Homogénéisation des matériaux hétérogènes élastoviscoplastiques basée sur la technique des « champs translatés » : extension « affine » au cas non linéaire pour des composites biphasés

Dans cette contribution, on passe tout d’abord en revue les principales étapes de la technique à « champs translatés » pour déterminer le comportement effectif de matériaux élasto-viscoplastiques dont les phases suivent un couplage spatio-temporel de type Maxwellien et dont le comportement est supposé dans un premier temps linéaire. L’application de l’approche à « champs translatés » au problème de l’inclusion d’Eshelby viscoélastique linéaire est également présentée dans cette première partie. Le traitement de cas particuliers montre la pertinence de l’approche en viscoélasticité linéaire par rapport aux solutions obtenues classiquement par transformées de Laplace-Carson. Ensuite, l’extension de la méthode à « champs translatés » au comportement local élasto-viscoplastique avec une viscoplasticité non linéaire est résolue par le biais d’une linéarisation du comportement viscoplastique des phases de type « affine ». Cette extension couplée à un schéma d’homogénéisation à champs moyens (Mori-Tanaka ou schéma autocohérent) pour le problème hétérogène élastoviscoplastique donne une nouvelle loi d’interaction qui contient les interactions mécaniques entre les champs moyens par phase et les grandeurs macroscopiques. Dans le but de situer la validité de l’approximation à « champs moyens », les réponses mécaniques du modèle sont reportées pour des composites biphasés et sont comparées aux résultats d’autres approches d’homogénéisation de type analytique ou numériques existants dans la littérature

A time-incremental Eshelby-based homogenization scheme for viscoelastic heterogeneous materials

International audienceA time-incremental Eshelby-based homogenization scheme for Maxwellian heterogeneous materials is proposed and discussed. This is based on the exact solution of the heterogeneous Eshelby ellipsoidal inclusion problem obtained in the time domain. In contrast with hereditary methods, the effective behavior as well as the evolution laws of the averaged stresses per phase are solved incrementally in the time domain without need to inverse Laplace or Laplace-Carson transforms. This is made through a time-differential equation to exactly solve a volume term in the integral equation that was generally approximated in previous internal variable methods. The present formulation works for any arbitrary anisotropic ellipsoidal Maxwellian inclusion embedded in an isotropic Maxwellian matrix without any other restrictive assumptions. In order to show the interest of the present approach, a Mori-Tanaka homogenization scheme is applied to two-phase composites using the developed strain rate concentration equations. The results are reported and discussed in comparisons with other existing methods, including hereditary approaches and more recent internal variable approaches, in order to show the efficiency of the present time-incremental homogenization scheme

Fast Fourier transform-based micromechanics of interfacial line defects in crystalline materials

International audienceSpectral methods using Fast Fourier Transform (FFT) algorithms have recently seen a surge in interest in the mechanics community. The present contribution addresses the critical question of determining local mechanical fields using the FFT method in the presence of interfacial defects. Precisely, the present work introduces a numerical approach based on intrinsic discrete Fourier transforms for the simultaneous treatment of material discontinuities arising from the presence of disclinations, i.e., rotational discontinuities, and inhomogeneities. A centered finite difference scheme for differential rules are first used for numerically solving the Poisson-type equations in the Fourier space to get the incompatible elastic fields due to disclinations and dislocations. Second, centered finite differences on a rotated grid are chosen for the computation of the modified Fourier-Green's operator in the Lippmann-Schwinger-Dyson type equation for heterogeneous media. Elastic fields of disclination dipole distributions interacting with inhomogeneities of varying stiffnesses, grain boundaries seen as DSUM (Disclina-tion Structural Unit Model), grain boundary disconnection defects and phase boundary "terraces" in anisotropic bi-materials are numerically computed as applications of the method

Des bi-cristaux aux inclusions sphériques : contraintes internes en présence de gradients de déformation plastique

