Journal of Materials Science

, Volume 55, Issue 5, pp 2186–2192 | Cite as

Approximate rotation vector expressions to consider crystal orientation changes in plastically deformed materials

  • Susumu OnakaEmail author
Metals & corrosion


Orientation changes are important factors when considering microstructural changes in materials caused by plastic deformation because these are related to the arrangement and density of dislocations. The three elements of a logarithm of a rotation matrix are called log angles, which compose a rotation vector that can be used as a parameter to represent crystal orientations. Herein, approximate expressions of a rotation vector are derived for a product of two small-angle rotations. The degrees of the first-order, second-order and third-order approximations are discussed. The non-commutativity of two rotations is graphically illustrated using the approximate expressions. Experimental results of orientation changes in a cold-rolled Cu bicrystal are considered to evaluate the validity of the approximate expressions. It is shown that a linear approximation for the product of rotation vectors is valid to treat small-angle orientation changes in plastically deformed crystals with dislocation structures.



The author thanks Koki Hamano for his contributions on Cu bicrystals. This work was supported by JSPS KAKENHI Grant No. 19K04985.

Compliance with ethical standards

Competing interests

The author declares no conflict of interests regarding the publication of this paper.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Materials Science and Engineering, School of Materials and Chemical TechnologyTokyo Institute of TechnologyYokohamaJapan

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