9-1 General mapping equations “ellipsoid-of-revolution to plane” 261
9-16 Deformation of the first kind
We first consider the deformation of the first kind: the deformation tensor of the first kind is based
upon the first fundamental form of differential geometry.
I := ds
2 =
2
k,l=1
g kl du
k du
l =
2
K,L=1
c KL dU
K dU
L , c KL :=
2
k,l=1
g kl
∂u
k
∂U K
∂u
l
∂U L .
(9.19)
The first invariant differential form I := ds
2 =
2
K,L=1 c KL dU
K dU
L is to be computed next. In
Box 9.3, the various steps of computing the matrix c KL are outlined.
Box 9.3 (The matrix c KL ).
∂u
1
∂U 1 =
∂λ
∂Λ
= a and
∂u
1
∂U 2 =
∂λ
∂Φ
= 0 ,
∂u
2
∂U 1 =
∂φ
∂Λ
= 0 and
∂u
2
∂U 2 =
∂φ
∂Φ
= f
(Φ)
(9.20)
⇔
∂u
k
∂U K =
"
∂λ/∂Λ ∂λ/∂Φ
∂φ/∂Λ ∂φ/∂Φ
#
=
"
a
0
0 f
(Φ)
#
.
(9.21)
If g 12 = 0, then
c 11 = c ΛΛ =
2
X
k,l=1
g kl
∂u
k
∂Λ
∂u
l
∂Λ
= g 11
„
∂λ
∂Λ
« 2
+ g 22
„
∂φ
∂Λ
« 2
,
c 12 = c ΛΦ =
2
X
k,l=1
g kl
∂u
k
∂Λ
∂u
l
∂Φ
= g 11
„
∂λ
∂Λ
« „
∂λ
∂Φ
«
+ g 22
„
∂φ
∂Λ
« „
∂φ
∂Φ
«
,
c 22 = c ΦΦ =
2
X
k,l=1
g kl
∂u
k
∂Φ
∂u
l
∂Φ
= g 11
„
∂λ
∂Φ
« 2
+ g 22
„
∂φ
∂Φ
« 2
(9.22)
⇒
c 11 = c ΛΛ = a
2 r
2 cos
2 φ , c 12 = c ΛΦ = c 21 = c ΦΛ = 0 , c 22 = c ΦΦ = r
2 f
2 (Φ) .
(9.23)
The matrix elements c KL of the first Cauchy–Green deformation tensor are summarized according to
(9.24). The principal stretches of the first kind amount to (9.25).
c KL =
a
2 r
2 cos
2 φ
0
0
r
2 f
2 (Φ)
,
(9.24)
Λ 1 =
c 11 /G 11 =
a 2 r 2 cos 2 φ
A 2
1 cos 2 Φ
(1 − E 2 sin
2 Φ)
=
ar cos φ
N cos Φ
,
Λ 2 =
c 22 /G 22 =
r 2 f 2 (Φ)
A 2
1 (1 − E 2 ) 2 (1 − E 2 sin
2 Φ) 3 =
rf
(Φ)
M
=
r
M
dφ
dΦ
.
(9.25)