Biomedical Engineering Reference
In-Depth Information
1
4
R
P
2
G
1
4
R
P
0
J
(
t
)
→
h
=
h
(
t
)
=
(5-29)
3
P
2
G
3
h
3/2
→
h
3/2
(
t
) =
=
P
0
J
(
t
)
(5-30)
8
R
8
R
2
2
γ
P
2
G
γ
→
h
2
=
h
2
(
t
)
=
P
0
J
(
t
)
(5-31)
π
tan
ψ
π
tan
ψ
For “real” loading conditions, there is a finite ramping time or
ramping rate up to the peak load. In this case, a hereditary integral
formulation is used, where integration takes place over the creep function
and the applied loading history
P
(
t
). For the general indentation,
indentation and load-control, this amounts to replacing
P
/2
G
with an
integral:
1
4
R
P
2
G
→
h
(
t
) =
1
4
R
u
)
dP
(
u
)
du
t
(5-32)
h
=
J
(
t
−
du
0
2/3
3
P
2
G
3
u
)
dP
(
u
)
du
→
h
(
t
) =
t
(5-33)
h
3/2
=
J
(
t
−
du
8
R
8
R
0
1/2
γ
2
P
2
G
γ
2
u
)
dP
(
u
)
du
→
h
(
t
) =
t
h
2
=
J
(
t
−
du
(5-34)
π
tan
ψ
π
tan
ψ
0
J
(
t
−
u
)
dP
(
u
)
du
t
term
is identical, and as such a single set of solutions
du
0
for commonly used experiments, such as ramping at constant loading
rate, or a creep test following a ramp, can be used in each geometry. For
a ramp test where
P
(
t
) =
kt
the integral is simply over the creep function
k
t
and the exponential representation of the creep function
indentation tests, such as a stair-step creep test at several (increasing)
peak loads, involves a substantial book-keeping effort but relatively
simple calculus
30
: solutions maintain the exponential functional form
of the creep function and are thus relatively easy to implement
J
(
t
−
u
)
du
0
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