6.6 General Theory for Growth in 3D
287
P = P r e r e r + P θ e θ e θ + P φ e φ e φ
S = S r e r e r + S θ e θ e θ + S φ e φ e φ .
(6.65)
Kinematic Relations
r = r(R),
θ = ,
φ =
λ r =
∂r
∂R
= G r λ
∗
r
λ θ =
r
R
= G θ λ
∗
θ
λ φ =
r
R
= G φ λ
∗
φ .
(6.66)
E
∗
i =
1
2 (λ
∗
i − 1)
(6.67)
J = J
∗ J G
J = λ r λ θ λ φ
J
∗
= λ
∗
r λ
∗
θ λ
∗
φ
J G = G r G θ G φ
(6.68)
Equilibrium
∂σ r
∂r
+
2σ r − σ θ − σ φ
r
= 0
(6.69)
Constitutive Relations
σ i = ¯
σ i − p
(i = r, θ, φ)
¯
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
(6.70)
Incompressibility
J
∗
= 1
p = ¯
σ r +
( ¯
σ θ + ¯
σ φ − 2 ¯
σ r )
dr
r
(6.71)