Civil Engineering Reference
In-Depth Information
d
y
µ
λ
d
x
1
Figure 4.12
A unit circle is mapped to an ellipse by the metric tensor M.
in which
ˆ
ˆ
λ
0
e
e
1
1
FUR tretch matrix U
=
,
=
,
rotationm
atrixR=
0
λ
2
2
and
ˆ , ˆ
e
12
are unit eigenvectors of M.
dr
2
= d
x
⋅M⋅d
x
= d
x
⋅F
T
⋅F⋅d
x
= (F⋅d
x
)⋅(F⋅d
x
) = d
y
⋅d
y
with d
y
= F⋅d
x
As shown in Figure 4.12, a unit circle is mapped onto an ellipse of stretches λ and μ along
the principal directions along with a rotation, and d
y
represents the transformed vector of
d
x
by metric M. Along the principal directions, there is no change in the direction for vector
d
x
but just a change in length by a scalar factor λ
i
; however, in any other directions, there
are rotation as well as scaling for vector d
x
. The transformation of vector d
x
into d
y
= F⋅d
x
=
U⋅(R⋅d
x
) can be interpreted geometrically as a rotation R to the direction of d
x
followed by
a stretching to the length of d
y
by U. Alternatively, d
x
can also be transformed to d
y
by first
a stretch followed by a rotation.
4.2.7.1 Change in length by metric M
Define fibre extension α by the ratio of the deformed length to the original length, i.e.
1
2
1
2
dr
d
dMd
dd
x
⋅
x
xx
⋅
d
d
x
x
α=
=
=⋅
[
uu
M
⋅
]
here
u
=
x
We can see that the change in length depends on M and the direction of the fibre
u
.
4.2.7.2 Change in area by metric M
Let M be the metric tensor at point P. The area of the triangle spanned by vectors d
x
1
and
d
x
2
measured with respect to metric M is given by
area A
M
= (F⋅d
x
1
) × (F⋅d
x
2
)⋅
n
= det(F)(d
x
1
× d
x
2
⋅
n
) = (λμ)A
E
det(F) = det(UR) = det(U)det(R) = λμ, A
E
= d
x
1
⋅d
x
2
⋅
n
= area in Euclidean metric
