Resolution of the Wicksell's equation by Minimum Distance Estimation

The estimation of the grain size in granular materials is usually performed by 2Dobservations. Unfolding the grain size distribution from apparent 2D sizes is commonly referred as the corpuscle problem. For spherical particles, the distribution of the apparent size can be related to that of the actual size thanks to the Wicksell’s equation. The Saltikov method, which is based on Wicksell’s equation, is the most widely used method for resolving corpuscle problems. This method is recursive and works on the finite histogram of the grain size. In this paper, we propose an algorithm based on a minimizing procedure to numerically solve the Wicksell’s equation, assuming a parametric model for the distribution (e.g. lognormal distribution). This algorithm is applied on real material and the results are compared to those found using Saltikov or Saltikov-basedstereology techniques. A criterion is proposed for choosing the number of bins in the Saltikov method. The accuracy of the proposed algorithm, depending on the sample size, is studied

Caractérisation et modélisation du procédé de fluotournage inverse à froid du TA6V

Flowforming is a means to produce seamless tubes by plastic deformation at room temperature. It consists in a tubular part, mounted on a mandrel, and a couple of rollers. While the mandrel is rotating, the roller translate along the tube axis. Because of the conical shape of the roller, the tube thickness reduces by cold plastic deformation, thus the tube elongates. Thanks to the high compressive stresses and to the incremental nature of the deformation process, flowforming can lead to a high thickness reduction and thus to high elongation of the deformed tubes. Ti-6Al-4V (also known as Ti64) is an α-β titanium alloy. Because of its high strength and its low density, Ti64 is the most widely used titanium alloy in aerospace industry. Nevertheless, flowforming of this alloy is a great challenge, considering its well known poor ductility at room temperature. In order to increase the "tube spinnability" of Ti64, first the flowformed material has been extensively investigated at microscopic scale (SEM, EBSD) as well as at macroscopic scale (mechanical properties). In addition, an experimental flowforming machine, equipped with load sensors, has been designed. It has provided valuable informations about the strain mechanism occuring during flowforming; furthermore, this experimental set-up has allowed to buid an adequate constitutive law for the plastic flow. Then, this law has been used to optimize the processing parameter, in order to avoid failure during the process.Le fluotournage est un procédé de fabrication de tubes sans soudures par déformation plastique à froid. Il consiste à réduire l'épaisseur d'une préforme tubulaire montée sur un mandrin en utilisant plusieurs molettes en translation parallèlement au tube. Grâce à la conicité des molettes, l'épaisseur du tube diminue, donc celui-ci s'allonge dans la direction de fluotournage. Le TA6V est un alliage de titane de type α-β. Ses très bonnes propriétés mécaniques et sa faible densité en font l'alliage de titane le plus utilisé dans l'industrie aéronautique. Toutefois, le fluotournage d'un tel alliage représente un vrai défi, car il est connu pour avoir une faible ductilité à température ambiante. Afin d'optimiser les conditions opératoires, permettant ainsi de très grandes déformations du TA6V, le matériau fluotourné a d'abord été analysé, tant du point de vue microscopique (MEB, EBSD) que macroscopique (propriétés mécaniques). De plus, un fluotour expérimental instrumenté a été conçu afin d'analyser les mécanismes de déformation lors de la mise en forme. Celui-ci a permis d'identifier les phénomènes microstructuraux à l'origine de la déformation et de déterminer une loi de comportement valide pour la simulation numérique. Cette dernière a enfin servi à l'optimisation des conditions opératoires

Calculs Éléments Finis à l’échelle des grains depuis des données EBSD

For the sake of understanding the behaviour of a polycrystal at its grain scale, Finite Element (FE) numerical simulations can be performed, taking into account the physical properties of each grain. Those simulations must take into account the phase heterogeneities, the local anisotropies and the crystalline orientations as well. In addition, the actual grain morphologies must be well described in order to model localization phenomena. An algorithm, named MTEX2Gmsh, is proposed to automatically generate an EF mesh from EBSD data. The resulting mesh gives smooth and accurate descriptions of the grain boundaries together with reduced Degrees of Freedom (DoF), hence limited computational times. Using this algorithm, the fragile-elastic behaviour of a ceramic and micro-plasticity phenomena in a nitrided steel are studied as application cases. The latter is based on a crystal-plasticity model for the ferrite phase

