9.2 Principle of Virtual Displacement
529
σ
p
i j u
p
i, j = σ
p
i j ε
p
i j
(9.2.19)
Substituting Eq. (9.2.19) into the expression (9.2.18), then substituting the
expression (9.2.18) into the expression (9.2.17), we get
˚
V
F i u
p
i dV +
¨
S
X i u
p
i dS =
˚
V
σ
p
i j ε
p
i j dV
(9.2.20)
The expression (9.2.20) is called the principle of generalized virtual displacement, principle of generalized virtual work or principle of possible work of the
elastic body, it is also called the Gauss formula. It is worth pointing out that because
the derivation process of the principle of generalized virtual displacement is not
involved in the constitutive relation, the possible stress σ
p
i j and possible strain ε
p
i j in
the expression (9.2.20) can be independent of each other, they independently change
respectively under the respective admissible conditions.
The principle of generalized virtual work denoted by the expression (9.2.20) can
be expressed as: The work done by the external forces (volume and surface forces)
on the possible displacement is equal to the work done by the static possible stress
on the possible strain corresponding to the possible displacement. The principle of
generalized virtual work is a concrete manifestation of the principle of conservation
of energy in elasticity.
Using the differential equations of equilibrium (9.2.14) and the geometric
Eqs. (9.1.3) to eliminate F i and ε
p
i j in the expression (9.2.20), we obtain
−
˚
V
σ
p
i j, j u
p
i dV +
¨
S
X i u
p
i dS =
1
2
˚
V
σ
p
i j (u
p
i, j + u
p
j,i )dV
(9.2.21)
where, it does not need to satisfy any relationship between σ
p
i j and u
p
i . This is another
representation of the principle of generalized virtual displacement.
9.2.3 Principle of Generalized Virtual Displacement
for Elastic Body
The principle of virtual displacement can be generalized to the deformed body. 下
面the expression of the principle virtual displacement for the elastic body is derived.
Let the elastic body satisfy the Eqs. (9.2.3), at a position of equilibrium to apply a
set of arbitrary infinitesimal virtual displacements to the elastic body δu, δv and δw,
then any point in the elastic body will produce the virtual displacement satisfying
compatibility conditions of deformation
δu =
δu δv δw
T
(9.2.22)
529
σ
p
i j u
p
i, j = σ
p
i j ε
p
i j
(9.2.19)
Substituting Eq. (9.2.19) into the expression (9.2.18), then substituting the
expression (9.2.18) into the expression (9.2.17), we get
˚
V
F i u
p
i dV +
¨
S
X i u
p
i dS =
˚
V
σ
p
i j ε
p
i j dV
(9.2.20)
The expression (9.2.20) is called the principle of generalized virtual displacement, principle of generalized virtual work or principle of possible work of the
elastic body, it is also called the Gauss formula. It is worth pointing out that because
the derivation process of the principle of generalized virtual displacement is not
involved in the constitutive relation, the possible stress σ
p
i j and possible strain ε
p
i j in
the expression (9.2.20) can be independent of each other, they independently change
respectively under the respective admissible conditions.
The principle of generalized virtual work denoted by the expression (9.2.20) can
be expressed as: The work done by the external forces (volume and surface forces)
on the possible displacement is equal to the work done by the static possible stress
on the possible strain corresponding to the possible displacement. The principle of
generalized virtual work is a concrete manifestation of the principle of conservation
of energy in elasticity.
Using the differential equations of equilibrium (9.2.14) and the geometric
Eqs. (9.1.3) to eliminate F i and ε
p
i j in the expression (9.2.20), we obtain
−
˚
V
σ
p
i j, j u
p
i dV +
¨
S
X i u
p
i dS =
1
2
˚
V
σ
p
i j (u
p
i, j + u
p
j,i )dV
(9.2.21)
where, it does not need to satisfy any relationship between σ
p
i j and u
p
i . This is another
representation of the principle of generalized virtual displacement.
9.2.3 Principle of Generalized Virtual Displacement
for Elastic Body
The principle of virtual displacement can be generalized to the deformed body. 下
面the expression of the principle virtual displacement for the elastic body is derived.
Let the elastic body satisfy the Eqs. (9.2.3), at a position of equilibrium to apply a
set of arbitrary infinitesimal virtual displacements to the elastic body δu, δv and δw,
then any point in the elastic body will produce the virtual displacement satisfying
compatibility conditions of deformation
δu =
δu δv δw
T
(9.2.22)