Platonic and Archimedean bodies as the basis of the structure of self-accommodating complexes of martensite crystals in alloys with shape memory effects

Abstract

The aim of the work is to analyze the relationship of the architecture of self-accommodation complexes (SC) with the lattice syngony of martensite crystals. Self-accommodating complexes consist of a set of pairwise twinned domains — crystals of martensite belonging to crystallographically equivalent variants of the orientation relationship between the lattices of austenite and martensite. The simplest SC are calculated for tetragonal, orthorhombic, rhombohedral and monoclinic distortion of the cubic lattice of austenite. It is shown that complete self-accommodation is possible only in complexes containing simultaneously all variants of the orientation relation. The issue of external faceting of complexes is discussed. The reason for the formation of SC is the minimization of elastic energy, i.e. the appearance regulated by the energy of the interphase boundary. On the other hand, if the outer surface of the SC is a polyhedron, then its symmetry should "fit"into the anisotropy of the elastic properties of austenite. For reasons of symmetry, it is clear that the polyhedron must be correct and have the same symmetry elements as the cubic lattice of austenite, while the axes of symmetry of the cubic lattice of austenite must coincide with the axes of symmetry of the polyhedron. Similar polyhedra are some of the bodies of Platon and Archimedes, which have axes of symmetry of the 2nd, 3rd and 4th order. A number of examples calculated in the work confirms the possibility of the existence of complexes in the form of these polyhedra.