Biomedical Engineering Reference
In-Depth Information
sphere and flat surface are taken to introduce an inelastic internal energy
component Γ, the excess surface energy density relative to the contacting
interface.
The system is regarded as a Hertzian contact perturbed by an
additional traction distribution, representing the surface-force effects,
derived from that for rigid flat-punch contact. The problem is formally
modeled as the sequential superimposition on the flat surface of two
pressure distributions describable as purely elastic systems, one non-
linear, one linear. In the first (Hertzian) element of the sequence, the load
is increased to
P
1
, leading to a load-point displacement and contact
radius,
h
1
and
a
, respectively. In the second (flat punch) element of the
sequence, the contact radius is held fixed at
a
and the load perturbed by
Δ
P
, leading to a displacement perturbation of
Δ
h
. The final load and
displacement are
P
and
h
, where
P
=
P
1
+Δ
P
(4-43)
and
h
=
h
1
+Δ
h
(4-44)
The quantities associated with the initial Hertzian loading are all
positive, whereas those associated with the following flat-punch loading
are all negative. (The lack of internal self-consistency comes from the
fact that the local surface displacement fields interior and exterior to the
contact cannot simultaneously meet the assumed loading conditions.)
a
2
R
+Δ
h
h
=
(4-45)
and
4
M
3
R
P
=
a
3
+Δ
P
,
(4-46)
where the
Δ
P
and
Δ
h
terms are related by
Eq. 4-33
by
Δ
P
= 2
aM
Δ
h
, (4-47)
The expression for the elastic internal energy of the system,
U
E
, (energy
associated with deformation of the sphere and surface) is obtained from
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