Determination of elastic strain fields and geometrically necessary dislocation distributions near nanoindents using electron back scatter diffraction

Abstract

The deformation around a 500-nm deep Berkovich indent in a large grained Fe sample has been studied using high resolution electron back scatter diffraction (EBSD). EBSD patterns were obtained in a two-dimensional map around the indent on the free surface. A cross-correlation-based analysis of small shifts in many sub-regions of the EBSD patterns was used to determine the variation of elastic strain and lattice rotations across the map at a sensitivity of ∼±10⁻⁴. Elastic strains were smaller than lattice rotations, with radial strains found to be compressive and hoop strains tensile as expected. Several analyses based on Nye's dislocation tensor were used to estimate the distribution of geometrically necessary dislocations (GNDs) around the indent. The results obtained using different assumed dislocation geometries, optimisation routines and different contributions from the measured lattice rotation and strain fields are compared. Our favoured approach is to seek a combination of GND types which support the six measurable (of a possible nine) gradients of the lattice rotations after correction for the 10 measurable elastic strain gradients, and minimise the total GND line energy using an L¹ optimisation method. A lower bound estimate for the noise on the GND density determination is ∼±10¹² m⁻² for a 200-nm step size, and near the indent densities as high as 10¹⁵ m⁻² were measured. For comparison, a Hough-based analysis of the EBSD patterns has a much higher noise level of ∼±10¹⁴m⁻² for the GND density.

Keywords:

• EBSD
• dislocation mechanics
• dislocations
• electron diffraction
• electron microscopy
• mechanics of materials
• nanoindentation

Acknowledgements

Thanks to Dr Mark Vaudin (NIST, Gaithersburg) for discussion of EBSD results on indents in Si prior to publication and initial MatLab script for plotting maps from CrossCourt output, to Dr Graham Meaden (BLG Productions Ltd) and Professor David Dingley (University of Bristol) for continuing discussion of all aspects of EBSD, to Professor Fionn Dunne (University of Oxford) and Dr Hong Liang (University of Oxford) for insightful discussions on their crystal plasticity simulations of indentation, and finally to Ben Britton (University of Oxford) for discussions on GND analysis and extensive input into Matlab coding.