Characterization of random stress fields obtained from polycrystalline aggregate calculations using multi-scale stochastic finite elements

The spatial variability of stress fields resulting from polycrystalline aggregate calculations involving random grain geometry and crystal orientations is investigated. A periodogram-based method is proposed to identify the properties of homogeneous Gaussian random fields (power spectral density and related covariance structure). Based on a set of finite element polycrystalline aggregate calculations the properties of the maximal principal stress field are identified. Two cases are considered, using either a fixed or random grain geometry. The stability of the method w.r.t the number of samples and the load level (up to 3.5 % macroscopic deformation) is investigated

Statistical inference of 2D random stress fields obtained from polycrystalline aggregate calculations

International audienceThe spatial variability of stress fields resulting from polycrystalline aggregate calculations involving random grain geometry and crystal orientations is investigated. A periodogram-based method is proposed to identify the properties of homogeneous Gaussian random fields (power spectral density and related covariance structure). Based on a set of finite element polycrystalline aggregate calculations the properties of the maximal principal stress field are identified. Two cases are considered, using either a fixed or random grain geometry. The stability of the method w.r.t the number of samples and the load level (up to 3.5 \% macroscopic deformation) is investigated

Ensemble averaging stress-strain fields in polycrystalline aggregates with a constrained surface microstructure-Part 1: Computational tools and application to anisotropic elastic behaviour

International audienceThe effect of three-dimensional grain morphology on the deformation at a free surface in polycrystalline aggregates is investigated by means of a large scale finite element and statistical approach. For a given 2D surface at z=0 containing 39 grains with given lattice orientations, 17 random 3D polycrystalline aggregates are constructed having different 3D grain shapes and orientations except at z=0, based on an original 3D image analysis procedure. They are subjected to overall tensile loading conditions. The resulting stress-strain fields at the free surface z=0 are analysed. Ensemble average and variance maps of the stress field at the observed surface are computed. In the case of an anisotropic elastic behaviour of the grains, fluctuations ranging between 5% and 60% are found in the equivalent stress level at a given material point of the observed surface from one realization of the microstructure to another. The obtained fields are compared to the prediction based on the associated columnar grain microstructure, often used in literature. These results have important implications in the way of comparing finite element simulations and surface strain field measurements in metal polycrystals

Ensemble averaging stress-strain fields in polycrystalline aggregates with a constrained surface microstructure-Part 2 : Crystal plasticity

International audienceThe effect of three-dimensional grain morphology on the deformation at a free surface in polycrystalline aggregates is investigated by means of a large scale finite element and statistical approach. For a given 2D surface at z=0 containing 39 grains with given lattice orientations, eight 3D random polycrystalline aggregates are constructed having different 3D grain shapes and orientations except at z=0, based on an original 3D image analysis procedure. They are subjected to overall tensile loading conditions. The continuum crystal plasticity framework is adopted and the resulting plastic strain fields at the free surface z=0 are analysed. Ensemble average and variance maps of the plastic strain field at the observed free surface are computed. In the case of elastoplastic copper grains, fluctuations ranging between 2% and 80% are found in the equivalent plastic slip level at a given material point of the observed surface from one realization of the microstructure to another. The obtained fields are compared to the prediction based on the associated columnar grain microstructure, often used in literature. The presented results have important implications in the way of comparing finite element simulations and experimental strain or lattice rotation field measurements in metal polycrystals

Cartan's spiral staircase in physics and, in particular, in the gauge theory of dislocations

In 1922, Cartan introduced in differential geometry, besides the Riemannian curvature, the new concept of torsion. He visualized a homogeneous and isotropic distribution of torsion in three dimensions (3d) by the "helical staircase", which he constructed by starting from a 3d Euclidean space and by defining a new connection via helical motions. We describe this geometric procedure in detail and define the corresponding connection and the torsion. The interdisciplinary nature of this subject is already evident from Cartan's discussion, since he argued - but never proved - that the helical staircase should correspond to a continuum with constant pressure and constant internal torque. We discuss where in physics the helical staircase is realized: (i) In the continuum mechanics of Cosserat media, (ii) in (fairly speculative) 3d theories of gravity, namely a) in 3d Einstein-Cartan gravity - this is Cartan's case of constant pressure and constant intrinsic torque - and b) in 3d Poincare gauge theory with the Mielke-Baekler Lagrangian, and, eventually, (iii) in the gauge field theory of dislocations of Lazar et al., as we prove for the first time by arranging a suitable distribution of screw dislocations. Our main emphasis is on the discussion of dislocation field theory.Comment: 31 pages, 8 figure

