COMPUTATIONAL MODELING OF INDENTATION - Nanoindentation with Biological Applications - page 161

An example of the reverse analysis result is given in Fig. 6-5, which
is taken from Ref. 7. In Fig. 6-5a , several ideal power-law materials are
assumed, whose properties correspond to that of brass, gold, and
aluminum, respectively. The uniaxial stress-strain curves of the input
(true constitutive) properties are given as thick solid lines in Fig. 6-5a.
The knowledge of their ( E , ν σ , n ) is employed in a numerical

indentation simulation using FEM, and the numerical P − curves are
obtained. The simulations are carried out for three indenter angles, with
half apex angle equals to 63.14, 70.3, and 75.79 degrees, respectively.
Finally, the procedure outlined above, Eqs. 6-9 to 6-14 are used to
identify three flow stress-total strain points (symbols) for each material
in Fig. 6-5a , from which the plastic properties can be determined as the
thin solid lines.
In Fig. 6-5b , two copper materials are studied and their uniaxial
stress-strain tests (lab experiments) are shown in thick solid curves.
These materials do not obey the power-law model exactly. Nevertheless,
such stress-strain curve may still be employed in a numerical simulation
using FEM, and the corresponding Ph

,

y

data can lead to the results of
reverse analysis. The results of the identified material from the reverse
analysis (symbols) are reasonably close to the input values, and the
discrepancy mainly arises from the fact that these two materials do not
conform to the power-law assumption exactly.
Other researchers have also proposed to measure the elastic modulus
of the material from the unloading curves with a fixed ,

−

with
approaches similar to Eq. 6-3 . 49,54,55,73,75 When combined, the elastic and
plastic properties of a specimen could be determined from the reverse
analysis of the dual (or plural) indenter method, however, uniqueness of
reverse analysis emerges as the main issue.

ν

4. Uniqueness of Indentation Analysis
It is well known that single sharp indentation cannot yield unique
solution. Based on extensive FEM simulations over a large range of
materials, Alkorta et al . 76 and Tho et al . 77 found that only one shape
factor of the P − curve can be claimed independent from unloading
(such as the unloading work) and another one from loading (such as the