J. Malzbender et al.: The P–h2 relationship in indentation
Also, the model is partly based on the unloading-
Eq. (12) with the appropriate values substituted for all
quantities. Clearly, there is good agreement between the
computational results and the model, even though the
computations showed a slight pile-up, and this effect is
not included in our model (see Fig. 1).
stiffness model given by Oliver and Pharr (1992);5 thus
all the assumptions made there, including for example
frictionless contact, are also present implicitly in
our model.
Note that Eq. (12) is valid only for hc > d. In practice,
a good indenter has a tip radius of about 100 nm, which
corresponds to a d of less than 6 nm when C in Eq. (11)
equals 24.5. Almost always, indentations will be deeper
in practice, so the condition hc > d is not restrictive in
practical situations.
As mentioned before, the model can be used to esti-
mate either E or H of the indented material if one or the
other is known. If we assume that E is known and equal
to 150 GPa, then a fit to the computed data gives 8.2
GPa. This is close to the value of 8.0 GPa as found
directly from the computation.
For comparison, Fig. 5 shows the FEM results together
with the model proposed by Hainsworth et al., Eq. (2).
Clearly the comparison is not as good as that for the
present model. An estimate of H with E fixed to 150 GPa
gave H ס
9.5 GPa, which is a less accurate estimate of
the true value of 8.0 GPa than the value found with the
present model. We have to bear in mind, however, that
the empirical factors ⌽ and ⌿ found by Hainsworth et al.
are valid only for the specific indenter they used, as
mentioned already before.
III. FINITE ELEMENT CALCULATION
To verify the model, we carried out finite element
calculations. Here the indenter was modeled as a cone
with a rounded tip. The half-included angle of the cone
was 70.3°, whereas the tip radius was 100 nm. The pa-
rameters in Eq. (11) corresponding to this shape are C ס
24.51 and ס
6.22 nm. The indenter was assumed to be
rigid, and the contact was assumed to be frictionless. The
indented material was an elastic–perfectly plastic mate-
rial with Young’s modulus E ס
150 GPa, Poisson’s ratio
ס
0.3, and yield stress Y ס
2.5 GPa. The calculations
were carried out with the FEM package MARC. A plot of
the mesh used is shown in Fig. 3. The size of the full
mesh was 17.5 × 17.5 m. We used 8761 axi-
symmetric quadrilateral linear elements. the size of the
elements in the contact area was 1.1 nm. The contact was
modeled with the direct constraint method as described
in Ref. 6.
A calculation was made up to an indentation depth of
187 nm. From the curves obtained, we computed the
hardness as H ס
8.0 GPa. This value was found to be
constant within 0.2 GPa variation over the whole inden-
tation depth. Figure 4 shows the computed indentation
load–displacement curve, along with the relation given in
FIG. 4. Indentation load–displacement behavior as computed from
FEM and that predicted by the model given by Eq. (12) with the
appropriate values substituted for all quantities.
FIG. 5. Indentation load–displacement behavior as computed from
FEM and that predicted by the model proposed by Hainsworth et al.
given by Eq. (2). The appropriate values were substituted for all quan-
tities.
FIG. 3. (a) Full view and (b) detail of the finite element mesh used in
the computation. The symmetry axis is on the left side; part of the
indenter is visible at the top of the figures.
J. Mater. Res., Vol. 15, No. 5, May 2000
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