An elasto-viscoplastic formulation based on fast Fourier transforms for the prediction of micromechanical fields in polycrystalline materials
Highlights
► A small-strain FFT-based model for elasto-viscoplastic polycrystals is presented. ► Local fields and effective behavior can be computed from a microstructure image. ► Implicit time discretization and an augmented Lagrangian iterative scheme are used. ► Benchmarks for mechanical behavior, boundary conditions and convergence are presented. ► The model is used to study the influence of crystal anisotropy on stress hot-spots.
Introduction
Polycrystalline materials play a fundamental role as structural and functional materials in current and future technological applications. The mechanical properties of plastically deforming polycrystals are dictated, on the one hand, by the structure and dynamics of crystalline defects, like vacancies and interstitials, dislocations, grain boundaries, voids, and, on the other hand, by the size, morphology, spatial distribution and orientation of the constituent single crystal grains, i.e. in a broad sense, by the texture of the polycrystal. From an experimental point of view, powerful techniques are emerging to fully characterize polycrystal textures in three dimensions (3-D) and follow its in situ evolution during thermo-mechanical processing. For example, serial-sectioning by Focus-Ion-Beam (FIB) combined with Electron Back-Scattering Diffraction (EBSD) is by now a well established tool to characterize (destructively) local orientations in 3-D (e.g. Uchic et al., 2006) with nanometric spatial resolution. Also, synchrotron-based X-ray diffraction can now be used for in situ measurement of the positions, shapes, and crystallographic orientations (e.g. Lauridsen et al., 2006) and local elastic strains of bulk grains in an aggregate (Oddershede et al., 2010, Oddershede et al., 2011), with micrometric and sub-micrometric resolution, in a non-destructive fashion.
From a modeling perspective, the challenge arising from these novel experimental techniques, which produce very large 3-D digital images of the microstructure (i.e. crystal orientation and/or the phase identification given on a regular grid of points with intragranular resolution) is to devise new, robust and very efficient numerical formulations for interpretation and exploitation of the massive amount of data generated by these measurements.
As a contribution to face this challenge, we present here an extension to the most general elasto-viscoplastic (EVP) deformation regime of a modeling technique originally developed by Suquet and co-workers (Moulinec and Suquet, 1994, Moulinec and Suquet, 1998, Michel et al., 2000, Michel et al., 2001) as an efficient method to compute the micromechanical fields of periodic heterogeneous materials directly from an image of their microstructure. Under this technique, which in the past we have extended, implemented and applied to polycrystals deforming in the more restricted elastic (Brenner et al., 2009) and the rigid-viscoplastic (Lebensohn, 2001, Lebensohn et al., 2008, Lebensohn et al., 2009, Lee et al., 2011) regimes, the input microstructural image is treated using the Fast Fourier Transform (FFT) algorithm to solve the corresponding micromechanical problem. The present extension of the FFT-based model to the EVP regime is a necessary step towards expanding its applicability to open problems involving polycrystal plasticity, like the prediction of internal stresses with intragranular resolution for interpretation of either space-resolved or global measurements, or the modeling of the role played by texture and microstructure on the distribution of extreme values of the micromechanical fields, which in turn determine the macroscopic mechanical behavior of polycrystalline aggregates during cyclic deformation, to mention some of the most challenging and relevant problems.
While the finite element method (FEM) has been extensively used to deal with problems involving crystal plasticity (CP) in the elastoplastic regime (e.g. Mika and Dawson, 1998, Barbe et al., 2001a, Barbe et al., 2001b; Raabe et al., 2001, Bhattacharyya et al., 2001, Diard et al., 2005, Delannay et al., 2006; and the comprehensive review of Roters et al., 2010), the large number of degrees of freedom required by FEM calculations with direct input from microstructural images constitutes an objective limitation for the size of the polycrystalline aggregates that can be treated with CP-FEM, within reasonable computing times. Conceived as an efficient alternative to FEM, the FFT-based methodology can account for fine-scale microstructural information with a level of fidelity, in practice, unreachable with FEM. One disadvantage of FFT-based formulations is the requirement of periodic microstructures, making them less general than FEM.
The plan of the paper is as follows. In Section 2 we present the extension of the FFT-based formulation to the case of polycrystals deforming in the elasto-viscoplastic regime. In Section 3 we benchmark the EVP–FFT formulation, including: (a) the assessment of the corresponding elastic (EL) and viscoplastic (VP) limits by comparison with earlier elastic (EL-FFT) and viscoplastic (VP-FFT) implementations, (b) the correct treatment of hardening, rate-sensitivity and boundary conditions, and (c) a detailed convergence analysis. In Section 4 we show an application of the EVP–FFT model to study how the single crystal elastic and plastic directional properties determine the distribution of local stresses in the initial elastic loading, the elastoplastic transition and the fully plastic regime, and, consequently, how this crystal anisotropy impacts the aforementioned transition at macroscopic level. In Section 5 we summarize and provide some perspectives for future applications of the EVP–FFT formulation.
Section snippets
Model
The FFT-based formulation is conceived for periodic unit cells and provides an exact solution (within the limitations imposed by the required discretization and the iterative character of the numerical algorithm) of the governing equations of equilibrium and compatibility, in such a way that the final (converged) equilibrated stress and compatible strain fields fulfill the required constitutive relation, at every discrete material point. The method was originally developed for linear (elastic) (
Benchmarks
The first benchmark for the proposed EVP–FFT numerical scheme is to verify that the elastic and viscoplastic limits match the previous simpler EL-FFT and VP-FFT implementations. For this we have chosen the case of a copper polycrystal, represented by a periodic unit cell generated by Voronoi tessellation, consisting of 100 grains with randomly chosen orientations. The unit cell was discretized using a 128 × 128 × 128 grid (same unit cell, discretized in the same fashion will be adopted throughout
Application
After assessing our model with the above relatively simple but indispensable benchmarks, in this Section we show a first application of the EVP–FFT formulation to study the interplay between elastic and plastic anisotropy, and its effect on the effective behavior and local micromechanical response during the elastoplastic transition. Similar analysis of the role of texture and microstructure on the distribution of stress “hot spots” (i.e. local values of stress significantly above the average)
Summary and perspectives
In this paper we have described the formulation, benchmarked our numerical implementation and presented a first application to stress hot-spot analysis, of the small-strain version of the EVP–FFT model for polycrystalline materials. The EVP–FFT formulation has also been extended to large strains and it is being reported elsewhere (Eisenlohr et al., in preparation). Depending on the specific need, the small-strain or large-strain versions of the model can be further utilized in a suite of
Acknowledgements
Ricardo A. Lebensohn wishes to thank Prof. Pierre Suquet (LMA, Marseille) for fruitful discussions. RAL also thanks the Humboldt Foundation for supporting his stay in Max-Planck-Institut für Eisenforschung (MPIE), Düsseldorf, through the Humboldt Research Award, as well as support from Joint DoD/DOE Munitions Technology Program and ASC Physics & Engineering Models, Materials Project. The work of Anand K. Kanjarla is supported by the US Department of Energy, Office of Basic Energy Sciences,
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