Twin nucleation and variant selection in Mg alloys: An integrated crystal plasticity modelling and experimental approach

Extension twin nucleation and variant selection in magnesium alloy WE43 is investigated in experimentally characterised and deformed microstructures replicated in crystal plasticity models. Total stored (dislocation) energy density is found to identify the experimentally observed locations of twins which are not otherwise explained by global Schmid factors or local resolved shear stress criteria. A critical total stored energy of the order 0.015 Jm-2 is determined below which twin nucleation does not occur. The total stored energy density explains the locations of the observed twins and the absence of twins in parent grains anticipated to be favourable for twin nucleation. Twin variant selection has been shown to be driven by minimising locally stored shear energy density, while the geometric compatibility and strain compatibility factors only aid in partial prediction. All experimentally observed variants were correctly determined

Effet de la morphologie tri-dimensionnelle et de la taille de grain sur le comportement mécanique d'agrégats polycristallins

Les modèles continus de plasticité cristalline appliqués aux calculs de polycristaux métalliques sont efficaces pour voir le comportement mécanique global du polycristal, à partir des lois de comportement du monocristal. On obtient ainsi également les champs de contraintes et déformations locaux dans les grains. Ceci est rendu possible à l'aide de simulations par éléments finis sur un volume élémentaire représentatif d'agrégat et les modèles d'homogénéisation numérique. Ces approches échouent néanmoins pour décrire les effets d'échelle, classiquement observés en métallurgie physique et dont l'archétype est l'effet de taille de grain. Un modèle de plasticité cristalline de Cosserat appliqué dans le cas du comportement élastoplastique d'aciers IF ferritiques est proposé dans ce travail. Il introduit dans sa formulation une loi de durcissement supplémentaire associé à la courbure de réseau. La simulation d'agrégats polycristallins permet de reproduire numériquement un effet analogue à la loi de Hall-Petch. Une autre limite de la plasticité cristalline est liée à l'étape clé de validation expérimentale locale. L'information expérimentale est, en général, disponible à la surface de l'éprouvette.La géométrie des grains sous la surface est cependant inconnue. Cette information est le plus souvent introduite dans les calculs par extension des joints de grains perpendiculairement à la surface. L'écart, fréquemment observé entre les résultats du calcul et les résultats expérimentaux, peut être expliqué par l'erreur qu'introduit ce choix. On donne ici un minorant de cette erreur en considérant plusieurs agrégats ayant la même morphologie granulaire à la surface libre mais des morphologies tridimensionnelles distinctes. En élasticité la dispersion des contraintes, en un point donné de la surface, avec différentes morphologies de grains sous-jacents est de l'ordre de 30%. En élastoplasticité la dispersion peut aisément atteindre 50% de la valeur de la contrainte, ce qui amène à considérer avec prudence l'identification d'une loi de comportement à partir des seules mesures de surface

Effet de la mophologie tri-dimensionnelle et de la taille de grain sur le comportement mécanique d agrégats polycristallins

PARIS-MINES ParisTech (751062310) / SudocSudocFranceF

Characterization of random stress fields obtained from polycrystalline aggregate calculations using multi-scale stochastic finite elements

The spatial variability of stress fields resulting from polycrystalline aggregate calculations involving random grain geometry and crystal orientations is investigated. A periodogram-based method is proposed to identify the properties of homogeneous Gaussian random fields (power spectral density and related covariance structure). Based on a set of finite element polycrystalline aggregate calculations the properties of the maximal principal stress field are identified. Two cases are considered, using either a fixed or random grain geometry. The stability of the method w.r.t the number of samples and the load level (up to 3.5% macroscopic deformation) is investigated

Characterization of random stress fields obtained from polycrystalline aggregate calculations using multi-scale stochastic finite elements

The spatial variability of stress fields resulting from polycrystalline aggregate calculations involving random grain geometry and crystal orientations is investigated. A periodogram-based method is proposed to identify the properties of homogeneous Gaussian random fields (power spectral density and related covariance structure). Based on a set of finite element polycrystalline aggregate calculations the properties of the maximal principal stress field are identified. Two cases are considered, using either a fixed or random grain geometry. The stability of the method w.r.t the number of samples and the load level (up to 3.5% macroscopic deformation) is investigated.ISSN:2095-2430ISSN:2095-244