Towards gigantic RVE sizes for 3D stochastic fibrous networks

The size of representative volume element (RVE) for 3D stochastic fibrous media is investigated. A statistical RVE size determination method is applied to a specific model of random microstructure: Poisson fibers. The definition of RVE size is related to the concept of integral range. What happens in microstructures exhibiting an infinite integral range? Computational homogenization for thermal and elastic properties is performed through finite elements, over hundreds of realizations of the stochastic microstructural model, using uniform and mixed boundary conditions. The generated data undergoes statistical treatment, from which gigantic RVE sizes emerge. The method used for determining RVE sizes was found to be operational, even for pathological media, i.e., with infinite integral range, interconnected percolating porous phase and infinite contrast of propertie

Some links between Cosserat, strain gradient crystal plasticity and the statistical theory of dislocations

International audienceA link is established between a phenomenological Cosserat model of crystal plasticity due to (Forest et al. 1997) and recent results obtained in the statistical theory of dislocations by (Groma et al. 2003). The existence of a back-stress related to the divergence of the couple stress tensor is derived. According to several dislocation--based models of single slip, the kinematic hardening modulus is found to be inversely proportional to dislocation density. Phenomenological extensions to multislip situations can be proposed based on these generalized continuum approaches

Construction d'opérateurs de régularisation à partir de l'approche micromorphe de la plasticité et de l'endommagement à gradient

International audienceThe micromorphic approach to gradient plasticity and damage is used to construct new regularisation operators arising in the mechanics of non linear material laws or large deformations.L’approche micromorphe de la plasticité et de l’endommagement à gradient est utilisée pour produire de nouveaux opérateurs de régularisation dans le contexte de la mécanique non linéaire géométrique ou matérielle

Thermodynamical Frameworks for Higher Grade Material Theories with Internal Variables or Additional Degrees of Freedom

Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG geförderten) Allianz- bzw. Nationallizenz frei zugänglich.This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.The objective of the present work is to compare several thermomechanical frameworks, taking into account the influence of strain gradient, internal variables, gradient of internal variables, and temperature gradient on the constitutive behavior of materials. In particular, the restrictions by the second law of thermodynamics are derived. The method of exploitation consists of two steps: an application of the well-known method by Liu and a new method of exploiting the residual inequality. The first example introduces an enlarged set of variables for the constitutive functions including in particular the strain gradient, an internal variable, its gradient, and the temperature gradient. In the second example, the power of internal forces is enriched to incorporate generalized stress measures. In the third example, the classical thermomechanical setting is complemented by a balance-type differential equation for an additional variable. Finally, material theories of grade n are envisaged. It is shown that the free energy density may depend on gradients only in the case that an additional balance equation is introduced. We also demonstrate that for isotropic materials the second law of thermodynamics implies for a large class of state spaces that the entropy flux equals the heat flux divided by temperature

Le calcul de structures en présence de vieillissement statique ou/et dynamique d'alliages métalliques

Colloque avec actes et comité de lecture. Internationale.International audienceLes phénomènes de Portevin Le Chatelier (effet PLC) et de Lüders sont très fréquents, dans les aciers et les alliages d'aluminium. Ils se présentent sous la forme d'instabilités sur la courbe de traction (Jaoul, 2008) et sont le résultat de l'interaction d'atomes de solutés avec les dislocations mobiles ou immobilisées au sein du matériau. Ils se manifestent lorsque les constantes de temps de la diffusion des solutés et de la plasticité microscopique sont du même ordre de grandeurs. Ils ont fait l'objet d'une quantité considérable de travaux en métallurgie physique dans les 60 dernières années, conduisant à des modélisations physiques satisfaisantes. Au contraire, les ingénieurs en calculs de structures ne prennent généralement pas ces phénomènes en compte dans l'établissement de la loi de comportement, préférant introduire un lissage des courbes de traction ou des courbes minimales. Une croyance commune consiste à penser que l'effet PLC ou le pic de Lüders n'existent que pour les essais de traction en raison de l'´etat apparemment homogène de contrainte ou de déformation imposée. En fait, les instabilités viscoplastiques de type PLC se produisent également dans les zones de concentration de contraintes que sont les perforations, les inclusions et les fissures. Si le développement des bandes de localisation reste confiné dans une telle zone de l'´eprouvette, on n'observe pas nécessairement de perturbations sur la courbe de charge globale. Les effets de la localisation de la déformation existent toutefois bel et bien dans certaines zones de la pièce. Le vieillissement dynamique ne se caractérise pas nécessairement par la présence d'irrégularités sur la courbe de traction. Il peut se traduire également par une sensibilité faible voire négative à la vitesse de déformation dans un certain domaine de température. Il se manifeste également de manière plus insidieuse par des réponses en fluage ou en relaxation inattendues. En fait, la plupart des alliages métalliques et même certains métaux purs sont concernés dans un certain domaine de température et de vitesse : alliages de titane et de zirconium, superalliages à base de nickel ou de cobalt, tantale, etc. Il est donc devenu urgent de le prendre en compte en calcul des structures de façon à comprendre ou éviter certaines fragilisations apparemment inexplicables.Le calcul de structures en présence de vieillissement statique ou/et dynamique d'alliages métallique

Coherence of Gray Categories via Rewriting

Over the recent years, the theory of rewriting has been extended in order to provide systematic techniques to show coherence results for strict higher categories. Here, we investigate a further generalization to low-dimensional weak categories, and consider in details the first non-trivial case: presentations of tricategories. By a general result, those are equivalent to the stricter Gray categories, for which we introduce a notion of rewriting system, as well as associated tools: critical pairs, termination orders, etc. We show that a finite rewriting system admits a finite number of critical pairs and, as a variant of Newman\u27s lemma in our context, that a convergent rewriting system is coherent, meaning that two parallel 3-cells are necessarily equal. This is illustrated on rewriting systems corresponding to various well-known structures in the context of Gray categories (monoids, adjunctions, Frobenius monoids). Finally, we discuss generalizations in arbitrary dimension

