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Cyril Cayron

Publications and source records attributed to Cyril Cayron.

At least 19 recordsLinked to original sources
arXiv2026

The crossmetric tensor and the geometrical meaning of the imaginary numbers

The product of two Cartesian quaternions can be written as a quadratic form based on a 4x4 matrix made of the symbols 1,i,j,k where i,j,k are imaginary numbers introduced by Hamilton. We generalized this matrix to non-Cartesian bases, and showed that the matrix is made of the metric and the cross tensors. We called it crossmetric tensor. Its symbols are s,a,b,c; they are the elementary crystallographic quaternions. We determined the crossmetric tensors for the six crystal families. We also showed that any unit crystallographic quaternion can be geometrically represented by an infinity of pairs of oriented planes intersecting along the vectorial component of the quaternion such that the angle between them is the semiangle of the rotation. The composition of quaternions follows the intuitive source-target rule. The elementary quaternions a,b,c are geometrically represented by pair of perpendicular and oriented planes intersecting along the axis a,b,c, respectively. Other complementary quaternions noted a',b',c' were also introduced. They are the pairs of planes ma, mb, mc. The elementary quaternions a,b,c follow Hamilton rules on the squares of imaginary numbers. The complementary quaternions follow Hamilton rules on the bi and tri-products. For Cartesian basis, the quaternions i,j,k appear as a specific case of crystallographic quaternions. They are formed by the pairs of perpendicular faces of the cube mx, my, mz. Since these planes intersect along the x, y and z axis, respectively, the elementary and complementary quaternions are equal, i.e. i=i',j=j',k=k', which explains why the square, bi and tri-product Hamilton rules are satisfied all together with only three quaternions i,j,k.

cond-mat.mtrl-sci↗
arXiv2026

The crystallographic quaternions and their product law

Unit quaternions are widely used in science to encode rotations because the quaternion product is more efficient than matrix product and more stable than Rodrigues product to calculate the composition of two rotations. However, quaternions in their usual form refers to a Cartesian basis; they cannot be used in crystallography as they are. The usual way to solve this issue to apply back-and-forth coordinate changes from the crystal basis to a Cartesian basis attached to the crystal with the help of the structure tensor. Here, we show that actually quaternions can be used directly in the crystal basis by generalizing the quaternion product law. In that aim, we introduced a matrix that we called cross tensor. It allows the calculation of the cross product in the crystal basis, a bit like the metric tensor allows it for the scalar product. We also show the cross tensor is proportional to the inverse of the metric tensor. The formula of crystallographic quaternion product is then given; it depends uniquely on the metric tensor. The application of the crystallographic quaternions to Electron Back Scatter Diffraction is discussed.

cond-mat.mtrl-sci↗
arXiv2025

Compatibilities and supercompatibility conditions in shape memory alloys determined from correspondence, metrics and symmetries

The phenomenological theory of martensite crystallography (PTMC) developed in the 1950s explains the main crystallographic and microstructural features of martensite in shape memory alloys, such as the habit planes of bi-variant laminate martensite product, and the transformation twins between the variants. It also permits to determine the austenite and martensite lattice parameters that allow supercompatibility, which has driven important research and development of new shape memory alloys with low hysteresis and high cyclability. Supercompatibility takes the form of three mathematical equations called cofactor conditions. The calculations are in great part based mathematical tools from continuum mechanics (polar decompositions and stretch tensors). They were recently replaced by pure crystallographic tools (metric tensors, group of symmetries and correspondence) in an alternative approach called correspondence theory (CT). The CT allows for the direct calculation of the transformation twins and their generic and non-generic characteristics. These twins ensure the compatibility at martensite/martensite (M/M) junction planes. Here, we show that the CT can also be used to determine the conditions of austenite/martensite (A/M) compatibility, and A/M/M supercompatibility.

cond-mat.mtrl-sci↗
arXiv2024

Hard-sphere model of the B2 to B19' phase transformation, and its application to predict the B19' structure in NiTi alloys and the B19 structures in other binary alloys

