Biomedical Engineering Reference
InDepth 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 onedimensional
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 (nonobservable) 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