A yield criterion for porous single crystals based on limit analysis

The ductile fracture of crystalline materials is a classical subject in mechanics of materials which still presents important challenges. It is driven by the process of growth and coalescence of voids and thus calls for constitutive models of the plastic response of porous crystalline materials. In this context, the consideration of the anisotropy, related to the single crystal response and the morphological and crystallographic textures of polycrystals, is a key issue. As compared to the studies on polycrystals, relatively few studies exist on the constitutive response of plastic single crystals containing voids. However, the importance of the crystalline anisotropy to describe the stress state surrouding intragranular voids has been clearly evidenced analytically, experimentally and numerically. The objective of this study is to derive a Gurson-type yield function for porous single crystals. Because of the widely spread finite-element (FE) implementations of the Gurson model, it would present an obvious interest to consider the case of single crystals deforming by crystallographic slip. With this aim in view, use is made of a regularization of the Schmid law. This key ingredient allows us to obtain a single yield function defining the plastic strength domain of voided single crystals. This feature is a definite advantage with respect to earlier proposals. The proposed criterion is assessed by comparison with results from the literature on unit-cell FE computations for Face-Centered Cubic single crystals with various orientations

Numerical computation of the effective properties of smart composite materials

The design of composite materials requires microstructurally based computational approaches to optimize the shape and the spatial arrangement of the constituents with respect to a sought-after effective property. This topic remains widely opened especially in the case of heterogeneous media which exhibit multifield couplings in their constitutive relations. Up to now, the numerical homogenization schemes widely used in this framework rely on the use of the finite-element method. We present results obtained with an alternative scheme based on fast Fourier transforms which has become popular for the study of the mechanical properties (elasticity, viscoplasticity, among others) of polycrystals and composite materials. It allows to consider unit-cell problems presenting complex microstructure with fine-scale information within reasonable computing times. Besides, it does not require a specific meshing of the microstructure. In this discussion, attention is paid to the linear properties of smart composite materials, used as sensors or actuators, with piezoelectric and magnetoelastic constituents. The numerical scheme proposed consists in the iterative resolution of periodic coupled Lippmann–Schwinger equations using the uncoupled Green operators (relative to elasticity, permittivity, and permeability) of a reference medium. The numerical results, on the effective response and the local fields, for various multifield couplings and microstructural cases, are compared with analytical solutions and finite element results

Investigation of the effective response of 2-1-2 piezoelectric composites

International audienceThe question of the effective response of two-phase hybrid "fibrous-laminate" piezoelectric composites, with periodic microstructure, is adressed with two homogenization approaches: a full-field numerical scheme based on Fourier transform and a simplifying approach relying on a decoupled two-step homogenisation process. In the case of a two-phase epoxy/PZT composite, this latter is shown to overestimate out-of-plane effective piezoelectric coefficients

Numerical implementation of static Field Dislocation Mechanics theory for periodic media

This paper investigates the implementation of Field Dislocation Mechanics theory for media with a periodic microstructure (i.e. the Nye dislocation tensor and the elastic moduli tensor are considered as spatially periodic continuous elds). In this context, the uniqueness of the stress and elastic distortion elds is established. This allows to propose an e cient numerical scheme based on Fourier transform to compute the internal stress eld, for a given spatial distribution of dislocations and applied macroscopic stress. This numerical implementation is assessed by comparison with analytical solutions for homogeneous as well as heterogeneous elastic media. A particular insight is given to the critical case of stress-free dislocation microstructures which represent equilibrium solutions of the Field Dislocation Mechanics theory

Etude du comportement élastoplastique d'agrégats polycristallins par homogénéisation numérique FFT

Dans ce travail nous proposons une méthode d'homogénéisation en champ complet basée sur la transformée de Fourier rapide (FFT) pour étudier le comportement mécanique élastique et plastique des polycristaux. Pour cela , différentes microstructures artificielles ont été générées afin d'étudier la sensibilité à l'échelle global et à l'échelle des grains en élasticité , et en plasticité il s'agira d'introduire des nouveaux critères et étudier leurs influence

