One of the most significant errors introduced by the finite element simulation technique lies in the inability of the method to determine the contact radius to better than the width of the elements. Because contact between the indenter and the specimen is given only at the nodes, it cannot be determined more closely than the element width. Therefore, error is introduced and the error increases with decreasing contact area, as shown below.
If hardness is defined as:
where is the hardness,
is the applied load, and
is the
radius of the projected contact area (a circle for conical indenters),
then an error (
) in the determination of the radius gives:
If the error in the radius is fixed by the element width in the region
neighboring the contact area, then the error in the calculated
hardness is a maximum when is a minimum, that is for smaller loads
(smaller contact areas).
The error due to this phenomenon can be bounded by using the full
element width for the error in the radius. The results of two such
calculations in a homogeneous material are shown in
Fig. 1. One result is for a relatively coarse
mesh (1144 nodes and 1066 elements), while the other is for a finer
mesh (2579 nodes and 2466 elements). Two results of note can be
extracted from this curve. First, the error decreases for increasing
load, due to the effects discussed above. This supports the
conclusions of Yost (1983), who used Monte Carlo simulations to
generate similar results for the depth-dependence of errors incurred
during hardness tests. The second conclusion exhibited by this curve
is that within the error bars, the hardness of this homogeneous
material is independent of depth. Assuming a constant hardness, the
relative hardness here lies somewhere between 0.91 and 1.05 for the
coarse mesh and between 0.96 and 1.02 for the finer mesh. The
conclusion that the hardness is independent of depth agrees with the
conclusion of previous simulations [(Bhattacharya and Nix, 1988a)]. For reference, the
smallest elements in the coarse-mesh calculation have a width of 0.083
microns, while the width of the smallest elements in the fine-mesh
calculation is 0.041 microns. This is significantly smaller than the
largest indentation depth, which is on the order of 10 microns. These
element widths were reduced even further for the rest of the
simulations presented in this paper. This reduction was such that the
ratio of the smallest element width to the peak indentation depth was
held roughly constant ().
It should be noted that this discretization is not the sole error encountered in such simulations. Large-strain plastic behavior is difficult to model, and much error is introduced by this fact. The above discussion merely indicates the importance of the choice of the element size in the neighborhood of the contact region. The curves in Fig. 1 do seem to indicate that this discretization error dominates the random errors produced by the inability of the code to simulate the contact at resolutions finer than the element width.