MTEX2Gmsh: a tool for generating 2D meshes from EBSD data

In material sciences applied to crystalline materials, such as metals or ceramics, the grain morphology (size and shape) and the crystallographic texture are of great importance for understanding the macroscopic behaviour of the materials. Micromechanics of polycrystalline aggregates consists in evaluating the thermo-mechanical behaviour of the aggregates at their grain scale. If the investigated material is subjected to macroscopic deformation, the local strain can be obtained either experimentally, thanks to full-field measurement methods such as microgrid technique or Digital Image Correlation (DIC), or thanks to numerical simulation of the microstructure. The latter needs to take into account the mechanical heterogeneities (due to the different constituents) and the anisotropy of each phase, depending on its crystalline orientation. Orientation Imaging Microscopy (OIM), usually made from Electron Backscatter Diffraction (EBSD), is now widely used as a characterization technique. Indeed, it is in great interest for investigating the grain morphology and local crystal orientations in crystalline materials. Raw EBSD data can be considered as matrices of measurements of crystallographic data: each dot contains information about the phase and its orientation at the corresponding position. In order to perform Finite Element Analysis (FEA) on a polycrystal, one needs to first generate a mesh based on either EBSD or reconstructed grains. In this mesh, the Grain boundaries (GBs) must be accurately described since they play an important role in the overall behaviour of aggregates. To the best authors' knowledge, it appears that no existing tool for generating meshes from EBSD data is able to provide a robust grain description (e.g. suitable for any kind of phase and geometry) together with customizable features (e.g. variable element sizes). The proposed software, named MTEX2Gmsh works regardless the number of phases and the symmetries of those phases. In addition, it provides a smooth and accurate definition of the GBs. It is based on the MTEX toolbox for Matlab and the Gmsh software. MTEX2Gmsh allows to mesh the volume with a couple of options, such as: - increasing element size with increasing distance from the grains boundaries; - element type (tetrahedron, wedge or brick elements); - nesting the Region of Interest (ROI) into a larger medium

Submicrocristalline structure and dynamic recovery of cold flowformed ELI grade Ti-6Al-4V

Flowforming is a means to produce seamless tubes by plastic deformation at room temperature. It consists in reducing the thickness of a tubular part mounted on a mandrel by deforming it using several rollers translating along the tube axis, while the tube is rotating along its axis. Thanks to the high compressive stresses, and to the incremental nature of the deformation process, flowforming can lead to a high thickness reduction and thus to high elongation of the deformed tubes. Ti-6Al-4V (Extra Low Interstitial grade) tubes have been deformed by cold flowforming, with a thickness reduction ratio higher than 60%, and their microstructures have been investigated using light optical microscopy (LOM), scanning electron microscopy (SEM) and electron backscattered diffraction (EBSD). Based on EBSD data, a post-processing analysis has been performed in order to study the texture of the flowformed parts. Optical Microscopy showed that the material could be deformed without displaying flow instability such as adiabatic shear banding, despite the fact that it has been processed out of the stable processing maps (high strain rate and low temperature). It also evidenced a major deformation along the tube axis accompanied with a slight twist due to torsion stress. EBSD analysis indicated the occurrence of continuous dynamic recrystallization, which is rarely reported in the α-β domain of such alloys. The recovery/ recrystallization effects resulted in a submicrocrystalline equiaxed structure, which is consistent with that previously reported for Ti-6Al-4V subjected to severe plastic deformation (SPD). The texture of the hexagonal α-phase appeared to be similar to that obtained on extruded Ti-6Al-4V, with a basal component perpendicular to the tube axisAgence Nationale pour la Recherche (ANR

On the damage criteria and their critical values for flowforming of ELI grade Ti64

