INDENTATION AND ADHESION AT SMALL LENGTH SCALES - Nanoindentation with Biological Applications - page 27

respectively. The stiffness of the indentation system, k , after contact is
given by

d F
d s

k = (2-3)

and is related, via elements in series, to the spring and contact stiffness
values by

1
k =

1
k S +

1
k a

(2-4)

The stiffness of the probe spring, k S , is given by

d F
d( s

k S =

w )

(2-5)

−

and may be estimated by considering the spring as a simple column, for
which

E S A S / L S , (2-6)
where E S is the Young's modulus of the column material and A S and L S
are the cross-sectional area and length of the column, respectively.
Taking the column modulus as E S = 200 GPa (steel) and the column to
be a 50 mm long circular rod of radius 2.5 mm gives

k S =

k S 8 × 10 7 N m −1 .

The stiffness of the contact, k a , is given by

d F
d w

k a = (2-7)

and for a circular contact is related to the contact radius by

2 aM (2-8)
where M is the indentation modulus of the tip-surface contact. Similar to
an elements-in-series concept, the indentation modulus takes into
account deformation in both the tip and the surface during contact and is
given by

k a =

1
M =

1
M tip +

1
M surface

(2-9)

where M tip and M surface are the indentation moduli of the tip and surface,
respectively. For elastically isotropic materials, the isotropic indentation