Biomedical Engineering Reference
In-Depth Information
Fig. 3.22 The principle of
observer indifference
material properties at each material point (material inhomogeneity). Aging effects
are observed, for example, in the form of creeping at decreasing velocity in con-
junction with concrete. The principle of determinism thus accommodates only the
remote past (including the present time) of the particular process, which is referred to
as a memory of a material. Future effects, however, are not accounted for (which can
not be strictly excluded from a natural philosophical perspective!) (Fig.
3.21
).
Restricted to homogenous and non-aging materials, the explicit dependence of
X and t no longer applies, such that (
3.151
) takes the form
t
w
ð
X
;
t
Þ¼
f
h
v
ð
Y
;
s
Þi:
ð
3
:
152
Þ
s
¼1
X
;
Y
2
K
Principle of material objectivity (observer or frame indifference, observer
change). A material equation must not depend on the choice of the reference frame
or observer. A sef, thus, must provide the same value of w for two observers in
relative motion if both observers trace the motion history of one material point X
relative to different points of reference O and O
*
(cf. Fig.
3.22
). When v(X, t) and
v*(X, t), respectively, are the configurations (motions) of the same body
K
; from
the point of view of two different observers O and O
*
, respectively, and
y* = v*(Y,t) and y = v(Y, t), respectively, are the corresponding motion histories.
The latter are related using the E
UKLIDIAN
transformation (rigid body motion) (in
(
3.153
)
2
the definition (
3.47
) was used)
y
¼
Q
ðÞ
y
þ
c
ðÞ
v
Y
; ðÞ¼
Q
ðÞ
v Y
; ðÞþ
c
ðÞ
with
Q
ð
t
¼
0
Þ¼
I
c
ð
t
¼
0
Þ¼
0
:
and
ð
3
:
153
Þ
In (
3.153
), Q(t) is a (proper) orthogonal time-dependent rotation tensor defined
by
