732
10 Variational Problems of Functionals with Vector, Tensor …
⎧
⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎩
∂ F
∂u
+
∂ F
∂u AB
−
∂ F
∂∇u AB
· ∇ −
∂ F
∂∇·u AB
∇ +
∂ F
∂∇×u AB
× ∇
m + n
· AB = 0
∂ F
∂ A
+ u B n
·
∂ F
∂u AB
−
∂ F
∂∇u AB
· ∇ −
∂ F
∂∇·u AB
∇ +
∂ F
∂∇×u AB
× ∇
= 0
∂ F
∂ B
+ u
∂ F
∂u AB
−
∂ F
∂∇u AB
· ∇ −
∂ F
∂∇·u AB
∇ +
∂ F
∂∇×u AB
× ∇
m
· A = 0
(8)
The natural boundary conditions are
∂ F
∂∇u AB
· n
S
= 0,
∂ F
∂∇ · u AB
S
= 0,
∂ F
∂∇ × u AB
× n
S
= 0
(9)
Example 10.7.10 Let A be a tensor of order m, B is a tensor of order n, with m ≤ n,
derive the Euler equations and natural boundary conditions of the functional
J =
V
F(u, A, B, u A m
× B, ∇u A m
× B, ∇ · u A m
× B, ∇ × u A m
× B)dV (1)
Solution Taking the variation to the functional, we obtain
δ J =
V
⎛
⎜
⎝
∂ F
∂u
δu +
∂ F
∂ A
m
· δ A +
∂ F
∂ B
n
· δ B +
∂ F
∂u A m
× B
n
· δu A m
× B +
∂ F
∂∇u A m
× B
n + 1
· δ∇u A m
× B
+
∂ F
∂∇ · u A m
× B
n − 1
· δ∇ · u A m
× B +
∂ F
∂∇ × u A m
× B
n
· δ∇ × u A m
× B
⎞
⎟
⎠dV
(2)
Let C =
∂ F
∂u A m
×
B
, D =
∂ F
∂∇u A m
×
B
, E =
∂ F
∂∇·u A m
×
B
, F =
∂ F
∂∇×u A m
×
B
, for the latter
four variations of Eq. (2), using Eq. (10.1.13), there are
C n
· δu A m
× B = C n
· A m
× Bδu + u[B n m×
(n − m)· C n−m C
T
n−m+1,n ] m
· δ A + uC n−m C
T
n−m+1,n
m
× A
T n
· δ B
(3)
D n + 1
· ∇δu A m
× B = ∇ · (δu A m
× B n
· D) − D · ∇ n
· δu A m
× B
= ∇ · (δu A m
× B n
· D) − D · ∇ n
· ( A m
× Bδu + uδ A m
× B + u A m
× δ B)
= ∇ · (δu A m
× B n
· D) − D · ∇ n
· A m
× Bδu
− u[B m×
(n − m)· ( D · ∇) n−m ( D · ∇) T
n−m+1,n ] m
· δ A
− u( D · ∇) n−m ( D · ∇) T
n−m+1,n
m
× A T n
· δ B
(4)
E n − 1
· ∇ · δu A m
× B = ∇ · (δu A m
× B n − 1
· E) − E∇ n
· ( A m
× Bδu + uδ A m
× B + u A m
× δ B)
= ∇ · (δu A m
× B n − 1
· E) − E∇ n
· A m
× Bδu