Fluid–Structure Interaction in Hemodynamics: Modeling, Analysis, and Numerical Simulation - Fluid-Structure Interaction and Biomedical Applications - page 92

Fig. 2.4 Asketchofan
axially symmetric fluid
domain with radial
displacement

The elastic energy of the Koiter membrane is given by the following:

Z L
0 A

h
2

E mem

el ./ D

G ./ G ./ R dz (2.27)

We consider the dynamics of the Koiter membrane with fixed end points, modeled
by the boundary conditions

.0/ D .L/ D 0:

Following ( 2.10 ), the variational formulation for the nonlinearly elastic Koiter
membrane problem is given by the following:
Z L Z L

0 A

Z L

h
2

K h@ t R dz C G ./ G 0 ./ R dz D fR dz ; 8 2 H 0 .0;L/;

(2.28)

0 0

where G 0 is Gateux derivative of G given by:

@ z @ z 0
0.RC /

:

G 0 ./ D

mem
el

This defines the following (nonlinear) differential operator L :

Z L
0 A

ǝ L

./; Ǜ WD

h
2

mem
el

G ./ G 0 ./ R dz ; 8 2 C c .0;L/:

Integration by parts yields the following formula:

./ D@ z h hE

2R 2 2 @ z i 1 2

hE 1

R C

1

mem
el

2.1 2 / .@ z / 2

L

C

hE

2.1 2 / .@ z / 2 1

1 2 1 2R 2 2 C R 2 ; 2 W 2; 0 .0;L/:

(2.29)

1 hE 1

C

R C R C

With this notation, the corresponding differential formulation of ( 2.28 ) can be
written as:

K h@ t C L

mem
el ./ D f:

(2.30)