L ¼ D þ W ¼ Ω D þ W
ð
ÞΩ
T
þ _
ΩΩ
T
ð7:53Þ
which implies that
D = ΩDΩ
T
ð7:54Þ
W = ΩWΩ
T
þ _
ΩΩ
T
ð7:55Þ
Therefore, stretch rate tensor (D) is objective, while spin tensor (W) is not
objective. Using multiplicative decomposition definition of deformation gradient
given in Eq. (7.7), we can write
F ¼ F
e F
p
¼ ΩF
e F
p
ð7:56Þ
Hence we can write the following relations:
F
e
¼ ΩF
e
ð7:57Þ
F
p
¼ F
p
ð7:58Þ
Therefore, elastic and plastic deformation gradients are not objective. Since
plastic deformation gradient refers to original and intermediate frame of reference,
plastic deformation gradient is not objective but invariant to changes in current frame
of reference. Using definitions for elastic and plastic rotation tensors, left stretch
tensor, right stretch tensor, Cauchy tensor, and Almansi tensor in Eqs. (7.20)–(7.27),
the following relations can be written as:
F
e
¼ R
e U
e
¼ ΩR
e U
e
ð7:59Þ
R
e
¼ ΩR
e
ð7:60Þ
U
e
¼ U
e
ð7:61Þ
F
e
¼ V
e R
e
¼ ΩV
e
Ω
T
ΩR
e
ð7:62Þ
V
e
¼ ΩV
e
Ω
T
ð7:63Þ
C
e
¼ U
e U
e
¼ U
e U
e
¼ C
e
ð7:64Þ
B
e
¼ V
e V
e
¼ ΩV
e
Ω
T
ΩV
e
Ω
T
¼ ΩB
e
Ω
T
ð7:65Þ
F
p
¼ R
p U
p
¼ ΩR
p U
p
ð7:66Þ
R
p
¼ ΩR
p
ð7:67Þ
U
p
¼ U
p
ð7:68Þ
7.2 Frame of Reference Indifference
349