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
As can be seen from the forms of Eqs. 5-32 to 5-34 is that the integral
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
( Eq. 5-27 ) makes the integration trivial. Solutions for multiple-stage
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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