MECHANICS OF ADHESION - Nanoindentation with Biological Applications - page 76

linear-elastic (the probe from Chapter 2) . Many of the required concepts
and definitions pervasive in analyses of adhesion and fracture will be
illustrated in consideration of this simple system: equilibrium—and the
interplay of forces associated with surface interactions and solid
deformations in determining characteristic system scales; and stability—
and the central role played by stiffness fields in determining observed
behavior.
The interaction between the bodies is described by a logarithmic
potential, Φ A ( z ):

Φ A = A ln( z / z A ) , (4-1)
where z is the separation of the bodies, and A and z A are amplitude and
length-scale parameters of the potential, respectively. Such a potential
describes the interaction between a point charge and an infinite line of
charge. 1 For convenience, then, it is useful to view the first body—the
“surface”—as flat with a long, narrow, charged strip on the surface and
the second body—the “probe”—as containing a small charged area. (The
two charges are of course of opposite sign, such that A is positive, and
the small area and the strip are assumed to be separated by z measured
normal to the surface.) The surface is taken to be rigid, given by z = 0,
thus defining one boundary of the system. (As no deformation of the
surface is assumed here, it is convenient to set the boundary of the
system to focus exclusively on the deformation of the probe; we will see
that in other systems, other choices for z = 0 are appropriate.) The probe
is taken to be linear elastic with a harmonic deformation potential,
Φ S ( z ) , given by

k

2

Φ=

S

( z − s ) . (4-2)

S

2

It is convenient to view the probe (as in Chapter 2 ) as consisting of
two parts: (i) a “tip,” in this case containing a small charged area,
attached to (ii) a “spring,” the other end of which is (position-) controlled
by the indentation system. k S is then the stiffness of the spring and s is
the position of the tip imposed by the indentation system in the absence
of other influences (such as Φ A above) and serves to define the other

boundary of the system. A schematic diagram of the system and