Biomedical Engineering Reference
In-Depth Information
mapping of three nonplanar d X 's, d X ( q ), d X ( r ), and d X ( s ), into their three-image
d x 's, d x ( q ), d x ( r ), and d x ( s ), a representation of the volumetric deformation may be
obtained. This algebra uses the triple scalar product (Section A.8, equation (A116))
to calculate the volume associated with the parallelepiped defined by three vectors
coincident with the three edges of the parallelepiped that come together at one
vertex. The element of volume d V in the undeformed configuration is given by
d V
¼
d X ( q )
(d X ( r )
d X ( s )), and in the deformed configuration by d v
¼
d x ( q )
(d x ( r )
d x ( s )). Substituting d x
¼
F
d X into d v
¼
d x ( q )
(d x ( r )
d x ( s )) three
times, it follows that
dv
¼fF
dXð
q
Þg ðfF
dXð
r
Þg fF
dXð
s
ÞgÞ;
(11.35)
which may be expanded and, using the fact that a determinant of a product of
matrices is the product of the determinants, rewritten as
dv
¼
JdV
;
(11.36)
where J
d X ( s )) are both determinants. Thus the
element of volume d V in the undeformed configuration is deformed into a volume
d v in the deformed configuration according to the rule d v
¼
Det F and d V
¼
d X ( q )
(d X ( r )
Det F is
called the Jacobian of the deformation. In order that no region of positive finite
volume be deformed into a region of zero or infinite volume it is required that
0
¼
J d V where J
¼
.
Consider now the question of the deformation of differential elements of area
where similar formulas for area change can be constructed. Let d A be a differential
vector representation of area in the material reference frame obtained by taking the
cross product of two different material filaments d X ,d A
<
J
< 1
d X ( s ). Simi-
larly, let d a be a differential vector representation of area in the spatial reference
frame, representing the deformed shape of the same material area, obtained by
taking the cross product of the deformed images d x ,d a
¼
d X ( r )
d x ( s ) of the
material filaments d X ( r ) and d X ( s ). The relationship between d a and d A is
constructed by twice substituting d x ¼ F d X into d a ¼ d x ( r ) d x ( s ),
¼
d x ( r )
da
¼f
F
dX
ð
r
Þg f
F
dX
ð
s
Þg:
(11.37)
The vector d a is then dotted with the deformation gradient F from the left, thus
da
F
¼f
F
dX
ð
r
Þg f
F
dX
ð
s
Þg
F
:
(11.38)
The right-hand side of this equation may be expanded, as the one for volume was
above, and, using the fact that a determinant of a product of matrices is the product
of the determinants (Section A.8, page 372), rewritten as
da
F
¼
JdA
:
(11.39)
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