Cold flowforming is a chipless forming process that deforms tubular parts by reducing their outer diameter and thickness while increasing their length. It consists of a rotating mandrel and one or more rollers that are translated along the tube axis, thus plastically deforming it. Flowforming of Ti-6Al-4V (also known as Ti64) is of great interest for improving the mechanical properties of the material, such as yield stress and fatigue strength. However this alloy is known to have poor ductility at room temperature. Therefore, flowforming of Ti64 without failure or crack is a great challenge. In this present paper, the authors have attempted to predict the different failure modes occurring during flowforming. An experimental machine has been built at the Center for Material Forming (CEMEF) in order to monitor the force on the single roller, the torque on the mandrel and the actual rotation speed of the roller as well. Numerous flowforming tests have been performed using different processing parameters, such as working depth, roller feed and initial geometry, in order to investigate the critical values which lead to the failure of the flowformed tube. In addition, numerical simulations of the process have been performed using the FORGE FEM solver. The results of the simulations have been used to evaluate the relevance of usual failure criteria (Crockford-Latham, Rice-Tracey and Oyane).Agence Nationale pour la Recherche (ANR

Radical Voronoï tessellation from random pack of polydisperse spheres: Prediction of the cells’ size distribution

This paper investigates the relevance of the representation of polycrystalline aggregates using Radical Vorono (RV) tessellation, computed from Random Close Packs (RCP) of spheres with radius distribution following a lognormal distribution. A continuous relationship between the distribution of sphere radii with that of RV cell volumes is proposed. The stereology problem (deriving the 3D grain size distributions from 2D sections) is also investigated: two statistical methods are proposed, giving analytical continuous relationships between the apparent grain size distribution and the sphere radius distribution. In order to assess the proposed methods, a 3D aggregate has been generated based on a EBSD map of a real polycrystalline microstructure

Thermomechanical modelling of a NiTi SMA sample submitted to displacement-controlled tensile test

AbstractShape Memory Alloys (SMAs) undergo an austenite–martensite solid–solid phase transformation which confers its pseudo-elastic and shape memory behaviours. Phase transformation can be induced either by stress or temperature changes. That indicates a strong thermo-mechanical coupling. Tensile test is one of the most popular mechanical test, allowing an easy observation of this coupling: transformation bands appear and enlarge giving rise to a large amount of heat and strain localisation. We demonstrate that the number of transformation bands is strongly associated with the strain rate. Recent progress in full field measurement techniques have provided accurate observations and consequently a better understanding of strain and heat generation and diffusion in SMAs. These experiments bring us to suggest the creation of a new one-dimensional thermomechanical modelling of the pseudo-elastic behaviour. It is used to simulate the heat rise, strain localisation and thermal evolution of the NiTi SMA sample submitted to tensile loading

Thermomechanical modelling of a NiTi SMA sample submitted to displacement-controlled tensile test

Shape Memory Alloys (SMAs) undergo an austenite–martensite solid–solid phase transformation which confers its pseudo-elastic and shape memory behaviours. Phase transformation can be induced either by stress or temperature changes. That indicates a strong thermo-mechanical coupling. Tensile test is one of the most popular mechanical test, allowing an easy observation of this coupling: transformation bands appear and enlarge giving rise to a large amount of heat and strain localisation. We demonstrate that the number of transformation bands is strongly associated with the strain rate. Recent progress in full field measurement techniques have provided accurate observations and consequently a better understanding of strain and heat generation and diffusion in SMAs. These experiments bring us to suggest the creation of a new one-dimensional thermomechanical modelling of the pseudo-elastic behaviour. It is used to simulate the heat rise, strain localisation and thermal evolution of the NiTi SMA sample submitted to tensile loading

Modélisation polycristalline du comportement d’un Alliage `a M´emoire de Forme (AMF) de type Ni-Ti sous sollicitations multiaxiales

Les propriétés des AMF sont dues à une transformation de phase solide solide réversible induite par la contrainte ou la température. Pour les AMF de type Ni49,75%at-Ti, c’est une transformation isochore qui, d’une phase cubique (austénite), forme une phase monoclinique (martensite) avec 24 orientations possibles (variantes). A partir des géométries locales et des énergies par variante, nous proposons un modèle polycristallin multiaxial qui rend compte du fort couplage thermomécanique lors de la transformation de phase. Des essais quasi statiques permettent de le valider