Elsevier

Acta Materialia

Volume 57, Issue 6, April 2009, Pages 1777-1784
Acta Materialia

Crystal plasticity simulations using discrete Fourier transforms

https://doi.org/10.1016/j.actamat.2008.12.017Get rights and content

Abstract

In this paper, we explore efficient representation of all of the functions central to crystal plasticity simulations in their complete respective domains using discrete Fourier transforms (DFTs). This new DFT approach allows for compact representation and fast retrieval of crystal plasticity solutions for a crystal of any orientation subjected to any deformation mode. The approach has been successfully applied to a rigid–viscoplastic Taylor-type model for face-centered cubic polycrystals. It is observed that the novel approach described herein is able to speed up the conventional crystal plasticity computations by two orders of magnitude. Details of this approach are described and validated in this paper through a few example case studies.

Introduction

Crystal plasticity theories are used extensively [1], [2], [3], [4], [5], [6], [7], [8], [9] in understanding and predicting the evolution of the underlying microstructure (mainly texture related aspects) and the concomitant anisotropic stress–strain response in polycrystalline metals subjected to finite plastic strains. Such physics-based constitutive theories are highly desirable for conducting more accurate simulations of various metal manufacturing/fabrication processes, since they provide better understanding and predictions of the material behavior [10], [11], [12]. The main deterrent in the more widespread use of these theories (in place of the highly simplified phenomenological isotropic plasticity theories typically used) is the fact that the implementation of the crystal plasticity theories in a finite element modeling framework demands substantial computational resources and highly specialized expertise.

A number of strategies are being explored currently to speed up the crystal plasticity calculations. The most promising of these strategies appear to be those that seek efficient spectral representations combined with a database approach that stores the main characteristics of the crystal plasticity solutions. Li et al. [13] and Kalidindi and Duvvuru [14] have demonstrated the viability of the Bunge–Esling approach [15], [16] using generalized spherical harmonics (GSH) for texture evolution. In this approach, the important details of texture evolution are captured in a database of streamlines for a selected deformation process. A process plane concept, based on proper orthogonal decomposition in Rodrigues–Frank space [17], has been presented by Sundararaghavan and Zabaras [18], again for selected deformation modes. Both the streamline approach and the process plane approach have not yet been successfully generalized for arbitrary deformation modes. Moreover, these models are based on conservation principles in the orientation space that do not presently resolve the differences in the strain hardening responses of differently oriented crystals.

In recent work [19], [20], we have identified that the crystal plasticity solutions for face-centered cubic (fcc) polycrystals experiencing rigid–viscoplastic deformations can be organized efficiently as a set of functions that describe the dependence of the stresses, the lattice rotations and the total slip rates in individual crystalline regions on their lattice orientation and the imposed velocity gradient tensor (quantifying the deformation mode) on those regions. The domain of these functions was defined to be the product space comprising all possible crystal orientations and all possible isochoric deformation modes. Even though the viability of this framework using GSH representations of texture was demonstrated [19], it did not produce the expected dramatic improvements in the computational speed. Recognizing that the relatively high computational cost of evaluating the GSH was the main limitation in this approach, we recently explored the use of discrete Fourier transforms (DFTs), in place of the GSH, in the development of appropriate spectral databases [20]. In this most recent work, we stored the values of the various desired functions on a discrete grid in their respective domains and used local DFT-based interpolations for evaluating the values of the function at any other desired location. In doing so, we recognized that the major advantage of using DFTs in place of GSH coefficients is that the DFTs can be computed extremely fast using well-established, highly efficient algorithms [21], [22], [23], [24], [25]. Indeed, the database of discretized function values together with the local DFT-based interpolation method was found to speed up the crystal plasticity calculations by an order of magnitude when compared to the traditional computations in fcc metals [20].

Building on these prior efforts, we have now explored the representation of all of the functions capturing crystal plasticity solutions in their complete respective domains using DFTs. The most remarkable discovery in this new direction was the recognition that only a limited number of the dominant DFTs were needed to recover the functions of interest across their entire respective domains. This approach is akin to a global spectral interpolation as opposed to the local spectral interpolation used in our earlier approach [20]. Also, in our previous approach, the global spectral database constituted the set of discretized function values, which can be thought of as a primitive Fourier representation. By using DFTs directly to build our spectral database, we are transforming from a primitive Fourier representation to the classical Fourier representation (with discrete frequencies). This new approach was found to be able to speed up the crystal plasticity computations by another order of magnitude compared to our most recent approach (i.e. about two orders of magnitude faster than the traditional approaches used in crystal plasticity computations). The details of this new approach are presented and discussed in this paper. A particularly attractive feature of this new approach is that it provides the user with tremendous flexibility in making trade-offs between accuracy and computational speed. In other words, the new spectral database described in this paper will allow the user to perform a large number of very quick simulations at a lower than desired accuracy, identify the specific ones that appear to produce promising results and redo them much more accurately (at a higher computational cost).

In the present study, we have confined our attention to rigid–plastic Taylor-type (full constraints) model calculations for fcc polycrystals with equal hardening of all slip systems. Application of this approach to more sophisticated constitutive models (such as elastic–viscoplastic models with latent hardening) and higher-order homogenization theories (such as LAMEL [8] or self-consistent [26] models) will be explored in future work. The new approach presented in this work was validated by comparing the solutions obtained by the conventional computational approaches against those obtained from the spectral approach described here for a few selected examples of deformation processes.

Section snippets

Crystal plasticity framework

The rigid–viscoplastic crystal plasticity model [1] used in this work can be described by the following set of equations:D=αγ˙αPα,Pα=0.5(mαnα+nαmα)γ˙α=γ˙oταsα1/msgn(τα),τα=σ·PαIn Eqs. (1), (2), D is the applied isochoric stretching tensor, and mα and nα are the unit vectors identifying the slip direction and the slip plane normal, respectively, for slip system α. For the fcc crystals studied in this paper, the family of twelve {111}11¯0 slip systems were considered as potential slip

Crystal plasticity using DFTs

The crystal plasticity computations typically demand significant computational resources because of the low value of m (which makes the resulting system of algebraic equations numerically extremely stiff). Moreover, the same computations are likely to be repeated several times in simulations performed by the conventional approach, because the results of computations in any one time step are immediately forgotten when the computations advance to the next time step. In the spectral crystal

Plane strain compression

In order to demonstrate the validity of the new DFT-based spectral approach described in this work, we simulated plane strain compression on polycrystalline OFHC copper to a true strain of ε = −1.0 along the compression axis. The polycrystal was assumed to possess a random initial texture that was captured by a set of 1000 discrete crystal orientations. We computed the deformed textures and the anisotropic stress–strain curves using the Taylor-type model, both by the traditional approach and the

Conclusions

It has been demonstrated that it is possible to speed up the crystal plasticity calculations by two orders of magnitude in fcc metals using a compact database of discrete Fourier transforms (DFTs). It was seen that a limited set of dominant transforms adequately captured the dependence of the stresses, the lattice spins and the strain hardening in individual crystals as a function of their lattice orientation and the applied deformation mode. A computationally efficient spectral interpolation

Acknowledgements

The authors gratefully acknowledge financial support received for this work from NSF Grants CMS-0654179 and CMS-0727931.

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