9.3 Principle of the Minimum Potential Energy
535
Obviously, W is the work done by the external force (include the volume force and
surface force) on the displacement u i . If the external force is the force in a potential
field, then
V = −W = −
˚
V
F i u i dV −
¨
S
X i u i dS
(9.3.3)
where, V is called the potential energy of external force.
Taking the variation to the expression (9.3.3), there is
δV = −
˚
V
F i δu i dV −
¨
S
X i δu i dS
(9.3.4)
and
Π = U + V =
˚
V
U 0 (ε i j )dV −
˚
V
F i u i dV −
¨
S
X i u i dS
(9.3.5)
is the sum of the elastic potential energy and potential energy of external force, it is
called the total potential energy of an elastic body. Thus Eq. (9.3.1) can be written
as
δΠ = 0
(9.3.6)
Equation (9.3.6) shows that when an elastic body is in the equilibrium position, the total potential energy has extremum. The research result indicates that at a
stable equilibrium position the potential energy of an elastic body has the minimum.
Thus, for an elastic body that the displacement and deformation are both small, the
principle of minimum potential energy can be expressed as: For an elastic body
under the action of the given external force, in all possible displacements satisfying the conditions of deformation compatibility and conditions of displacement boundary, the real displacement makes the total potential energy of the
elastic body obtain the minimum. According to the principle of minimum potential
energy, the boundary value problem of finding displacement differential equation can
be transformed into the variational problem of finding total potential energy functional. Extracting the displacement of the elastic body, the stress can be obtained, to
analyze the strength of the elastic body.
Now proving the principle of minimum potential energy.
Proof Using u, v, w and u
p , v
p , w
p respectively to represent a set of displacements
of a real solution and a set of possible displacements, and putting
u
p
= u + δu, v
p
= v + δv, w
p
= w + δw
535
Obviously, W is the work done by the external force (include the volume force and
surface force) on the displacement u i . If the external force is the force in a potential
field, then
V = −W = −
˚
V
F i u i dV −
¨
S
X i u i dS
(9.3.3)
where, V is called the potential energy of external force.
Taking the variation to the expression (9.3.3), there is
δV = −
˚
V
F i δu i dV −
¨
S
X i δu i dS
(9.3.4)
and
Π = U + V =
˚
V
U 0 (ε i j )dV −
˚
V
F i u i dV −
¨
S
X i u i dS
(9.3.5)
is the sum of the elastic potential energy and potential energy of external force, it is
called the total potential energy of an elastic body. Thus Eq. (9.3.1) can be written
as
δΠ = 0
(9.3.6)
Equation (9.3.6) shows that when an elastic body is in the equilibrium position, the total potential energy has extremum. The research result indicates that at a
stable equilibrium position the potential energy of an elastic body has the minimum.
Thus, for an elastic body that the displacement and deformation are both small, the
principle of minimum potential energy can be expressed as: For an elastic body
under the action of the given external force, in all possible displacements satisfying the conditions of deformation compatibility and conditions of displacement boundary, the real displacement makes the total potential energy of the
elastic body obtain the minimum. According to the principle of minimum potential
energy, the boundary value problem of finding displacement differential equation can
be transformed into the variational problem of finding total potential energy functional. Extracting the displacement of the elastic body, the stress can be obtained, to
analyze the strength of the elastic body.
Now proving the principle of minimum potential energy.
Proof Using u, v, w and u
p , v
p , w
p respectively to represent a set of displacements
of a real solution and a set of possible displacements, and putting
u
p
= u + δu, v
p
= v + δv, w
p
= w + δw