6.11 Case Study: Functional Adaptation in Arteries
321
where 0 < K θ ≤ 1. For reasons to be discussed below, the normalized target stresses
are assumed to increase with pressure during development from zero at t = 0 to one
at maturity via the relation
[ ˆ
σ θ0 , ˆ
τ 0 ] = P /P m .
(6.128)
6.11.3 Analysis of Growing and Contracting Artery
The equations for growth in principal cylindrical coordinates are given in Sect. 6.6.4.
Extending these relations to include contraction, as in Sect. 6.10, provides the equations listed below. Since contraction occurs only in the circumferential direction, we
set K r = K z = 1 and σ ra = σ za = 0.
Kinematic Relations
λ r =
∂r
∂R
= λ
∗
rp G r
λ θ =
r
R
= λ
∗
θp G θ = λ
∗
θa G θ K θ
λ z = λ = λ
∗
zp G z
E
∗
ip =
1
2 (λ
∗2
ip − 1)
(6.129)
J = J
∗
p J G
J = λ r λ θ λ z
J
∗
p = λ
∗
rp λ
∗
θp λ
∗
zp
J G = G r G θ G z
(6.130)
Equilibrium
∂σ r
∂r
+
σ r − σ θ
r
= 0.
(6.131)
N = 2π
b
a
σ z r dr = πa
2 P →
b
a
(2 ¯
σ z − ¯
σ r − ¯
σ θ ) r dr = 0.
(6.132)
Note: The expressions for the axial force N come from Eqs. (5.63) and (5.66).