Biomedical Engineering Reference
In-Depth Information
the work performed by the applied loading during this sequence (by
integrating the load-displacement path) and is given by
2
5
P
1
h
1
+
P
1
Δ
h
+
1
2
Δ
P
Δ
h
.
U
E
=
(4-48)
5
3
a
2
3
ha
2
UM
=
−
+
ha
.
(4-49)
E
5
R
2
R
The inelastic internal energy of the system,
U
I
, (energy not associated
with elastic deformation processes but with the surface interactions via
the increase in the contact area
π
a
2
) is given by
a
2
. (4-50)
Equilibrium of the system at a given imposed displacement is determined
by
U
I
(
a
)
=−Γπ
∂
(
U
E
+
U
I
)
∂
= 0
(4-51)
a
Solving this equation gives the displacement and load increments
associated with the second element of the contact sequence:
1/2
2
πΓ
a
Δ=−
h
(4-52)
and
(
)
1/2
Δ=−
P
8
π
Γ
Ma
3
,
(4-53)
and thus
1/2
2
a
2
πΓ
a
ha
()
=−
(4-54)
RM
and
3
4
Ma
(
)
1/2
3
Pa
()
=
−
8
π
Γ
Ma
,
(4-55)
3
R
which form a parametric set for
P
(
h
). Clearly, the Hertzian solutions are
recovered for
= 0. In addition, it is clear that for a given load or
displacement, the contact radius is greater than the Hertzian value
Γ
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