Biomedical Engineering Reference
In-Depth Information
Material of Grade N. Constitutive equations of matters of grade N arise from
(
3.163
) by truncating the T
AYLOR
series expansion after the N-th term. The func-
tional f thus takes gradients of F up the N-th grade into account, involving an
estimation of the remainder terms. Based on this, gradient theories or non-classical
continuum models can be developed, which describe micro-mechanical material
effects (Silber 1986; Alizadeh 2001; Trostel 1985, 1988).
Principle of Local Action - Materials of Grade 1 (Simple Matter). The
constitutive equation referring to simple matter (materials of grade 1) and going
back to (Coleman and Noll 1959) arises from (
3.163
) for N = 1 such that the
following form results
t
w
ð
X
;
t
Þ¼
f
h
F
ð
X
;
s
Þi:
ð
3
:
164
Þ
s
¼1
The state of w at the material point X thus is influenced by its immediate
(infinitesimal) neighbourhood only, considered from a spatial point of view
(extreme case of the principle of local action), however, fully considering the
history of the deformation gradient F(X,s).
Since the argument F in (
3.164
) is not an objective tensor, the principle of
material objectivity must once more be applied to w to obtain a material equation
that satisfies all principles of rational mechanics. If, analogue to (
3.152
), the
constitutive equation for w with respect to the fixed observer in O is given by
(
3.164
) and, analogue to (
3.155
), the constitutive equation of the moving observer
is given by
t
w
ð
X
;
t
Þ¼
f
h
F
ð
X
;
s
Þi;
ð
3
:
165
Þ
s
¼1
the following must hold based on the assumption that the objectivity condition
(
3.156
)
1
and (
3.156
)
2
, respectively, formally applies if the argument is F
t
t
w
¼
!
h
F
ð
X
;
s
Þi¼
!
w
h
F
ð
X
;
s
Þi:
or
f
f
ð
3
:
166
Þ
s
¼1
s
¼1
The ''E
UKLIDIAN
transformed argument'' F
*
in (
3.165
) and (
3.166
) is obtained
through substitution of (
3.153
)
1
and (
3.153
)
2
, respectively, in (
3.49
) and by
substituting x = y
*
as well as considering the time-dependence of Q and c:
F
¼
y
r
0
¼½
Q
ð
t
Þ
y
þ
c
ð
t
Þr
0
¼½
Q
ð
t
Þ
y
r
0
þ
c
ð
t
Þr
0
¼
Q
ð
t
Þð
y
r
0
Þ
|{z}
F
¼
Q
ð
t
Þ
F
;
i.e. F
¼
Q
ð
t
Þ
F (which immediately implies that F is not an objective tensor).
Equation (
3.166
) thus is transformed into the following objectivity condition
