C. Cayron, A. Baur, R. Logé
The atoms in face-centered cubic, body-centered cubic and hexagonal close-packed structures are geometrically represented with hard-spheres. This representation is classically used in the classrooms of engineering schools to show to the students the creation of staking faults or twins; however, the hard-sphere hypothesis is ignored by the classical crystallographic theories of phase transformation and mechanical twinning. These theories consider only the lattices and transformation between the lattices by simple shears. That’s unfortunate because hard-spheres can’t be sheared. For the last years we have rehabilitated the hard-sphere hypothesis and composed it with linear algebra in order to calculate the continuous atomic paths and lattice distortion from the initial state to the final state for the main displacive phase transformations: fcc→bcc, bcc→hcp, fcc→hcp [1]. The transformations are described with only one unique angular parameter without other ad-hoc assumption. In martensitic iron alloys, the same distortion matrix (the one associated to the Kurdjumov-Sachs orientation relationship) and simple variant selection rules are sufficient to deduce
a) the {225} habit planes in the high carbon steels [2]
b) the {557} habit planes in the low carbon steels [3].
The approach also applies to deformation twinning. The cases of fcc→fcc [1] twinning, and hcp→hcp extension twinning in magnesium [4] are treated. The volume change during the lattice distortion, expected from the Kepler’s conjecture, is calculated. Some twinning modes experimentally observed but not predicted by the classical Bevis and Crocker’s theory are calculated; and a solution to the apparent abnormality of the Schmid factor is proposed.
[1] C. Cayron, Acta Mater. 111 (2016) 417-441.
[2] A. Baur, C. Cayron, R. Logé, under review.
[3] C. Cayron, A. Baur, R. Logé, https://arxiv.org/abs/1606.04257
[4] C. Cayron, https://arxiv.org/abs/1608.07037
FCC-BCC continuous distortion with final Bain orientation
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Transformation from FCC to BCC structure by angular distortion with a hard-sphere model. The distortion occurs by contracting one <100> direction and letting elongate the two other <100> directions. This is Bain’s model (1924), with a hard-sphere hypothesis.
FCC-BCC continuous distortion with final Kurdjumov-Sachs (KS) orientation
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Transformation from FCC to BCC structure by angular distortion with a hard-sphere model. The angle between two <110> directions opens from 60° to 70.5°, such that the (111) plane containing these directions remains untilted and one of the two directions remains invariant. The final orientation is Kurdjumov-Sachs (1930). The initial structure is a single FCC cell.
FCC-HCP continuous distortion with final Shoji-Nishiyama orientation
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Transformation from FCC to HCP structure by angular distortion with a hard-sphere model. The habit plane (111)fcc remains fully invariant during the distortion and becomes (001)hcp when the process is completed.
Continuous distortion associated with extension twinning in magnesium
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Extension twinning viewed as a transformation from a HCP to a HCP phase by continuous angular distortion with a hard-sphere model. The distortion occurs by rotating a magnesium atom around a <100> direction such that, by steric effect, the basal plane is transformed into a prismatic plane, and a prismatic plane is transformed into the basal plane, while maintaining untilted the (10-12) plane. The maximum volume change during the intermediate states is +3%. The new orientation is induced by the work performed by the atomic displacements in the external stress field.