10.9 Introduction to the Famous Scientists
749
−
S σ
u · (T
· R · N − P)dS −
S u
u · T
· R · N dS
Given that the right polar decomposition of deformation gradient F is F =
R·U, write out the Euler equations and boundary conditions of the functional.
10.50 Find the Euler equations and boundary conditions of the functional
Π =
V
Σ
c
(T ) + (F − I − u∇) : τ +
1
2
T : (I − F
T
· F)
dV
+
A τ
u ·
◦
T
N
d A +
A u
(u −
◦
u) · τ · Nd A
where, T is the symmetric tensor.
10.51 The total complementary energy functional of the Hellinger-Reissner generalized variational principle is
Π H R =
V
1
2
σ i j (u i, j + u j,i ) − U c
0 (σ i j ) − ¯
F i u i
dV
−
S u
n j σ i j (u i − ¯
u i )dS −
S σ
¯
X i u i dS
Find its Euler equations and boundary conditions.
10.52 The total potential energy functional of the Hu Haichang-Kyuichiro Washizu
generalized variational principle is
Π H −W =
V
U 0 (ε i j ) + σ i j [ε i j −
1
2
(u i, j + u j,i )] − ¯
F i u i
dV −
Su
σ i j n j (u i − ¯
u i ) · dS
−
Sσ
¯
X i u i dS
Find its Euler equations and boundary conditions.
10.53 Find the Euler equations and boundary condition of the functional Π
P
O OC O =
V [A(e i j ) − e i j σ i j − (σ i j, j + ¯
F i )u i ]dV +
S u
σ i j n j ¯
u i dS, where, A(e i j ) is
the strain energy density, the strain tensor e i j and the stress tensor σ i j are
both symmetric tensors.
10.54 Find the Euler equations and boundary condition of the functional Π
C
O O Od =
V [B(σ i j ) − σ i j u i, j + ¯
F i u i ]dV +
S p
¯
p i u i dS, where, B(σ i j ) is the complementary energy density, the stress tensor σ i j is the symmetric tensor.
10.55 Find the Euler equations and boundary condition of the functional Π
P
O Ocd =
V [A(e i j ) − e i j σ i j − (σ i j, j + ¯
F i )u i ]dV +
S u
σ i j n j ¯
u i dS, where, A(e i j ) is
the strain energy density, the strain tensor e i j and the stress tensor σ i j are
both symmetric tensors.
10.56 Discuss the variational problem of the functional J =
V F(a, ∇ a · a∇)dV .