MECHANICS OF ADHESION - Nanoindentation with Biological Applications - page 106

3. Adhesion and Indentation Coordinate Systems
In the previous two sections, attention has been focused on the adhesive
and separation interactions between two bodies, a surface and a probe,
with the recognition that the indentation interaction occurs between
these two interactions in a contact sequence. In consideration of
the adhesive and separation interactions it was convenient to use a
coordinate system with force and displacement taken as positive when
directed outward from the surface. In indentation processes this is not so
convenient. In this section we consider the relationship between the
coordinate system used so far in consideration of adhesion and separation
and the coordinate system more conventionally used in consideration of
indentation.
The force interaction between the surface and the probe, say
described by the Mie potential, F M , can be balanced in (neutral)
equilibrium by an applied force, F app, say exerted by an indenter:

F M + F app = 0 (4-29)

or, more physically,

F app =− F M . (4-30)

A diagram of these forces and this force balance is shown in Fig. 4-
16 . Provided the force applied by the indenter matches that exerted by
the surface on the probe, the system can be maintained in stable
equilibrium for any tip location, z . In particular, for z < z 0 , F app < 0 and
this describes an indentation process. Obviously, when attention is
focused on indentation processes, especially if surface interactions are
weak or negligible, and the probe spring is very stiff such that the
equilibrium tip position differs very little from the imposed position, s ,
this double negative coordinate system is not very convenient. Hence,
contact mechanics systems are usually described by a coordinate
system in which force and position are taken as positive when directed
inwards from the surface, into the indented material. 3 The reference
configuration, in which both the applied load and conjugate load-point
displacement are zero, is taken as the quiescent point. In terms of the
variables above, the load, P , and displacement, h , in this coordinate
system are given by