590
9 Variational Principles in Mechanics and Their Applications
=
¨
S
n j (λ i j δu i )dS −
˚
V
λ i j, j δu i dV
=
¨
S σ
n j (λ i j δu i )dS +
¨
S u
n j (λ i j δu i )dS −
˚
V
λ i j, j δu i dV
(9.9.8)
Similarly, Using integration by parts and the Gauss formula to the second term
of the last integral in Eq. (9.9.7), note that the symmetry of λ i j , the result is still
Eq. (9.9.8).
The variation of the functional (9.9.3) is
δ J 2 =
¨
S u
(u i − ¯
u i )δμ i dS +
¨
S u
μ i δu i dS
(9.9.9)
Substituting Eq. (9.9.8) into Eq. (9.9.7), then Substituting Eqs. (9.9.6), (9.9.7) and
(9.9.9) into Eq. (9.9.5), after arrangement, we get
δ J
∗
=
˚
V
∂U 0 (ε i j )
∂ε i j
− λ i j
δε i j dV −
˚
V
(λ i j, j + F i )δu i dV
+
˚
V
1
2
(u i, j + u j,i ) − ε i j
δλ i j dV +
¨
S σ
(n j λ i j − X i )δu i dS
+
¨
S u
(u i − ¯
u i )δμ i dS +
¨
S u
(n j λ i j + μ i )δu i dS = 0
(9.9.10)
From the arbitrariness of δε i j , δλ i j and δu i , we obtain
λ i j =
∂U 0 (ε i j )
∂ε i j
(9.9.11)
λ i j, j + F i = 0
(9.9.12)
n j λ i j − X i = 0
(9.9.13)
n j λ i j + μ i = 0
(9.9.14)
ε i j =
1
2
(u i, j + u j,i )
(9.9.15)
u i = ¯
u i
(9.9.16)
It follows from Eqs. (9.9.12) and (9.9.13) that λ i j is the stress conponent, namely
λ i j = σ i j
(9.9.17)