9.9 The Hu Haichang-Kyuichiro Washizu Generalized Variational Principles
591
From Eqs. (9.9.14) and (9.9.17), we get
μ i = −n j σ i j
(9.9.18)
Substituting λ i j = σ i j and μ i = −n j σ i j into the auxiliary functional (9.9.5),
and replacing J
∗ by J H −W , the functional of the Hu Haichang-Kyuichiro Washizu
variational principle can be obtained
J H −W =
˚
V
U 0 (ε i j ) − F i u i +
1
2
(u i, j + u j,i ) − ε i j
σ i j
dV
−
¨
S σ
X i u i dS −
¨
S u
n j σ i j (u i − ¯
u i )dS
(9.9.19)
The independent functions of the functional (9.9.19) are the displacement u i ,
stress σ i j and strain ε i j , although there are the independent function u i, j (or u j,i ),
but because there is the geometric Eq. (9.9.15), namely u i, j (or u j,i ) and ε i j have
the restrictive relationship, when constructing the functional (9.9.2) the relationship
has been applied, such that u i, j (or u j,i ) can not be considered to be the fourth group
independently functions. According to the theory of the calculus of variations, of
course it can also take the displacement u i , stress σ i j and derivative u i, j (or u j,i )
of the displacement with respect to the coordinates as the independent functions.
Therefore, the Hu Haichang-Kyuichiro Washizu variational principle is a generalized
variational principle of three groups of independent functions or variables of the three
kinds.
The Hu Haichang-Kyuichiro Washizu generalized variational principle can be
expressed as: In all the displacement fields, stress fields and strain fields, the real
displacement field, stress field and strain field should make the first variation
of the functional (9.9.19) be δ J H −W = 0.
The variation of the functional for the Hu Haichang-Kyuichiro Washizu generalized variational principle gives all the fundamental equations and boundary
conditions of boundary value problem of elastic mechanics.
Substituting Eq. (9.4.16) into Eq. (9.9.19), eliminating the volume integral of the
term
1
2
(u i, j + u j,i )σ i j , and multiplied by −1, we obtain
J H W = −J H −W =
˚
V
[σ i j ε i j − U 0 (ε i j ) + (σ i j, j + F i )u i ]dV
−
¨
S σ
(n j σ i j − X i )u i dS −
¨
S u
n j σ i j ¯
u i dS
(9.9.20)
This shows that the functional (9.9.20) and functional (9.9.19) are completely
equivalent, they differ just a negative sign. The functional (9.9.20) appears to have
four groups of independent functions namely σ i j , ε i j , σ i j, j and u i , but this time the
independent functions σ i j and ε i j are subjected to the limit of the Eq. (9.4.16), only a
kind of independent function is independent between them, therefore, the functional
591
From Eqs. (9.9.14) and (9.9.17), we get
μ i = −n j σ i j
(9.9.18)
Substituting λ i j = σ i j and μ i = −n j σ i j into the auxiliary functional (9.9.5),
and replacing J
∗ by J H −W , the functional of the Hu Haichang-Kyuichiro Washizu
variational principle can be obtained
J H −W =
˚
V
U 0 (ε i j ) − F i u i +
1
2
(u i, j + u j,i ) − ε i j
σ i j
dV
−
¨
S σ
X i u i dS −
¨
S u
n j σ i j (u i − ¯
u i )dS
(9.9.19)
The independent functions of the functional (9.9.19) are the displacement u i ,
stress σ i j and strain ε i j , although there are the independent function u i, j (or u j,i ),
but because there is the geometric Eq. (9.9.15), namely u i, j (or u j,i ) and ε i j have
the restrictive relationship, when constructing the functional (9.9.2) the relationship
has been applied, such that u i, j (or u j,i ) can not be considered to be the fourth group
independently functions. According to the theory of the calculus of variations, of
course it can also take the displacement u i , stress σ i j and derivative u i, j (or u j,i )
of the displacement with respect to the coordinates as the independent functions.
Therefore, the Hu Haichang-Kyuichiro Washizu variational principle is a generalized
variational principle of three groups of independent functions or variables of the three
kinds.
The Hu Haichang-Kyuichiro Washizu generalized variational principle can be
expressed as: In all the displacement fields, stress fields and strain fields, the real
displacement field, stress field and strain field should make the first variation
of the functional (9.9.19) be δ J H −W = 0.
The variation of the functional for the Hu Haichang-Kyuichiro Washizu generalized variational principle gives all the fundamental equations and boundary
conditions of boundary value problem of elastic mechanics.
Substituting Eq. (9.4.16) into Eq. (9.9.19), eliminating the volume integral of the
term
1
2
(u i, j + u j,i )σ i j , and multiplied by −1, we obtain
J H W = −J H −W =
˚
V
[σ i j ε i j − U 0 (ε i j ) + (σ i j, j + F i )u i ]dV
−
¨
S σ
(n j σ i j − X i )u i dS −
¨
S u
n j σ i j ¯
u i dS
(9.9.20)
This shows that the functional (9.9.20) and functional (9.9.19) are completely
equivalent, they differ just a negative sign. The functional (9.9.20) appears to have
four groups of independent functions namely σ i j , ε i j , σ i j, j and u i , but this time the
independent functions σ i j and ε i j are subjected to the limit of the Eq. (9.4.16), only a
kind of independent function is independent between them, therefore, the functional