710
10 Variational Problems of Functionals with Vector, Tensor …
= −∇ ·
F
×
(m + n − 1)· δu AB
+ ∇ × F m + n
· ABδu
+ u(∇ × F n
· B) m
· δ A + u A m
· ∇ × F n
· δ B
(6)
Substituting Eqs. (3)–(6) into Eq. (2), we obtain
δ J =
V
∂ F
∂u
+ (C − ∇ · D − ∇ E + ∇ × F) m + n
· AB
δu
+
∂ F
∂ A
+ u(C − ∇ · D − ∇ E + ∇ × F) n
· B
m
· δ A
∂ F
∂ B
+ u A m
· (C − ∇ · D − ∇ E + ∇ × F)
n
· δ B
dV
+
V
∇ · ( D m + n
· δu AB) + ∇ · (δu AB m + n − 1
·
E) − ∇ ·
F
×
(m + n − 1)· δu AB
dV (7)
The Euler equations of the functional are
⎧
⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎩
∂ F
∂u
+
∂ F
∂u AB
− ∇ ·
∂ F
∂∇u AB
− ∇
∂ F
∂∇·u AB
+ ∇ ×
∂ F
∂∇×u AB
m + n
· AB = 0
∂ F
∂ A
+ u
∂ F
∂u AB
− ∇ ·
∂ F
∂∇u AB
− ∇
∂ F
∂∇·u AB
+ ∇ ×
∂ F
∂∇×u AB
n
· B = 0
∂ F
∂ B
+ u A m
·
∂ F
∂u AB
− ∇ ·
∂ F
∂∇u AB
− ∇
∂ F
∂∇·u AB
+ ∇ ×
∂ F
∂∇×u AB
= 0
(8)
The natural boundary conditions are
n ·
∂ F
∂∇u AB
S
= 0,
∂ F
∂∇ · u AB
S
= 0, n ×
∂ F
∂∇ × u AB
S
= 0
(9)
Example 10.6.19 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 get
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)