Page 101 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

78 BIOMECHANICS OF THE HUMAN BODY

equilibrium of a tube element in its deformed state is expressed in a lagrangian (material) coordi-
ˆ
ˆ n
t
nate system ( , , q), which is attached to the surface of the tube (Fig. 3.7).

FIGURE 3.7 Mechanics of the arterial wall: (a) axisym-
metric wall deformation; (b) element of the tube wall under
biaxial loading. The Ts are longitudinal and circumferential
internal stresses.

The first-order approximations for the axial (u ) and radial (v ) components of the fluid velocity,
1 1
and the pressure (P ) as a function of time (t) and space (r, z), are given by
1
⎡ ⎛ r ⎞ ⎤
⎢ J α 0 ⎟ ⎥
0 ⎜
A 1 ⎢ ⎝ R ⎠ ⎡ ⎛ z ⎞ ⎤
0 ⎥
ur z t) = 1+ m exp iω t − ⎠ ⎥ (3.25)
,
(,
⎢
1
cρ F ⎣ ⎢ J (α 0 ) ⎥ ⎦ ⎦ ⎣ ⎝ c ⎦
0
⎡ ⎛ r ⎞ ⎤
⎢ J α 0 ⎟ ⎥
1 ⎜
A β r ⎝ R ⎠ ⎡ ⎛ z ⎞ ⎤
0 ⎥
vr z t) = 1 i ⎢ + m exp iω t − (3.26)
,
(,
1 ⎢ ) ⎥ ⎢ ⎝ ⎠ ⎥
cρ R α J (α ⎣ c ⎦
F ⎣ ⎣ 0 0 0 ⎦
⎡ ⎛ z ⎞ ⎤
Pz t) = A exp ⎢ iω t − (3.27)
(,
1
1
⎣ ⎝ c ⎠ ⎥ ⎦
The dimensionless parameters m, x, k, t , and F are related to the material properties and defined
q 10
as
2 + x 2 [ σ − 1 ( −τ θ )] Eh ρ h
T
m = x = k =
ρ
−
x 1 τ θ ) F − 2 ]σ 1 ( −σ 2 ) R ρ c 2 ρ R 0
[(
F
0 F
10
/ 12
⎛
Eh ⎞
c = 2 • c 0 c =
0
2
/
2 12 12
0 F ⎠
+ )
+ )
{( k 2 +[( k 2 − k 8 1− σ )]]} / ⎜ ⎝ 2R ρ ⎟
(
(3.28)
Eh 2 J (α )
1
τ θ = T 0 θ 1−σ T = PPR F = α 00 0 0 )
0 θ
10
00
J (α
ω R 2 ω R
3
α = 0 α 0 2 = i α β = 0
c c c