The pseudoelastic and pseudoplastic properties of NiTi alloys result from the closeness of the structures between the B2 cubic austenite and the B19' monoclinic martensite, and the facility to transform one into each other. Until now, the paths followed by the atoms during the B2 to B19' transformation were imagined as independent shears and shuffles. Here, we propose a simplified hard-sphere atomistic model of phase transformation decomposed into three distinct types of atomic movements. The model's inputs are the Ti and Ni atomic or ionic diameters and the monoclinic angle beta. The outputs are the lattice parameters of the B19' phase and the atomic positions. The results are remarkably close to those reported in the literature from X-ray diffraction experiments or DFT simulations. The hard-sphere model explains the change of enthalpy by the formation of short Ti-Ti bonds in B19'. It is also shown that the value of the monoclinic angle beta close to 97.9 degrees corresponds to the highest molar volume among all the possible hard-sphere monoclinic B19' structures; which suggests that it could be a consequence of a maximization of the vibrational entropy. The hard-sphere model was applied in the special case of absence of monoclinicity to predict the B19 structure. The calculations do not agree well with the B19 structure reported in NiTi alloys; they are however in excellent agreement with the B19 structures reported in other binary alloys, such as in AuTi, PdTi, and AuCd.

cond-mat.mtrl-sci↗
arXiv2021

Axial heterotwins

The current theory of twin crystallography is based on the concept of invariant plane. The present paper extends the theory to encompass the cases of heterotwins in which the composition plane is only quasi-invariant.

cond-mat.mtrl-sci↗
arXiv2021

Lattice reduction by cubification

Lattice reduction is a NP-hard problem well known in computer science and cryptography. The Lenstra-Lenstra-Lovasz (LLL) algorithm based on the calculation of orthogonal Gram-Schmidt (GS) bases is efficient and gives a good solution in polynomial time. Here, we present a new approach called cubification that does not require the calculation of the GS bases. It relies on complementary directional and hyperplanar reductions. The deviation from cubicity at each step of the reduction process is evaluated by a parameter called lattice rhombicity, which is simply the sum of the absolute values of the metric tensor. Cubification seems to equal LLL; it even outperforms it in the reduction of columnar matrices. We wrote a Python program that is ten times faster than a reference Python LLL code. This work may open new perspectives for lattice reduction and may have implications and applications beyond crystallography.

cs.DS↗
arXiv2021

Solving constrained optimization problems without Lagrange multipliers

Constrained optimization problems exist in many domains of science, such as thermodynamics, mechanics, economics, etc. These problems are classically solved with the help of the Lagrange multipliers and the Lagrangian function. However, the disadvantage of this approach is that it artificially increases the dimensionality of the problem. Here, we show that the determinant of the Jacobian of the problem (function to optimize and constraints) is null. This extra equation transforms any equality-constrained optimization problem into a solving problem of same dimension. We also introduced the constraint matrices as the largest square submatrices of the Jacobian of the constraints. The boundaries of the constraint domain are given by the nullity of their determinants. The constraint matrices also permit to write the function to be optimized as a Taylor series of any of its variable, with its coefficients algebraically determined by an iterative process of partial derivation.

math.OC↗
arXiv2018

The transformation matrices (distortion, orientation, correspondence), their continuous forms, and their variants

The crystallography of displacive phase transformations can be described with three types of matrices: the lattice distortion matrix, the orientation relationship matrix, and the correspondence matrix. The paper gives some formula to express them in crystallographic bases, orthonormal bases, and reciprocal bases, and it explains how to use them to deduce the matrices of inverse transformation. In the case of hard-sphere assumption, a continuous form of the distortion matrix can be determined, and its derivative is identified to the velocity gradient used in continuum mechanics. The distortion, the orientation and the correspondence variants are determined by coset decomposition with intersection groups that depend on the point groups of the phases and on the type of transformation matrix. The stretch variants required in the phenomenological theory of martensitic transformation should be distinguished from the correspondence variants. The orientation variants and the correspondence variants are also different; they are defined from the geometric symmetries and algebraic symmetries, respectively. The concept of orientation (ir)reversibility during thermal cycling is briefly and partially treated by generalizing the orientation variants with n-cosets and graphs. Some simple examples are given to show that there is no general relation between the numbers of distortion, orientation and correspondence variants, and to illustrate the concept of orientation variants formed by thermal cycling.

