Variant selection of primary–secondary extension twin pairs in magnesium: an analytical calculation study

Twining is an important deformation mode in magnesium. In a deformed magnesium sample, an extension twin crystal, i.e., {101¯2} twin, can form inside a {101¯2} primary twin, which is named {101¯2}–{101¯2} secondary twin. These secondary twins often appear at the intersection of two primary twins, and form primary–secondary twin pairs. Experimental observations show that the most frequently observed primary–secondary twin pairs have a unique misorientation, i.e., twin variant selection exists. Such variant selection of the primary–secondary twin pairs is studied in this work. The crystallographic analysis reveals that the twin planes of the primary and secondary twins that form a twin pair have coincident intersection lines with the boundary where the twin pair adjoins. An analytical calculation method based on Eshelby's inclusion theory is developed, and the calculation results show that only for this unique misorientation, the stress fields concentrated at the rims of the primary and the secondary twins are mutually favoured. The analysis is further extended to the incoming–outgoing twin pairs across ordinary grain boundaries, and compared with the commonly used geometrical compatibility factor m′. It is found that m′ only gives good prediction for twin transmission when the shear stress component on the twin plane along the twin shear direction of the incoming twin is the major contributor to the resolved shear stress of the outgoing twin. When other stress components play a dominant role, m′ becomes ineffective in prediction, which is the case for the primary–secondary twin pairs.