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

Limiting fiber extensibility as parameter for damage in venice walls

The End