9.9 The Hu Haichang-Kyuichiro Washizu Generalized Variational Principles
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The functional (9.9.1) has two independent functions namely the strain ε i j and
displacement u i . In elastic mechanics, the strain ε i j and displacement u i are required
to satisfy the geometrical Eq. (9.1.3), meanwhile the displacement is also required to
satisfy the boundary conditions (9.2.7) on S u . Thus, the principle of minimum potential energy is also a conditional extremum principle. In order to transform a functional
(9.9.1) with constraint conditions into the functional without constraint conditions,
Nine Lagrange multipliers λ i j and geometric Eqs. (9.1.3) need be introduced to
construct the functional
J 1 =
˚
V
1
2
(u i, j + u j,i ) − ε i j
λ i j dV
(9.9.2)
Moreover introducing three Lagrange multipliers μ i and the displacement
boundary conditions (9.2.7) on S u , we construct the functional
J 2 =
¨
S u
(u i − ¯
u i )μ i dS
(9.9.3)
Adding the above three functionals, we get a generalized functional J
∗ , namely
J
∗
=
˚
V
[U 0 (ε i j ) − F i u i ]dV −
¨
S σ
X i u i dS
+
˚
V
1
2
(u i, j + u j,i ) − ε i j
λ i j dV +
¨
S u
(u i − ¯
u i )μ i dS
(9.9.4)
The variation of the functional (9.9.4) is
δ J
∗
= δ J + δ J 1 + δ J 2 = 0
(9.9.5)
The variation of the functional (9.9.1) is
δ J =
˚
V
∂U 0 (ε i j )
∂ε i j
δε i j dV −
˚
V
F i δu i dV −
¨
S σ
X i δu i dS
(9.9.6)
The variation of the functional (9.9.2) is
δ J 1 =
˚
V
1
2
(u i, j + u j,i ) − ε i j
δλ i j dV −
˚
V
λ i j δε i j dV +
1
2
˚
V
(λ i j δu i, j + λ i j δu j,i )dV
(9.9.7)
Using integration by parts and the Gauss formula to the first term of the last
integral in Eq. (9.9.7), we have
˚
V
λ i j δu i, j dV =
˚
V
[(λ i j δu i ) , j − λ i j, j δu i ]dV