2015 • 85 Pages • 30.24 MB • English

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Micromechanical Analysis of Stress-Strain Inhomogeneities with Fourier transforms (MASSIF) 1 5 Anthony Rollett, Ricardo Lebensohn , Sukbin Lee , 2 3 4 Seth Wilson , Benjamin Anglin , Sean Donegan , Reeju 1 1 Pokharel , Evan Lieberman and Tugce Ozturk Materials Science & Engineering, Carnegie Mellon Univ. 1 Materials Science & Technology, Los Alamos Natl. Lab. 2 3 4 5 Ames Lab., Bettis Lab., Bluequartz, UNIST, Korea Support from HPCMO/PETTT, NDSEG, BES, NSF, and CMSN acknowledged 1 Carnegie Mellon

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2 3D Workshop 8-10 July + GRC Phys. Metallurgy 19-25 July For those interested in the future of the discipline, come to the next Gordon Conference on Physical Metallurgy, July th 19-24 2015, at the Univ. New England, Maine.

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3 Outline § Status § Mathematical basis for the “FFT method”, which discretizes a microstructure on a regular grid in order to use Fast Fourier Transforms to solve the (partial) differential equations for stress equilibrium. § Wide range of applications of a spectral method to calculating microstructure-property relationships in polycrystals using Fast Fourier transforms (FFTs). § The method was originated by Moulinec & Suquet (for elastic loading) with a focus on composite materials; it was further developed by Lebensohn (for viscoplastic deformation) with a focus on polycrystalline materials (metals, ceramics, ice). § Rollett’s group adopted the FFTW package and made it MPI parallel (within FFTW). An alternative parallel FFT scheme (3d instead of slab) is also available, based on work by Yang Wang (PSC). § Recent developments include thermoelastic, elasto-viscoplastic and dual grid (e.g. void growth); Roters and Eisenlohr have incorporated the FFT as a solver inside FE (damask).

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4 Status, Big Data, RVE, Validation § Status of FFT code(s): on request. § Scalability: – The computational part of the runtime scales as well as the FFT i.e. nearly linear in the number of gridpoints. – (Thermo-)Elastic calculations are fastest. – Viscoplastic calculations require more time because of the need to solve a 5x5 non-linear equation to get slip rates (each step, each iteration). – Elasto-viscoplastic calculations require yet more time because of the need for small strain increments (at least through yield). § Potential connection to big data: although not yet exercised, every reason to expect to be able to iterate back and forth with Dream3D to design microstructures and test their micromechanical response. Depending on domain size, step count etc., rapid accumulation of data. § Domain size: appears to be small (as a number of grains) for elastic, much larger for viscoplastic. § Validation: for both FFT and FE with crystal plasticity, this is not complete.

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5 Topics, Examples § Misorientation development in polycrystalline copper: 2D (from EBSD) and 3D (from HEDM) comparisons. § Elastic response of low volume fraction foams. § Stress hot spots during (visco-)plastic deformation in relation to microstructure. § Analysis (elastic) of twinning in tensile deformation of Zr polycrystal. § Dependence of strain (rate) distribution in a metal-metal composite with hard particles in a soft matrix, motivated by studies of W-Ni-Fe. • Use of the thermoelastic (eigenstrain) method for computing stress fields between dislocations • Analysis (thermoelastic) of stress concentration in thermal barrier coatings and the role of interface roughness • Analysis (thermoelastic) of driving forces for whisker growth from thin films. • Fatigue crack initiation – comparison with SEM, EBSD, HEDM data. • Comparisons with crystal plasticity finite element calculations. • Analysis (elasto-viscoplastic) of a shock experiment leading to incipient spalling in copper.

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6 Voids from Post-Shock Image Image of the surface of a polycrystalline Cu sample subjected to a mild shock; resulting voids superimposed on image. Elastoviscoplastic FFT being used to obtain micromechanical fields and learn about void nucleation. Evan Lieberman, David Menasche, Bob Suter, Ricardo Lebensohn, Curt Bronkhorst, Ed Kober, ADR

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7 Microstructure-Property Simulation with FFT Faster than FEM for large problems (order N log[N]) Requires periodic boundary conditions Proposed by Moulinec & Suquet for linear (1994) and non-linear composites (1998) Extended by Lebensohn for viscoplasticity for polycrystals (2001), on a suggestion by Canova about using FFTs Solving Stress Equilibrium ε(X f ) and σ (X f ) → Elasticity Periodic Simulation Domain • ε (X f ) and σ ( X f ) → Viscoplasticity Moulinec & Suquet, Comput. Methods Appl. Mech. Engrg. 157 69-94 (1998). Michel, Moulinec & Suquet CMES-Comput. Mod. Eng. Sci. 1 79-88 (2000). Lebensohn, Acta Mater. 49 2723-273€7 (2001); Acta Mater. 56 3914-3926 (2008). Rollett, et al., MSMSE, 18 074005 (2010); Anglin et al., Comp. Matls. Sci. 87 209 (2014). Lebensohn et al., Intl. J. Plasticity, 32-33, 59 (2012).

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8 Advantages & Disadvantages of the FFT Method § Caveat: intended for materials problems, not for solving problems with load- bearing structures. FE widely understood whereas FFT approach ~unknown. § Advantage: no need to make a mesh. 3D meshes, especially conforming to microstructure notoriously time consuming and difficult to make a mesh that is free of element quality problems. Nevertheless, commercial solutions exist, e.g., Simpleware (also Jessica Zhang, MechE/CMU). § Advantage: direct instantiation with 3D images from serial sectioning, 3D x-ray microscopy, or other sources. § Drawbacks: periodic structure required in at least one direction out of three; With buffer zones, however, many materials testing situations can be modeled. § Advantage: model microstructures can be easily generated, enabling microstructural design to be investigated. § Elastic and viscoplastic versions of the model have been devised to date: an elasto-plastic model has been developed and published. A thermo-elastic version has been published. § Comparisons to Finite Element method calculations show good agreement. § Advantage: Time required for equivalent calculation is much less for the FFT method, thanks to the Nlog(N) scaling. E.g., for 10 millions degrees of freedom, th viscoplastic, vpFFT requires of order 1/10 time as FE (with crystal plasticity).

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9 Finite Element vs. FFT Fcc Rolling to 40% See also: Eisenlohr et al., IJP 46 37-53 (2013)

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11 Inputs § Any code that computes micro-mechanical fields needs to know: – what type of calculation (elastic, thermoelastic, viscoplastic …) – boundary conditions (type of strain to be imposed, magnitude …) – materials properties (elastic moduli, slip systems, twinning …) § The (FFT) image-based method, not surprisingly, needs an image of the material (as opposed to a mesh). Think of sampling the material on a regular grid (uniform point spacing). Each gridpoint can be a different material but, in practice, we are interested in gradients across bulk features (grains, particles, lamellae …); therefore apply the rule of thumb that 10 points across a feature of interest. § For the FFT codes, there are 3 input files: i) the image (generally 1 line per gridpoint with orientation + grain ID + phase ID); ii) the constitutive properties (e.g. fcc.sx); iii) the control file (e.g. options.in, fft.in).

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