dx ¼ F Á dr
Hence, we can write
ds
ð Þ
2 ¼ dr Á F
T
À
Á
F Á dr
ð
Þ¼dr Á F
T
Á F
Â
à Á dr
Similarly, the square of the original length (dS)
2 can be written as
dS
ð Þ
2 ¼ dr Á dr
Because dr = F
21
Á dx, we can write
dS
ð Þ
2 ¼ dx Á F
À1
À
Á T
h
i
Á F
À1
Á dx
Â
à ¼ dx Á F
À1
À
Á T F
À1
À
Á
h
i
dx
Let’s compare these last equations with our strain definition equations where
Green deformation tensor, C is given by
ds
ð Þ
2 ¼ dr Á C Á dr
Therefore, Green deformation tensor is also defined by
C ¼ F
T
Á F
and Cauchy deformation tensor was defined by
ds
ð Þ
2 ¼ dx B
2 1 dx
Therefore, we can write
B
À1
¼ F
À1
À
Á T Á F
À1
À
Á
h
i
in indicial notation, we can write the following relation:
C ij ¼
∂x k
∂r i
∂x k
r j
and
B
À1
ij ¼
∂r k
∂x i
∂r k
∂x j
Summation on repeated indices applies in both of these equations.
2.10 Finite Strain and Deformation
51