
c
e
r
o
the vessel wall as:
and axial force F
; λ
z
c
e
r
; λ
z
o
z
are obtained by integration across
256 4 Constitutive Models of Coronary Vasculature
https://t.me/med1917
Z
e
r
e
e
σ
σ
rr
θθ
dr and F
r
e
are the components of the “extra stress” σecalculated by
θθ
e
rr
o
e
r
i
and σ
Pc¼
where σ
e
σ
¼ RTeRTas detailed below. The second relation is because the axial force
Z
e
r
o
c
¼
z
2πσzzrdr þπPer
e
r
i
2
e
, ð4:105Þ
i
measured by force transducer includes contribution from both the tissue and the
pressure (Wang et al., 2006). Similar to previous works (Wang et al., 2006; Zhang,
Wang, et al., 2007) or Appendix 7), a search for {n, E
, Ez, Er} is made that
θ
minimizes an error function given by:
hi
P
c
e
P
ðÞ
P
err1 n; E
θ
r
; Ez; E
ðÞ¼
P
P
where summation is conducted over all stretch–inflation data points, A
area (calculated from r
tion). It is found that w
e
and the zero-stress geometry using incompressible condi-
o
¼ 0.4 gives overall good fit for the data. In all measurements
F
and calculations, the unit is kPa for pressure, mN for the axial force, and mm
2
þ wFF
e2
þ wFFe2=A
c
z
F
2
2
e
=A
z
l
2
l
, ð4:106Þ
is the luminal
l
2
for the
luminal area. It should be noted that a combined fitting error is used, while in Wang
et al. (2008) the fitting errors for pressure and force are treated separately. A limitedmemory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) quasi-Newton minimization
method (Liu & Nocedal, 1989) is used for the data curve fitting.
Identification of Shear Parameters
The shear parameter Gθzis identified from torsion data with twist angle α 6¼ 0. Given
a trial G
, the deviatoric stress Teis calculated via Eq. (4.104), and the deviatoric
θz
Cauchy stress σ
as:
where r
e
o
and r
i
the shear stress σ
of D and that R 6¼ I in torsion. Therefore, σ
E
, Er}. Finally,Gθzis identified by minimizing an error function:
z
e
is obtained as σe¼ RTeRT. The total torque Mcis then calculated
Z
e
r
c
¼ 2π
M
e
are the outer and inner radius of the deformed vessel. It is noted that
is coupled with the normal stresses as indicated by the definition
θz
o
σθzr2dr ð4:107Þ
e
r
i
e
depends on normal parameters {n, Eθ,
Appendix 8: Incompressibility in the Generalized Hooke’s Law (Li ... 257
https://t.me/med1917
P
err2 GθzðÞ¼
c
ðÞ
M
P
M
M
2
e
e2
ð4:108Þ
Deformation
As detailed in Appendix 7, the deformation gradient F and the right Cauchy-Green
deformation tensor C of the vessel with respect to the zero-stress state are given by:
0
λ
r
@
F ¼
0 λ
00λ
where ξ
¼ λzαr L1, with r being the current radius of the material point, and L the
θ
current length of the vessel. The eigenvalues (Λ
C are calculated numerically as C ¼ ∑
F ¼ RU is conducted as:
1
00
A
ξ
θ
θ
z
, C ¼
U ¼
i¼1,3
¼ F
0
2
λ
00
r
0 λ
2
θ
@
0 λθξθλ
i ¼ 1, 3Λiui
X
ffiffiffiffiffi
p
Λ
u
uiand R ¼ FU
i
i
!
X
i¼1,3
p
ffiffiffiffiffi
Λ
1
i
1
A
λθξ
2
þ ξ
θ
) and eigenvectors {ui}(i ¼ 1,2,3) of
i
in coordinate e
θ
2
θ
ui. Then, spectral decomposition
1
ui u
i
; eθ; e
fg
r
z
ð4:109Þ
ð4:110Þ
Finally, the logarithmic-exponential strain tensor D defined in Eqs. (4.94) and
(4.95) is:
1
Q ln C ¼ exp n Λ
D ¼
2
Stress
It follows from Eq. (4.104) together with the condition tr(σe) ¼ 0 that:
!
