Biomedical Engineering Reference
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P =− F app
(4-31)
and
h =− s . (4-32)
The indentation process for the Mie potential in ( P , h ) coordinates is
shown in Fig. 4-17 ; adhesive processes are not excluded in this notation,
they simply occupy the P < 0 and h < 0 quadrant.
Indentation of a flat surface by the flat punch considered in Chapter 2
is thus described by
P = 2 aMh (4-33)
in contact mechanics coordinates, and the contact stiffness, k , is given by
d P
d h =
k
=
2 aM
(4-34)
where a is the punch radius and M is the indentation modulus. Flat punch
indentation is a linearly-elastic system as the stiffness is invariant.
Nonlinear systems, in which the stiffness varies with indentation load or
displacement, include the Mie system considered above in which the
non-linearity derives from the non-linearity of the interaction forces
assumed to exist between the materials of the tip and surface; the
materials are non-linearly elastic. Nonlinear indentation systems can also
occur when the materials deform in a linearly elastic manner but the
geometry of the contact causes stiffness to vary with indentation load and
displacement. Common examples of this are indentation by spheres and
cones. In spherical indentation of a flat surface, often referred to as
Hertzian indentation, the load and displacement are described by
4 MR 1/2
3
h 3/2
P =
(4-35)
where R is the radius of the indenter. The crux of the non-linearity is that
the contact radius varies with indentation displacement as
a 2
=
Rh
(4-36)
such that the contact stiffness is still given by
d P
d h =
k =
2 aM
(4-37)
 
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