10.7 Variational Problems of Functionals with Series-Type …
727
∂ F
∂ A
+
∂ F
∂ A
T
+ I + (−1)
k
∇
c
t
∂ F
∂∇ t A t ∇
c
t ∇ = 0
(10.7.31)
where, k is the sum of the Hamiltonian operator quantities in the Hamiltonian operator trains, in this case, the sign of the conjugate Hamiltonian operator train is not
considered anymore. The operational order of the Hamiltonian operator trains can
be determined by practical situation.
Proof The last term of Eq. (10.7.31) are proved as follows:
When taking the variation to ∇ t A, for the Hamiltonian operator train ∇ t , every
time only one Hamiltonian operator can be retained on the left-hand side of the
variational symbol δ, the right-hand side of δ is regarded as a tensor, according to
the leftmost Hamiltonian operator being the gradient, the divergence or the rotation,
substituting ∇ t δ A into the above corresponding variational formulas respectively,
such that the Hamiltonian operators in ∇ t δ A reduce in quantity, the left-hand side of
∇ t δ A every time reduces a gradient operator, the right-hand side of its adjoint tensor
increases a divergence operator, the left of ∇ t δ A every time reduces a divergence
operator, the right-hand side of its adjoint tensor increases a gradient operator, the
left-hand side of ∇ t δ A every time reduces a rotation operator, the right-hand side of
its adjoint tensor increases a rotation operator, finally δ∇ t A can be changed into the
form of δ A, meanwhile the corresponding boundary conditions can also be obtained.
Similarly, δ A t ∇ can also be changed into the form of δ A. Taking note that the gradient
operator, divergence operator and rotation operator are operated one time, their signs
are just changed one time. Therefore the last term of Eq. (5.2) can be obtained. Quod
erat demonstrandum.
Example 10.7.5 The generalized variational problem of three-dimensional theory
of elasticity. Let u = (u, v, w) be the displacement vector, Φ = (Φ i j ), Φ
0
= (Φ
0
i j ),
T = (σ i j ), ˜
T = ( ˜
σ i j ), E = (ε i j ), ˜
E = (˜ ε i j ), Q = (Q i j ) are the seven threedimensional symmetric tensors of order two, taking the eight groups of unknown
quantities u, Φ, Φ
0 , T, ˜
T , E, ˜
E and Q as the independent functions, the energy
functional proposed by professor Wu Jike is
Π =
V
[W (E) + V ( ˜
T ) − T : E − ˜
T : ˜
E − Φ : Q − (∇ · T + f ) · u + ( Q − ∇ × ˜
E × ∇) : Φ 0 ]dV
+
∂ uV
n · T · u 0 dS +
∂vV
(n · T − p 0 ) · udS +
∂ uV
˜
E × ∇ : R 1 dS −
∂ uV
˜
E × n : R 2 dS
+
∂vV
( ˜
E × ∇ − P 1 ) : Φ × n dS +
∂vV
( ˜
E × n − P 2 ) : Φ × ∇dS
(1)
where, W and V are the scalar functions defined on E and ˜
T respectively, f =
( f 1 , f 2 , f 3 ) is the volumetric force vector acting on the elastic body, u
0 and p 0 are
respectively the given tensor functions defined on the displacement boundary ∂ uV
and the stress boundary ∂vV , R 1 , R 2 , P 1 and P 2 are respectively the tensor functions
727
∂ F
∂ A
+
∂ F
∂ A
T
+ I + (−1)
k
∇
c
t
∂ F
∂∇ t A t ∇
c
t ∇ = 0
(10.7.31)
where, k is the sum of the Hamiltonian operator quantities in the Hamiltonian operator trains, in this case, the sign of the conjugate Hamiltonian operator train is not
considered anymore. The operational order of the Hamiltonian operator trains can
be determined by practical situation.
Proof The last term of Eq. (10.7.31) are proved as follows:
When taking the variation to ∇ t A, for the Hamiltonian operator train ∇ t , every
time only one Hamiltonian operator can be retained on the left-hand side of the
variational symbol δ, the right-hand side of δ is regarded as a tensor, according to
the leftmost Hamiltonian operator being the gradient, the divergence or the rotation,
substituting ∇ t δ A into the above corresponding variational formulas respectively,
such that the Hamiltonian operators in ∇ t δ A reduce in quantity, the left-hand side of
∇ t δ A every time reduces a gradient operator, the right-hand side of its adjoint tensor
increases a divergence operator, the left of ∇ t δ A every time reduces a divergence
operator, the right-hand side of its adjoint tensor increases a gradient operator, the
left-hand side of ∇ t δ A every time reduces a rotation operator, the right-hand side of
its adjoint tensor increases a rotation operator, finally δ∇ t A can be changed into the
form of δ A, meanwhile the corresponding boundary conditions can also be obtained.
Similarly, δ A t ∇ can also be changed into the form of δ A. Taking note that the gradient
operator, divergence operator and rotation operator are operated one time, their signs
are just changed one time. Therefore the last term of Eq. (5.2) can be obtained. Quod
erat demonstrandum.
Example 10.7.5 The generalized variational problem of three-dimensional theory
of elasticity. Let u = (u, v, w) be the displacement vector, Φ = (Φ i j ), Φ
0
= (Φ
0
i j ),
T = (σ i j ), ˜
T = ( ˜
σ i j ), E = (ε i j ), ˜
E = (˜ ε i j ), Q = (Q i j ) are the seven threedimensional symmetric tensors of order two, taking the eight groups of unknown
quantities u, Φ, Φ
0 , T, ˜
T , E, ˜
E and Q as the independent functions, the energy
functional proposed by professor Wu Jike is
Π =
V
[W (E) + V ( ˜
T ) − T : E − ˜
T : ˜
E − Φ : Q − (∇ · T + f ) · u + ( Q − ∇ × ˜
E × ∇) : Φ 0 ]dV
+
∂ uV
n · T · u 0 dS +
∂vV
(n · T − p 0 ) · udS +
∂ uV
˜
E × ∇ : R 1 dS −
∂ uV
˜
E × n : R 2 dS
+
∂vV
( ˜
E × ∇ − P 1 ) : Φ × n dS +
∂vV
( ˜
E × n − P 2 ) : Φ × ∇dS
(1)
where, W and V are the scalar functions defined on E and ˜
T respectively, f =
( f 1 , f 2 , f 3 ) is the volumetric force vector acting on the elastic body, u
0 and p 0 are
respectively the given tensor functions defined on the displacement boundary ∂ uV
and the stress boundary ∂vV , R 1 , R 2 , P 1 and P 2 are respectively the tensor functions