þ Λ2þ Λ3 3ðÞ½
1
X
ln Λiui u
i¼1,3
i
ð4:111Þ
258 4 Constitutive Models of Coronary Vasculature
https://t.me/med1917
1
00
C
C
C
C
1
C
ffiffiffi2p
0
C
C
C
1
C
ffiffiffi2p
0
C
A
p
2
6
6
6
4
3
2E
r
p
ffiffi
3
1
2
E
θ
ffiffi
3
2
1
1
E
þ
E
z
θ
001=2G
1
1
E
E
θ
z
1
E
2E
z
0
1
0
r
0
B
B
B
B
@
0
ffiffiffi
r
2
B
1
e
T
θθ
C
e
T
C
zz
C
¼
C
e
T
A
rr
ffiffiffi2p
e
T
θz
B
B
B
B
B
B
B
B
B
@
3
1
ffiffiffi6p
1
ffiffiffi6p
001
0
2Drr Dθθ D
B
B
B
Dzz D
B
B
@
ffiffiffi2p
D
ffiffiffi6p
ffiffiffi2p
1
zz
C
C
C
θθ
C
C
A
θz
Stress Components in Axisymmetric Deformation (Eq. 4.105)
In axisymmetric deformation, R ¼ I such that σe¼ Te. Therefore,
1
0
C
2
C
C
1
C
ffiffiffi2p
C
C
A
1
ffiffiffi2p
3
2E
4
r
p
ffiffi
3
1
2
E
θ
E
p
2
1
1
E
z
θ
ffiffi
3
1
E
θ
1
þ
E
z
0
3
2Drr Dθθ D
1
1
B
E
z
B
5
@
1
2E
Dzz D
r
ffiffiffi6p
θθ
ffiffiffi2p
0@1
e
σ
rr
e
A
σ
θθ
e
σ
zz
0
r
ffiffiffi
2
B
3
B
B
1
B
¼
B
B
@
ffiffiffi6p
1
ffiffiffi6p
3
1
7
7
7
5
θz
ð4:112Þ
1
zz
C
C
A
The integration of the radial equilibrium dσ
boundary condition σ
with σ
σθθ¼ σ
rr
/dr +(σrr σθθ)/r ¼ 0, subject to
rr
¼ 0 on the external surface of the vessel (r ¼ r
rr
Z
r
1
e
σ
rðÞ¼
rr
e
e
σ
rr
. In addit ion, the axial stress σzzcan be derived as:
θθ
σ
zz
σ
rr
s
r
out
¼ σrrþ σ
sðÞσ
e
σ
zz
e
sðÞ
ds ð4:114Þ
θθ
e
rr
), gives
out
ð4:113Þ
ð4:115Þ
Appendix 9: Viscoelasticity 259
https://t.me/med1917
Stress Components in Axial Torsion (Eq. 4.107)
In axial torsion, the rigid rotation tensor R 6¼ I. Therefore, the shear stress σ
involves all components in Tethrough the relation σe¼ RTeRT. Specificall y, the
shear stress in Eq. (4.107) is given by:
σ
e
¼ σ
θz
¼ RθθRzθT
θz
Table 4.20 Material parameters of porcine left anterior descending arteries (LAD)
Intact
vessel nE
LAD 1 1.26 30.50 43.07 15.49 6.45 0.13 1.13 1.26 3.32 4.66 0.95 0.95 0.91
LAD 2 1.55 18.97 32.29 9.06 5.72 0.25 1.25 1.55 2.25 4.13 0.97 0.96 0.92
LAD 3 1.26 28.43 35.94 12.43 8.56 0.25 1.25 1.26 4.22 2.50 0.93 0.95 0.95
LAD 4 1.68 32.06 40.75 15.57 4.16 0.14 1.14 1.68 2.94 1.87 0.97 0.98 0.97
LAD 5 1.43 21.88 35.35 10.27 5.27 0.26 1.26 1.43 2.08 3.48 0.98 0.98 0.94
mean 1.44 26.37 37.48 12.56 6.04 0.20 1.20 1.44 –––––
SD 0.18 5.67 4.35 2.97 1.65 0.07 0.07 0.18 –––––
Media
LAD-M11.96 10.86 13.76 4.95 4.62 0.20 1.20 1.25 1.93 2.33 0.99 0.99 0.96
LAD-M21.52 20.34 22.82 9.65 5.66 0.11 1.11 1.12 4.31 3.08 0.94 0.93 0.95
LAD-M31.88 17.61 20.25 7.82 3.03 0.19 1.19 1.22 4.22 3.87 0.95 0.93 0.95
LAD-M41.75 39.64 32.26 16.88 6.74 0.06 1.06 1.05 5.26 4.35 0.93 0.91 0.93
LAD-M52.54 10.42 13.86 5.17 3.28 0.13 1.13 1.18 4.02 4.66 0.94 0.92 0.92
θ
e
þ RθzRzθþ RθθR
ðÞT
θθ
E
E
z
r
Gθzv
e
þ RθrRzrT
zz
θz
vθrvzrerr1 err2
θz
e
þ RθzRzzT
rr
e
ð4:116Þ
zz
2
R
2
F
z
2
R
R
P
M
θz
mean 1.93 19.77 20.59 8.90 4.67 0.14 1.14 1.16 –––––
SD 0.38 11.90 7.64 4.87 1.57 0.06 0.06 0.08 –––– –
M refers to media layer of the vessel.nis nonlinearity parameter, E
