Biomedical Engineering Reference
In-Depth Information
!
o
D
oN
N
N
f
¼
f
ð
10
Þ
where N
f
denotes the cycle at failure.
For high cycle fatigue under purely elastic strain it is possible to apply Chab-
oche's model expressed by [
6
]:
b
h
i
k
r
eqM
r
M
Þ
b
þ
1
oD
¼
1
1
D
ð
oN
ð
11
Þ
ð
1
D
Þ
where k is a parameter depending mainly on the loading conditions given by:
*
+
r
eqM
þ
r
eqm
2r
D
2r
u
r
eqM
þ
r
eqm
k
¼
1
g
ð
12
Þ
g, b and M are material parameters which can be obtained from experimental
fatigue tests from S-N curves [
6
].
r
eqM
, r
eqm
, r
D
and r
u
are respectively the equivalent maximum peak stress, the
equivalent minimum peak stress, the fatigue limit and the ultimate stress.
Operator ..
hi
denotes
hi¼
x if x
0 and
hi¼
0ifx\0.
Integration of Eq. (
11
) leads to the Chaboche model expressed by [
6
]:
2
3
!
1
1
a
1
1
þ
b
N
N
f
4
5
D
¼
1
1
ð
13
Þ
Furthermore, to distinguish between the differences in fatigue accumulation in
tension and compression in bone [
36
,
42
], the cycle at failure N
f
can be computed
using the experimental results of Martin et al. [
36
]:
N
f
¼
1
:
479
10
21
De
10
:
3
for compressive loads
ð
14
Þ
N
f
¼
3
:
630
10
32
De
14
:
1
for tensile loads
ð
15
Þ
where De is the amplitude of applied microstrain.
The material properties for bone used for the simulation are given in Table
4
.
A practical and sufficiently accurate way to represent fracture propagation at
continuum level is the so-called ''kill element'' technique. When the damage
parameter reaches its critical value D
c
inside an element, its stiffness matrix is set
to zero, leading to the redistribution of the stress state in the vicinity of the
fractured zone (crack initiation). Once a crack is initiated, the propagation
direction is simulated by the propagation of the broken elements of the mesh. At
continuum level, the local critical damage value in tension is generally equal to 1
(D
c
1
:
0) [
42
,
54
]. To avoid numerical convergence problems, the critical
damage value at fracture was set to D
c
¼
0
:
95. When the kill element method is
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