Biomedical Engineering Reference
In-Depth Information
d
s
¼
W
d
G
H
A
j
:
¼
b
j
J
a
j
b
j
with
and
2
4
3
5
;
2
4
3
5
;
2
4
3
5
k
a
j
1
þ
A
j
ds
1
ds
2
ds
3
dG
1
dG
2
dG
3
A
j
A
j
W
:
¼
2
X
N
k
a
j
2
þ
A
j
d
G
H
d
s
:
¼
:
¼
l
j
A
j
A
j
k
a
3
þ
A
j
j
¼
1
A
j
A
j
ð
3
:
400
Þ
where
W
is the stiffness matrix.
According to D
RUCKER
'
S
stability criterion, material stability requires the matrix
W
to be positive definite, which leads to certain restrictions on the values of the
material constants. One possible check for positive definiteness of the symmetric
n 9 n matrix
W
is using the determinant criteria
det
ð
W
k
;
l
j
k
;
l
p
Þ
[ 0
8
p
2
1
;
...
;
n
or
det
ð
W
k
;
l
j
k
;
l
p
Þ
[ 0
8
p
2
1
;
...
;
n
ð
3
:
401
Þ
which test all leading principal minors to be positive. This leads to the following
conditions to be satisfied
ð
i
Þ
W
11
[ 0
ð
ii
Þ
W
11
W
22
W
12
W
21
[ 0
ð
iii
Þ
det
W
[ 0
:
ð
3
:
402
Þ
Alternatively, as noted in A
BAQUS
, positive definiteness of the symmetric
3 9 3 matrix
W
exists if the tensor invariants, i.e. the coefficients of the charac-
teristic polynomial of
W
, P
ð
k
Þ¼
III
W
II
W
k
þ
I
W
k
2
k
3
, are positive
ð
i
Þ
I
W
¼
tr
W
[ 0
ð
ii
Þ
II
W
¼
1
2
[ 0
tr
2
W
tr
W
2
ð
3
:
403
Þ
ð
iii
Þ
III
W
¼
det
W
[ 0
Using the example of the uniaxial deformation mode with the 1-direction being
the loading direction in conjunction with (
3.395
)
2
, the lateral deformation k
3
¼
k
2
is implicitly given through
ð
s
2
¼
s
3
¼
0
J
¼
k
1
k
2
Þ
and
h
i
¼
0
ð
3
:
404
Þ
g k
1
;
k
ð Þ
X
¼
X
N
N
k
a
j
2
k
1
k
2
a
j
b
j
l
j
a
j
l
j
a
j
k
a
j
2
J
a
j
b
j
j
¼
1
j
¼
1
In A
BAQUS
, for given sets of material parameters, the stability check is per-
formed. The above conditions (i) to (iii) are tested by incrementing k
1
at intervals
of Dk
1
¼
0
:
01 in the range 0
:
1
k
1
1
:
0 (compression) and 1
:
01
k
1
10
:
0
(tension). For given k
1
and the order of series expansion N [ 1, (
3.404
) cannot be