Incremental homogenization approach for ageing viscoelastic polycrystals

An approximate self-consistent modelling is proposed to estimate the effective viscoelastic response of polycrystals presenting an ageing constitutive behaviour. This approach makes use of the equivalence between the Dirichlet series approximation of the viscoelastic functions and an internal variables formulation. An illustrative application is performed for model 2D polycrystals for which the exact expression of the continuous effective relaxation spectrum is give

Periodic smoothing splines for FFT-based solvers

The aim of this paper is to develop a periodic smoother based on splines for FFT-based solvers. Spurious oscillations in FFT-based methods are shown to be due to pseudo-spectral differentiation of discontinuous fields. An automatic smoother based on polynomial splines is developed, which permits to add smoothness to initial material properties. The method, which is applied in various problems including conductivity, elasticity and field dislocation mechanics, improves significantly the local fields and reduces spurious oscillations

Understanding the thermomechanical behavior of a TATB-based explosive via microstructure-level simulations. Part I: Microcracking and viscoelasticity

International audienceIn view of a better understanding of the thermomechanical behavior of pressed explosives, a Fourier-based computational tool is used to perform numerical homogenization and compare predictions to experimental macroscopic properties. This is first done in a purely thermoelastic context on simplified polycrystalline virtual microstructures, then extended to cracked polycrystalline ones. A further extension is proposed, aiming at predicting the nucleation and propagation of (micro)-cracks. Besides, a mean-field (self-consistent) approach is also followed, providing accurate thermoelastic predictions. It is currently being extended to account for linear (non-ageing) viscoelasticity of the binder. The study of irreversible deformation mechanisms of the TATB crystal, in view of their incorporation in the full-field tool, is the subject of the companion paper

A self-consistent estimate for linear viscoelastic polycrystals with internal variables inferred from the collocation method

The correspondence principle is customarily used with the Laplace–Carson transform technique to tackle the homogenization of linear viscoelastic heterogeneous media. The main drawback of this method lies in the fact that the whole stress and strain histories have to be considered to compute the mechanical response of the material during a given macroscopic loading. Following a remark of Mandel (1966 Mécanique des Milieux Continus(Paris, France: Gauthier-Villars)), Ricaud and Masson (2009 Int. J. Solids Struct. 46 1599–1606) have shown the equivalence between the collocation method used to invert Laplace–Carson transforms and an internal variables formulation. In this paper, this new method is developed for the case of polycrystalline materials with general anisotropic properties for local and macroscopic behavior. Applications are provided for the case of constitutive relations accounting for glide of dislocations on particular slip systems. It is shown that the method yields accurate results that perfectly match the standard collocation method and reference full-field results obtained with a FFT numerical scheme. The formulation is then extended to the case of time- and strain-dependent viscous properties, leading to the incremental collocation method (ICM) that can be solved efficiently by a step-by-step procedure. Specifically, the introduction of isotropic and kinematic hardening at the slip system scale is considered

Étude expérimentale et simulation numérique, au moyen de modèles de plasticité cristalline, de chargements non proportionnels.

Le comportement plastique de matériaux à structure cubique est ici étudié sous chargements complexes et à température ambiante. Les résultats expérimentaux multi-échelles sont confrontés aux simulations numériques par éléments finis du comportement d'agrégats polycristallins, et aux modélisations par homogénéisation. Des combinaisons de trajets de chargement simple sont considérées, pour différentes orientations des éprouvettes. Des mesures locales de champs de déformation sont effectuées par corrélation d'images sous microscope électronique à balayage tandis que la déformation macroscopique est obtenue par extensométrie classique. Les simulations éléments finis correspondantes sont effectuées sur un agrégat polycristallin prenant en compte la microstructure du matériau. La texture, mesurée par diffraction aux rayons-X, est représentée. Plusieurs modèles de monocristaux, introduisant les douze systèmes de glissement de la famille octaédrique, sont considérés. Plusieurs options pour la prise en compte de l'auto-écrouissage et de l'écrouissage latent sont étudiées. Enfin, la comparaison entre simulations et essais expérimentaux s'effectue sur trois types de variables~: à l'échelle globale grâce aux réponses macroscopiques, à l'échelle locale avec les champs de déformation obtenus, et en moyenne par phase. Des informations sur la pertinence des règles de transition d'échelle sont déduites