Three-dimensional modeling of indent-induced plastic zone at a mesoscale1
Introduction
In the last few years, there has been much interest in using contact techniques such as nano/micro-hardness testing to assess the mechanical properties of materials1, 2, 3, 4, 5, 6. However, controversy still exists for the discrepancy often observed between nanoindentation and bulk hardness values, an effect which could either be due to genuine size mechanisms or from the presence of passive surface films7, 8. When coupling loading curves with Transmission Electron Microscopy (TEM) observations of the associated indent-induced plastic zones, it has been shown that information derived from the analysis of indentation curves alone could lead to quite misleading conclusions regarding the involved deformation mechanisms8, 9, 10. These TEM observations have also provided useful complementary information regarding the activated slip systems and the extent of the plastic zone. It is observed that the indent dislocation microstructure results from complicated dynamic combinations of dislocation sequences including nucleation, multiplication, cross-slipping and junction formation. Therefore, only numerical modeling can simultaneously keep track of these mechanisms and of their effects on the loading–unloading indentation curves. While the two-dimensional dislocation simulations11, 12, 13are unable to deal with such mechanisms, these sequences can now all be modeled thanks to the recent development of three-dimensional mesoscale dislocation simulations14, 15, 16. In order to reproduce the particular case of nanoindentation testing, suitable boundary conditions have to be implemented in the three-dimensional dislocation code. In this way, a procedure using the finite element method has recently been developed17, 18. The resulting simulated dislocation microstructures can then be compared to TEM micrographs of actual indent-induced plastic zones, provided all the experimental preparation and observation steps are also numerically reproduced.
In Section 2, the experimental nanoindentation conditions and sample preparation are described. The corresponding mesoscale modeling is discussed and a complete set of rules defining both the boundary conditions and the full nucleation process is given. The detailed loading algorithm is also described.
In Section 3, an application of the whole method to the case of a [001] copper single crystal indentation is presented. The simulated dislocation microstructure and the indent-induced plastic volume are directly compared to the experimentally observed ones.
Section snippets
Experimental
In order to observe in TEM an indent-induced plastic zone produced on a surface perpendicular to a given crystallographic orientation, a multi-step and delicate sample preparation is needed. In this way, small cubes are first taken from a Cu single crystal bar and installed onto a two axis goniometric stage. The initial crystallographic orientation of the top surface of one cube is checked by X-ray diffraction and the desired orientation is obtained by tilting the goniometric stage
Experimental
At the loading rate of 0.08 mN/s, highly reproducible load–displacement curves showing a clear slope discontinuity at load Pd=(0.4±0.2) mN have been observed (see Fig. 5), for two distinct maximum penetration depths: 50 and 100 nm. Monotonic loading–unloading sequences with Pmax<Pd yield no residual displacement. Thus, we assume that the portion of the loading curve with P<Pd corresponds to the Hertz curve for a spherical contact on a bi-material consisting of a rigid film of copper oxide passive
Conclusions and perspectives
A model describing the formation of indent-induced plastic zones combining three-dimensional discrete dislocation simulation and FEM has been worked out. A complete set of nucleation rules has been specified with the help of experimental data coming from both nanoindentation load–displacement curves and TEM observations of the plastic zone.
A nanoindentation simulation at a depth of 50 nm on a [001] oriented Cu single crystal has been performed. The simulated microstructure has been directly
Acknowledgements
The participation of A. de Gayffier from LAMS/DMT of CEA Saclay has been appreciated for help in the coupling between the dislocation simulation and the finite element code CASTEM2000. The authors are also grateful for the help provided by Ecole de Chimie de Paris in the sample preparation, as well as to S. Poissonnet from SRMP for the indentation experiments and S. Belliot from Ecole Centrale de Paris for the AFM observations of the indenter tip. We also thank R. Phillips from Brown University
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This paper is dedicated to Gilles Canova whose untimely death occurred on 28 July 1997 at the age of 43.