Les solutions analytiques des contraintes dans un bi-cristal infini avec interface plane et plasticité uniforme par morceaux sont tout d’abord étendues au cas où des variations de distorsion plastique le long de la normale à l’interface sont autorisées. Il est ensuite montré que le champ de contrainte associé au problème de l’inclusion sphérique d’Eshelby peut être retrouvé par une sommation appropriée des solutions précédentes. Cette méthode de sommation se généralise pour traiter les problèmes d’inclusion où la déformation libre de contraintes (ou “eigenstrain”) est non-uniformément spatialement distribuée. En particulier, la méthode est adaptée pour aborder aisément les problèmes de déformation libre polynomiale avec des exposants pairs ou toute déformation libre développable en série entière contenant seulement des exposants pairs comme cos(x), cosh(x), sin(x)/x ou sinh(x)/x. L’expression analytique explicite du tenseur intérieur des contraintes est alors déterminée pour un problème d’inclusion où la déformation libre varie en loi de puissance avec un exposant pair quelconque. Enfin, on s’intéresse au cas d’une déformation libre qui varie suivant sinh(r/l)/(r/l), r étant la distance au centre de l’inclusion et l une longueur caractéristique. En fonction de la valeur du paramètre l, cette description permet en effet de rendre compte de manière versatile des gradients de distorsion plastique à l’intérieur d’un grain. De très fortes concentrations de contrainte sont trouvées proches de l’interface, ces dernières pouvant même dépasser localement les contraintes liées à un comportement purement élastique. En outre, la méthode permet aussi d’obtenir une expression analytique explicite de l’énergie élastique interne. Dans un diagramme log-log, cette dernière a l’allure d’une fonction tanh en fonction du rapport l/a (a étant le rayon de l’inclusion), mettant ainsi en évidence une loi d’échelle sur un certain intervalle. A partir de ces résultats, de nouvelles lois d’interaction polycristalline dépendantes de la taille de grains peuvent être formulées

A fast Fourier transform-based mesoscale field dislocation mechanics study of grain size effects and reversible plasticity in polycrystals

International audienceA numerical implementation of a non-local polycrystal plasticity theory based on a mesoscale version of the field dislocation mechanics theory (MFDM) of Acharya and Roy (2006) is presented using small-strain elasto-viscoplastic fast Fourier transform-based (EVPFFT) algorithm developed by Lebensohn et al. (2012). In addition to considering plastic flow and hardening only due to SSDs (statistically stored dis-locations) as in the classic EVPFFT framework, the proposed method accounts for the evolution of GND (geometrically necessary dislocations) densities solving a hyperbolic-type partial differential equation, and GND effects on both plastic flow and hardening. This allows consideration of an enhanced strain-hardening law that includes the effect of the GND density tensor. The numerical implementation of a reduced version of the MFDM is presented in the framework of the FFT-based augmented Lagrangian procedure of Michel et al. (2001). A Finite Differences scheme combined with discrete Fourier transforms is applied to solve both incompatibility and equilibrium equations. The numerical procedure named MFDM-EVPFFT is used to perform full field simulations of polycrystal plasticity considering different grain sizes and their mechanical responses during monotonic tensile and reversible tension-compression tests. Using Voronoi tessellation and periodic boundary conditions , voxelized representative volume elements (RVEs) with different grain sizes are generated. With MFDM-EVPFFT, a Hall-Petch type scaling law is obtained in contrast with the conventional crystal plasticity EVPFFT. In the case of reversible plasticity, a stronger Bauschinger effect is observed with the MFDM-EVPFFT approach in comparison with conventional EVPFFT. The origin of these differences is analyzed in terms of heterogeneity, GND density and stress evolutions during the compression stage

Lattice strain measurements using synchrotron diffraction to calibrate a micromechanical modeling in a ferrite–cementite steel

In situ tensile tests were performed at room temperature on a ferrite–cementite steel specifically designed for this study. The evolution of the average stress in ferrite during loading was analyzed by X-ray diffraction.Lattice strain measurements were performed with synchrotron ring diffraction in both ferrite and cementite.These in situ tests were complemented by macroscopic tensile and reversible tensile-compression tests to study the Bauschinger effect. In order to reproduce stresses in ferrite and cementite particles,a recently developed micromechanical Internal Length Mean Field (ILMF) model based on a generalized self-consistent scheme is applied. In this designed ferrite–cementite steel,the third ‘‘phase’’of the model represents finite intermediate‘‘layers’’in ferrite due to large geometrically necessary dislocation (GND) densities around cementite particles. The assumed constant thickness of the layers is calibrated thanks to the obtained experimental data.The ILMF model is validated by realistic estimates of the Bauschinger stress and the large difference between mean stresses in ferrite and in cementite phases.This difference cannot be reproduced by classic two-phase homogenization schemes without intermediate GND layers

A Fast Fourier Transform-based approach for Generalized Disclination Mechanics within a Couple Stress theory

