588
9 Variational Principles in Mechanics and Their Applications
=
¨
S u
n j σ i j u i dS +
¨
S σ
n j σ i j u i dS −
1
2
˚
V
σ i j (u i, j + u j,i )dV
(9.8.15)
Substituting Eq. (9.8.15) into Eq. (9.1.14), and multiplied by −1, we get
J H −R = −J 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
(9.8.16)
The functional (9.8.16) is the very original functional posed by the HellingerReissner generalized variational principle, it is completely equivalent to the functional (9.8.14), and just differs by a negative sign. The independent functions of the
functional (9.8.16) are the stress σ i j , displacement u i and derivatives u i, j (or u j,i ) of
the displacement with respect to the coordinates. The functional (9.8.16) is a kind of
generalized expression of the total potential energy of an elastic body, it is called the
generalized potential energy with three groups of independent functions, but traditionally it is called the generalized potential energy with the variables of the two
kinds, at this time the independence functions of the functional-the derivatives u i, j
(or u j,i ) of the displacement with respect to the coordinates is not taken into account,
in other words, the displacement u i and derivatives u i, j (or u j,i ) of the displacement
with respect to the coordinates are regarded as the variables of the same kind.
9.9 The Hu Haichang-Kyuichiro Washizu Generalized
Variational Principles
Hu Haichang and Kyuichiro Washizu in 1954 and 1955 successively generalised the
principle of minimum potential energy, an unconditional variational principle taking
the displacement u i , stress σ i j and strain ε i j as independent function be obtained, the
principle is called the Hu Haichang-Kyuichiro Washizu generalized variational
principle, it is called the Hu Haichang-Kyuichiro Washizu variational principle
or Hu-Kyuichiro principle for short.
Like the Hellinger-Reissner generalized variational principle, method the functional of the Hu Haichang-Kyuichiro Washizu variational principle can still be
established with the Lagrange multiplier.
The functional established by the principle of minimum potential energy is
J =
˚
V
[U 0 (ε i j ) − F i u i ]dV −
¨
S σ
X i u i dS
(9.9.1)
9 Variational Principles in Mechanics and Their Applications
=
¨
S u
n j σ i j u i dS +
¨
S σ
n j σ i j u i dS −
1
2
˚
V
σ i j (u i, j + u j,i )dV
(9.8.15)
Substituting Eq. (9.8.15) into Eq. (9.1.14), and multiplied by −1, we get
J H −R = −J 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
(9.8.16)
The functional (9.8.16) is the very original functional posed by the HellingerReissner generalized variational principle, it is completely equivalent to the functional (9.8.14), and just differs by a negative sign. The independent functions of the
functional (9.8.16) are the stress σ i j , displacement u i and derivatives u i, j (or u j,i ) of
the displacement with respect to the coordinates. The functional (9.8.16) is a kind of
generalized expression of the total potential energy of an elastic body, it is called the
generalized potential energy with three groups of independent functions, but traditionally it is called the generalized potential energy with the variables of the two
kinds, at this time the independence functions of the functional-the derivatives u i, j
(or u j,i ) of the displacement with respect to the coordinates is not taken into account,
in other words, the displacement u i and derivatives u i, j (or u j,i ) of the displacement
with respect to the coordinates are regarded as the variables of the same kind.
9.9 The Hu Haichang-Kyuichiro Washizu Generalized
Variational Principles
Hu Haichang and Kyuichiro Washizu in 1954 and 1955 successively generalised the
principle of minimum potential energy, an unconditional variational principle taking
the displacement u i , stress σ i j and strain ε i j as independent function be obtained, the
principle is called the Hu Haichang-Kyuichiro Washizu generalized variational
principle, it is called the Hu Haichang-Kyuichiro Washizu variational principle
or Hu-Kyuichiro principle for short.
Like the Hellinger-Reissner generalized variational principle, method the functional of the Hu Haichang-Kyuichiro Washizu variational principle can still be
established with the Lagrange multiplier.
The functional established by the principle of minimum potential energy is
J =
˚
V
[U 0 (ε i j ) − F i u i ]dV −
¨
S σ
X i u i dS
(9.9.1)