Biomedical Engineering Reference
In-Depth Information
Q (visual line of the observer in O
*
) (cf. the addition of vectors in Fig.
3.22
). The
position of the material point Y with respect to the observer in O
*
(at fixed y)
follows from (
3.151
) and yields (cf. the addition of vectors in Fig.
3.22
)
Q
T
ð
t
Þ
y
¼
y
þ
Q
T
ð
t
Þ
c
ð
t
Þ:
ð
3
:
160
Þ
Non-local Materials. Following the principle of determinism, the physical
action at the (material) point X is determined by the action at all other points Y of
the body, whereby the region of influence extends over the entire body. Restricting
this region of influence, following the principle of local action, it seems reasonable
to replace the difference in motion history (
3.158
) with its T
AYLOR
series, which
can then be truncated, depending on the quality of description (modelling). Using
the distance vector DX
¼
Y
X of positions X and Y of two material points X and
Y in the ICFG (cf. Fig.
3.21
), the motion history of Y may formally be written as
v
ð
Y
;
s
Þ¼
v
ð
X
þ
DX
;
s
Þ:
Analogous to the T
AYLOR
series, expansion of a scalar-
valued function of multiple variables, the T
AYLOR
series expansion of v(Y, s) about
the position X of the material point X yields
2
v
ð
X
;
s
Þ
oX
2
v
ð
Y
;
s
Þ¼
v
ð
X
þ
DX
;
s
Þ¼
v
ð
X
;
s
Þþ
1
1!
o
v
ð
X
;
s
Þ
oX
DX
þ
1
2!
o
ð
DXDX
Þþ
...
ð
3
:
161
Þ
Considering the definition of the deformation gradient (
3.49
) and the thus
defined
2
v
ð
X
;
s
Þ=
oX
2
¼
oF
=
oX
¼
x
rr¼
F
ð
X
;
s
Þr
as well as o
3
v
ð
X
;
s
Þ=
oX
3
¼
o
2
F
=
oX
2
¼
x
rrr¼
F
ð
X
;
s
Þrr
etc.,
(
3.161
) can be rewritten as
v
ð
Y
;
s
Þ
v
ð
X
;
s
Þ¼
F
DX
þ
1
higher
gradients
of
F,
namely
o
2
ð
F
rÞð
DXDX
Þþ
.........
þ
......
ð
3
:
162
Þ
Substitution of (
3.162
)in(
3.158
) leads to the following constitutive equation
for w (the infinite expansion occurring in the second form of (
3.163
) may hereby
be thought of as being ''smeared'' into the functional directive)
*
+
X
1
t
1
k!
½
F
ð
X
;
s
Þrr
...
r
w
ð
X
;
t
Þ¼
f
|{z}
k fold scalar
ð
DXDX...DX
Þ
|{z}
k times
|{z}
k
1 times
s
¼1
k
¼
1
*
+
ð
3
:
163
Þ
t
¼
f
F
ð
X
;
s
Þ;
F
ð
X
;
s
Þr;
......
;
F
ð
X
;
s
Þrr
...
r
|{z}
N
1 times
|{z}
Material of Grade N
;
.........
s
¼1
Equation (
3.163
) represents the exact constitutive equation of (
3.158
) since the
T
AYLOR
series expansion has not been truncated and all gradients of F have been
accounted for. The term non-local theories implies that non-local actions (at a
distance) are taken into account. Based on this (
3.163
), the following special cases
may be extracted:
