Three Measures of Strain

Engineering strain can be considered in three ways:

Type 1 Macroscopic strain across the whole component. External constraints impose this type of strain, whether as a consequence of being a component of a machine, construction or device, or as a result of an imposed strain during mechanical testing.

Type 2 Strain which is generated internally and ranges over several grains.  For example, strain produced by a fibre embedded in a metal matrix; the effects will vary from grain to grain because of the metal’s elastic anisotropy and the grains’ varying orientation.

Type 3 Microscopic strain varying from point to point within each grain. Microscopic strain  components are any residual strain from mechanical loading, from Type 2 strains and from strain due to incompatible lattice matching at grain boundaries, to precipitates within the grains and from the remaining dislocation structure. Microscopic strain occurs at the interaction of microstructure and the strain field and useful for analysis of fatigue and crack formation and also functional materials (semi-conductors, photo-voltaic cells, piezo-electric devices etc.) where the strain field can have a marked effect on a materials’ desired properties.

Uniquely, CrossCourt allows the study and mapping of Type 3 Microscopic elastic strains, and  the corresponding stress field; data is collected using a standard EBSD system attached to a regular Scanning Electron Microscope. CrossCourt provides the strain distribution at each point within a grain relative to a selected point in the grain; usually the strain distribution with the smallest KAM (Kernel Average Misorientation) value is recorded and, therefore, the most likely point of minimum or zero strain.

Strain Measurement and HREBSD

High Resolution EBSD (HREBSD) compares high quality EBSD patterns from very similar regions of a crystalline material, e.g. from within the same grain of a metal, and differences between patterns due to lattice distortion are measured using cross-correlation based methods. These differences are used to calculate the distortion matrix required to match the measured changes.

The basis for the distortion matrix calculation is presented in the work,

A J. Wilkinson, G. Meaden, D. J. Dingley, “High-resolution elastic strain measurement from electron backscatter diffraction patterns: New levels of sensitivity”, Ultramicroscopy 106 2006 https://doi.org/10.1016/j.ultramic.2005.10.001

Infinitesimal strain theory Professor John Nye, Physical Properties of Crystals, Google. (n.d.)  is used to separate the distortion matrix into:

• The elastic strain matrix.
• A set of rotations about the reference axes describing how the lattice has rotated.

An explanation of Plastic Strain and HR-EBSD is given below.

The elastic strain tensor is a 3×3 matrix comprising the normal strains (tensile or compressive) which are parallel to three reference directions in the crystal and the shear strains acting along the three planes normal to the reference directions.

Refinements

Remapping

Where there is a significant amount of lattice rotation in samples (i.e. greater than 1 degree), infinitesimal strain theory starts to break down and the elastic strain components cannot be extracted easily as the figures are compromised by the larger rotations.

CrossCourt4 achieves cleaner strain measurement by performing a second data analysis, this is called the remapping pass. In the remapping pass, CrossCourt4 uses the infinitesimal rotations measured in the first analysis to back rotate the recorded EBSD pattern, around the pattern centre, and remove the measured rotations. The remapped patterns can then be analysed, mostly, as before and the elastic strains extracted without the interference of the rotations.

Robust Fitting

Calculating the distortion matrix requires 4 measurements of the pattern distortion but usually more are made, around 20 to 40, than necessary; a least square’s solution is generally found as the problem is over determined.

However, the quality of the measures of the pattern distortion within a pattern may be variable where there is:

• Gross blurring of some Kikuchi bands in the pattern due to the presence of large number of statistically stored dislocations.
• Marks or other phosphor defects.
• Over saturation of the pattern in parts of the image.

In these cases, the application of a robust fitting approach, iteratively weighted least squares, to the calculation of the distortion matrix, both before and after remapping, can vastly improve the measurement of elastic strain.

Effects of Plastic Strain

Plastic strain occurs when the stress is high enough to operate dislocation sources, either from grain boundaries or from Frank Read sources, leading to a rapid increase in the total number of dislocations distributed within the grains. The material’s yield point is defined as where the multiplication of dislocations is very rapid; however, prior to this macroscopic yield point, dislocation glide can occur. Consequently, the study of this early dislocation activity, along with what occurs at macroscopic yielding, has for many years been of both academic and engineering interest.

HREBSD is able to measure this dislocation activity because the strain field associated with a dislocation produces a lattice rotation which can be extracted from the deformation tensor. HREBSD does not have the sensitivity to measure the rotations of individual dislocations, but, is sensitive enough to detect the resulting rotation of a cluster of dislocations with a local density as small as ~10¹² m^-2. Where dislocations within this cluster have Burgers vectors of opposite sign, this results in a zero rotation and so they do not contribute to the observed rotation. Therefore, where HREBSD measurements are made, these relate only to those dislocations within the clusters where rotations are not mutually cancellable; these dislocations are called Geometrically Necessary Dislocations (GNDs). The highest densities that HREBSD can measure are in the ~10¹⁶ m^-2 range which corresponds to dislocations 10nm apart. However, HREBSD’s spatial resolution is the same as EBSD and is of the order 50nm which means this is the finest resolution that dislocation density distribution can be measured. The HREBSD method:

• Yields the average value for GNDs within the area covered by the x and y scan step sizes.
• Can give a measure of the GND distribution of each slip component if the slip systems are known.

Angus J. Wilkinson & David  Randman 2010, “Determination of elastic strain fields and geometrically necessary dislocation distributions near nanoindents using electron back scatter diffraction” Phil. Mag. 90, (2010) http://dx.doi.org/10.1080/14786430903304145

Stress measurement

Residual strain results in distortion of the EBSD pattern and this is what is measured experimentally; from these measurements it is possible to calculate the stress state through the standard stress-strain relationship X Young’s modulus. As both stress and strain are tensors, the full set of elastic stiffness coefficients needs to be used in the calculation. A set of stresses equivalent to the strain set is calculated, i.e. three stress components acting parallel to the three references axes of the crystal and three shear stresses acting in the directions of the planes normal to the reference axes.

Engineering Applications

In engineering, it is usually more common to express strains with reference to principal axes and combined stresses in the form of the von Mises’ stress yield criterion. The principal strains are those parallel to the three directions which remain unrotated under the influence of the residual strain, one of which will be parallel to the maximum residual normal stress. The von Mises’ yield criterion is a scalar value where the normal and shear stresses are combined in a way that, when the criterion reaches the yield stress of the material in simple tension, it is then likely the sample under the measured stress configuration will yield.