Comparison of methods of algebraic strain estimation from Rf/ɸ data: A unified theory of 2D strain analysis

A unified development of the subject of the algebraic strain analysis methods using Rf/ɸ data is outlined, embodying the main features the theories of Shimamoto and Ikeda, Mulchrone et al. and Yamaji. It is shown that the theories yields an identical strain ellipse from the same data set. However, error estimation in that of Shimamoto and Ikeda is difficult owing to the distortion of its parameter space: Resolution of their method depends on the choice of a reference orientation in the plane where strain markers are observed. In this respect, the remaining two theories have advantages. The hyperbolic vector mean method was developed in the Minkowski 3-space, thereby linked seamlessly with the visualizing methods of Rf/ɸ data, optimal strain and its confidence region. In addition, the residuals of the optimal strain ellipse determined by this method have clear physical meanings concerning logarithmic strains needed to transform a unit circle to given ellipses.

► Theories of strain analysis are formulated consistently in a Minkowski 3-space.
► The strain ellipses determined by algebraic methods are shown to be identical.
► The Shimamoto–Ikeda method, the most popular one, has anisotropic resolution.
► Therefore, the method is inconvenient for error estimation.