Quantifying the Micromechanical Failure Processes of LSHR:

Characterization, Microstructure Generation, & Simulation Framework

Research Sponsor AFOSR FA9550-10-1-0213

Dr. David Stargel

Albert Cerrone, Joseph Tucker, Clayton Stein, Anthony Rollett, Anthony Ingraffea

6th Int. Conference on Multiscale Materials ModelingBiopolis, SingaporeOctober 16th, 2012

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Overview• LSHR (low solvus, high refractory) disk alloy

o nickel-based superalloyo low solvus

contributes to resistance to crack quenchingo high refractory

high tensile strength and creep resistanceo processed via powder metallurgyo used in disks of gas turbine engines

• Methodologyo quantify microstructurally small fatigue cracks (MSFCs)

improve safe life design aid in development of next generation materials

1. by producing high fidelity, 3D finite element models of microstructures

2. from advanced characterization techniques (3D non-destructive orientation mapping)

3. and simulated in a HPC environment using a crystal plasticity framework

20μm

LSHR

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Outline• The Workflow

o characterization EBSD HEDM

o microstructure generation reconstruction synthetic generation

o meshing surface volume

o constitutive modeling crystal plasticity

o simulation

• Case Study: Crack Initiation Induced by Coherent Σ3 Twin Boundaries

• Future Work

600μm

200μm

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Workflow

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Characterization• Scanning Electron Microscopy (w/ EBSD)

o electron backscatter diffractiono used to detect crystallographic orientationso often coupled with serial sectioning, a destructive method

• High Energy X-Ray Diffraction Microscopy (HEDM)o orientation mapping in 3Do spatial information, as wello nondestructiveo the high energy x-rays are uniquely able to penetrate high-Z, fully dense materials

CCD camera lens

X-raybeam

Rotation

stage

Detector stage Sample

stage

EBSD Map Pole Figures

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wire EDM’d specimen sample

loading direction

EBSD-HEDMComparison

100μm

3mm

100μm

EBSD

HEDM

10μm

Characterization

to our knowledge, first time MSFC located within a 3D non-destructive orientation map

IPFColoration

voxelated grains containing crack entire volume

• Two Optionso reconstruction (authentic representation)

align sections segment grains

o synthetic generation (statistically representative) DREAM.3D

• The voxelated microstructure is then meshed for simulation.

600μm

Microstructure Generation

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Meshing• Surface Meshing

o multiple-material marching cubes algorithmo generated with the constraint of conformal boundarieso each grain’s mesh contained in STL fileo must volume mesh each grain for 3D FE analysis

• Volume Meshingo meshing algorithm

1. octree generation2. advancing front procedure3. mesh improvement

a) back-trackingb) smoothing

o mesh quality gauged with tetrahedron shape metric *o Parallelized Polycrystal Mesher (PPM)

exposes FRANC3D and ABAQUS meshing routines used to mesh synthetic micros of nacre, R88DT, LSHR, AA7075-T651 available at http://www.cfg.cornell.edu/~arc247/PPM/

* Freitag and Knupp, 1999, 8th International Meshing Roundtable.

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5 x 105

Shape Metric

Num

ber o

f Ele

men

ts

degenerate = 0equilateral tet = 1

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Crystal Plasticity• rate, temperature, and grain size sensitive *

• 12 FCC octahedral and 6 FCC cubic (high temperature) slip systems

• resolved shear stress

• flow rule

• hardness evolution

• smaller grains implicitly hardened via ΔiJ term in hardness evolution

• smaller grains explicitly hardened via Hall-Petcho dislocations pile up at grain boundarieso in smaller grains, greater stress required to move dislocations across boundarieso higher applied stress, higher yield strength

𝜏𝛼=𝑠𝛼 [ (𝐹 𝑒 )𝑇 𝐹 𝑒𝑆 ]𝑚𝛼

𝛾𝛼=𝛾𝑜𝜏𝛼

𝑔𝛼|𝜏𝛼

𝑔𝛼|1𝑚− 1

𝑔𝛼=𝐻 𝑜𝛽2𝜇2𝑏

2 (𝑔𝛼−𝑔𝑜❑𝛼 )∑𝛼=1

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|Δ𝑖𝐽𝑚 𝐽𝛼||𝛾𝛼|+𝐺𝑜( 𝑔𝑠−𝑔

𝛼

𝑔𝑠−𝑔𝑜𝛼 )∑𝛼=1

1 8

|𝛾𝛼|

* Matouš and Maniatty, 2004, IJNME

Δ𝑖𝐽=𝜖 𝐽𝐾𝐿𝐹𝑖𝐿 ,𝐾𝑝

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Case Study:Crack Initiation Induced by

Coherent Σ3 Twin Boundaries

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MSFC Characteristics• Nucleation

o microcracks nucleate close to coherent Σ3 twin boundarieso twin boundaries are in large, high Schmid factor (soft) grains

• Propagationo microcracks propagate along Σ3 twin boundarieso predominant mechanism is transgranularo cracks arrest at highly misoriented grains

MSFCs confined to pockets of low misoriented grains

twin = 1, matrix = 2René88DT

20μm

image from Miao, Pollock, Jones, 2009, Acta Mater.

