286
6 Growth
λ z = λ = G z λ
∗
z
(6.59)
E
∗
i =
1
2 (λ
∗
i − 1)
(6.60)
J = J
∗ J G
J = λ r λ θ λ z
J
∗
= λ
∗
r λ
∗
θ λ
∗
z
J G = G r G θ G z
(6.61)
Equilibrium
∂σ r
∂r
+
σ r − σ θ
r
= 0
(6.62)
Constitutive Relations
σ i = ¯
σ i − p
(i = r, θ, z)
¯
σ i =
λ ∗2
i
J ∗
∂W ∗
∂E ∗
i
=
λ ∗
i
J ∗
∂W ∗
∂λ ∗
i
P i = J σ i /λ i
S i = J σ i /λ
2
i
(6.63)
Incompressibility
J
∗
= 1
p = ¯
σ r +
( ¯
σ θ − ¯
σ r )
dr
r
(6.64)
6.6.5 Growth Equations in Principal Spherical Coordinates
Although symmetry demands that λ φ = λ θ , etc., the equations listed below keep
circumferential and meridional quantities separate. Thus, we set
F = λ r e r e r + λ θ e θ e θ + λ φ e φ e φ
F
∗
= λ
∗
r e r e r + λ
∗
θ e θ e θ + λ
∗
φ e φ e φ
G = G r e r e r + G θ e θ e θ + G φ e φ e φ
σ = σ r e r e r + σ θ e θ e θ + σ φ e φ e φ