ρl ¼ _
w int
ð7:94Þ
_
w int ¼ Γ
e
I : L
e
I þ Γ
e
M : L
e
M þ J
À1
Γ
p
I : L
p
I þ J
À1
Γ
p
M : L
p
M
ð7:95Þ
where, Γ
e
I , Γ
e
M , Γ
p
I , and Γ
p
M are stress measures conjugate to rate of deformation
measures L
e
I , L
e
M , L
p
I , and L
p
M , which were defined in Eqs. (7.12) and (7.13). Noting
that J
À1
¼ J
e
À1 , the J
À1 multiplier in front of last two terms in Eq. (7.95) recovers
work definitions from intermediate configuration to deformed configuration.
Requirement of frame indifference for internal work can be given by the following
relation:
_
w int ¼ w
:
int
ð7:96Þ
which implies that
Γ
e
I : L
e
I þ Γ
e
M : L
e
M
þJ
À1
Γ
p
I : L
p
I þ J
À1
Γ
p
M : L
p
M
!
¼
Γ
e
I : L
e
I þ Γ
e
M : L
e
M
þJ
À1
Γ
p
I : L
p
I þ J
À1
Γ
p
M : L
p
M
!
ð7:97Þ
Using transformation rules for elastic and plastic velocity gradients in Eqs. (7.73)
and (7.74) we can write the following relations:
Γ
e
I : L
e
I þ J
À1
Γ
p
I : L
p
I
þΓ
e
M : L
e
M þ J
À1
Γ
p
M : L
p
M
!
¼
Γ
e
I : ΩL
e
I Ω
T
þ _
ΩΩ
T
À
Á þ J
À1
Γ
p
I : L
p
I
þΓ
e
M : ΩL
e
M Ω
T
þ _
ΩΩ
T
À
Á þ J
À1
Γ
p
M : L
p
M
"
#
ð7:98Þ
or
Γ
e
I : L
e
I þ J
À1
Γ
p
I : L
p
I
þΓ
e
M : L
e
M þ J
À1
Γ
p
M : L
p
M
!
¼
Ω
T
Γ
e
I Ω
À
Á : L
e
I þ Ω
T
Γ
e
M Ω
À
Á : L
e
M
þJ
À1
Γ
p
I : L
p
I þ J
À1
Γ
p
M : L
p
M
þΓ
e
I : _
ΩΩ
T
þ Γ
e
M : _
ΩΩ
T
2
6
6
6
4
3
7
7
7
5
ð7:99Þ
The first two terms on the right-hand side of Eq. (7.99) indicate that stress
measures corresponding elastic work Γ
e
I , Γ
e
M
À
Á
are objective as shown in
Eqs. (7.100) and (7.101). Since _
ΩΩ
T is a skew symmetric tensor, Eqs. (7.55) and
(7.77), the last two terms on the right-hand side of Eq. (7.99) imply that stress
measures corresponding elastic work Γ
e
I , Γ
e
M
À
Á
are symmetric (Eqs. (7.102) and
(7.103)). The third and fourth terms on the right-hand side of Eq. (7.99) show that
stress measures corresponding plastic work Γ
p
I , Γ
p
M
ð
Þare not objective but invariant
to changes in current frame of reference as shown in Eqs. (7.104) and (7.105):
354
7 Unified Micromechanics of Finite Deformations
w int
ð7:94Þ
_
w int ¼ Γ
e
I : L
e
I þ Γ
e
M : L
e
M þ J
À1
Γ
p
I : L
p
I þ J
À1
Γ
p
M : L
p
M
ð7:95Þ
where, Γ
e
I , Γ
e
M , Γ
p
I , and Γ
p
M are stress measures conjugate to rate of deformation
measures L
e
I , L
e
M , L
p
I , and L
p
M , which were defined in Eqs. (7.12) and (7.13). Noting
that J
À1
¼ J
e
À1 , the J
À1 multiplier in front of last two terms in Eq. (7.95) recovers
work definitions from intermediate configuration to deformed configuration.
Requirement of frame indifference for internal work can be given by the following
relation:
_
w int ¼ w
:
int
ð7:96Þ
which implies that
Γ
e
I : L
e
I þ Γ
e
M : L
e
M
þJ
À1
Γ
p
I : L
p
I þ J
À1
Γ
p
M : L
p
M
!
¼
Γ
e
I : L
e
I þ Γ
e
M : L
e
M
þJ
À1
Γ
p
I : L
p
I þ J
À1
Γ
p
M : L
p
M
!
ð7:97Þ
Using transformation rules for elastic and plastic velocity gradients in Eqs. (7.73)
and (7.74) we can write the following relations:
Γ
e
I : L
e
I þ J
À1
Γ
p
I : L
p
I
þΓ
e
M : L
e
M þ J
À1
Γ
p
M : L
p
M
!
¼
Γ
e
I : ΩL
e
I Ω
T
þ _
ΩΩ
T
À
Á þ J
À1
Γ
p
I : L
p
I
þΓ
e
M : ΩL
e
M Ω
T
þ _
ΩΩ
T
À
Á þ J
À1
Γ
p
M : L
p
M
"
#
ð7:98Þ
or
Γ
e
I : L
e
I þ J
À1
Γ
p
I : L
p
I
þΓ
e
M : L
e
M þ J
À1
Γ
p
M : L
p
M
!
¼
Ω
T
Γ
e
I Ω
À
Á : L
e
I þ Ω
T
Γ
e
M Ω
À
Á : L
e
M
þJ
À1
Γ
p
I : L
p
I þ J
À1
Γ
p
M : L
p
M
þΓ
e
I : _
ΩΩ
T
þ Γ
e
M : _
ΩΩ
T
2
6
6
6
4
3
7
7
7
5
ð7:99Þ
The first two terms on the right-hand side of Eq. (7.99) indicate that stress
measures corresponding elastic work Γ
e
I , Γ
e
M
À
Á
are objective as shown in
Eqs. (7.100) and (7.101). Since _
ΩΩ
T is a skew symmetric tensor, Eqs. (7.55) and
(7.77), the last two terms on the right-hand side of Eq. (7.99) imply that stress
measures corresponding elastic work Γ
e
I , Γ
e
M
À
Á
are symmetric (Eqs. (7.102) and
(7.103)). The third and fourth terms on the right-hand side of Eq. (7.99) show that
stress measures corresponding plastic work Γ
p
I , Γ
p
M
ð
Þare not objective but invariant
to changes in current frame of reference as shown in Eqs. (7.104) and (7.105):
354
7 Unified Micromechanics of Finite Deformations