MECHANICS OF INDENTATION - Nanoindentation with Biological Applications - page 142

experimentally. For example, 30,31 for spherical indentation creep
following a ramp to peak load P max in time t R (such that the loading rate
is constant and equal to k = P max / t R ) the solution for displacement-time
can be written in the compact form:

3 P max
8

( )

(5-35) h 3/2 ( t ) = C 0 − C i exp − t / τ i RCF i

R

τ

[ ]

( )

(5-36) RCF =

i

exp t / τ − 1

i R i

R

where RCF i is a dimensionless “ramp correction factor” 30 ; comparison of
Eqs. 5-35 and 5-30 illustrates that this factor fully accounts for a finite
ramping time instead of an assumption of step loading.
Recall that the restriction on this simple mathematical approach 27 is
that the contact area is non-decreasing such that unloading is not a focus.
A different approach to the same problem 28 a few years later yielded a
complementary solution that does not carry the restriction and allows for
full unloading analysis in a linearly viscoelastic material. A further
discussion on these two approaches in the context of indentation
viscoelasticity is available. 32
For displacement controlled testing, under a step displacement h ( t ) =
h 0 H( t ) the load-time relationship is again trivial by the substitutions
paralleling those used in Eqs. 5-29 to 5-31 :

P = 4 Rh 0 G ( t ) (5-37)

8 R

3

h 3/2 G ( t ) P ( t ) = (5-38)

P ( t ) = π

tan ψ

h 2 G ( t ) (5-39)

2

γ

However, the hereditary integral approach is slightly more complicated.
The form of the integral depends on the linearity or nonlinearity of the
load-displacement relationship—integration is conducted over 2 Gh m
where m is the power law exponent for the indentation load-displacement
relationship: