Relation between elastic-constant tensors of hexagonal and cubic structures

Abstract

The third-order elastic constants of hexagonal and cubic structures are related using an extension of a transformation derived by Martin. The transformation is used to calculate the zinc-blende second- and third-order elastic constants of CdS from the experimental wurtzite elastic constants. Also, the transformation is shown to give good results for some face-centered-cubic hexagonal closed-packed polymorphic metals. In particular, the second-order elastic constants of cobalt and thallium in the hexagonal structure are compared with those calculated from cubic elastic constants extrapolated from alloy data, and second- and third-order hexagonal elastic constants of Co-32% Ni are compared with those calculated from cubic elastic constants.

Inapplicability of Martin transformation to elastic constants of zinc-blende and wurtzite group-III nitride alloys

“…3 Despite these two assumptions, the Martin transformation was successfully used to many material systems, including II-VI and III-V semiconductors, metals, solid rare gases, and hard sphere crystals. 3,[5][6][7][8][9][10][11] Together with Vegard-like law, it was also applied to predict the elastic constants in semiconductor and metal alloys. 12,13 Moreover, it was used to determine pressure derivatives of the elastic constants and was also extended to third-order elasticity relating the third-order elastic constants of the ZB and WZ crystal structures.…”

“…12,13 Moreover, it was used to determine pressure derivatives of the elastic constants and was also extended to third-order elasticity relating the third-order elastic constants of the ZB and WZ crystal structures. 9,13 In the case of group-III nitride semiconductors, the Martin transformation was used by Shermin and Drummond to estimate the ZB elastic constants in GaN, InN, and AlN from the experimental results obtained for the WZ phase. 14 Kim et al combined the Martin transformation with density functional theory (DFT) calculations in order to determine the elastic constants of the binary nitride compounds in the ZB and WZ phases.…”

Mechanical derivation of elastic constants of ε-martensite

“…The first term of the sum in Eq.11 is a simple average of elastic stiffnesses whereas the last term is called an internal strain contribution [Mar72,FW74]. This expression shares some similarities with the one of the effective compliance tensor (Eq.5), which is also composed of a first average term and a second one that depends on differences.…”

“…It is noteworthy that a previous procedure relating the elasticity tensors of HCP and FCC structures already existed. It was developed by Martin [Mar72] for wurtzite (HCP) and zinc-blende (FCC) structure materials and then applied by Fuller and Weston to metallic structures [FW74]. The starting point of their method is the construction of wurtzite and zinc-blende structures from tetrahedral building blocks whose corners lie along 111 directions [Rob68]: for zinc-blende structure, tetrahedra are aligned in equivalent orientations and for wurtzite structure, tetrahedra are alternatively aligned in two mirror orientations [Mar72,FW74] which can be related to the frame e 1 , e 2 , e 3 and the associated twinned phase previously defined.…”

“…It was developed by Martin [Mar72] for wurtzite (HCP) and zinc-blende (FCC) structure materials and then applied by Fuller and Weston to metallic structures [FW74]. The starting point of their method is the construction of wurtzite and zinc-blende structures from tetrahedral building blocks whose corners lie along 111 directions [Rob68]: for zinc-blende structure, tetrahedra are aligned in equivalent orientations and for wurtzite structure, tetrahedra are alternatively aligned in two mirror orientations [Mar72,FW74] which can be related to the frame e 1 , e 2 , e 3 and the associated twinned phase previously defined. Then, according to the procedure of Martin [Mar72], the elastic constants of the wurtzite structure are deduced from the minimization of its elastic strain energy density U (computed as the average density of the two trigonal orientations) with respect to the difference of strains in the two trigonal orientations [[ε i ]]:…”