322
6 Growth
Wall Stress
σ i = ¯
σ i − p
¯
σ r = φ p ¯
σ rp
¯
σ θ = φ p ¯
σ θp + φ a σ θa
¯
σ z = φ p ¯
σ zp
(6.133)
Fluid Shear Stress
τ =
4μQ
πa 3
(6.134)
Constitutive Relations
¯
σ rp = λ
∗2
rp
∂W ∗
p
∂E ∗
rp
¯
σ θp = λ
∗2
θp
∂W ∗
p
∂E ∗
θp
¯
σ zp = λ
∗2
zp
∂W ∗
p
∂E ∗
zp
σ θa = λ
∗
θa
∂W ∗
a
∂λ ∗
θa
(6.135)
Note: W ∗
p = W ∗ med
p
(E ∗
ip ) or W ∗ adv
p
(E ∗
ip ) and W a = W ∗ med
a
(λ ∗
θa ); φ p = 1 and
φ a = 0 in the adventitia.
Incompressibility
J
∗
p = 1 → J = J G
r
2
= a
2
+
2
λ
R
a 0
J G (R) R dR
(6.136)
Boundary Conditions
r = a :
σ r = −P
r = b :
σ r = 0.