L
e
I ¼ ÀF
e
I D
p
I F
e
À1
I
ð7:119Þ
L
e
M ¼ ÀF
e
M D
p
M F
e
À1
M
ð7:120Þ
Accordingly, principal of virtual work can be rewritten for this special case using
Eqs. (7.106) and (7.197) as
_
W int ¼
Z
V
Γ
e
I : ÀF
e
I D
p
I F
e
À1
I
þ J
À1
Γ
p
I : D
p
I
þΓ
e
M : ÀF
e
M D
p
M F
e
À1
M
þ J
À1
Γ
p
M : D
p
M
2
6
4
3
7
5dV
ð7:121Þ
or
_
W int ¼
Z
V
J
À1
Γ
p
I À F
e
T
I Γ
e
I F
e
ÀT
I
: D
p
I þ J
À1
Γ
p
M À F
e
T
M Γ
e
M F
e
ÀT
M
: D
p
M
h
i
dV
ð7:122Þ
Since velocity gradient is assumed to be zero for this special case, velocity field
will be also equal to zero. Therefore, external work will be equal to zero _
W ext ¼ 0
À
Á
.
As a result, individual terms inside parenthesis in Eq. (7.122) should be equal to zero
for arbitrary selection of V, D
p
I , and D
p
M .
J
À1
Γ
p
I À F
e
T
I Γ
e
I F
e
ÀT
I
¼ 0
ð7:123Þ
J
À1
Γ
p
M À F
e
T
M Γ
e
M F
e
ÀT
M ¼ 0
ð7:124Þ
Using definitions of stress measures in Eqs. (7.116) and (7.117), it can be shown
that
Γ
p
I ¼ JF
e
T
I T I F
e
ÀT
I
ð7:125Þ
Γ
p
M ¼ JF
e
T
M T M F
e
ÀT
M
ð7:126Þ
which form definition of elastic symmetric Mandel stress in intermolecular structure
and molecular network structure as
M
e
I ¼ JF
e
T
I T I F
e
ÀT
I
ð7:127Þ
M
e
M ¼ JF
e
T
M T M F
e
ÀT
M
ð7:128Þ
whereas since trace of plastic stretch rate is equal to zero due to incompressible
plastic flow, assumption stress conjugate to plastic stretch rate should be a deviatoric
tensor. Therefore, we can write
7.3 Unified Mechanics Theory Formulation for Finite Strain
357