is shear modulus, vθz, vθr, and vzrare Poisson’s ratios, err1 and err2 (in %) are fitting error as defined in
Eqs. (4.106) and (4.109), R
Liu, Zhang, et al. (2011) with permission
Appendix 9: Viscoelasticity
Coronary Arteries (Zhang, Chen, et al., 2007)
Figure 4.21 is a schematic diagram of the generalized Maxwell model, where a linear
spring with the elastic modulus (stiffness constant) μ
elements in parallel. In the i-th Maxwell body, the spring has an elastic modulus μ
, Ez, and Er(in kPa) are Young’s moduli, G
θ
2
2
, and R
P
2
are the R2values of axial force, pressure, and torque. Reproduced from
M
is connected with m Maxwell
0
, R
F
z
θz
i
260 4 Constitutive Models of Coronary Vasculature
https://t.me/med1917
and the dashpot has a viscous coefficient ηi(i ¼ 1, ..., m). The stress in the i-th
Maxwell body is σ
¼ μiεi¼ ηid(ε εi)/dt.
i
Since the components are connected in parallel in Fig. 4.21, all elements have the
same strain (deformation) equal to the overall strain ε(t). The total stress of the whole
system is the sum of the stresses in each element as given by Fung (1993):
!
m
X
i¼1
D=μ
D
þ 1=η
i
ε tðÞ, ð4:117Þ
i
σ tðÞ¼
m
X
σitðÞ¼ μ0þ
i¼0
where D ¼ d/dt denotes differentiation with respect to time.
To reduce the number of model parameters, ω
is denoted as the characteristic
i
frequency of i-th Maxwell element and assume that the characteristic frequencies
form a geometric series, namely:
μ
i
ω
¼
i
i1
¼ ρ
η
i
=τ i ¼ 1; ...; mðÞ, ð4:118Þ
where ρ is a nondimensional real constant characterizing the “gap” between successive frequencies, τ represents the characteristic relaxation time of the first Maxwell
element (inverse of the characteristic frequency ω
).
1
Substituting Eq. (4.118) into Eq. (4.117), the following differential form of the
constitutive equation can be obtained:
!
σ tðÞ¼ μ
m
X
þ
0
i¼1
μiD
D þ ρ
ε tðÞ: ð4:119Þ
i1
=τ
The constitutive relation in Eq. (4.119) still has the main shortcoming of a
generalized Maxwell model, i.e., the number of material constants increases with
the number of Maxwell elements. To acquire a series of μ
capture rate-insensitive hysteresis behavior, the cyclic loading condition must be
considered.
Response to Oscillatory Loading
The response of a linear viscoelastic material to an oscillating load is typically
studied with complex variable functions (Fung, 1993). If the applied load is assumed
to be an oscillatory strain containing a single angular frequency ω (the real part
corresponds to the actual load), the oscillating strain can be written as:
ε tðÞ¼ε
that allow Eq. (4.119)to
i
jωt
e
, ð4:120Þ
0
Appendix 9: Viscoelasticity 261
https://t.me/med1917
Fig. 4.21 A generalized
Maxwell viscoelastic model
σ, ε
σ
0
σ
1
σ
2
, ε
μ
0
, ε
μ
1
μ
2
, ε
η
, ε–ε
1
1
1
η
, ε–ε
2
2
σ, ε
2
…
η
μ
, ε
m
m
σ
m
ffiffiffiffiffiffiffi
where ε0is the amplitude of strain and j ¼
1pdenotes the imaginary number.
