520
9 Variational Principles in Mechanics and Their Applications
Fig. 9.1 The strain energy
and complementary strain
energy
O
ε x
σ x
A
B
C
The equation describing the material property is called the constitutive equation of
the material. Because the mathematical description is represented in mathematical
model or mathematical equation, the constitutive relation and the constitutive equation have the same meaning, in many cases it does not distinguish the constitutive
relation from the constitutive equation. In the mechanics, the relation between the
mechanical parameters (stress, stress rate etc.) and the kinematics parameters (strain,
strain rate etc.) is called the constitutive relation or constitutive equation.
If a particle at any location within a certain space is subjected to a force that
the magnitude and direction are completely determined, then the space is called the
force field or field of force. If a particle moves in the force field, the work done by
applied force to the particle in the force field has nothing to do with the path passed
through by the particle, but only has to do with the positions of the starting point and
end point of the particle, then the force field is called the conservative field of force
or potential field. The force acting on the particle in the potential field is called the
potential force or conservative force. In the potential field, a particle moves from
a point M to any chosen point M 0 , the work done by the potential force is called the
potential energy of the particle at point M relative to point M 0 , it is often denoted
by U or V. The mechanical system in the conservative field of force is called the
conservative system.
Consider the system consisting of n particles, let r i be the position vector of the
i-th particle relative to a fixed coordinate origin, the total kinetic energy of the system
is the sum of the kinetic energies of various particles, it can be expressed as
T =
1
2
n
i=1
m i ˙
r
2
i =
1
2
n
i=1
m i ˙
r i · ˙
r i =
1
2
n
i=1
m i v
2
i
(9.1.23)
9 Variational Principles in Mechanics and Their Applications
Fig. 9.1 The strain energy
and complementary strain
energy
O
ε x
σ x
A
B
C
The equation describing the material property is called the constitutive equation of
the material. Because the mathematical description is represented in mathematical
model or mathematical equation, the constitutive relation and the constitutive equation have the same meaning, in many cases it does not distinguish the constitutive
relation from the constitutive equation. In the mechanics, the relation between the
mechanical parameters (stress, stress rate etc.) and the kinematics parameters (strain,
strain rate etc.) is called the constitutive relation or constitutive equation.
If a particle at any location within a certain space is subjected to a force that
the magnitude and direction are completely determined, then the space is called the
force field or field of force. If a particle moves in the force field, the work done by
applied force to the particle in the force field has nothing to do with the path passed
through by the particle, but only has to do with the positions of the starting point and
end point of the particle, then the force field is called the conservative field of force
or potential field. The force acting on the particle in the potential field is called the
potential force or conservative force. In the potential field, a particle moves from
a point M to any chosen point M 0 , the work done by the potential force is called the
potential energy of the particle at point M relative to point M 0 , it is often denoted
by U or V. The mechanical system in the conservative field of force is called the
conservative system.
Consider the system consisting of n particles, let r i be the position vector of the
i-th particle relative to a fixed coordinate origin, the total kinetic energy of the system
is the sum of the kinetic energies of various particles, it can be expressed as
T =
1
2
n
i=1
m i ˙
r
2
i =
1
2
n
i=1
m i ˙
r i · ˙
r i =
1
2
n
i=1
m i v
2
i
(9.1.23)