L ¼ L
e
þ F
e L
p F
e
À1
ð7:11Þ
where
L
e
¼ _
F
e F
e
À1
ð7:12Þ
L
p
¼ _
F
p F
p
À1
ð7:13Þ
Stretching rate tensor (D) and spin tensor W for elastic and plastic parts can be
derived similarly from Eq. (7.10), where superscripts “e” and “p” represent elastic
and plastic parts of corresponding quantity, respectively
sym L
e
ð Þ ¼ D
e ; asym L
e
ð Þ ¼ W
e ; L
e
¼ D
e
þ W
e
ð7:14Þ
sym L
p
ð Þ ¼ D
p ; asym L
p
ð Þ ¼ W
p ; L
p
¼ D
p
þ W
p
ð7:15Þ
Right and left stretch tensors (U, V) and rotation tensor (R) can be found from
right and left polar decompositions of deformation gradient as follows:
F ¼ RU
ð7:16Þ
F ¼ VR
ð7:17Þ
U and V stretch tensors are positive definite symmetric tensors and R is a proper
orthogonal tensor. Cauchy (C) and Almansi (B) tensors can be formulated as
follows:
C ¼ F
T F ¼ UU
ð7:18Þ
B
À1
¼ F
ÀT F
À1
¼ V
À1 V
À1
ð7:19Þ
In the same fashion, the following relations can be written for elastic and plastic
parts of the deformation gradient.
F
e
¼ R
e U
e
ð7:20Þ
F
e
¼ V
e R
e
ð7:21Þ
C
e
¼ F
e
T F
e
¼ U
e U
e
ð7:22Þ
B
e
À1 ¼ F
e
ÀT F
e
À1 ¼ V
e
À1 V
e
À1
ð7:23Þ
F
p
¼ R
p U
p
ð7:24Þ
F
p
¼ V
p R
p
ð7:25Þ
7.1 Introduction to Finite Deformations
345