Appendix: All Solutions to the Problems
963
(τ · N −
◦
T N )
A τ
= 0, (u −
◦
u)
A u
= 0
( 2 )
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.
Solution The original functional can be written as
Π H −R =
˚
V
[σ : ∇u − U
c
0 (σ ) − ¯
F · u]dV −
¨
S u
(u − ¯
u)n : σ dS −
¨
S σ
¯
X · udS
(1)
Putting F = σ : ∇u − U
c
0 (σ ) − ¯
F · u, the various partial derivatives are
∂ F
∂σ
= ∇u −
∂ F
∂U c
0 (σ )
dU c
0 (σ )
dσ
= ∇u −
dU c
0 (σ )
dσ
= ∇u − ε,
∂ F
∂∇u
= σ ,
∂ F
∂
= − ¯
F
Taking note that the strain tensor ε is the symmetric tensor, the Euler equations
of the functional are
∇ · σ + ¯
F = 0
ε =
1
2
(∇u + u∇)
(2)
The boundary conditions are
(u − ¯
u)| S u = 0, (σ · n − ¯
X)
S σ
= 0,
(3)
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
−
¨
S u
σ i j n j (u i − ¯
u i ) · dS −
¨
S σ
¯
X i u i dS
Find its Euler equations and boundary conditions.