E ij ¼
1
2
∂x k
∂r i
∂x k
∂r j
À δ ij
!
E 11 ¼
1
2
1 þ
∂u
∂r
1 þ
∂u
∂r
þ
∂v
∂r
∂v
∂r
þ
∂w
∂r
∂w
∂r
À 1
!
or
E 11 ¼
∂u
∂r
þ
1
2
∂u
∂r
2
þ
∂v
∂r
2
þ
∂w
∂r
2
"
#
Similarly
E 22 ¼
∂v
∂s
þ
1
2
∂u
∂s
2
þ
∂v
∂s
2
þ
∂w
∂s
2
"
#
E 33 ¼
∂w
∂t
þ
1
2
∂u
∂t
2
þ
∂v
∂t
2
þ
∂w
∂t
2
"
#
off-diagonal (shear strain terms)
E 12 ¼
1
2
1 þ
∂u
∂r
∂u
∂s
þ
∂v
∂r
1 þ
∂v
∂s
þ
∂w
∂r
∂w
∂s
À 0
!
or
E 12 ¼
1
2
∂u
∂s
þ
∂v
∂r
þ
1
2
∂u
∂r
∂u
∂s
þ
∂v
∂r
∂v
∂s
þ
∂w
∂r
∂w
∂s
!
A
D
B
B’
v
w
u
s
t
r
Fig. 2.33 Spatial and local
(material) coordinates. x i ¼
spatial coordinate, r i ¼ local
(material) coordinate.
displacement u i is expressed
in local (material)
coordinates, u i ¼ u i (r, s, t,
time)
2.10 Finite Strain and Deformation
53