10.6 Variational Problems of Functionals with Parallel-Type Inner …
699
∂ F
∂∇u
= K + K · ∇u = K · (I + ∇u) = K · F
T
(10)
where, F
T the transposition of the deformation gradient F, F
T
= (I + u∇)
T
=
(I + ∇u).
The first equation of Eq. (7) can be written as
∇ · (K · F
T
) + ρ 0 b = 0 (in V )
(11)
Example 10.6.6 The unconstrained complementary energy functional of the generalized Levinton principle is
J [τ , u] =
V
[
∼
Σ c (τ ) − trτ + u · (τ · ∇)]dV −
A u
◦
u ·τ · ndA −
A t
u · (τ · n −
◦
T n )d A
(1)
where, ˜
Σ
c
= τ : F(τ ) − Σ[F(τ )], with F(τ ) =
d ˜
Σ
c
dτ
. Find the Euler equations and
the boundary conditions of the variations of the stress τ and the displacement u.
Solution Let the integrand F = ˜
Σ
c
(τ ) − trτ + u · (τ · ∇), where the variation of
the term ˜
Σ
c
(τ ) is δ ˜
Σ
c
(τ ) =
∂ ˜
Σ
c (τ )
∂τ
: δτ = F : δτ , thus we obtain
∂ F
∂τ
= F,
∂ F
∂trτ
∂trτ
∂τ
= I,
∂ F
∂τ · ∇
= u,
∂ F
∂ u
= τ · ∇
(2)
Combine the variation
A (u · δτ ) · ndA of the surface integral corresponding to
the term u · (τ · ∇) with the variation −
A u
◦
u ·δτ · ndA of the surface integral on
the displacement boundary into
A u
(u −
◦
u) ·δτ · ndA, the variation of the surface
integral on the stress boundary is −
A t
(τ · n −
◦
T n ) · δudA, therefore, the Euler
equations and boundary conditions of the functional are
u∇ = F − I (in V )
(3)
τ · ∇ = 0 (in V )
(4)
u =
◦
u (on A u )
(5)
τ · n =
◦
T n (on A t )
(6)
Example 10.6.7 The total complementary energy functional of the generalized
Veubeke (Baudouin M. Fraeijs de Veubeke, 1917.8.3–1976.9.16) principle is