An Explanation for the Shape of Nanoindentation Unloading Curves based on Finite Element Simulation
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where P is load, h is displacement, hf is the final displacement after complete unloading, and A and m are power-law parameters. The values of A, m and hf were determined from regression analysis of the experimental data. An important observation in the work of Oliver and Pharr is that the values of the power law exponent, while slightly material dependent, generally fall in the range 1.25 - 1.51 (1.25 for fused silica; 1.51 for tungsten). As such, observed experimental behavior is consistent with neither the flat punch indenter geometry (m = 1), nor the conical indenter (m = 2). The disagreement with the latter geometry is particularly disconcerting given that a cone most closely approximates the self-similar geometry of the Berkovich indenter used to obtain the experimental data. Curiously, an indenter with the geometry of a parabola of revolution, for which m = 1.5, comes closest to predicting the experimental observations. However, how a sharp Berkovich
675 Mat. Res. Soc. Symp. Proc. Vol. 356 01995 Materials Research Society
indenter approximates a parabola of revolution, which is neither sharp at its tip nor properly describes the relationship between depth and contact area, is not evident. In this paper, results of a finite element simulation are presented which reproduce the experimentally observed unloading behavior and from which an understanding of the origin of the power law exponents is obtained. The understanding is based on the concept of an effective indenter shape whose geometry is determined by the shape of the plastic hardness impression formed during indentation. FINITE ELEMENT SIMULATION PROCEDURE Elastic/plastic indentation was simulated using the axisymmetric capabilities of the ABAQUS finite element code. As shown in Fig. 1, the indenter was modeled as a rigid cone with a half-included angle of 70.3'. This angle gives the same area-to-depth ratio as the Berkovich
triangular pyramid used in the nanoindentation experiments of Oliver and Pharr. The specimen was modeled as a large cylinder using the finite element mesh shown in Fig. 1. The dark lines along the centerline and the upper surface are a very fine mesh which cannot be resolved in the figure. The fine surface mesh was needed to carefully establish the surface profile. The material examined in the simulation was aluminum alloy 8009, which was studied previously by the authors to explore the effects of residual stress on nanoindentation data analysis [4-6]. The elastic/plastic constitutive behavior for the alloy was approximated by piecewise fitting the results of a tension test, as shown in Fig.2, and using the elastic constants E = 82.12 GPa and v = 0.31. The von Mises yield criterion was used to model the plastic behavior, and no friction was assumed between the punch and material. RESULTS The first finite element simulation was undertaken to establish how well the method works in simulating real experimental behavior. Fig.3 compares the indentation load-displacement data generated in the simulation to the results of an actual
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