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10 Variational Problems of Functionals with Vector, Tensor …
∂ F
∂∇ × ∇ × sin kua
× n
S
= 0,
∇ ×
∂ F
∂∇ × ∇ × sin kua
× n
S
= 0 (6)
Example 10.5.15 The functional with the rotation of dyad of two vectors. Let the
functional
J =
V
F(a, b, ∇ × (ab))dV
(1)
where, the double inner product of the tensors of second order conforms to the law
of parallel operation, Derive its Euler equations and natural boundary condition.
Solution Taking the variation to the functional, we get
δ J =
V
∂ F
∂ a
· δa +
∂ F
∂ b
· δb +
∂ F
∂∇ × (ab)
: δ∇ × (ab)
dV
(2)
Putting T =
∂ F
∂∇×(ab)
, the integral of the last term in Eq. (2) can be written as
T : δ∇ × (ab) = T : ∇ × δ(ab) = ∇ · [δ(ab) ×
· T ] + (∇ × T ) : δ(ab)
= ∇ · [δ(ab) ×
· T ] + (∇ × T ) : (δab + aδb)
= ∇ · [δ(ab) ×
· T ] + [(∇ × T ) · b] · δa + [a · (∇ × T )] · δb (3)
Substituting Eq. (3) into Eq. (2), we get
δ J =
V
∂ F
∂ a
+ (∇ × T ) · b
· δa +
∂ F
∂ b
+ a · (∇ × T )
· δb + ∇ · [δ(ab) ×
· T ]
dV
(4)
The Euler equations of the functional are
⎧
⎪ ⎨
⎪ ⎩
∂ F
∂ a
+ ∇ ×
∂ F
∂∇ × (ab)
· b = 0
∂ F
∂ b
+ a · ∇ ×
∂ F
∂∇ × (ab)
= 0
(5)
The natural boundary condition is
n ×
∂ F
∂∇ × (ab)
S
= 0
(6)
Except the meanings of individual examples are similar, various functionals of the
Sect. 10.3 to the Sect. 10.5 are all be incorporated into the functional (10.2.12), the
Euler equations and the corresponding natural boundary conditions with the more
contents can be obtained.