x ¼ χ r, t
ð Þ
ð7:1Þ
F ¼ ∇ X χ
ð Þ ¼
∂χ
∂r
ð7:2Þ
J ¼ det F
ð Þ > 0
ð7:3Þ
v x, t
ð Þ ¼ _
x x, t
ð Þ
ð7:4Þ
L ¼ grad v
ð Þ ¼
∂ _
x
∂r
ð7:5Þ
_
F ¼ L ∙ F
ð7:6Þ
Deformation gradient (F) can be decomposed into elastic part (F
e
) and plastic part
(F
p ), and information related to thermal deformation is included in the elastic part of
deformation gradient. Plastic deformation gradient (F
p ) defines a deformation with
respect to initial configuration (Σ o ) to an intermediate, “relaxed” or “natural”
configuration (Σ
0 ) which is followed by an elastic transformation (F
e
) to the final
configuration (Σ). Symmetric part of velocity gradient (L) is defined as stretching
rate tensor (D) and asymmetric part of velocity gradient is defined as spin tensor (W)
F ¼ F
e F
p ; J ¼ J
e J
p
ð7:7Þ
J
e
¼ det F
e
ð Þ > 0
ð7:8Þ
J
p
¼ det F
p
ð Þ > 0
ð7:9Þ
sym L
ð Þ ¼ D; asym L
ð Þ ¼ W; L ¼ D þ W
ð7:10Þ
Using Eq. (7.7), elastic and plastic part of velocity gradient can be defined as
Fig. 7.1 Deformation from original configuration to current configuration
344
7 Unified Micromechanics of Finite Deformations