MECHANICS OF ADHESION - Nanoindentation with Biological Applications - page 111

supercritical system of a surface and flat-punch probe that are rigid.
Under this condition, separation of the tip from the surface leads to a
load-displacement relation that is a reflection only of the adhesive
interactions as considered in the last section. The work of separation per
punch area, Γ , for the flat punch is then given by

2

a
d

1

~ Φ 0 Γ=Φ 0

d 2 ,

(4-42)

a 2 π

using the notation and scaling arguments of Chapter 2 and the previous
sections (the first term in parentheses is the number of bonds in the
contact area and the second term is the punch area). It is clear that Γ
reflects only the properties of the tip and surface materials, or at least
their interface. Consideration of the area exterior to the contact of the flat
punch prior to separation suggests that can be considered as the excess

energy of the system per area associated with uncontacted surface
relative to the contacting interface. For this reason Γ is usually referred to
as the “surface energy.” Γ figures prominently in the next section, in
which adhesive surface interactions and indentation deformation
interactions are combined in a simple adhesive indentation model.

Γ

4. The JKR Model of Adhesive Indentation Contact

In this section we consider a simple model of adhesion due to Johnson,
Kendall, and Roberts (JKR) 4 that, although not fully self-consistent,
encapsulates all the essential elements of adhesive indentation, and
introduces surface energy explicitly. As in previous sections, we shall see
that the stiffness of the system is sub-critical, although here the
compliant element is the elastic deformation of tip and surface; the probe
spring is taken as rigid. The tip is taken as spherical with radius R and the
tip displacement, h , is imposed at the sphere center (it is convenient to
consider the load, P , measured here, too). The contact radius, a , between
the tip and surface is taken to be very small such that R >> a (the usual
Hertzian contact assumption). At an even smaller length scale, the tip
and flat surface interact via infinitesimally short-range attractive surface
forces. The sphere and surface are taken to be elastic (with different, but
isotropic elastic properties) and the free (non-contacting) surfaces of the