Biomedical Engineering Reference
In-Depth Information
Fig. 11.3
An illustration
of Example 11.2.1, a set of
parallel lines deforming into
a set of parallel lines
a
3
2
1
-2
-1
1
2
-1
b
8
6
4
2
-2
-1
1
2
-2
where the constants
a
,
b
, and
c
represent the intersection points of the ellipsoid on
the Cartesian coordinates axes. In material coordinates an ellipsoid has the
representation
X
A
X
¼
1
;
(11.6)
where
A
is a constant second rank tensor. To see that (
11.5
) and (
11.6
) are
equivalent representations of an ellipsoid, let
A
be in its principal coordinate system
and set
A
11
¼
a
2
,
A
22
¼
b
2
,
A
33
¼
c
2
. Substituting the inverse of (
11.2
),
X
¼
L
1
x
, into (
11.6
) yields
x
A*
x
¼
1
;
(11.7)
where a new constant second rank tensor
A*
has been defined by the transformation
A*
L
1
. Eq. (
11.7
) is also the equation of an ellipsoid, an ellipsoid in
the spatial coordinate system, thus permitting us to conclude that an ellipsoid in the
material system is deformed into an ellipsoid in the spatial system by a
(
L
1
)
T
¼
A
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