A double-shear theory is introduced that predicts the commonly observed {5 5 7} habit planes in low-carbon steels. The novelty of this theory is that the shearing systems are chosen in analogy to the original (single-shear) phenomenological theory of martensite crystallography as those that are macroscopically equivalent to twinning. Out of all the resulting double-shear theories, the ones leading to certain {h h k} habit planes naturally arise as those having small shape strain magnitude and satisfying a condition of maximal compatibility, thus making any parameter fitting unnecessary.An interesting finding is that the precise coordinates of the predicted {h h k} habit planes depend sensitively on the lattice parameters of the face-centered cubic and body-centered cubic phases. Nonetheless, for various realistic lattice parameters in low-carbon steels, the predicted habit planes are near {5 5 7}. The examples of Fe–0.252C and Fe–0.6C are analyzed in detail along with the resulting orientation relationships which are consistently close to the Kurdjumov–Sachs model. Furthermore, a MATLAB app Lath Martensite is provided which allows the application of this model to any other material undergoing an fcc to bcc transformation.

The wisdom of the crowd is a valuable asset in today’s society. It is not only important in predicting elections but also plays an essential role in marketing and the financial industry. Having a trustworthy source of opinion can make forecasts more accurate and markets predictable. Until now, a fundamental problem of surveys is the lack of incentives for participants to provide accurate information. Classical solutions like small monetary rewards or the chance of winning a prize are often not very attractive for participants. More attractive solutions, such as prediction markets, face the issue of illegality and are often unavailable. In this work, we present a solution that unites the advantages from classical polling and prediction markets via a customizable incentivization framework. Apart from predicting events, this framework can also be used to govern decentralized autonomous organizations.

The identification of orientation relationships (ORs) plays a crucial role in the understanding of solid phase transformations. In steels, the most common models of ORs are the ones by Nishiyama-Wassermann (NW) and Kurdjumov-Sachs (KS). The defining feature of these and other OR models is the matching of directions and planes in the parent face-centred cubic γ-phase to ones in the product body-centred cubic/tetragonal α/α′-phase.In this paper, a novel method that identifies transformation strains with ORs is introduced and used to develop a new strain-based approach to phase transformation models in steels. Using this approach, it is shown that the transformation strains that leave a close packed plane in the γ-phase and a close packed direction within that plane unrotated are precisely those giving rise to the NW and KS ORs when a cubic product phase is considered. Further, it is outlined how, by choosing different pairs of unrotated planes and directions, other common ORs such as the ones by Pitsch (PT) and Greninger-Troiano (GT) can be derived.One of the advantages of our approach is that it leads to a natural generalisation of the NW, KS and other ORs for different ratios of tetragonality r of the product bct α′-phase. These generalised ORs predict a sharpening of the transformation textures with increasing tetragonality and are thus in qualitative agreement with experiments on steels with varying alloy concentration.

This article provides a rigorous proof of a conjecture by E. C. Bain in 1924 on the optimality of the so-called Bain strain based on a criterion of least atomic movement. A general framework that explores several such optimality criteria is introduced and employed to show the existence of optimal transformations between any two Bravais lattices. A precise algorithm and a graphical user interface to determine this optimal transformation is provided. Apart from the Bain conjecture concerning the transformation from face-centred cubic to body-centred cubic, applications include the face-centred cubic to body-centred tetragonal transition as well as the transformation between two triclinic phases of terephthalic acid.

A mathematical framework is proposed to predict the features of the (5 5 7) lath transformation in low-carbon steels based on energy minimisation. This theory generates a one-parameter family of possible habit plane normals and a selection mechanism then identifies the (5 5 7) normals as those arising from a deformation with small atomic movement and maximal compatibility. While the calculations bear some resemblance to those of double shear theories, the assumptions and conclusions are different.

Using a variational model based on non-linear elasticity we investigate whether in a cubic-to-tetragonal phase transformation it is energetically preferable to nucleate martensite within austenite. Under minimal growth assumptions on the free energy density W away from the wells, we derive explicit upper bounds on W^qc(id), i.e. on the macroscopic free energy density of a region that has been macroscopically deformed by the identity map. The bounds only depend on material parameters, the temperature difference and the growth of W away from the wells. By comparing W^qc(id) and W(id) we conclude that nucleation is always energetically preferable and are able to provide quantitative upper bounds for the gain in negative energy due to nucleation.