COMPUTATIONAL MODELING OF INDENTATION - Nanoindentation with Biological Applications - page 157

( )

PEh =Ψ

2

σνα / E n ,,,

. (6-4)

y

is a minor factor in
sharp indentation 1,64 and in spherical indentation 48 ; and thus ν can taken
to be a constant (typically about 0.3). Although a recent study by Chen
et al . 65 showed that the effect of ν is not negligible, by temporarily
neglecting its influence the analysis could become simpler. The
dimensionless function Ψ in Eq. 6-4 , which is the normalized loading
curvature and primarily depends on the material parameter set

Many researchers argue that the Poisson's ratio ν

( )

σ / En ,

y

for a given α , can then be calculated by using forward analysis. When
the material properties are varied in a large range, a smooth functional
form of Ψ may be obtained by fitting of the numerical forward analyses.
A similar dimensionless function for the unloading curve was also
proposed. 1,64 Finally, in order to obtain the constitutive behavior
( )

of the bulk specimen from the indentation load-displacement
curve, an effective reverse analysis algorithm must be developed. For
example, Wang et al . 50,51 have developed numerical methods to solve for
(

σ / En ,

y

)

σ from several dimensionless functions analogous to Eq. 6-4.
A mathematical trick known as the representative strain may be
employed to take advantage of the self-similarity of the loading Ph

/ En ,

y

−

curves and reduce the number of apparent unknown variables in Ψ down
to one. The concept of the representative strain was introduced by Atkins
and Tabor, 66 which was later extended by Dao et al ., 4 and generalized by
Ogasawara et al . 5,7 For a given indenter angle α , a representative strain

σ (which depends on n )
can be identified such that the new dimensionless loading function

and the corresponding representative stress ε

R R

C E

(6-5) =Π

σ σ

R R

depends essentially only on . There are various formulations of the E / σ

R

). Ogasawara et al . 5 proposed that
the representative strain is related with the plastic strain of equi-biaxial
deformation, and on the uniaxial stress-strain curve, the corresponding
representative stress is ( Fig. 6-2 ) :

representative strain-stress pair (

ε

,

σ

R R

n

σ

. (6-6) σ = R 2

R

+ 2 ε

R R