9.8 The Hellinger-Reissner Generalized Variational Principles
587
From arbitrariness of ¯
v, we obtain
λ i, j =
∂U
c
0
∂σ i j
(in V )
(9.8.11)
λ i = μ i (on S σ )
(9.8.12)
λ i = ¯
u i (on S u )
(9.8.13)
Again by the arbitrariness of δλ i and δμ i , the equations of equilibrium (9.2.3) and
the stress boundary conditions (9.2.10) on S σ can be achieved. It follows from Eq.
(9.8.11) and (9.8.12) that the introduced Lagrange multipliers λ i and μ i are really
the displacement u i . Using the displacement u i to replace the Lagrange multipliers
λ i and μ i in Eq. (9.8.4), and replacing J
∗ with J H R , we obtain
J H R =
˚
V
[U c
0 (σ i j ) + (σ i j, j + F i )u i ]dV −
¨
S u
n j σ i j ¯
u i dS −
¨
S σ
(n j σ i j − X i )u i dS (9.8.14)
Equation (9.8.14) is the functional of the Hellinger-Reissner generalized variational principle. The independence functions of the functional are the stress σ ij ,
derivative σ ij,j of that stress with respect to the coordinates and displacement u i ,
according to the theory of variational methods, it should be a generalized variational
principle with three groups of independent functions, but traditionally it is considered the generalized variational principle with two kinds of variables, at this time,
the independence functions of the functional-the derivatives σ ij,j of the stress with
respect to the coordinates are not included in the inside, in other words, the stress σ ij
and derivatives σ ij,j of the stress with respect to the coordinates are regarded as the
variables of the same kind. The functional (9.8.14) is a kind of generalized expression
of the total complementary energy of an elastic body, it is called the generalized
complementary energy with three groups of independent functions, but traditionally it is called the generalized complementary energy with the variables of the
two kinds. It follows from first variation of the functional (9.8.14) that δ J H R = 0
gives all the fundamental equations of elastic mechanics.
The Hellinger-Reissner generalized variational principle can be expressed as: In
all the displacement fields and stress fields, the real displacement field and stress
field should make the first variation of the functional (9.8.14) be δ J H R = 0.
According to the principle of generalized virtual displacement (9.2.21), and using
Eq. (9.2.12), there is
˚
V
σ i j, j u i dV =
¨
S
X i u i dS −
1
2
˚
V
σ i j (u i, j + u j,i )dV
=
¨
S u
X i u i dS +
¨
S σ
X i u i dS −
1
2
˚
V
σ i j (u i, j + u j,i )dV
587
From arbitrariness of ¯
v, we obtain
λ i, j =
∂U
c
0
∂σ i j
(in V )
(9.8.11)
λ i = μ i (on S σ )
(9.8.12)
λ i = ¯
u i (on S u )
(9.8.13)
Again by the arbitrariness of δλ i and δμ i , the equations of equilibrium (9.2.3) and
the stress boundary conditions (9.2.10) on S σ can be achieved. It follows from Eq.
(9.8.11) and (9.8.12) that the introduced Lagrange multipliers λ i and μ i are really
the displacement u i . Using the displacement u i to replace the Lagrange multipliers
λ i and μ i in Eq. (9.8.4), and replacing J
∗ with J H R , we obtain
J H R =
˚
V
[U c
0 (σ i j ) + (σ i j, j + F i )u i ]dV −
¨
S u
n j σ i j ¯
u i dS −
¨
S σ
(n j σ i j − X i )u i dS (9.8.14)
Equation (9.8.14) is the functional of the Hellinger-Reissner generalized variational principle. The independence functions of the functional are the stress σ ij ,
derivative σ ij,j of that stress with respect to the coordinates and displacement u i ,
according to the theory of variational methods, it should be a generalized variational
principle with three groups of independent functions, but traditionally it is considered the generalized variational principle with two kinds of variables, at this time,
the independence functions of the functional-the derivatives σ ij,j of the stress with
respect to the coordinates are not included in the inside, in other words, the stress σ ij
and derivatives σ ij,j of the stress with respect to the coordinates are regarded as the
variables of the same kind. The functional (9.8.14) is a kind of generalized expression
of the total complementary energy of an elastic body, it is called the generalized
complementary energy with three groups of independent functions, but traditionally it is called the generalized complementary energy with the variables of the
two kinds. It follows from first variation of the functional (9.8.14) that δ J H R = 0
gives all the fundamental equations of elastic mechanics.
The Hellinger-Reissner generalized variational principle can be expressed as: In
all the displacement fields and stress fields, the real displacement field and stress
field should make the first variation of the functional (9.8.14) be δ J H R = 0.
According to the principle of generalized virtual displacement (9.2.21), and using
Eq. (9.2.12), there is
˚
V
σ i j, j u i dV =
¨
S
X i u i dS −
1
2
˚
V
σ i j (u i, j + u j,i )dV
=
¨
S u
X i u i dS +
¨
S σ
X i u i dS −
1
2
˚
V
σ i j (u i, j + u j,i )dV