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10 Variational Problems of Functionals with Vector, Tensor …
10.6.3 Some Examples of the Functionals with Parallel-Type
Inner Product Tensors and Hamiltonian Operators
In various variational operations, the following second order tensor identities are
often used
A : B · C = A
T
· B : C
T
= B · C : A = C · A
T
: B
T
= B
T
: C · A
T
= C
T
: A
T
· B = B : A · C
T
= C : B
T
· A
(10.6.42)
Example 10.6.1 Let V be the region of the elastic body, S(S = S u + S σ ) is the
boundary surface of V, N is the exterior normal unit vector on the S, the displacement
is given on the displacement boundary S u : u| S u = u, the force per unit area is given
on the stress boundary S σ : X| S σ = X, f is the force per unit body, ρ is the mass
density, F is the deformation gradient, T is the Kirchhoff stress tensor, τ (τ = F · T )
is the Piola (Count Gabrio Piola Daverio, 1794.7.15–1850) stress tensor. The total
(or full) energy functional Φ of the elastic body is defined as follows
Φ =
V
[(F · T ) : (u∇) − ¯
ρ f · u]dV −
S u
¯
u · (F · T ) · N dS −
S σ
u · XdS
=
V
[τ : (u∇) − ¯
ρ f · u]dV −
S u
¯
u · (F · T ) · N dS −
S σ
u · XdS
(1)
Find the Euler equation of the displacement u variation.
Solution Let the integrand F = (F · T ) : (u∇) − ¯
ρ f · u, taking the variation to the
terms with displacement u, and get the partial derivatives
∂ F
∂ u
= − ¯
ρ f ,
∂ F
∂ u∇
= F · T = τ
(2)
The Euler equation of the functional namely the equilibrium equation is
τ · ∇ + ¯
ρ f = 0 (in V )
(3)
or
(F · T ) · ∇ + ¯
ρ f = 0 (in V )
(4)
On the stress boundary S σ there is
(F · T ) · N = X (on S σ )
(5)
or