Field Dislocation Mechanics for heterogeneous elastic materials: A numerical spectral approach

Abstract

International audienceSpectral methods using Fast Fourier Transform (FFT) algorithms have recently seen a surge in interest in the mechanics of materials community. The present contribution addresses the critical question of determining accurate local mechanical fields using FFT methods without artificial fluctuations arising from materials and defects induced discontinuities. 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 dislocations and from elastic stiffness heterogeneities. To this end, the elasto-static equations of the field dislocation mechanics theory for periodic heterogeneous materials are numerically solved with FFT in the case of dislocations in proximity of inclusions of varying stiffness. An optimal intrinsic discrete Fourier transform method is sought based on two distinct schemes. A centered finite difference scheme for differential rules are used for numerically solving the Poisson-type equation in the Fourier space, while centered finite differences on a rotated grid is chosen for the computation of the modified Fourier Green's operator associated with the Lippmann Schwinger-type equation. By comparing different methods with analytical solutions for an edge dislocation in a composite material, it is found that the present spectral method is accurate, devoid of any numerical oscillation, and efficient even for an infinite phase elastic contrast like a hole embedded in a matrix containing a dislocation. The present FFT method is then used to simulate physical cases such as the elastic fields of dislocation dipoles located near the matrix/inclusion interface in a 2D composite material and the ones due to dislocation loop distributions surrounding cubic inclusions in 3D composite material. In these configurations, the spectral method allows investigating accurately the elastic interactions and image stresses due to dislocation fields in the presence of elastic inhomogeneities. (C) 2016 Elsevier B.V. All rights reserved

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

“…CP-EVPFFT (Lebensohn et al, 2012;Grennerat et al, 2012;Suquet et al, 2012); large-strain elasto-viscoplasticity (Eisenlohr et al, 2013;Shanthraj et al, 2015;Kabel et al, 2016;Vidyasagar et al, 2018;Lucarini and Segurado, 2019); dilatational plasticity ; lower-order (Haouala et al, 2020) and higher-order (Lebensohn and Needleman, 2016) straingradient crystal plasticity; curvature-driven plasticity (Upadhyay et al, 2016); transformation plasticity (Richards et al, 2013;Otsuka et al, 2018); twinning (Mareau and Daymond, 2016;Paramatmuni and Kanjarla, 2019), fatigue (Rovinelli et al, 2017(Rovinelli et al, , 2018Lucarini and Segurado, 2019); and quasi-brittle damage (Li et al, 2012;Sharma et al, 2012). FFT-based methods were also applied to field dislocation mechanics (FDM) and field disclination mechanics (Brenner et al, 2014;Berbenni et al, 2014;Djaka et al, 2015;Berbenni et al, 2016;Djaka et al, 2017;Berbenni and Taupin, 2018), and discrete dislocation dynamics (DDD) problems (Bertin et al, 2015;Graham et al, 2016;Bertin and Capolungo, 2018), providing the efficiency needed for the implementation of these powerful and numerically-demanding formulations.…”

Multiscale modelling of precipitation hardening in Al–Cu alloys: Dislocation dynamics simulations and experimental validation

“…To this end, a rectangular prismatic loop parallel to one cube faces was introduced in the cubic domain and two opposite sides of the loop were moved in opposite directions until they reached the boundaries of the domain and annihilate each other leading to two straight dislocations forming a dipole within the domain. One of the dislocations was fixed during the simulation and the Field Dislocation Mechanics method was used to cancel the stress field created in the domain by the fixed dislocation following the methodology presented in [46,47,48].…”

Fourier‐based numerical approximation of the Weertman equation for moving dislocations

“…We shall also use (31) with m replaced by 2m (keeping fixed the step size h that defines the spatial resolution). As is well known, conducting by means of DFTs the convolution of a translation-invariant discretized kernel O with a vector v requires using the injection and projection operators (29). This approach, called zero padding, prevents DFT-induced periodization artifacts (sometimes called aliasing artifacts 35 ) from showing up near both extremities of the interval of interest in the direct space.…”