Limiting fiber extensibility as parameter for damage in venous wall
Lukas Horny, Rudolf Zitny, Hynek Chlup, Tomas Adamek and Michal Sara
Faculty of Mechanical Engineering Czech Technical University in Prague
Czech Republic
Introduction Constitutive modeling
of a wall of human vena cava inferior.
Constitutive model is fundamental information about material behavior
Constitutive model is needed for every engineering simulation which includes displacements and deformations
Goals Suggestion of new constitutive model
incorporation of structural informationpossibility of clear interpretation for parameters
Description of mechanical responsewithin elastic behaviorwithin inelastic behavior
Blood vessel mechanics Geometric nonlinearity
large strains
Physical nonlinearity nonlinear stress – strain relationship large strain stiffening
Uniaxial tension - aorta
-20
30
80
130
180
230
0 0.05 0.1 0.15 0.2 0.25 0.3
stretch e [1]
stre
ss s
[kP
a]
circumferential axial Anisotropy
Blood vessel mechanics Inelastic behavior
Preconditioning
Viscoelasticity
Relaxation test - aorta
0
0.2
0.4
0.6
0.8
1
0 100 200 300 400 500 600 700
Time t [s]
Axia
l fo
rce
[N]
Uniaxial tension - aorta
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.05 0.1 0.15 0.2 0.25 0.3
stretch e [1]
stre
ss s
[M
Pa] 1st cycle
2nd cycle
10th cycle
Loading
Unload
ing
PseudoelasticityLoading and unloading curves are always different
Constitutive equations Stored energy function ψ
1st law of Thermodynamics
Zero energy under reference configuration
Zero stress under reference configuration
T
JF F
C
2s
bIa
be
1 32 1
Qce Q b E b E b E 2 2 2
1 11 2 22 9 3112
Constitutive equation
Terms for stored energy
Limiting fiber extensibility
Reference configuration
Deformed configuration
Limiting configuration
str
es
s
stretch
Limiting fiber extensibility mimics idea of limiting chain extensibility in polymer physics
Limiting fiber extensibility can capture large strain stiffening
Refe
rence
Defo
rmed
Lim
itin
g
L l L max maxl L
Limiting chain extensibility
mm
IJ
Jln
1 31
2
Gent model - isotropy
AN Gent 1996 - rubbers
mI Jlim
1 3
mJ
I 2 2 21 1 2 3
…material parameter - shear modulus
…material parameter – limiting extensibility parameter
…1st invariant of a deformation tensor
stre
ss
I1I1 = Jm + 3
Lim
itin
g
mI J 1 3
Limiting fiber extensibility Suggested model – local orthotropy
lnm
m
IJ
J
2
4
2
11
cos sint z fI 2 2 2 2 24
mJ…shear modulus
…limiting extensibility parameter
…4th invariant of a deformation tensor
…angle between fibers and circumferential axis
;m mI J J 4 1 1
Blood vessel wall as a fiber reinforced composite
stre
ssI4
I4 = Jm + 1
Lim
itin
g
z
t
Material parameters estimationMaterial parameters must be identified experimentally
Inflation–extension test
Computational model
Thick walled – tube
Hyperelastic fiber reinforced material
Matrix + fibers
Incompressibility
No shear strains
No residual strains
Regression Measured data Model predictions
lnm
m
Ic I J
J
2
41 2
13 1
o
i
r
ttr
drp
r
o
i
r
z tz tr
F rdr
2
Radial displacement(image analysis of photographs)
Axial displacement(image analysis of photographs)
Internal pressure(pressure probe recording)
Axial force(defined weight + pressure onto bottom)
p …internal pressure
F …axial force
r …deformed radius
t, z …stretch – z axial; t circumferential
Results – vena cava inferior
0
2
4
6
8
10
12
14
16
18
20
22
1.000 1.025 1.050 1.075 1.100 1.125 1.150 1.175 1.200 1.225
0
2
4
6
8
10
12
14
Inte
rnal p
ressu
re p
[kP
a]
Circumferential and axial stretch t, z [1]
Axial force F [N]
1st overloading cycle2nd overloading cycle3rd overloading cycle4th overloading cycle
Axial force
• Internal pressure
o Physiological loading• Supra-physiological
Results – vena cava inferior
0
2
4
6
8
10
12
14
16
18
20
22
1.000 1.025 1.050 1.075 1.100 1.125 1.150 1.175 1.2000
2
4
6
8
10
12
Axial force F [N]
Inte
rnal p
ressu
re p
[kP
a]
Representative cycle of supra-physiological
loading
Circumferential and axial stretch t, z [1]
Axial force
• Internal pressure
Model predictions
lnm
m
Ic I J
J
2
41 2
13 1
0 2564.c kPa 43 9. kPa
0 2945.mJ
27 29.
0
2
4
6
8
10
12
14
16
18
20
22
1.000 1.025 1.050 1.075 1.100 1.125 1.150 1.175 1.200 1.2250
2
4
6
8
10
12
Damage evolution
1st overloading cycle2nd overloading cycle3rd overloading cycle4th overloading cycle
Axial force
• Internal pressure
Supra-physiological loading only
Axial force F [N]
Inte
rnal p
ressu
re p
[kP
a]
Circumferential and axial stretch t, z [1]
Model predictions
0 2945.mJ
0 2923.mJ
0 3177.mJ
0 3282.mJ
Conclusion Stored energy function based on limiting fiber extensibility assumption fits
experimental data (inflation –extension test) successfully
Damage can be related to evolution of the limiting extensibility parameter Jm
The only history dependent parameter Jm is capable to explain different trends of
stretches in axial and circumferential directions
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