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Box 1.1 (Left and right Cauchy-Green deformation tensor).
Left CG :
Right CG :
d s 2 =
d S 2 =
∂u μ
∂U M
∂u ν
∂U M
∂u μ
= g μν {f λ ( U L ) }
∂U N d U M d U N == G MN {F L
{u λ ) }
∂U N
∂u ν d u μ d u ν =
= c MN ( U L )d U M d U N ,
= C μν ( u λ ) du μ du ν ,
(1.16)
c MN ( U L )=
C μν ( u λ )=
= g μν ( U L ) ∂u μ
∂U M ( U L ) ∂u ν
= G MN ( U λ ) ∂U M
∂u μ ( u λ ) ∂U N
∂U N ( U L ) .
∂u ν ( u λ ) .
2
2
Example 1.3 (Cauchy-Green deformation tensor, f :
E
A 1 ,A 1 ,A 2 S
r ).
The embedding of an ellipsoid-of-revolution M
l = E
A 1 ,A 1 ,A 2 and a sphere M
r = S
r into a three-
dimensional Euclidean space { R
3 , I 3 } with respect to a standard Euclidean metric I 3 (where I 3 is
the 3
×
3 unit matrix) is governed by
E 2 )sin Φ
X ( Λ, Φ )= E 1 A 1 cos Φ cos Λ
+ E 2 A 1 cos Φ sin Λ
+ E 3 A 1 (1
1
1
1
=
E 2 sin 2 Φ
E 2 sin 2 Φ
E 2 sin 2 Φ
cos Φ cos Λ
cos Φ sin Λ
A 1
,
1
=[ E 1 , E 2 , E 3 ]
(1.17)
E 2 sin 2 Φ
E 2 )sin Φ
(1
E 2 := ( A 1 − A 2 ) /A 1 =1 ( A 2 /A 1 ) , ( A 2 /A 1 )=1 − E 2 ,
and by
x ( λ, φ )= e 1 r cos φ cos λ + e 2 r cos φ sin λ + e 3 r sin φ =
r cos φ cos λ
r cos ψ sin λ
r sin φ
,
=[ e 1 ,e 2 ,e 3 ]
(1.18)
2
A 1 ,A 1 ,A 2
respectively. The coordinates ( X,Y,Z )and( x, y, z ) of the placement vectors X ( Λ, Φ )
E
2
and x ( λ, φ )
S
r are expressed in the left and right orthonormal fixed frames
{
E 1 , E 2 , E 3 |O}
and
.
Next, we are going to construct the left tangent space T
{
e 1 ,e 2 ,e 3 |O}
at their origins
O
and
2
M
l as well as the right tangent space
2
T
M
r , respectively. The vector field X ( Λ, Φ ) is locally characterized by the field of tangent vectors
X
∂Λ , ∂Φ }
,the Jacobi map with respect to the “surface normal ellipsoidal longitude Λ ”andthe
“surface normal ellipsoidal latitude Φ ”, namely
{
 
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