7.8 Case Study: Changing Fiber Orientation During Cyclic Stretch
391
Combining the above equations, setting q n (t i+1 , t i+1 ) = 1, and solving for ˙
J + (t i+1 )
yield
˙
J
+ (t i+1 ) NNt = 1 −
N
n=1
⎡
⎣ J
n
0 q
n (t i+1 , 0) +
i
j =1
˙
J
+ (t j ) q
n (t i+1 , t j ) )t
⎤
⎦ ,
(7.110)
which is then substituted back into the first relation for J n (t i+1 ). As a check, the
computations verify that J = 1 holds for all time points.
7.8.4 Illustrative Results
Two types of cyclic stretch are considered: classic uniaxial and strip biaxial. 6 For
both, we specify
λ x (t) = 1 + a sin
2 πf t,
where a is the amplitude and f the frequency. For classic uniaxial stretch in the xdirection, λ y = 1/
√
λ x follows from incompressibility (λ x λ y λ z = 1) and symmetry
(λ y = λ z ). In strip biaxial stretch, the lateral edges of the membrane are constrained
to move only in the x-direction, giving λ y = 1. All results are based on a = 0.1
and f = 1 Hz, which are typical values used for these types of experiments. For
K = 1 × 10 4 , results are shown for k 0 = 1 s −1 and 0.1 s −1 . [Note that λ 0 cancels
out in Eq. (7.101).] While these turnover rates are much faster than those observed
for SFs (Kaunas et al. 2006), they let us focus more closely on details of the system
dynamics.
For classic uniaxial cyclic stretch (Fig. 7.18a), J n is plotted as a function of the
reference fiber angle α n
0 (Fig. 7.18b). At t = 0, all SF families have the same
value (J n = φ n
0 = 1/N = 0.0270). During loading, the fibers tend to reorient
toward ±55 ◦ (Fig. 7.17c), as indicated by the peaks and valleys in J n at steady state
(Fig. 7.18b).
In contrast, strip biaxial stretch induces fibers to reorient perpendicular to the
stretch direction (Fig. 7.19). Both results agree with observed behavior (Kaunas and
Hsu 2009). Therefore, for the same set of parameters, the predicted response is
generally consistent with 2D experiments.
A closer look at the dynamic (time-dependent) behavior for α n
0 = 0 ◦ and ±55 ◦
reveals that the fiber volume for each family oscillates about a mean that increases
or decreases toward a steady-state value (Fig. 7.18c). A family reaches steady state
6 Classic uniaxial stretch is actually uniaxial loading, as the directions normal to the direction of
stretch become shorter.
391
Combining the above equations, setting q n (t i+1 , t i+1 ) = 1, and solving for ˙
J + (t i+1 )
yield
˙
J
+ (t i+1 ) NNt = 1 −
N
n=1
⎡
⎣ J
n
0 q
n (t i+1 , 0) +
i
j =1
˙
J
+ (t j ) q
n (t i+1 , t j ) )t
⎤
⎦ ,
(7.110)
which is then substituted back into the first relation for J n (t i+1 ). As a check, the
computations verify that J = 1 holds for all time points.
7.8.4 Illustrative Results
Two types of cyclic stretch are considered: classic uniaxial and strip biaxial. 6 For
both, we specify
λ x (t) = 1 + a sin
2 πf t,
where a is the amplitude and f the frequency. For classic uniaxial stretch in the xdirection, λ y = 1/
√
λ x follows from incompressibility (λ x λ y λ z = 1) and symmetry
(λ y = λ z ). In strip biaxial stretch, the lateral edges of the membrane are constrained
to move only in the x-direction, giving λ y = 1. All results are based on a = 0.1
and f = 1 Hz, which are typical values used for these types of experiments. For
K = 1 × 10 4 , results are shown for k 0 = 1 s −1 and 0.1 s −1 . [Note that λ 0 cancels
out in Eq. (7.101).] While these turnover rates are much faster than those observed
for SFs (Kaunas et al. 2006), they let us focus more closely on details of the system
dynamics.
For classic uniaxial cyclic stretch (Fig. 7.18a), J n is plotted as a function of the
reference fiber angle α n
0 (Fig. 7.18b). At t = 0, all SF families have the same
value (J n = φ n
0 = 1/N = 0.0270). During loading, the fibers tend to reorient
toward ±55 ◦ (Fig. 7.17c), as indicated by the peaks and valleys in J n at steady state
(Fig. 7.18b).
In contrast, strip biaxial stretch induces fibers to reorient perpendicular to the
stretch direction (Fig. 7.19). Both results agree with observed behavior (Kaunas and
Hsu 2009). Therefore, for the same set of parameters, the predicted response is
generally consistent with 2D experiments.
A closer look at the dynamic (time-dependent) behavior for α n
0 = 0 ◦ and ±55 ◦
reveals that the fiber volume for each family oscillates about a mean that increases
or decreases toward a steady-state value (Fig. 7.18c). A family reaches steady state
6 Classic uniaxial stretch is actually uniaxial loading, as the directions normal to the direction of
stretch become shorter.