cond-mat.mtrl-sci↗
arXiv2017

Over the shear paradigm

Deformation twinning and martensitic transformations are displacive transformations; they are defined by high speed collective displacements of the atoms, the existence of a parent/daughter orientation relationship, and plate or lath morphologies. The current crystallographic models of deformation twinning in metals are based on the 150 year-old concept of simple shear. For martensitic transformations, a generalized version of simple shear called invariant plane strain takes into account the volume change; it is associated with one or two simple shears in the phenomenological theory of martensitic crystallography built more than 60 years ago. As simple shears would involve unrealistic stresses, dislocation/disconnection-mediated versions of the usual crystallographic models of displacive transformations have been developed over the last decades. However, fundamental questions remain unsolved. How do the atoms move? How could dislocations be created and propagate in a coordinated way at the speed of sound? In order to solve these issues an approach that is not based on simple shear nor on dislocation/disconnection has been applied to different displacive transformations over the last years. It assumes that the atoms are hard-spheres, which permits for any specific orientation relationship to determine the atomics trajectories, the lattice distortion and shuffling (if required) as analytical functions of a unique angular parameter. The aim of the present paper is to give a brief historical review of the current models based on the shear concept and of their dislocation-mediated versions, and to introduce the new paradigm of angular distortion. Examples will be taken by using some recent publications. The possibilities offers by this approach in mechanics and thermodynamics are briefly discussed.

cond-mat.mtrl-sci↗
arXiv2017

Evidence of new twinning modes in magnesium questioning the shear paradigm

Twinning is an important deformation mode of hexagonal close-packed metals. The crystallographic theory is based on the 150-years old concept of simple shear. The habit plane of the twin is the shear plane, it is invariant. Here we present Electron BackScatter Diffraction observations and crystallographic analysis of a millimeter size twin in a magnesium single crystal whose straight habit plane, unambiguously determined both the parent crystal and in its twin, is not an invariant plane. This experimental evidence demonstrates that macroscopic deformation twinning can be obtained by a mechanism that is not a simple shear. Beside, this unconventional twin is often co-formed with a new conventional twin that exhibits the lowest shear magnitude ever reported in metals. The existence of unconventional twinning introduces a shift of paradigm and calls for the development of a new theory for the displacive transformations

cond-mat.mtrl-sci↗
arXiv2017

The (11-22) and (-12-16) twinning modes modelled by obliquity correction of a (58deg, a+2b) prototype stretch twin

The {11-22} and {11-26} twinning modes were recently put in evidence by Ostapovets et al. (Phil. Mag, 2017)and interpreted as {101-2}-{101-2} double-twins formed by a simultaneous action of two twinning shears. We propose another interpretation in which the twinning modes result from a one-step mechanism based on the same (58deg, a+2b) prototype stretch twin. . The two twins differ from the prototype twin by their obliquity correction. The results are compared with the classical theory of twinning and with Westlake-Rosenbaum model of {11-22} twinning. An unconventional twinning mode recently discovered in a magnesium single crystal based on the same prototype twin will be the subject of a separate publication.

cond-mat.mtrl-sci↗
arXiv2017

A crystallographic model of the {557} habit planes in low-carbon martensitic steels

Low-alloy steels are constituted of twenty-four variants of lath martensite that exhibit gradients of orientations from Kurdjumov-Sachs (KS) to Nishiyama-Wassermann (NW). They are structured into four packets on each of the common close-packed plane {111}fcc// {110}bcc; and each packet is composed of three blocks constituted by pairs of low-misoriented variants. The habit planes reported in literature for this type of martensite are {557}fcc, but it is not clear whether they correspond to the laths or to the blocks. In this paper, we present crystallographic calculations proving that the average of the two KS distortions associated with the variants in a block is exactly a NW distortion. A new method of averaging distortion matrices was introduced for this purpose. It is also shown that the {575}fcc planes are let untilted by this NW distortion, and are thus good theoretical candidates for the observed habit planes. The predicted {575}fcc planes, however, do not contain any of the common close-packed directions of the two variants in the block, which is in apparent contradiction with the current view. In order to clarify this point, some Electron BackScatter Diffraction (EBSD) maps were acquired on different low-carbon steels; the prior austenitic grains were automatically reconstructed and the traces of the habit planes predicted by the different models were analyzed and compared to the morphologies. This experimental work shows that {575}fcc planes are the habit plane of the blocks, and that the habit plane of one block often dominates the others, which impedes to discriminate the different models. The advantages of our model are its simplicity, the absence of fitting parameters, and the symmetric role played by the variants the blocks.

cond-mat.mtrl-sci↗
arXiv2016

Displacive model of deformation twinning in hexagonal close-packed metals. Case of the (90 deg, a) and (86 deg, a) extension twins in magnesium

