INDENTATION AND ADHESION AT SMALL LENGTH SCALES - Nanoindentation with Biological Applications - page 28

modulus, M iso , is related to the material Young's modulus, E , and
Poisson's ratio, ν, by

E
1 − ν

M iso =

2 ,

(2-10)

which may be recognized as the isotropic “plane-strain” modulus. (For
elastically anisotropic materials, calculations involving all the elastic
constants are required.) For a typical (cubic crystal) diamond indenter
used in nanoindentation, M tip 1150 GPa . Taking the surface as

isotropic and the material Young's modulus and Poisson's ratio as E =
100 GPa and ν = 0.25 gives M surface 107 GPa . The indentation modulus

in this case is thus 98 GPa . For a typical nanoindentation contact

radius of a = 100 nm, this gives the contact stiffness as

M

k a 2 × 10 4 N m −1 (this is about as stiff as a diving board at a swimming
pool).
The above result reflects the usual conditions during nanoindentation:
The contact stiffness is much less than the spring stiffness, k a k S , and

hence most of the displacement imposed on the system is taken up by the
contact, with negligible deformation of the spring, i.e ., ws . Similarly,

within the contact, the surface indentation modulus is much less than the
probe indentation modulus, M tip , and most of the contact

displacement appears as surface deformation. The nature of this surface
deformation may be inferred from comparison of the bulk-like contact
stiffness just calculated, k a , and the interatomic bond stiffness, k bond ,
estimated above. If the contact stiffness were purely determined by the
deformation of bonds in the circular area directly beneath the contact (in
the manner of a Winkler foundation), the contact stiffness would be
given by

M surface

k a, Winkler k bond ( d /3 a )( a / d ) 2 . (2-11)
The first term in parentheses accounts for the bonds in series
under the contact; such strings of bonds are taken to extend about three
times the contact radius, a typical extent for an indentation deformation
field. The second term accounts for the number of such strings in parallel
within the contact area. Using the values given above for k bond , a , and, d