Biomedical Engineering Reference
In-Depth Information
Fig. 3.13 On the
transformation of line-, area-
and volume elements
sometimes preferred) which provides a linear transformation between the directed
area elements dA
0
and dA in the ICFG and CCFG (transformation of area ele-
ments). The term J
¼
det F is referred to as J
ACOBI
determinant which is obtained
by forming the determinant of matrix (
3.52
).
Furthermore, regarding the volume elements dV
0
and dV in the ICFG and
CCFG (cf. Fig.
3.13
) transformation (of volume elements), yields (without proof)
dV
¼
JdV
0
with
J
¼
det F
:
ð
3
:
55
Þ
Remarks: Since dV and dV
0
are always positive, it follows from (
3.55
) that the
determinant of F is always positive, namely J
¼
det F [ 0
:
Motions where J
¼
1
lead according to (
3.55
)todV
¼
dV
0
and are referred to as volume preservative or
isochore.
Displacement Gradient. With respect to linearization of strain tensors (see
Sect. 3.2.3.6
), it is necessary to introduce the material displacement gradient H
given as follows
H
ð
X
;
t
Þ
:
¼
u
ð
X
;
t
Þr:
ð
3
:
56
Þ
The relationship between the deformation gradient and the displacement gra-
dient is established by substituting (
3.48
)in(
3.49
) which leads to F
¼
x
r¼
ð
u
þ
X
Þr¼
u
rþ
X
r:
With respect to Cartesian coordinates (
3.46
) and (
3.50
),
the second term of the right side reads X
r¼ð
X
i
e
i
Þð
o
=
oX
j
e
j
Þ¼
oX
i
=
oX
j
e
i
e
j
¼
d
ij
e
i
e
j
¼
e
i
e
i
¼
I
;
i.e. the identity tensor I (the same can also be deduced vecto-
rially using the operation X
r¼
X
ð
o
=
oX
Þ¼
oX
=
oX
¼
I where the partial deri-
vation of X with respect to X yields the ''tensorial one''). Together with (
3.56
)it
follows (compare with the one-dimensional form (
3.45
))
F
¼
I
þ
H
or
x
r¼
I
þ
u
r:
ð
3
:
57
Þ
Analogue to (
3.51
), the representation of H with respect to a OBS reads
H
¼
H
ij
e
i
e
j
¼
o
u
i
ox
j
e
i
e
j
:
ð
3
:
58
Þ