Under the strain loading of Eq. (4.120), the stress response is also an oscillation at
the same frequency with a leading phase angle δ (Findley, Lai, & Onaran, 1989):
jδ
¼ σ0e
jωt
, ð4:121Þ
e
where σ
σ tðÞ¼σ
is the amplitude of stress.
0
j ωtþδðÞ
e
0
According to Eq. (4.120), the differentiation of strain with respect to time can be
simply written as (Fung, 1993):
Dε tðÞ¼jωε tðÞ, ð4:122Þ
m
, ε–ε
m
which reveals that the differentiation D is equivalent to a multiplication by jω.
Given Eqs. (4.120)–(4.122), the differential constitutive equation (Eq. 4.119) can
be rewritten as:
jδ
σ
jωt
e
e
0
!
¼ μ0þ
m
X
i¼1
jωμ
jω þ ρ
i1
i
=τ
jωt
e
: ð4:123Þ
ε
0
From Eq. (4.123), the complex modulus (or dynamic modulus) (4.118, 4.122,
4.123) of the generalized Maxwell body can be obtained as:
m
E
ωðÞ¼
σ tðÞ
ε tðÞ
σ
¼
ε
0
ejδ¼ μ0þ
0
X
i¼1
jωμ
jω þ ρ
i
: ð4:124Þ
i1
=τ
The mechanical loss (a measure of the internal friction) is defined as the tangent
of the phase angle δ (Fung, 1993):
262 4 Constitutive Models of Coronary Vasculature
https://t.me/med1917
tan δðÞ¼
Im EωðÞðÞ
Re E
, ð4:125Þ
ωðÞðÞ
where Re and Im represent the real and imaginary parts of the complex variable,
respectively. The nondimensional tan(δ) in Eq. (4.125) is a function of frequency ω
and is proportional to the ratio of dissipated e nergy to stored energy in a dynamic
loading cycle (Lakes, 1999). Thus, rate insensitivity of the generalized Maxwell
model can be realized by selecting an appropriate elastic modulus for each Maxwell
element to make tan(δ) nearly independent of ω. The following series of elastic
moduli fits the criterion:
i1
μ0i ¼ 1; ...; mðÞ, ð4:126Þ
where β ¼ μ
μ
¼ β 1 þ βðÞ
i
is the ratio of the elastic modulus of the first Maxwell body (which
1/μ0
has a characteristic relaxation time τ as shown in Eq. (4.118)) to that of the first
spring (see Fig. 4.21). Using Eq. (4.126), the differential constitutive model
(Eq. 4.119) can be written as:
σ tðÞ¼μ
!
0
1 þ β
m
X
i¼1
i1
1 þ βðÞ
i1
D þ ρ
=τ
D
ε tðÞ, ð4:127Þ
and the complex modulus (Eq. 4.124 ) becomes:
ωðÞ¼μ01 þ β
E
!
m
X
jω 1 þ βðÞ
jω þ ρ
i¼1
i1
i1
: ð4:128Þ
=τ
There are five material constants in the model: μ
unit of stress, τ has the unit of time, and the other three parameters are
nondimensional. For given model parameters, the internal friction or the normalized
energy dissipation can be examined by plotting tan(δ) against frequency ω (the
normalized energy dissipation differs from tan(δ) by a multiplication factor (Lakes,
1999)).
The real part of the applied complex strain load in Eq. (4.120) is a cosine function
containing a single angular frequency ω:
The corresponding stress response is the real part of Eq. (4.121), which is another
cosine function with the same freque ncy (Lakes, 1999):
ε tðÞ¼ε
σ tðÞ¼σ
, τ, m, ρ, and β, where μ0has the
0
cos ωtðÞ: ð4:129Þ
0
cos ωt þ δðÞ: ð4:130Þ
0
Appendix 9: Viscoelasticity 263
https://t.me/med1917
The hysteresis loop can be obtained by plotting σ(t) versus ε(t), which should be
nearly independent of angular frequency ω in a wide range because the internal
friction tan(δ) is insensitive to the loading frequency.
