Biomedical Engineering Reference
In-Depth Information
may be interpreted as equilibrium stress and r
M
as the dissipative part of the total
stress r and equation (
3.301
)
4
may be interpreted as an evolution equation of r
M
as
long as r
M
is considered an internal variable. At constant stress, the P
OYNTING
-
T
HOMSON
model responds spontaneously elastically and then creeps (asymptoti-
cally) towards a strain value, cf. Fig.
3.24
b. Similarly, at constant strain, the stress
relaxes towards a final stress value, cf. Fig.
3.24
c.
Representation for Second P
IOLA
-K
IRCHHOFF
Stress Tensor. For the previ-
ously introduced class of materials, generally, a potential for the stress in form of a
strain energy function w is postulated which, according to the one-dimensional
rheological model (cf. Fig.
3.24
) and in extension to the hyperelastic case (
3.171
),
is formulated as a function of a strain tensor (C is used instead of G) and further
second order tensors C
k
(k = 1,2,….,n) [cf. e.g. (Simo 1987; Govindjee and Simo
1992; Holzapfel et al. 1996)]
w
¼
w C
;
C
1
;
C
2
; :::;
C
n
ð
Þ:
ð
3
:
302
Þ
The C
k
are intended to represent the dissipative effects of the viscoelastic
material and are referred to as (non-observable) internal variables, hidden vari-
ables or history variables in contrast to external (measurable) variables such as F.
In the case of dissipative media and isothermal processes, the C
LAUSIUS
-D
UHEM
-
inequality (
3.149
), q
0
D
JS
D
w
0, must be satisfied such that, considering
the equality of conjugated stress and strain measures JS
D
¼
P
II
G and 2 G
¼
C and using (
3.302
), it follows
C
X
n
1
2
P
II
o
w C
;
C
1
;
C
2
; :::;
C
n
ð
Þ
o
w C
;
C
1
;
C
2
; :::;
C
n
ð
Þ
C
k
0
:
oC
oC
k
k
¼
1
ð
3
:
303
Þ
From (
3.303
) and the arbitrary choice of the tensor C, the formulation for the
second P
IOLA
-K
IRCHHOFF
stress tensor P
II
and the residual inequality for non-
negative internal dissipation or local entropy production reads
and
X
n
P
II
¼
2
o
w C
;
C
1
;
C
2
; :::;
C
n
ð
Þ
o
w C
;
C
1
;
C
2
; :::;
C
n
ð
Þ
C
k
0
oC
oC
k
k
¼
1
ð
3
:
304
Þ
Following Simo (1987), the decoupled strain energy function with an elastic
part w
0
, cf. (
3.194
), and a dissipative part w
d
is introduced as follows with a
residual internal variable C
1
: C
w C
; ð Þ
:
¼
w
0
C
; ð Þþ
w
d
C
; ð Þ
w
d
C
; ð Þ
:
¼
1
w
0
C
; ð Þ
:
¼
w
0
2
C
C
ð Þþ
w
d
ðÞ
ð
3
:
305
Þ
ðÞþ
f
0
ðÞ
and
with