The role of the fluctuation field in higher order homogenization

International audienceHomogenization methods aim at replacing a composite material by a homogeneous equivalent medium endowed by effective properties [1]. When the size of the heterogeneities (inclusions, grains...) is much smaller than the wave length of the variation of the macroscopic stress and strain fields, i.e. under the assumption of separation of scales, the effective properties of a composite with periodic microstructures can be determined by considering a unit cell V and applying the following loading conditions: u i = E ij x j + v i , ∀x ∈ V with v i (x +) = v i (x −), t(x +) = −t(x −) (1) The displacement vector inside V is u. The constant tensor E ij is the average strain applied to the unit cell and v is a periodic fluctuation, meaning that it takes the same value at homologous points (x − , x +) of the boundary ∂V of the unit cell. The traction vector t = σ · n takes opposite values at homologous points of the boundary with normal vector n. When the macroscopic medium is subjected to slowly–varying mean fields so that strain gradients may influence the material response, the previous homogenization scheme must be extended. For that purpose, several authors have proposed to replace the composite material by a generalized continuum, like a Cosserat medium in [2], a couple stress continuum in [3], a strain gradient model in [4] and a micromorphic model in [5]. 1 Non–homogeneous loading conditions of the unit cell The identification of the generalized continuum model in the mentioned contributions goes through the extension of the classical loading conditions (1) to non–homogeneous conditions: u i = E ij + D ijk x j x k + v i , ∀x ∈ V (2) where D ijk is a constant third rank tensor, symmetric with respect to the two last indices. The coefficients of the quadratic polynomial can be related to the generalized strain measures of the effective continuum but, in this note, the attention is focused on the properties of the fluctuation v. In [5], the fluctuation is assumed to vanish, which may lead to the prediction of too stiff effective elastic properties. In [2], the fluctuation was considered periodic as in Eq. (1), which is generally not a valid assumption for such quadratic boundary conditions as discussed in [6]. In [4], the periodicity requirement for v is relaxed by an heuristic integral condition but the traction vector is still assumed to be anti–periodic, which cannot be expected in general because of the existence of stress gradient induced by the quadratic conditions. The objective of this note is to provide a precise characterization of the real fluctuation field under quadratic boundary conditions by means of a computational homogenization approach. 2 Determination of the fluctuation field For that purpose, numerical simulations have been performed for a composite material made of an elastic grid with soft square inclusions under plane strain conditions, see the unit cell of figure 1(a). The elastic properties of the grid and inclusion materials respectively are (E = 200000 MPa, ν = 0.3) and (E = 20000 MPa, ν = 0.3). The edge length of the unit cell is 1 mm. The quadratic boundary conditions (2) where applied to the boundary of one single unit cell, but also to on ensemble of cells containing 3×3 (see 1(b)), 5×5, up to 9 × 9 cells. Fixed values of the coefficients D ijk were chosen and the fluctuation v was set to zero at the outer boundary of the N × N ensemble of cells. The normalized elastic energy fields σ(x) : ε(x)/W with W = V9×9 are drawn in figure 1, where V N ×N denotes spatial averaging over the central unit cell of the N × N set of cells. The only non–vanishing component of the polynomial was D 112 = 1 mm −1. Strong boundary layer effects arise close to the outer boundary where the Dirichlet conditions are applied. In the central cell a deviation from the polynomial field appears that is such that its boundaries do not remain straight lines during deformation. We find that the elastic energy distribution in the central unit cell and the fluctuation v at its boundary converge toward fixed fields when the number of cells N increases. This means that there exists a representative volume element size for quadratic boundary conditions. For the considered material, convergence was reache

Large scale finite element simulations of polycrystalline aggregates: applications to X-ray diffraction and imaging for fatigue metal behaviour

International audienceLarge scale finite element simulations of the elastoviscoplastic behaviour of polycrystalline aggregates have become a standard technique to study the stress-strain heterogeneities that develop in grains during deformation. For a long time, comparison between continuum crystal plasticity and experimental field measurements was confined to the observation of surface behaviour. As for example the study of the development of intense deformation bands at the free surface of a polycrystal. Recent 3D experimental techniques open new perspectives in computational crystal plasticity. After reviewing how to define a representative volume element for polycrystal properties and showing that actual 3D computations, including grain shapes and orientations, are really needed to accurately determine the stress and strains distributions, two examples of applications of large scale simulations are described in this paper. First the simulation of 3D coherent X-ray diffraction in a polycrystalline gold sample is detailed. Based on the real geometry of the grains and their columnar nature, a 3D avatar is reconstructed. FE computations are then carried out to evaluate the effect of mechanical and thermal strain of the diffraction pattern resolved in the reciprocal space by complex FFT. Qualitative comparison with the experimental diffraction patterns shows that such computations can help understand the true nature of strain heterogeneities within the material. The second example of application deals with short fatigue crack propagation in polycrystals. One fundamental problem caused by short fatigue cracks is that despite decades of research, so far no reliable prediction of the crack propagation rates, comparable to the well-known Paris law in the long crack regime, could be established. This ``anomalous'' behaviour of short cracks is commonly attributed to factors like their complex three dimensional shapes and the influence of the local crystallographic environment affecting their propagation behaviour via a combination of physical mechanisms. Crystal plasticity computations based on the real grain shapes and orientations obtained thanks to diffraction contrast tomography are carried out using an ideal crack shape. The stress concentration at the crack tip is analysed with respect to possible crack growth directions