Mixed boundary conditions for FFT-based homogenization at finite strains

Abstract

In this article we introduce a Lippmann-Schwinger formulation for the unit cell problem of periodic homogenization of elasticity at finite strains incorporating arbitrary mixed boundary conditions. Such problems occur frequently, for instance when validating computational results with tensile tests, where the deformation gradient in loading direction is fixed, as is the stress in the corresponding orthogonal plane. Previous Lippmann-Schwinger formulations involving mixed boundary can only describe tensile tests where the vector of applied force is proportional to a coordinate direction. Utilizing suitable orthogonal projectors we develop a Lippmann-Schwinger framework for arbitrary mixed boundary conditions. The resulting fixed point and Newton-Krylov algorithms preserve the positive characteristics of existing FFT-algorithms. We demonstrate the power of the proposed methods with a series of numerical examples, including continuous fiber reinforced laminates and a complex nonwoven structure of a long fiber reinforced thermoplastic, resulting in a speed-up of some computations by a factor of 1000

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

“…Let us note that Eq. 15is consistent with the Lippmann-Schwinger formulation recently derived by Kabel et al [40] with mixed boundary conditions in the case where only traction boundary conditions are prescribed. Then, the field solution of Eq.…”

“…The conjugate gradient method was later adapted to non-linear elastic behavior using the Newton-Raphson algorithm [38]. Efficient fixed-point and Newton-Krylov solvers for FFT-based homogenization of elasticity at finite strains (hyperelasticity) were recently developed by Kabel and co-workers [39,40]. Furthermore, the Lippmann-Schwinger formulation for arbitrary mixed boundary conditions solved by FFT-based homogenization at finite strains were also reported [40].…”

On Quasi‐Newton methods in fast Fourier transform‐based micromechanics

“…The material parameters according to Doghri et al 63 are given in Table 1. We apply mixed boundary conditions, 66 corresponding to a uniaxial extension of 5% perpendicular to the fiber direction, in a single load step. The L-BFGS scheme and Anderson acceleration are investigated for depths from 1 to 200.…”

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

“…For polycrystals, the different constitutive regimes solved with FFT-based methods include: linear elasticity (Lebensohn, 2001;Brenner et al, 2009); linear viscosity (Lebensohn et al, 2005); thermoelasticity (Vinogradov and Milton, 2008;Anglin et al, 2014;Donegan and Rollett, 2015); rigid-viscoplasticity (Lebensohn, 2001;Lebensohn et al, 2008Lebensohn et al, , 2009Lee et al, 2011;Rollett et al, 2010); smallstrain crystal plasticity elasto-viscoplasticity, i.e. CP-EVPFFT 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, 2018); dilatational plasticity ; lower-order (Lucarini and Segurado, 2018) and higher-order strain-gradient plasticity (Lebensohn and Needleman, 2016); curvature-driven plasticity (Upadhyay et al, 2016); transformation plasticity (Richards et al, 2013;Otsuka et al, 2018); fatigue (Rovinelli et al, 2017a,b); 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;…”