Relaxation and Creep Functions
Under a given constant strain load ε
, the relaxation function can be expressed by
0
Fung (1993):
m
GtðÞ¼
σ tðÞ
ε
0
¼ μ0þ
X
i¼1
ωit
μie
: ð4:131Þ
Using Eqs. (4.118) and (4.126), the reduced relaxation function (Fung, 1993) can
be obtained from Eq. (4.131) as:
gtðÞ¼
GtðÞ
G 0ðÞ
¼
1 þ β
P
1 þ β
m
1 þ βðÞ
i¼1
P
m
i¼1
1 þ βðÞ
i1
i1
ρ
t=τ
e
, ð4:132Þ
i1
from which one can easily show the following holds:
g 0ðÞ¼1, g 1ðÞ¼
1
: ð4:133Þ
m
1 þ βðÞ
Equation (4.133) indicates that the lower bound of the reduced relaxation function, g(1), depends on m and β only.
Under a constant stress load σ
, the strain of a viscoelastic body varies with time.
0
The creep function (compliance) of a linear viscoelastic material can be defined by
Fung (1993):
ε tðÞ
JtðÞ¼
: ð4:134Þ
σ
0
According to the linear viscoelastic theory, the creep function and the relaxation
function are related by the following convolution (Lakes, 1999 ):
Equation (4.135) can be used to obtain the creep function if the relaxation
function is known, or vice versa. Findley et al. (1989) provided a partial fraction
expansion method to solve Eq. (4.135) using the Laplace transform.
Z
t
Jt ξðÞE ξðÞdξ ¼ t: ð4:135Þ
0
264 4 Constitutive Models of Coronary Vasculature
https://t.me/med1917
Opening Angle (Zhang et al., 2008)
To simulate the viscoelasticity of opening angle, a generalized Maxwell body is
considered where a spring with elastic modulus μ
elements (springs in parallel with dashpots, Fig. 4.22). In the i-th Voigt body, the
spring has an elastic modulus μ
and the dashpot has a viscous coefficient η
i
(i ¼ 1, ..., m). Under a constant stress load σ0, the creep function J(t) is the
responsive strain ε(t) divided by σ
JtðÞ¼
(Fung, 1993; Findley et al. 1989):
0
ε tðÞ
σ
0
¼
X
1
þ
μ
0
i¼1
is connected serially with m Voigt
0
m
1
μ
i
1 e
t=τ
i
, ð4:136Þ
i
where t denotes time.
To reduce model parameters, it is assumed that τ
indicates a power) and μ
¼ μ0/[β(1 + β)
i
i1
¼ ηi/μi¼ ρ
i
] which imply that the characteristic
i1
τ (where i 1
frequencies form a geometric series and the elastic moduli of each Voigt element are
interrelated (see Appendix “Coronary Arteries (Zhang, Chen, et al., 2007)”). As a
result, the reduced creep function can be obtained as:
i1
1 e
m
1 þ βðÞ
ρ
1i
t=τ
, ð4:137Þ
jtðÞ¼
JtðÞ
J 1ðÞ
¼
1 þ β
P
m
i¼1
1 þ βðÞ
where τ is a characteristic relaxation time, ρ characterizes the gap between successive relaxation times, β ¼ μ
, and J(1) ¼ (1 + β)m/μ0.
0/μ1
For a ring cut from loaded state, the strain decreases first elastically as a step,
followed by a creep process. The creep recovery can be described by the superposition principal (Findley et al., 1989). Suppose the loaded artery is under a stress
σ
in the circumferential direction and the viscous stress has been fully relaxed (the
0
corresponding strain is (Eqs. 4.136 and 4.137):
¼ σ0J 1ðÞ¼σ01 þ βðÞm=μ0: ð4:138Þ
ε
0
Starting from t ¼ 0, the ring is cut open and a stress σ
induces a new strain
0
(using Eqs. (4.136) and (4.137)) given by:
0
tðÞ¼σ0JtðÞ¼
ε
0
"#
σ0J 1ðÞ
m
1 þ βðÞ
1 þ β
m
X
1 þ βðÞ
i¼1
i1
1 e
ρ
1i
t=τ
: ð4:139Þ
The total strain recovery ε
(t) after the radial cut (stress is completely released) is:
r
Соседние файлы в папке Библиотека им академика М.И. Перельмана