Biomedical Engineering Reference
In-Depth Information
Q
Q
T
¼
Q
T
Q
¼
I
Q
T
¼
Q
1
respectively
and
det Q
¼þ
1
; ð
3
:
154
Þ
and c(t) is an arbitrary (time-dependent) translation which is independent of Q.By
(
3.154
), the constitutive equation of w, referred to the stationary (fixed) observer in
O, is given and,
t
w
ð
X
;
t
Þ¼
f
h
v
ð
Y
;
s
Þi
ð
3
:
155
Þ
s
¼1
X
;
Y
2
K
denotes the constitutive equation of the moving observer in O
*
. A scalar (and thus
also w) is referred to as objective if the following objectivity condition holds
t
t
h
v
ð
Y
;
s
Þi¼
!
h
v
ð
Y
;
s
Þi
and
w
and
f
f
w
¼
!
s
¼1
X
;
Y
2
K
s
¼1
X
;
Y
2
K
ð
3
:
156
Þ
t
t
h
v
ð
Y
;
s
Þi
¼
!
f
f
h
Q
ð
t
Þ
v
ð
Y
;
t
Þþ
c
ð
t
Þi
s
¼1
X
;
Y
2
K
s
¼1
X
;
Y
where the value of the functional f must equal the argument v and its transformed
v
*
. Since both quantities in (
3.156
)
3
, c(t) and Q, may take arbitrary values and are
independent from each other, they may be chosen as
c
ð
t
Þ¼
v
ð
X
;
t
Þ
and
Q
ð
t
Þ¼
I
ð
3
:
157
Þ
such that, according to (
3.157
)
1
, the motion histories of all material points X, Y[
K
are equal and undergo a pure translation c(t) and, according to (
3.157
)
2
, the
rotation Q degenerates to identical mapping. Substitution of (
3.157
)in(
3.156
)
3
leads to the following reduced constitutive equation which satisfies the principles
of determinism and objectivity
t
w
ð
X
;
t
Þ¼
f
h
v
ð
Y
;
s
Þ
v
ð
X
;
s
Þi:
ð
3
:
158
Þ
s
¼1
X
;
Y
2
K
According to (
3.158
), the sef at the material point X at position X at (present)
time t in the ICFG is not determined by the absolute motion v(Y, t) but rather by
the difference in motion histories v
ð
Y
;
s
Þ
v
ð
X
;
s
Þ
and thus in addition, by the
''action'' of all other body points with position Y in the ICFG (cf. Fig.
3.21
).
Note: regarding Fig.
3.22
, using (
3.153
), (
3.154
)
1
can be rewritten as follows:
y
¼
Q
T
ð
t
Þ
y
Q
T
ð
t
Þ
c
ð
t
Þ
Q
T
ð
t
Þ½
y
c
ð
t
Þ:
ð
3
:
159
Þ
According to (
3.159
) (at fixed y
*
) the position y of the material point Y with
respect to the observer in O (visual line of the observer in O) is composed of the
translational motion -Q
T
c (O
*
to O) and the rotated position y
*
with respect to
