COMPUTATIONAL MODELING OF INDENTATION - Nanoindentation with Biological Applications - page 156

HPAc = / = σ

, (6-2)

by

and

S = 2 γβ E A / π

. (6-3)

( )

is the plane-strain modulus and the projected
contact area is A . The constant c in Eq. 6-2 is a constraint factor that
depends on indenter angle and material pro pe rties: c increases with
/

Here, EE ≡ 1 − ν

2

. 59,60 γ is a
correction factor and it depends on the indenter angle (for a Berkovich
indenter

and approaches a constant ( ) when E σ ≈ 3 E tan ασ > / 20

y y

β = . 62 From
dimensional analysis, 1 both hardness and stiffness are independent of the
indentation depth if the strain gradient effect 47 is ignored for the
continuum bulk.
The classic hardness-based indentation technique outlined above has
several disadvantages. First, as the indenter penetrates the specimen, the
material either produces plastic pile-up at the crater rim (when the yield
strain,

), 61 and for axisymmetric indenters γ ≈ 1.08 1

, is small), or exhibits the elastic sink-in effect (when σ / E σ / E

y y

is large) 60 - both strongly affect the projected contact area A and are
difficult to measure in practice (especially at the maximum penetration
when it needs to be measured), which makes it one of the primary error
sources. Moreover, this approach cannot account for hardening and
residual stress.
Since only the indentation load P and displacement h may be
accurately measured during experiments, the key issue of a successful
indentation analysis (theory) is to uniquely determine the material
elastoplastic properties (

E ,, ,

y

νσ n

) from the shape factors of the Ph

−

curves. The self-similarity of Ph curves of sharp indentation on bulk

materials makes the conical indenters more widely used in indentation
analysis: the resulting loading Ph

−

curve is quadratic and the unloading −

curve can be fitted into a power-law relationship. 63 Note that an
analytical solution for indentation on an elastoplastic work-hardening
solid (which involves finite plastic deformation) does not exist, and most
studies in this area are based on numerical analyses. From dimensional
analysis, Cheng and Cheng proposed the following relationship to
correlate the indentation load, displacement, elastoplastic property, and
indenter angle during loading 1,64 :

Ph −