d
dt
ds
ð Þ
2 ¼ 2dx Á D Á dx
Also, we have shown that dx ¼ F Á dr. Hence, we write the following relation:
d
dt
ds
ð Þ
2 ¼ 2 drF
T
À
Á Á D Á F Á dr
ð
Þ
¼ 2dr Á F
T
Á D Á F
À
Á Á dr
Subtracting strain rate equation from rate of deformation,
2dr Á
dE
dt
Á dr À 2dr Á F
T
Á D Á F
À
Á Á dr ¼ 0
dr Á
dE
dt
À F
T
Á D Á F
À
Á
!
Á dr ¼ 0
Of course dr ¼ 0 is the trivial solution. Hence, the following relation must be
satisfied:
dE
dt
¼ F
T
Á D Á F
or in indicial notation,
dE ij
dt
¼
∂x m
∂r i
D mn
∂x n
∂r j
The relationship between Green deformation rate C and strain rate tensor _
E can be
given by
dC
dt
¼ 2
dE
dt
When displacement gradient components are small compared to unity, strain rate
is equal to rate of deformation:
dE
dt
¼ D
or in indicial notation,
2.10 Finite Strain and Deformation
55