International audienceRecently, a small-distortion theory of coupled plasticity and phase transformation accounting for the kinematics and thermodynamics of generalized defects called generalized disclinations (abbreviated g- disclinations) has been proposed. Then, a first numerical spectral approach has been developed to solve the elasto-static equations of field dislocation and g-disclination mechanics set out in this theory for periodic media and for linear elastic media using the classic Hooke’s law. Here, given a spatial distribution of generalized disclination density tensors in a homogenous linear higher order elastic media described, a couple stress theory with elastic incompatibilities of first and second orders is developed. The incompatible and compatible elastic second and first distortions are obtained from the solution of Poisson and Navier-type equations in the Fourier space. The efficient Fast Fourier Transform (FFT) algorithm is used based on intrinsic Discrete Fourier Transforms (DFT) that are well adapted to the discrete grid to compute higher order partial derivatives in the Fourier space. Therefore, stress and couple stress fields can be calculated using the inverse FFT. The numerical examples are given for straight wedge disclinations and associated wedge disclination dipoles which are of importance to geometrically describe tilt grain boundaries at fine scales in polycrystalline solids

Self-consistent modelling of heterogeneous materials with an elastic-viscoplastic behavior: Application to polycrystalline agregates

The self-consistent scheme is a common homogenization method that was developed to connect local deformation mechanisms to the overall behavior of heterogeneous disordered materials. In the past decades, many efforts have been made to obtain extensions of the self-consistent approximation to the non-linear case. This work focuses on the specific case of heterogeneous materials with an elastic-viscoplastic behavior. For such materials, the overall behavior is strongly dependent on the space-time couplings originating from the differential form of the local constitutive law. Different approaches have thus been developed to describe the impact of such complex couplings on the overall behavior. (...

A FFT-based numerical implementation of mesoscale field dislocation mechanics: Application to two-phase laminates

International audienceIn this paper, we present an enhanced crystal plasticity elasto-viscoplastic fast Fourier transform (EVPFFT) formulation coupled with a phenomenological Mesoscale Field Dislocation Mechanics (MFDM) theory here named MFDM-EVPFFT formulation. In contrast with classic CP-EVPFFT, the model is able to tackle plastic flow and hardening due to polar dislocation density distributions or geometrically necessary dislocations (GNDs) in addition to statistically stored dislocations (SSDs). The model also considers GND mobility through a GND density evolution law numerically solved with a recently developed filtered spectral approach, which is here coupled with stress equilibrium. The discrete Fourier transform method combined with finite differences is applied to solve both lattice incompatibility and Lippmann-Schwinger equations in an augmented Lagrangian numerical scheme. Numerical results are presented for two-phase laminate composites with plastic channels and elastic second phase. It is shown that both GND densities and slip constraint at phase boundaries influence the overall and local hardening behavior. In contrast with the CP-EVPFFT formulation, a channel size effect is predicted on the shear flow stress with the present MFDM-EVPFFT formulation. The size effect originates from the progressive formation of continuous screw GND pileups from phase boundaries to the channel center. The effect of GND mean free path on local and global responses is also examined for the two-phase composite

Effect of anisotropic elasticity on dislocation pile-ups at grain boundaries

This study deals with in-situ micromechanical tests of micron-sized bi-crystals and observations coupling SEM, AFM and EBSD. Different FCC bi-crystals are obtained from FIB machining. A SEM in-situ compression test with a low strain is performed on a micron-sized bi-crystal in order to induce single slip deformation. Spatial variations in the surface step height due to dislocation activity in localized slip bands terminating at the grain boundary (GB) are measured by AFM. This allows the determination of the Burgers vector distribution and hence the dislocation positions in the pile-up as shown in Figure 1. Furthermore, local misorientation along slip bands is measured by high resolution EBSD in order to determine the deformation caused by the dislocation pile-ups. In parallel, an analytical approach based on the Leknitskii-Eshelby-Stroh (LES) formalism (1,2), which provides the elastic fields of straight dislocation pile-ups in anisotropic bi- and tri- materials (3) while considering (or not) free surface effects (4), are performed. The tri-material configuration allows considering a non-zero thickness in the nanometer range and a specific stiffness for the GB region. The configuration with two free surfaces could be used to study size effects. The effects of anisotropic elasticity, crystallographic orientation, GB stiffness and free surfaces are first studied in the case of a single dislocation in a Ni bi-crystal. Image forces may arise because of both dissimilar grain orientations, the presence of a finite grain boundary region and the presence of free surfaces. In particular, it is shown that the Peach-Koehler force projected along the dislocation glide direction can exhibit a change of sign with the dislocation position (5). The dislocation positions in a pile-up are calculated by an iterative relaxation scheme minimizing the Peach-Koehler force on each dislocation as shown in Figure 2. Both, GB stiffness and grains misorientation, influence pile-up length and the induced resolved shear stress in the neighboring grain, but the effect of misorientation is clearly predominant (5). Hence, the driving force for slip activation in the neighboring grain can be computed and compared to the observed GB resistance for slip transmission. Please click Additional Files below to see the full abstract