Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications

高等学校計算数学学報 = Numerical mathematics

A gradient-enhanced computational homogenization procedure, that allows for the modelling of microstructural size effects, is proposed within a general non-linear framework. In this approach the macroscopic deformation gradient tensor and its gradient are imposed on a microstructural representative volume element (RVE). This enables us to incorporate the microstructural size and to account for non-uniform macroscopic deformation fields within the microstructural cell. Every microstructural constituent is modelled as a classical continuum and the RVE problem is formulated in terms of standard equilibrium and boundary conditions. From the solution of the microstructural boundary value problem, the macroscopic stress tensor and the higher-order stress tensor are derived based on an extension of the Hill-Mandel condition. This automatically delivers the microstructurally based constitutive response of the higher-order macro continuum and deals with the microstructural size in a natural way. Several examples illustrate the approach, particularly the microstructural size effects.

http://www.mate.tue.nl/mate/pdfs/2091.pdf

Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme

https://hal.archives-ouvertes.fr/hal-01389078/document

A computational approach to handle complex microstructure geometries

Multi-scale second-order computational homogenization of multi-phase materials : a nested finite element solution strategy

This study presents a family of estimates for effective potentials in nonlinear polycrystals. Noting that these potentials are given as averages, several quadrature formulae are investigated to express these integrals of nonlinear functions of local fields in terms of the moments of these fields. Two of these quadrature formulae reduce to known schemes, including a recent proposition (Ponte Castaneda 2015 Proc. R. Soc. A471, 20150665 (doi:10.1098/rspa.2015.0665)) obtained by completely different means. Other formulae are also reviewed that make use of statistical information on the fields beyond their first and second moments. These quadrature formulae are applied to the estimation of effective potentials in polycrystals governed by two potentials, by means of a reduced-order model proposed by the authors (non-uniform transformation field analysis). It is shown how the quadrature formulae improve on the tangent second-order approximation in porous crystals at high stress triaxiality. It is found that, in order to retrieve a satisfactory accuracy for highly nonlinear porous crystals under high stress triaxiality, a quadrature formula of higher order is required.

https://hal.archives-ouvertes.fr/hal-01578482/document

Effective potentials in nonlinear polycrystals and quadrature formulae.

The problem of a spherical inclusion surrounded by an unbounded matrix loaded at infinity plays a prominent role in the mechanics of materials, and is addressed in this paper for power-law viscous materials via the finite element method. With respect to a previous analysis, the results of which are further developed, the influence of the necessarily finite size of the mesh is carefully analyzed (by using two types of boundary conditions, among other tests) and two finite element codes are compared. In addition to the localization rules for the strain rate and stress deviator in the inclusion, the analysis also covers the influence of a dilute concentration of deformable spheres in a power-law matrix, which generalizes previous studies on cavities or rigid inclusions. All the results are fitted by curves, and this is expected to assist in further practical applications.

Finite element solution of the problem of a spherical inhomogeneity in an infinite power-law viscous matrix

The objective of this study is to assess the variational definition of the effective fracture energy of a heterogeneous brittle material recently proposed by Schneider (2020) based on available mathematical homogenization results. First, similarities between the variational problems defining this effective surface energy and variational limit loads problems, are highlighted. These analogies allow us to compute the effective surface energy using the analytical and computational tools developed to determine the extremal surfaces of ideally plastic heterogeneous materials. Second, the same analogy opens the way to the derivation of variational bounds on the effective surface energy which appear to be in good agreement with the numerical simulations of other authors (Lebihain et al., 2021). Third, the status of this effective surface energy with respect to the effective toughness of heterogeneous materials is discussed. It is shown to always be a lower bound to the actual energy dissipated along the propagation of a crack in a heterogeneous material. Examples where both energies are different are exhibited. 

http://manuscript.elsevier.com/S0022509622000941/pdf/S0022509622000941.pdf

Merits and limits of a variational definition of the effective toughness of heterogeneous materials

A model of coalescence by internal necking of primary voids is developed which accounts for the presence of a second population of cavities. The derivation is based on a limit-analysis of a cylindrical cell containing a mesoscopic void and subjected to boundary conditions describing the kinematics of coalescence. The second population is accounted locally in the matrix surrounding the mesoscopic void through the microscopic potential of Michel and Suquet (1992) for spherical voids. The macroscopic criterion obtained is assessed through comparison of its predictions with the results of micromechanical finite element simulations on the same cell. A good agreement between model predictions and numerical results is found on the limit-load promoting coalescence.

/pdf/void-coalescence-in-porous-ductile-solids-containing-two-2o2duxf8bm.pdf

Void coalescence in porous ductile solids containing two populations of cavities

La repartition heterogene du plutonium dans le combustible nucleaire MOX implique que la
 modelisation de son comportement, durant l’irradiation en reacteur a eau pressurisee,
 passe par la construction de lois macroscopiques faisant intervenir les effets
 microstructuraux. Pour homogeneiser ce probleme local, nous avons choisi une methode
 d’analyse par champs de transformation non uniformes, abrege anglais NTFA (J.-C. Michel,
 P. Suquet, Int. J. Solids Struct. 40 (2003) 6937–6955). Nous presentons donc ici, les
 travaux menes sur ce modele pour l’adapter au comportement du MOX en service, ainsi qu’une
 comparaison des resultats globaux et locaux de ce modele a des calculs de reference.

/pdf/modelisation-du-comportement-effectif-du-combustible-mox-par-52qw064whh.pdf

Jean-Claude Michel

Papers

Effective potentials in nonlinear polycrystals and quadrature formulae.

Finite element solution of the problem of a spherical inhomogeneity in an infinite power-law viscous matrix

Merits and limits of a variational definition of the effective toughness of heterogeneous materials

Void coalescence in porous ductile solids containing two populations of cavities

Modélisation du comportement effectif du combustible MOX par une analyse micromécanique en champs de transformation non uniformes