248
5 Contraction
5.3 A rectangular block of incompressible muscle tissue contains contractile
elements (volume fraction φ a ) that are oriented in the y-direction and embedded
in an isotropic matrix (volume fraction φ p ). The passive and active strainenergy density functions are
W p = c p
λ
2
x + λ
2
y + λ
2
z − 3
W a = c a
λ
∗2
y − 1
2
.
With the width held fixed in the x-direction, the muscle undergoes contraction
of magnitude K. The surfaces normal to the y- and z-axes are free (Fig. 5.20).
(a) Given c p , c a , and K, determine an equation to solve for the stretch ratio λ y .
(b) Write the stress σ x in terms of λ y .
5.4 An epithelial cell is represented by a rectangular box filled with an incompressible fluid (Fig. 5.21a). The apex of the cell contains contractile elements
aligned in the x-direction. Assume that all sides of the cell remain planar, the
edge lengths b and c are constant, and the cell membrane has a thickness h that
is small compared to the other dimensions. In addition, the constitutive relations
for the membranes at the base (1) and apex (2) of the cell, respectively, are
P 1 = c p
λ
2
1 − 1
P 2 = c p (λ 2 − 1)
3
+ c a
λ
∗
2 − 1
,
Fig. 5.20 Contraction of a
block of muscle
(Problem 5.3)
x
y
Fig. 5.21 Cell deformed by
active contractile elements at
apex (Problem 5.4)
e
v
i
t
c
A
e
v
i
s
s
a
P
a
b
c
b
c
a 1
a 2
d
apex
base
x
)
b
(
)
a
(