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75μm

• twin embedded in ALA grain• ALA grain assigned high

Schmid factor (soft)• ALA grain – nearest neighbor

misorientations < 20o

• 35 steps of smoothing• 10-mil DOFs

Loading

Σ3 boundary

0.5% - 1.0%Applied Strain

Nucleation (Hot-Spot ID)

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Future Work

1. Predict nucleation event in microstructures using slip-based damage metrics which follow from crystal plasticity formulation.

2. Determine microstructural dependence on MSFC driving forces.

3. Determine microstructural dependence on MSFC propagation rate law.

• The Workflowo 3D, nondestructive characterizationo microstructure generation / reconstructiono surface and volume meshingo crystal plasticity model formulationo simulation

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BACKUP

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Microstructure Generation• DREAM.3D *

o microstructure processing and generation

o synthetic microstructure generation

1. input (from characterization) misorientation / orientation aspect ratio grain size distribution (GSD)

2. closed volume packed with ellipsoids representative of GSD

3. simulated annealing optimizes packing4. cellular automaton nucleates and grows

ellipsoids5. voxelated microstructure output

o reconstruction can be used to align, clean, and reconstruct slices

of data from serial sectioning and HEDM

42.5μm

* dream3d.bluequartz.net

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Computational Specifics• Finite Element Driver

o Finite Element All-Wheel Drive (FEAWD)o scales to 1,024+ coreso built on PETSc, HDF5, and FEMLib

• Computational Resourceo Ranger (XSEDE resource from Texas Advanced Computing Center)

• Performance

10 20 30 40 50 60 700.5

1

1.5

2

2.5

3

3.5

4

4.5

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Number of Nodes (16 cores per node)

Spee

dup

113-sec

394-sec

92-sec

Spee

dup

Number of Nodes (16 cores per node)

394-sec

113-sec

92-sec

speedup plot12.8-mil DOFs

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Shape Metric

𝜅𝑤 (𝐴𝑛)=|𝐴𝑛𝑊−1||(𝐴𝑛𝑊−1 )−1|𝐴𝑛= (−1 )𝑛 [𝑒𝑛+1 ,𝑛𝑒𝑛+2 ,𝑛𝑒𝑛+3 ,𝑛 ]

metric

ea,b is an edge vector from vertex a to vertex b of the tetrahedron

n denotes the vertex number

W is the Jacobian of the linear transformation between a unit equilateral tetrahedron and the reference configuration

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HEDM

1) 50-100 kilo-electron volt X-ray beam

2) Beam illuminates thin plan section of sample

3) Bragg spots are imaged on CCD detectors

4) Measuring a set of spots from multiple sample-to-detector distances yields the position of the diffracted grain

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Material State MappingCP model asserts volume preserving plastic deformation.

𝐹 𝑝=𝐹 𝑝

❑𝑚𝑎𝑝

(det (𝐹 𝑝❑𝑚𝑎𝑝 ) )1 /3

multiplicative decomposition

𝐹=𝐹 𝑝 ∙𝐹𝑒

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ΔCTD Criterion𝑑𝑎𝑑𝑁

=𝐺 ( Δ𝐶𝑇𝐷− Δ𝐶𝑇 𝐷𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 )

da = crack growth incrementdN = cycle incrementΔCTD = cyclic change in crack tip displacementΔCTDcritical = minimum crack tip displacement required for propagationG = material constant (0.3-0.5, dependent on material, strain, and strain ratio)

Crack opening is the dominant MSFC propagation rate mechanism in Stage II.

Stage I: sliding dominated: along the slip systems(s) most favorably aligned with the direction(s) of maximum shear stress

Stage II: opening dominated: in the direction normal to maximum tangential stress ahead of the crack front

blunting: the cyclic change in crack displacement near the crack tip (CTD)

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Schmid Factor

𝑚=𝜏𝜎

critical resolved shear stress applied stress

𝑚=cos (𝜅 )cos (𝜆 ) angle between loading direction and slip plane normal angle between loading direction and slip direction

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Voce-Kocks

𝜎=𝜎𝑠− (𝜎 𝑠−𝜎𝑜 ) exp (−𝜖𝜖𝑜 )σ macroscopic true stressε true plastic strainσs saturation stress extrapolated to zero work-hardening rateσo initial or threshold stress at which homogeneous plastic deformation begins to be appreciable

Voce Law

Kockstemperature and strain rate

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The Saltykov Method to Predict3D Grain Size Distributions• alternative to linear-intercept and sphere-equivalent methods

• predicts grains per unit volume from grains per unit area

• assumes all grains are spheres

• grains assumed equiaxed 2D map to estimate 3D grains

• grain sizes are binned based on intersection probability of a sphere with a section plane

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Volume Meshing• octree generation

o constructed around the graino refined to the element sizes of the surface mesh, and then to the largest cell size on

the boundary

• advancing fronto meshes inward from boundary, discretizing volume with quadratic tetrahedrao facets on grain boundary unchanged, preserving conformity between adjacent

grains

• mesh cleaningo smoothingo back-tracking