5.4 Mechanical Properties of Contractile Fibers
229
Solution
Taking advantage of previous work, we extend the analysis for extension and
inflation of a tube (Sect. 4.4) to include two layers and active contraction. Equations (4.52) and (5.3) provide the stretch ratios
λ r =
∂r
∂R
= λ
∗
r
λ θ =
r
R
= K(t) λ
∗
θ
λ z = λ = λ
∗
z ,
(5.38)
and the incompressibility condition λ r λ θ λ z = 1 leads to Eq. (4.54), i.e.,
r(R) =
a
2
+
1
λ
R
2
− a
2
0
1
2
.
(5.39)
For incompressible tissue, the total Cauchy stress components are written in the
form
σ r = ¯
σ r − p,
σ θ = ¯
σ θ − p,
σ z = ¯
σ z − p.
(5.40)
For contraction in the circumferential direction only, extending Eqs. (5.10)
and (5.11) to 3D yields the constitutive relations
¯
σ r = φ p ( ¯
σ r ) p = φ p λ r
∂W p
∂λ r
¯
σ θ = φ p ( ¯
σ θ ) p + φ a (σ θ ) a = φ p λ θ
∂W p
∂λ θ
+ φ a λ
∗
θ
∂W a
∂λ ∗
θ
¯
σ z = φ p ( ¯
σ z ) p = φ p λ z
∂W p
∂λ z
.
(5.41)
In the media, we set W p = W med
p , W a = W med
a
, φ p = φ med
p , and φ a = φ med
a . In the
adventitia, we set W p = W adv
p , φ p = 1, and φ a = 0.
The equilibrium equation (4.58) and boundary conditions (4.61) remain the same.
Thus, substituting (5.40) into (4.58) and integrating yield Eqs. (4.63) and (4.64), i.e.,
p(r) = ¯
σ r (r) +
b
r
( ¯
σ θ − ¯
σ r )
dr
r
p i =
b
a
( ¯
σ θ − ¯
σ r )
dr
r
.
(5.42)