Biomedical Engineering Reference
In-Depth Information
u
)
dh
(
u
)
du
t
P
=
4
R
2
Gh
→
P
(
t
)
=
4
RG
(
t
−
du
(5-40)
0
8
R
8
R
3
u
)
dh
3/2
(
u
)
du
t
P
=
2
Gh
3/2
→
P
(
t
)
=
G
(
t
−
du
(5-41)
3
0
u
)
dh
2
(
u
)
du
=
π
tan
ψ
γ
=
π
tan
ψ
γ
t
2
Gh
2
P
→
P
(
t
)
G
(
t
−
du
(5-42)
2
2
0
This nonlinearity can result in integral expressions with no closed form
solution, particularly in a spherical indentation configuration.
33
The expressions here considered the case of bulk material indentation
in that the elastic solutions used were applicable to an elastic half-space.
However, it is equally possible to use this correspondence approach in
the context of other elastic solutions, such as a layered system. Such an
approach has been extremely common in flat-punch indentation for a soft
layer on a stiff underlayer, such as articular cartilage on bone
21
or a
polymer thin film on a silicon substrate.
34
6.1.3
. Viscous-elastic-plastic materials
In sharp indentation problems, especially when the materials considered
are glassy polymers, plastic deformation may be present along with
viscoelastic deformation. Two approaches have been used to address
this plastic deformation in the viscoelastic data analysis. The first
mechanism is to remove the plastic deformation component from
consideration using a load-unload-reload protocol,
35
in which the
reloading segment is assumed to be purely viscoelastic. Alternately, the
plastic deformation can be modeled explicitly and thus included as an
additional parameter in the analysis. A viscous-elastic-plastic series
model with independent elements for each deformation mode has been
developed.
36
In its simplest incarnation, the model incorporates three
independent deformation components such that the total displacement
during loading is the sum of the component displacements:
h
P
(5-43)
There are three defining material properties, one associated with each
deformation component (a plane strain elastic modulus
E
h
=
h
V
+
h
E
+
, a resistance to
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