ds
ð Þ
2 ¼ dr Á C Á dr
or in indicial notation as
ds
ð Þ
2 ¼ dr i C ij dr j
In the same manner, Cauchy deformation tensor B
21 can be written as
dS
ð Þ
2 ¼ dx Á B
À1
Á dx
or in indicial notation as
dS
ð Þ
2 ¼ dx i B
À1
À
Á
ij
dx j
Comparing Lagrangian strain tensor, E, and Green deformation tensor, C, we
observe that
2E ¼ C À I or 2E ij ¼ C ij À δ ij
In the same manner, comparing Eulerian strain tensor E
à and Cauchy deformation
tensor B
21 , we observe that
2E
Ã
= I 2 B
2 1
or 2E
Ã
ij ¼ δ ij À B
À1
À
Á
ij
There is no special reason for defining Cauchy deformation tensor as B
21 . It
could be named just about any character. However, we are following the notation
used by Cauchy in 1827.
In the formulation given above, both Green deformation tensor, C, and Cauchy
deformation tensor, B
21
, reduce to unit tensor when the strain is zero (Malvern
1969).
2.10.2 Relation Between Deformation, Strain,
and Deformation Gradient Tensors
The aquare of the new length (ds)
2 can be written as
ds
ð Þ
2 ¼ dx Á dx
where dx is the vector in the deformed configuration. On the other hand, the relation
between dx and the undeformed vector dr in local (material) coordinates is given by
50
2 Stress and Strain in Continuum