7.8 Case Study: Changing Fiber Orientation During Cyclic Stretch
389
3. Fibers are added and subtracted parallel to existing fibers in each family.
Therefore, the growth ratio G n is unity in the fiber direction.
7.8.2 Analysis
The basic strategy is to track the changes in volume for each fiber family. A loss of
fiber volume in direction x 1 and a simultaneous gain in direction x 2 would indicate
reorientation of fibers from x 1 to x 2 .
Relative to the reference configuration, the deformation gradient tensor for the
membrane is given by
F(t) = λ x (t) e x + λ y (t) e y + λ z (t) e z .
(7.103)
Let
M
n
= e x cos α
n
0 + e y sin α
n
0
(7.104)
be the unit vector parallel to fibers in family n at t = 0. With Eqs. (3.53) and (3.76),
the corresponding stretch ratio (squared) is
(λ
n )
2
= M
n
·(F
T
·F)·M
n
= λ
2
x cos
2 α
n
0 + λ
2
y sin
2 α
n
0
(7.105)
in the fiber direction. With G n = 1 (assumption #3), Eq. (7.20) yields the elastic
stretch ratio
λ
n∗ (t, τ ) = λ 0
λ n (t)
λ n (τ )
(7.106)
for fibers created at time τ .
The volume ratio for the nth fiber family, given by Eq. (7.13), is
J
n (t) = J
n (0) q
n (t, 0) +
t
0
˙
J
+ (τ ) q
n (t, τ ) dτ,
(7.107)
where we have set ˙
J n + = ˙
J + , which is the same for each family (assumption
#2). Moreover, because k n − changes with time, the survival function is computed
using (7.70), i.e.,
q
n (t, τ ) = e
−
t
τ k n − ( ¯
τ ) d ¯
τ .
(7.108)