Biomedical Engineering Reference
In-Depth Information
w
¼
w
ð
k
i
¼
1
Þ¼
0
for
ð
i
¼
1
;
2
;
3
Þ
ð
3
:
392
Þ
(iii) w must be consistent with the linear theory of small strain isotropic elas-
ticity, i.e. classical S
AINT
-V
ENANT
-K
IRCHHOFF
material must result in the
limits.
As further criteria, w has to meet the so-called growth conditions, i.e.
(iv) a material cannot be compressed to have zero volume with the stresses
approaching
1
, nor can it be infinitely stretched with the volume and the
stresses approaching
þ1
; both would require infinite strain-energy, which
is physically impossible (see in
Sect. 3.2.6.1
Eq. (
3.196
))
lim
J
!
o
w
¼1
and
J
!1
w
¼1
lim
ð
3
:
393
Þ
where J is the ratio between the current volume change dV and the refer-
ence configuration dV
¼
JdV
0
and J
¼
det F
¼
k
1
k
2
k
3
the J
ACOBIAN
determinant of the deformation gradient F respectively (see in
Sect. 3.2.3.5
Eq. (
3.55
) and
Sect. 3.2.6.1
Eq. (
3.189
)).
These restrictions together with mathematical proofs regarding existence of
solutions of the related boundary value problems imply that the material parameters
employed in the particular strain-energy function cannot be chosen arbitrarily.
Rather, the parameters must be restricted such that the particular strain-energy
function ensures that the material modeled behaves in a physically acceptable
manner. Due to inaccurate parameter choice non-physical effects may result, as for
instance, a non-strictly increase of the stress-strain function at a displacement-driven
uniaxial tensile test. Such effects are commonly referred to as material instabilities.
Detailed overviews about material stability issues, including requirements of
convexity and polyconvexity, are presented in Baker and Ericksen (1954), Ciarlet
(1988, 1989), Reese (1994), Hartmann and Neff (2003), Marsden and Hughes
(1983), Ogden (1984), Rivlin (1980, 2004), Romanov (2001) or Wriggers (2008).
Restrictions on the material parameters as deduced in the following sections can
be implemented in a parameter optimization process as described in Sect. 3.5.4
and as depicted in the flow chart of Fig.
3.31
.
3.4.9 Drucker Stability
Based on Drucker's stability postulate (Drucker 1964), and implemented in a
special form in (Abaqus 2010), the infinitesimal change in K
IRCHHOFF
stress-tensor
ds following from an infinitesimal change in the corresponding H
ENCKY
or loga-
rithmic strain tensor dG
H
must hold
ds
ð
dG
H
Þ
T
[ 0
:
ð
3
:
394
Þ
