162
4 Problems in Soft Tissue Biomechanics
given by Eqs. (4.1) and (4.2) for an incompressible and a compressible membrane,
respectively. In Case 1, the fibers are parallel to the x-axis.
4.2.2 Solution
Much of the analysis from Sect. 4.1 also applies to this problem. The equilibrium
equations (4.3), constitutive relations (4.5), and strain invariants (4.8) are the same.
As in the uniaxial extension problem, the Cartesian axes are principal directions,
and the stresses are constant throughout the membrane. Here, all stress components
are zero except P x and P y .
Case 1: Incompressible Membrane With λ x and λ y specified, the incompressibility condition J = λ x λ y λ z = 1 gives
λ z =
1
λ x λ y
.
(4.19)
Again, finding the deformation field does not require knowing material properties.
Next, setting P z = 0 in Eq. (4.5) 3 yields p as given by Eq. (4.6). Putting this
result into Eqs. (4.5) 1,2 leads to
P x =
∂W
∂λ x
−
λ z
λ x
∂W
∂λ z
P y =
∂W
∂λ y
−
λ z
λ y
∂W
∂λ z
.
(4.20)
Substituting Eqs. (4.9) and (4.19), along with (4.1), into these relations yields
P x = 2λ x
1 −
1
λ 4
x λ 2
y
W 1 + λ
2
y W 2
+ W 4
= 2λ x
1 −
1
λ 4
x λ 2
y
c 1 + λ
2
y c 2 + c 3
λ
2
x − 1
e
c 4
λ 2
x −1
2
P y = 2λ y
1 −
1
λ 2
x λ 4
y
W 1 + λ
2
x W 2
= 2λ y
1 −
1
λ 2
x λ 4
y
c 1 + λ
2
x c 2
.
(4.21)
If we set λ y = λ
−1/2
x
as in Eq. (4.4), this solution should reduce to the uniaxial
solution of the previous section. In this case, λ 4
x λ 2
y = λ 3
x and λ 2
x λ 4
y = 1, and the
4 Problems in Soft Tissue Biomechanics
given by Eqs. (4.1) and (4.2) for an incompressible and a compressible membrane,
respectively. In Case 1, the fibers are parallel to the x-axis.
4.2.2 Solution
Much of the analysis from Sect. 4.1 also applies to this problem. The equilibrium
equations (4.3), constitutive relations (4.5), and strain invariants (4.8) are the same.
As in the uniaxial extension problem, the Cartesian axes are principal directions,
and the stresses are constant throughout the membrane. Here, all stress components
are zero except P x and P y .
Case 1: Incompressible Membrane With λ x and λ y specified, the incompressibility condition J = λ x λ y λ z = 1 gives
λ z =
1
λ x λ y
.
(4.19)
Again, finding the deformation field does not require knowing material properties.
Next, setting P z = 0 in Eq. (4.5) 3 yields p as given by Eq. (4.6). Putting this
result into Eqs. (4.5) 1,2 leads to
P x =
∂W
∂λ x
−
λ z
λ x
∂W
∂λ z
P y =
∂W
∂λ y
−
λ z
λ y
∂W
∂λ z
.
(4.20)
Substituting Eqs. (4.9) and (4.19), along with (4.1), into these relations yields
P x = 2λ x
1 −
1
λ 4
x λ 2
y
W 1 + λ
2
y W 2
+ W 4
= 2λ x
1 −
1
λ 4
x λ 2
y
c 1 + λ
2
y c 2 + c 3
λ
2
x − 1
e
c 4
λ 2
x −1
2
P y = 2λ y
1 −
1
λ 2
x λ 4
y
W 1 + λ
2
x W 2
= 2λ y
1 −
1
λ 2
x λ 4
y
c 1 + λ
2
x c 2
.
(4.21)
If we set λ y = λ
−1/2
x
as in Eq. (4.4), this solution should reduce to the uniaxial
solution of the previous section. In this case, λ 4
x λ 2
y = λ 3
x and λ 2
x λ 4
y = 1, and the