Biomedical Engineering Reference
In-Depth Information
its current form as an inverted delta
r
. The name N
ABLA
refers to William
Robertson Smith (1846-1894), a physicist and theologian, who was reminded of
an ancient harp (hebr. nével, gr. m 9bka nábla, lat. nablium) by the reversed delta
shape.
Since in (
3.49
) a vector (x) and a (vector-valued) vector function (v), are
differentiated, a tensor (of second order) (F) is generated and a dyadic product
(x
r
), where the order of both vectors, x and
r
, must not be changed (F is not a
symmetric tensor and is referred to as bi-vector or two-field-tensor). Obeying
(
3.46
), (
3.49
) and (
3.50
), F can be represented with respect to an OBS (following
the Einstein convention where the repetition of an index in a term denotes a
summation with respect to that index over its range, such that nine coordinates of
F are generated):
F
¼
F
ij
e
i
e
j
¼
ox
i
oX
j
e
i
e
j
¼
F
11
e
1
e
1
þ
F
12
e
1
e
2
þ
F
13
e
1
e
3
þ
F
21
e
2
e
1
þ
F
22
e
2
e
2
þ
F
23
e
2
e
3
þ
F
31
e
3
e
1
þ
F
32
e
3
e
2
þ
F
33
e
3
e
3
¼
ox
1
oX
1
e
1
e
1
þ
ox
1
oX
2
e
1
e
2
þ
ox
1
oX
3
e
1
e
3
þ
ox
2
oX
1
e
2
e
1
þ
ox
2
oX
2
e
2
e
2
þ
ox
2
oX
3
e
2
e
3
þ
ox
3
oX
1
e
3
e
1
þ
ox
3
oX
2
e
3
e
2
þ
o
x
3
oX
3
e
3
e
3
ð
3
:
51
Þ
In matrix notation (
3.51
) takes the following form
2
4
3
5
h
e
i
e
j
i¼
ox
1
=
oX
1
2
4
3
5
h
e
i
e
j
i:
F
11
F
12
F
13
ox
1
=
oX
2
ox
1
=
oX
3
½
F
¼
F
21
F
22
F
23
ox
2
=
oX
1
ox
2
=
oX
2
ox
2
=
oX
3
F
31
F
32
F
33
ox
3
=
oX
1
ox
3
=
oX
2
ox
3
=
oX
3
ð
3
:
52
Þ
Three important properties of the deformation gradient. The abstract def-
inition of the deformation gradient (
3.49
) requires clarification. Using the total
differential of motion x (
3.47
), at a fixed point in time t (analogue to scalar-valued
functions), the following term dx
¼
d
½
v
ð
X
;
t
Þ¼ð
ov
=
oX
Þ
dX is obtained, and
together with (
3.49
), it follows
dx
¼
F
dX
ð
x
rÞ
dX
:
ð
3
:
53
Þ
Imaging such that the directional line elements dx and dX represent an edge
length of a volume element in the ICFG and CCFG, respectively (cf. Fig.
3.13
),
the deformation gradient F maps the line element dX from the ICFG into the line
element dx in the CCFG (transformation of line elements).
With the (directional) areas elements dA
0
¼
dA
0
n
0
and dA
¼
dAn in the ICFG
and CCFG, respectively, with the area normal vectors n
0
and n orthogonal to the
areas dA
0
and dA, respectively (cf. Fig.
3.13
), it follows (without proof):
CofF :
¼
JF
T
:
dA
¼ð
CofF
Þ
dA
0
with
J
¼
det F
and
ð
3
:
54
Þ
In (
3.54
), CofF
ð
adjF
Þ
T
is referred to as area configuration tensor or
cofactor tensor
(the
transpose
operation
''adj''
adjugate of
the tensor
F
is