MECHANICS OF ADHESION - Nanoindentation with Biological Applications - page 114

for 0 —a consequence of the adhesive interaction. Figure 4-20 shows

the variation of P and h with a for the JKR model. Figure 4-21 shows the
variation in P with h for the JKR model and the Hertzian model; note
that in the limit of large loads and displacements the two models
converge.
As observable in Fig. 4-20 , the system is capable of supporting a
tensile load ( P < 0) at a finite contact radius. The critical radius, a c , at
which the greatest tensile load occurs, is obtained by setting
d P ( a )/d a = 0 to gain

Γ ≠

1/3

9 π Γ R

2

a = . (4-56)

c P

8 M

The conjugate load and displacement are thus

P =− 3 π Γ R / 2 (4-57)

c P

and

2/3

1/2

a

2

1 9
38

π Γ R

h =− =−

c

. (4-58)

c P

M R

These values define the system configuration at “pull-off ” under load
control, the d P ( h )/d h 0 condition in Fig. 4-21 . We are interested in

displacement control here, in which the indenter actuator imposes h . The
contact radius, load, and displacement at pull off under displacement
control, the d P ( h )/d h =−∞condition in Fig. 4-21 , are obtained by
setting d h ( a )/d a = 0 to gain

=

= (1 / 9 ) 1/ 3 a c P a c h ≈ 0.48 a c P , (4-59)

P c h = (5 / 9) P c P ≈ 0.55 P c P , (4-60)

and

9 1/ 3 h c P 2.08 h c P . (4-61)

Just as in the example Mie potential sub-critical example considered in a
previous section, the JKR model has a quiescent condition of zero load
and a snap-on condition intermediate to the quiescent condition and the
pull-off condition. The parameters associated with these conditions may
be obtained by setting P ( a )

h c h = ≈

= 0and h ( a ) = 0 , respectively and are given