T.B. Britton

The effect of crystal orientation on the indentation response of commercially pure titanium: experiments and simulations

This study combines nanoindentation, electron backscatter diffraction (EBSD) and crystal plasticity finite element analysis to examine the anisotropy in the indentation behaviour of individual grains within an α-Ti polycrystal. Nanoindentation is utilized to mechanically probe small volumes of material within grains for which orientations are known from prior EBSD mapping. Both indentation modulus and hardness decrease significantly as the indentation axis is inclined further from the c-axis; the plastic response showing the more marked anisotropy. Recently developed high angular resolution EBSD has been utilized to examine selected indents, providing maps of elastic strain variations and lattice rotations. From such maps lower bound solutions for the density of geometrically necessary dislocations (GNDs) have been established. Crystal plasticity modelling showed promise in capturing correctly the orientation dependence of load–displacement response and in lattice rotations local to the indenter, particularly for indentation into a basal plane which generated threefold rotational symmetry about an axis parallel with the indentation direction which was also observed in experiments.

High resolution electron backscatter diffraction measurements of elastic strain variations in the presence of larger lattice rotations

In this paper we explore methods of measuring elastic strain variations in the presence of larger lattice rotations (up to -11°) using high resolution electron backscatter diffraction. We have examined the fundamental equations which relate pattern shifts to the elastic strain tensor and modified them to a finite deformation framework from the original infinitesimal deformation one. We incorporate the traction free boundary condition into the minimisation problem for the finite deformation case (i.e. large rotations and small elastic strains). Numerical experiments show that this finite deformation kinematic analysis continues to work well, while the infinitesimal analysis fails, when the misorientation between test and reference pattern is made increasingly high. However, measurements on patterns simulated using dynamical diffraction theory indicated that this formulation is not sufficient to recover elastic strains accurately because the pattern shifts are not determined accurately when large rotations are present. To overcome this issue we remap the test pattern to an orientation that is close to that of reference pattern. This remapping was defined by a finite rotation matrix, which was estimated from the infinitesimal rotation matrix measured using cross-correlation. A second cross-correlation analysis between the reference pattern and the remapped test pattern allows the elastic strains to be recovered using the much simpler infinitesimal deformation theory. We have also demonstrated that accurate recovery of elastic strains requires accurate knowledge of the pattern centre if this remapping algorithm is used.

Measurement of geometrically necessary dislocation density with high resolution electron backscatter diffraction: effects of detector binning and step size.

Recent advances using cross-correlation analysis of full resolution high quality electron backscatter diffraction (EBSD) patterns have provided a method for quantitatively mapping the stored dislocation density at high spatial resolution. Larger areas could be mapped with image binning or coarser step sizes. We have studied the effects of image binning and step size on the recovery of GND density. Our results suggest that: (i) the measured lower bound GND density noise floor broadly agrees with Wilkinson and Randmans 2009 prediction, where a decrease in step size or an increase in misorientation uncertainty increases the noise floor; (ii) increasing the step size results in a lower GND density being recovered as some dislocations are now considered as statistically stored dislocations (SSDs); (iii) in deformed samples the average GND density stays relatively constant as the degree of pattern binning is increased up to 8×8. Pattern binning thus provides a means of increasing the data acquisition and analysis rate without unduly degrading the data quality.

Measurement of residual elastic strain and lattice rotations with high resolution electron backscatter diffraction

A set of dynamically simulated electron backscatter patterns (EBSPs) for α-Ti crystals progressively rotated by 1° steps were analysed using cross-correlation to determine image shifts from which strains and rotations were calculated. At larger rotations the cross-correlation fails in certain regions of the EBSP where large shifts are generated. These incorrect shifts prevent standard least square error procedures from obtaining a valid solution for the strain and rotation, where the applied rotation exceeds ∼ 8°. Using a robust iterative fitting routine reliable strains and rotations can be obtained for applied rotations of up to and including ∼ 11° even though some image shifts are measured incorrectly. Finally, high resolution electron backscatter diffraction has been used to analyse the residual elastic strain, lattice rotations and density of stored geometrically necessary dislocations in a sample of copper deformed to 10% total strain. The robust iterative fitting analysis allows reliable analysis of a larger proportion of the map, the difference being most obviously beneficial in regions where significant lattice rotations have been generated.

Factors affecting the accuracy of high resolution electron backscatter diffraction when using simulated patterns

High resolution EBSD directly compares electron backscattering patterns (EBSPs), generated in a scanning electron microscope, to measure relative strain and rotation to a precision of ∼ 10(-4) in strain and 10(-4)rad (0.006 °) in rotation. However the measurement of absolute strain and rotation requires reference EBSPs of known strain and orientation (or a far field region of known strain). Recent suggestions of using simulated EBSPs with known strain show much promise. However precise measurement of the experimental geometry (pattern centre) is required. Common uncertainties of 0.5% in pattern centre result in uncertainty of ∼ 10(-3) in strain state. Aberrations in the compact lenses used for EBSP capture can also result in image shifts that correspond to strains/rotations of ± 10(-3) between experimental and simulated EBSPs. Simulated EBSPs can be generated using dynamical or kinematic models (or a combination of the two). The choice in simulation model has a significant effect on the measured shifts, particularly at zone axis and high structure factor bands, due to large intensity variations, and for simple kinematic simulations can result in the measurement of rogue shifts and thus erroneous strain measurements. Calibrant samples of known strain provide a method of measuring the experimental geometry but imprecise stage movement combined with the high depth of field in the SEM could also result in uncertainties in strain of ∼ 10(-3).

Assessing the precision of strain measurements using electron backscatter diffraction – part 1: Detector assessment

We analyse the link between precision of pattern shift measurements and the resolution of the measurement of elastic strain and lattice rotation using high resolution electron backscatter diffraction (HR-EBSD). This study combines analysis of high quality experimentally obtained diffraction patterns from single crystal silicon; high quality dynamical simulations using Bloch wave theory; quantitative measurements of the detector Modulation Transfer Function (MTF) and a numerical model. We have found that increases in exposure time, when 1×1 binning is selected, are the primary reason for the observed increase in sensitivity at greater than 2×2 binning and therefore use of software integration and high bit depth images enables a significant increase in strain resolution. This has been confirmed using simulated diffraction patterns which provide evidence that the ultimate theoretical resolution of the cross correlation based EBSD strain measurement technique with a 1000×1000 pixel image could be as low as 4.2×10(-7) in strain based on a shift precision of 0.001 pixels.