Inflatation of latex tube – material parameter identification

BIOMECHANICS

NONLINEARITIES In continuum mechanics

Geometrical nonlinearity Large displacements

Large deformation

Material nonlinearity Nonlinear constitutive equation

Constraints-contact Boundary conditions

E

nE 0

ij ijkl klc

nE 0

Inflation test

Load

- Transmural pressure

- Axial force

Goal

- Constitutive equation of material

ModellingTensors of deformation

Gradient of deformation F

X(X1,X2,X3,t) – reference configuration x(X1,X2,X3,t) – loaded state.

Displacement U(X,t)=x(X,t)-X.

iij

j

x X X X tF

X

, , ,

1 2 3 tx X,

FX

In terms of displacements x=X+U

iij ij

j

UF

X

ModellingTensors of deformation

Green–Lagrange tenzor

TE F F I 12 ij ki kj ijE F F 1

2ji k k

ijj i i j

UU U UE

X X X X

12

Example:

One-dimenional homogeneous deformationí x1=X1, x2=X2, x3=X3.

Displacement U : U1=x1-X1, U1= X1-X1.

x xE E F F

X X

21 11 1 1 111 11 112 2 2 2

1 1

1 1 1 1

ModellingTensors of deformation

Engineering deformation

Example:

One-dimenional homogeneous deformationí x1=X1, x2=X2, x3=X3.

Displacement U : U1=x1-X1, U1= X1-X1.

jiij

j i

UUX X

12

X XU UX X X

1 11 11 1 111 2 2 2

1 1 1

2 2 1 1 1

Hyperelastic material

ijij

W

- true stress

- engineering deformation

Ronald S. Rivlin (1915-2005)

Melvin Mooney (1893-1968)

W c I c I Jd

2

10 1 01 2

13 3 1

I1, I2 first and second invariant of deformation tensor deviator, J change of volume, i main stretches

Function – density of deformation energy W.

Hyperelastic material

1. adiabatic

2. incompressible

i i

i

Wp

, ,

1 2 3

i…stretchesp…Lagrange multiplicator (pressure)

ExperimentInflation test

Measured quantities

1. Outer radius ro

2. Length of tube l3. Axial force F4. Internal pressure pi

Experimental setup

1. Sample2. Flanges (3.)

4. Weights5. Tank6. Syringe – pressure generator7. Syringe – weight adjustment8. T-cock9. Valve10. Pressure transducer11. Scale12. Stand13. Camera

Model - deformations Cylindrical coordinate system

, , , ,X Z R x z r Stretches i

t

o r rO R R

22 z

lL

r

hH

tangential axial radial

t

z

r

rR

lL

hH

F

0 00 0

0 0 0 0

0 00 0

Deformation gradient F

Model - deformations

o i o iV v R R L r r l 2 2 2 2

Incompressibility constraint

vJ

Vdet F 2 1

t

z t z r

r

det F

2

2 2 2 2

0 0

0 0 1

0 0

t z r

rlhRLH

1

Ro…initial outer radius Ri…initial inner radius ro…actual outer ri…and inner radius

L…initial length l…actual length

Model-membrane

i zz zzi

pr Gz F h r p r G

h rh:

20 2 0

2 2

j tt ttj

prt F hdz p rdz

h: 0 2 2 0

zz

tt dzG

Balance of forces

Stress from loads (p,G)

o i o o o t ztt

p r r p r h r rp rp p

h h h h H

2 1 12 2 2 2

tt tt tt ttz z

zzo i o i o

G GG Gr h r r h R R H H R H

2 2 2 2 2 2

i ot

i o

r rR R

o i o iR R L r r l 2 2 2 2

Outer radius is measured, inner radius is calculated from the incompressibility constraint

Stress from Const. Equat. Stresses from deformation energy

tt tt

W

zz zz

W

t z r t z r t z z r r tW W W c c, , , , 2 2 2 2 2 2 2 2 21 2 3 1 23 3

Mooney–Rivlin model W

Using incompressibility W is expressed as function of t, r

t r zJ 1 rt z

1 t zW W ,

Material parameter identification

Regrese

MODtt t

t

W c c,

1 2

MODzz z

z

W c c,

1 2

Model prediction Experiment

MOD o t ztt

rp

H

12

MOD tt z

zzo

GH R H

2 2

Goal function – least squares

n

EXP MOD EXP MODtt tt zz zz

j j

Q

2 2

1

Qmin

n is number of measurement (measured points)

Material parameter identification

Linear regression

n

EXP MOD EXP MODtt tt zz zz

j j

Q c c,

2 2

1 21

0

Stationary point (minimum) [c1*,c2

*]:

Q c c

c

,

1 2

1

0 Q c c

c

,

1 2

2

0 Q([c1*,c2

*])=minQ

Experiment

1. Assembly of measuring instruments

2. Measure dimensions in reference configuration (Ro,Ri,L,H,m)

3. Adjust camera

4. Flood pipe

5. Several test cycles (preconditioning) without records

6. At least 3 measuring cycles, recorded

7. Disassembly and cleaning

Experiment Measuring cycle

At least 6 measuring points ([ro,l,p,G]-loads and corresponding dimensions)

Upper limit for load ~ 20 kPa

Close the valve in each measuring point!

It is not necessary to measure in the region where the model assumptions are viaolated (buckling at higher loads)