544
9 Variational Principles in Mechanics and Their Applications
is the complementary stress energy of the elastic body. Putting
−V
c
= W
c
=
¨
S
¯
u i X i dS
(9.5.7)
where, V
c is called the complementary virtual energy of the surface forces, it and
the complementary virtual work W
c of the surface forces differ by a negative sign.
Π
c is called the total complementary energy of an elastic body.
Equation (9.5.5) expresses the extremum principle of complementary energy. In
fact, imitating the method of proof of the principle of minimum potential energy, it
can be proved that this complementary energy is the minimum, thus the principle of
minimum complementary energy be obtained:
In all the functions of stress components satisfying the equilibrium equations
and static boundary conditions, the real stress component can make the total
complementary energy of an elastic body takes the minimum.
Making use of the principle of minimum complementary energy, the boundary
value problem of differential equation for the stress can be converted into the variational problem of finding the total complementary energy functional. The same as the
principle of minimum potential energy, the principle of minimum complementary
energy only in the case of a conservative system being in the stable equilibrium holds.
The principle of minimum complementary energy and the principle of minimum
potential energy become the complementary form.
It needs to emphatically point out that the principle of minimum potential energy
although and the principle of complementary virtual work are similar even same in
the mathematical expression form, but their connotations are different, the principle
of complementary virtual work does not require the constitutive relation of material,
while the principle of minimum complementary energy gives the constitutive relation
of material.
9.6 The Hamilton Principles and Their Applications
9.6.1 The Hamilton Principle of System of Particles
Let there be a non-free particle system composed of the n particles, the resultant
force of the i-th particle P i can be divided into two classes: the active force F i and
constraint reactive force R i . If the sum of elementary work done by the reactions
force of constraint on any virtual displacements of the system is identically equal to
zero, namely
n
i=1
R i · δr i = 0
(9.6.1)
9 Variational Principles in Mechanics and Their Applications
is the complementary stress energy of the elastic body. Putting
−V
c
= W
c
=
¨
S
¯
u i X i dS
(9.5.7)
where, V
c is called the complementary virtual energy of the surface forces, it and
the complementary virtual work W
c of the surface forces differ by a negative sign.
Π
c is called the total complementary energy of an elastic body.
Equation (9.5.5) expresses the extremum principle of complementary energy. In
fact, imitating the method of proof of the principle of minimum potential energy, it
can be proved that this complementary energy is the minimum, thus the principle of
minimum complementary energy be obtained:
In all the functions of stress components satisfying the equilibrium equations
and static boundary conditions, the real stress component can make the total
complementary energy of an elastic body takes the minimum.
Making use of the principle of minimum complementary energy, the boundary
value problem of differential equation for the stress can be converted into the variational problem of finding the total complementary energy functional. The same as the
principle of minimum potential energy, the principle of minimum complementary
energy only in the case of a conservative system being in the stable equilibrium holds.
The principle of minimum complementary energy and the principle of minimum
potential energy become the complementary form.
It needs to emphatically point out that the principle of minimum potential energy
although and the principle of complementary virtual work are similar even same in
the mathematical expression form, but their connotations are different, the principle
of complementary virtual work does not require the constitutive relation of material,
while the principle of minimum complementary energy gives the constitutive relation
of material.
9.6 The Hamilton Principles and Their Applications
9.6.1 The Hamilton Principle of System of Particles
Let there be a non-free particle system composed of the n particles, the resultant
force of the i-th particle P i can be divided into two classes: the active force F i and
constraint reactive force R i . If the sum of elementary work done by the reactions
force of constraint on any virtual displacements of the system is identically equal to
zero, namely
n
i=1
R i · δr i = 0
(9.6.1)