MECHANICS OF INDENTATION - Nanoindentation with Biological Applications - page 126

ε i = 0.2( a / R ) = 0.2( h / R ) 1/2 (5-6)

At small strains, the yield stress has not been reached and the contact is
perfectly elastic. When the yield stress has been reached, the functional
form of the load-displacement response shifts due to dominant plastic
deformation, as will be discussed further in Section 4.1.

Figure 5-3. Perfectly elastic (Hertzian) contact on glass, in load-displacement ( P - h )
coordinates, showing the fit to Eq. 5-5 .

3.2. Flat punch elastic contact

Although not often used on stiff materials at the nano-scale, flat
punch indentation is occasionally utilized for extremely compliant
materials, including hydrogels and hydrated biological tissues. The load-
displacement response for a cylindrical indenter with radius R is:

E
(1 − ν

P = 2 R

) h

(5-7)

2

In contrast to the Hertzian case, in which the relationship between load
and displacement is not linear, the constant contact area means that in the
flat punch case, the load-displacement response is linear.