Biomedical Engineering Reference
In-Depth Information
Fig. 3.23 Material
symmetry of a material
In terms of a pre-operation, two reference configurations, i.e. ICFG 1 and ICFG
2, are considered (cf. Fig.
3.23
). On the basis of the modification of rigid motion
(
3.153
) introduced in the context of material objectivity where the (current)
positions of two moving observes are transformed, the following relation
X
ð
X
Þ¼
K
X
þ
c
ð
3
:
172
Þ
transforms the position of a material point from X (ICFG 1) to X (ICFG 2). In
(
3.172
), K is the tensor of the ''change in reference configurations'' (e.g. rotations)
and c represents a translation. According to (
3.47
), the material point X is trans-
formed from position X (ICFG 1) and X (ICFG 2), to position x (CCFG) by the
motion v and X, respectively, such that x
¼
v
ð
X
;
t
Þ¼
v
ð
X
;
t
Þ
. Substituting this
relation in (
3.53
) and using (
3.172
) as well as the chain rule yields
oX
¼
ov
~
X
ðÞ;
t
¼
ov
~
X
ðÞ;
t
o
X
oX
F
¼
ox
¼
F
K
:
ð
3
:
173
Þ
oX
oX
|{z}
K
|{z}
F
Scalar multiplication of the latter relation by K
1
from the right-hand side,
yields the following transformation rule
F
¼
F
K
1
ð
3
:
174
Þ
stating that both deformation gradients F and F emerge from the change-in ref-
erence-configuration tensor K.
