Because indentation causes large-strain plastic deformation, there is little one can do analytically to study the problem. Hence, finite element methods are used here to simulate hardness tests. In previous studies, several investigators [(Bhattacharya and Nix, 1988a)][(Shih et al., 1990)] have used the finite element method to study the indentation of homogeneous materials, and one study [(Bhattacharya and Nix, 1988b)] considered the hardness testing of coatings. These studies have established good agreement between the codes and hardness measurements in homogeneous materials, so this paper makes no attempt to reestablish the ability of the finite element method to accurately represent such tests. This work relies heavily on the results of these previous studies in modeling ion-implanted materials.
The simulations described in this paper were carried out using the finite element code NIKE2D [(Hallquist, 1986)]. This is an implicit, finite deformation, large strain code for plane strain, plane stress, and axisymmetric analyses. The simulations were run on a CRAY-II, with typical execution times on the order of 15 minutes of CPU time.
The model used for most of the analyses consisted of 2466 4-node
quadrilateral elements and 2579 nodes. The analysis was conducted in
axisymmetric mode, thus modeling a conical indenter. The indenters
used in hardness tests are rarely conical, but the use of conical
indenters to reduce simulation models to two-dimensions has been shown
to produce adequate results [(Bhattacharya and Nix, 1988a)]. The included angle of the
conical indenter was 136. (That is, the angle between the
indenter and the surface of the specimen was 22
.) The
boundary conditions are identical to those employed by Bhattacharya
and Nix (1988a,b), i.e. symmetry conditions were imposed on
the centerline of the model and sliding conditions were imposed on the
bottom and side of the indented specimen. It is presumed that the
bottom and side of the specimen are sufficiently removed from the
indent region as to not affect the hardness. This was verified by
conducting additional simulations for larger specimens and comparing
the hardness results. The contact between the indenter and the
specimen surface was modeled using a slideline (adopting the
terminology of the NIKE codes) and the indenter was modeled using 66
elements. (The author was forced to model the indenter because the
code could not be made to model the contact between a string of
coupled nodes and another surface). The contact surface was assumed
to be frictionless. The material model used for the specimen was
linearly elastic-perfectly plastic, as the modeling of strain
hardening has been shown to be relatively unimportant for hardness
simulation [(Bhattacharya and Nix, 1988a)]. The indenter was modeled as an elastic
material with a stiffness (elastic modulus) 40 times that of the
specimen. Higher stiffnesses tended to cause the simulation to fail.
The nominal properties used for the analysis are: GPa,
, and
MPa, where
is the elastic
modulus,
is Poisson's ratio, and
is the yield
strength. These properties were taken from an earlier
work [(Bhattacharya and Nix, 1988b)], in order to allow verification of our results for
indentation of unimplanted materials. The properties are for silicon.
In cases where an implanted surface is simulated, the effect of the
implantation is modeled by varying the yield strength of the specimen
with depth from the surface. For most of the analyses, the peak yield
stress in the implanted layer was chosen to be about 3.6 times that of
the unimplanted material in order to produce hardness effects that
exceed the noise resulting from errors in the simulation. This ratio
is significantly smaller than the ratio of nearly 100 found previously
in implanted aluminum [(Bourcier et al., 1991)], but it should be sufficient for
demonstration of the distributions expected in typical ion-implanted
samples.
In the ion-implanted materials, the yield stress was assumed to peak below the surface of the specimen. The actual distribution was bi-linear, such that the yield stress was equal to that of the unimplanted material everywhere except in a local region of high strength (induced by the implantation). In this region, the strength was assumed to rise linearly to a peak value, and then decrease linearly to the value for the unimplanted material. This distribution implicitly assumes that the property changes are dominated by the formation of hard phases, such as carbides or nitrides, so that the property distribution follows the implanted ion distribution. The peaks in the yield stress were produced using 8 sets of material properties, each with a different yield stress. The elastic modulus was not changed. Because 8 materials were used, implanted layers with peaks below the surface were 15 elements wide, while those with peaks at the surface were 8 elements wide.