158
4 Problems in Soft Tissue Biomechanics
Since λ y = λ z , setting P y = 0 yields the same result. Substitution into the above
relation for P x yields
P x =
∂W
∂λ x
−
λ z
λ x
∂W
∂λ z
.
(4.7)
The strain-energy density function of Eq. (4.1) requires the strain invariants
I 1 = λ
2
x + λ
2
y + λ
2
z
I 2 = λ
2
x λ
2
y + λ
2
y λ
2
z + λ
2
z λ
2
x
I 3 = λ
2
x λ
2
y λ
2
z
I 4 = λ
2
x
(4.8)
as provided by Eqs. (3.73) and (3.228) 1 , with the fiber stretch ratio being λ f = λ x .
(I 3 will be needed for Case 2.) The chain rule provides the derivatives
∂W
∂λ x
=
∂W
∂I 1
∂I 1
∂λ x
+
∂W
∂I 2
∂I 2
∂λ x
+
∂W
∂I 4
∂I 4
∂λ x
= 2λ x W 1 + 2λ x
λ
2
y + λ
2
z
W 2 + 2λ x W 4
∂W
∂λ y
=
∂W
∂I 1
∂I 1
∂λ y
+
∂W
∂I 2
∂I 2
∂λ y
+
∂W
∂I 4
∂I 4
∂λ y
= 2λ y W 1 + 2λ y
λ
2
x + λ
2
z
W 2
∂W
∂λ z
=
∂W
∂I 1
∂I 1
∂λ z
+
∂W
∂I 2
∂I 2
∂λ z
+
∂W
∂I 4
∂I 4
∂λ z
= 2λ z W 1 + 2λ z
λ
2
x + λ
2
y
W 2 ,
(4.9)
where
W i ≡
∂W
∂I i
.
(4.10)
Putting these relations and Eq. (4.4) into (4.7) and rearranging the resulting expression yield
P x = 2λ x
W 1 +
1
λ x
W 2
1 −
1
λ 3
x
+ W 4
.
(4.11)