MECHANICS OF ADHESION - Nanoindentation with Biological Applications - page 109

(as for all axisymmetric indentation processes), but in this case varies
with indentation displacement as

d P
d h = 2 MR 1/2 h 1/2 .

k = (4-38)

In indentation of a flat surface by an axisymmetric cone of included
angle 2 ψ , the load-displacement relation is

P = π M tan ψ
2

h 2 (4-39)

and the relation between contact radius and indentation displacement is

2tan ψ

a = h (4-40)

π

such that the contact stiffness varies with displacement as

ψ h . (4-41)
The load-displacement behavior for flat punch, spherical, and conical
indentation are shown in Fig. 4-18 . Schematic cross-sectional diagrams
of these elastic indentation processes are shown in Fig. 4-19.
In real life, of course, adhesive interactions described by some sort of
interaction potential (of the overall shape of the example Mie potential)
occur between the indentation tip and the surface, both interior and
exterior to the contact radius, which could then be defined by the locus of
points at which the separation between the surfaces is the quiescent
separation, z = z 0 . Frequently, z 0 << a and it is a good approximation that
adhesive interactions only occur within the contact. This is certainly true
of flat-punch loading. If the adhesive interactions are strong enough such
that the surface has sub-critical stiffness (small elastic modulus) and the
probe is very stiff, the contact can support a negative load. The load-
displacement behavior of an adhesive flat punch under negative ( P < 0)
load is shown in Fig. 4-18 ; a schematic diagram of such a process is
shown in Fig. 4-19 . The surface is displaced outwards ( h < 0).
In the next section, a simple model will be described that makes use
of adhesive flat punch loading to describe an adhesive indentation
contact. Before doing so, however, it is useful to consider the opposite

k = π M tan