A crystallographic displacive model is proposed for the extension twins in magnesium. It is based on a hard-sphere assumption previously used for martensitic transformations. The atomic displacements are established, and the homogeneous lattice distortion is analytically expressed as a continuous angular-distortive matrix that takes the usual form of shear when the distortion is complete. The calculations prove that a volume change of 3 percents occurs for the intermediate states and that the twinning plane, even if untilted and restored when the distortion is complete, is not fully invariant during the transient states. The crystallographic calculations also show that the (90 deg, a) twins observed in magnesium nano-pillars and the (86 deg, a) twins observed in bulk samples come from the same mechanism, the only difference being the existence of a slight obliquity angle (+/- 3.4 deg) required to reduce the strains in the latter case. Continuous features in the pole figures between the low-misoriented (86 deg, a) twin variants are expected; they are confirmed by EBSD maps acquired on a deformed magnesium single crystal. As the continuous mechanism of extension twinning is not a simple shear, a "virtual work" criterion using the value of the intermediate distortion matrix at the maximum volume change is proposed in place of the usual Schmid's law. It allows predicting the formation of extension twins for crystal orientations associated with negative Schmid factors.

cond-mat.mtrl-sci↗
arXiv2016

Crystallography of deformation twinning in hexagonal close-packed metals. Revisiting the case of the (56 deg, a) contraction twins in magnesium

Contraction twinning in magnesium alloys leads to new grains that are misoriented from the parent grain by a rotation of 56 deg around the a-axis. The classical theory of deformation twinning does not precise the atomic displacements and does not explain why contractions twinning is less frequent than extension twinning. The paper proposes a new model in the continuity of our previous works on martensitic transformations and extension twinning. A continuous angular distortion matrix that transforms the initial hcp crystal into a final hcp crystal is determined such that the atoms move as hard spheres and reach the final positions expected by the orientation relationship. The calculations prove that the distortion is not a simple shear when it is considered in its continuity. The (01-11) twin plane is untilted and restored, but it is not fully invariant because some interatomic distances in this plane evolve during the distortion process; the unit volume also increases up to 5% before coming back to its initial value when the twinning distortion is complete. Then, the distortion takes the form a simple shear on the twin plane with a shear direction along the direction [18,-5,-5] and a shear amplitude of 0.358. It is the first time that this twinning mode is reported. Experiments are proposed to validate or infirm the new model.

cond-mat.mtrl-sci↗
arXiv2015

Lattice distortion and atomic displacements during the fcc/bcc martensitic transformation

From our previous models of martensitic transformation, the continuous matrices of atomic displacements and lattice deformations from face-centred-cubic (fcc) to body centred-cubic (bcc) phases are calculated in agreement with different possible final orientation relationships, such as Bain, Pitsch and Kurdjumov-Sachs (KS). The angular distortion introduced in the calculations appears a natural order parameter of transition. The distortion corresponding to KS is the only one that respects the parallelism of a dense direction and of a dense plane of both the fcc and bcc phases. This paper gives an alternative to the classical crystallographic theories and shear concepts associated to martensitic transformations.

cond-mat.mtrl-sci↗
arXiv2015

Continuous atomic displacements and lattice distortions during martensitic transformations in fcc-bcc-hcp systems

This work generalizes our previous works on fcc-bcc martensitic transformations to the larger family of transformations in the fcc-bcc-hcp system and to fcc-fcc mechanical twinning. The analytical expressions of the atomic displacements and lattice distortions are calculated directly from the orientation relationships without any adjustment of free parameter; the unique assumption is that the atoms are hard-spheres that cannot interpenetrate themselves. The habit planes are predicted on the simple criterion that they are unrotated by the distortion, and the results are compared to experimental observations published in literature. It is shown that shuffle is required for transformations implying the hcp phase because the hcp primitive Bravais lattice contains two atoms, instead of one for the fcc and bcc phases. A simple encoding of the lattice distortions and shuffles permits to attribute a groupoid structure to the transformations in the fcc-hcp-bcc system. The analytical expressions of the intermediate states are given and could be used to calculate activation energies. The martensitic distortion occurs in one-step, without shearing, and its accommodation generates orientation gradients in the parent phase, independently of its glide modes. The concept of reversibility is detailed on the basis of crystallographic and morphological arguments. The possibility to apply this approach to diffusion-limited martensitic transformations is discussed.

cond-mat.mtrl-sci↗