Quantifying the Micromechanical
Failure Processes of LSHR:
Characterization, Microstructure
Generation, & Simulation Framework
Albert Cerrone, Joseph Tucker, Clayton Stein,
Anthony Rollett, Anthony Ingraffea
6th Int. Conference on Multiscale Materials Modeling
Biopolis, Singapore
October 16th, 2012
Research Sponsor
AFOSR FA9550-10-1-0213
Dr. David Stargel
Overview
• LSHR (low solvus, high refractory) disk alloy
o nickel-based superalloy
o low solvus
 contributes to resistance to crack quenching
o high refractory
 high tensile strength and creep resistance
o processed via powder metallurgy
o used in disks of gas turbine engines
20μm
• Methodology
o 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 nondestructive orientation mapping)
3. and simulated in a HPC environment using a crystal
plasticity framework
LSHR
2
Outline
• The Workflow
o characterization
 EBSD
 HEDM
o microstructure generation
 reconstruction
 synthetic generation
600μm
200μm
o meshing
 surface
 volume
o constitutive modeling
 crystal plasticity
o simulation
• Case Study: Crack Initiation Induced
by Coherent Σ3 Twin Boundaries
• Future Work
3
Workflow
4
Characterization
• Scanning Electron Microscopy (w/ EBSD)
o electron backscatter diffraction
o used to detect crystallographic orientations
o often coupled with serial sectioning, a
destructive method
EBSD Map
Pole Figures
• High Energy X-Ray Diffraction Microscopy (HEDM)
o
o
o
o
orientation mapping in 3D
spatial information, as well
nondestructive
the high energy x-rays are uniquely able to
penetrate high-Z, fully dense materials
Detector
stage
Rotation
stage
CCD
camera
lens
X-ray
beam
Sample
stage
5
Characterization
wire EDM’d
specimen sample
3mm
EBSD-HEDM
Comparison
HEDM
100μm
IPF
Coloration
EBSD
10μm
loading direction
to our knowledge, first time MSFC located within
a 3D non-destructive orientation map
100μm
6
Microstructure
Generation
• Two Options
o reconstruction (authentic representation)
 align sections
 segment grains
o synthetic generation (statistically representative)
 DREAM.3D
• The voxelated microstructure is then meshed for simulation.
voxelated grains containing crack
entire volume
600μm
7
Meshing
• Surface Meshing
o multiple-material marching cubes algorithm
o generated with the constraint of conformal boundaries
o each grain’s mesh contained in STL file
o must volume mesh each grain for 3D FE analysis
• Volume Meshing
o meshing algorithm
5
x 10
5
Number of Elements
1. octree generation
4
2. advancing front procedure
degenerate = 0
equilateral tet = 1
3. mesh improvement
3
a) back-tracking
2
b) smoothing
1
o mesh quality gauged with
tetrahedron shape metric *
0
0
0.2
0.4
0.6
0.8
o Parallelized Polycrystal Mesher (PPM)
Shape Metric
 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.
1
8
Crystal Plasticity
• rate, temperature, and grain size sensitive *
• 12 FCC octahedral and 6 FCC cubic (high temperature) slip systems
• resolved shear stress
𝜏 𝛼 = 𝑠 𝛼 𝐹 𝑒 𝑇 𝐹 𝑒 𝑆 𝑚𝛼
• flow rule
𝛾 𝛼 = 𝛾𝑜
𝛼
1
𝛼 𝑚−1
𝜏
𝜏
𝑔𝛼 𝑔𝛼
• hardness evolution
𝑔𝛼 = 𝐻𝑜
18
𝛽 2 𝜇2 𝑏
2 𝑔𝛼 − 𝑔𝑜
Δ𝑖𝐽 𝑚𝐽𝛼 𝛾 𝛼 + 𝐺𝑜
𝛼
𝛼=1
𝑔𝑠 − 𝑔𝛼
𝑔𝑠 − 𝑔𝑜𝛼
• smaller grains implicitly hardened via ΔiJ term in hardness evolution
18
𝛾𝛼
𝛼=1
𝑝
Δ𝑖𝐽 = 𝜖𝐽𝐾𝐿 𝐹𝑖𝐿,𝐾
• smaller grains explicitly hardened via Hall-Petch
o dislocations pile up at grain boundaries
o in smaller grains, greater stress required to move dislocations across boundaries
o higher applied stress, higher yield strength
* Matouš and Maniatty, 2004, IJNME
9
Case Study:
Crack Initiation Induced by
Coherent Σ3 Twin Boundaries
10
MSFC Characteristics
• Nucleation
o microcracks nucleate close to coherent Σ3 twin boundaries
o twin boundaries are in large, high Schmid factor (soft) grains
René88DT
twin = 1, matrix = 2
• Propagation
20μm
o microcracks propagate along Σ3 twin boundaries
o predominant mechanism is transgranular
o cracks arrest at highly misoriented grains
 MSFCs confined to pockets of low misoriented grains
image from Miao, Pollock, Jones, 2009, Acta Mater.
11
Nucleation (Hot-Spot ID)
Σ3 boundary
75μm
Loading
• twin embedded in ALA grain
• ALA grain assigned high
Schmid factor (soft)
• ALA grain – nearest neighbor
misorientations < 20o
0.5% - 1.0%
Applied Strain
• 35 steps of smoothing
• 10-mil DOFs
12
Future Work
• The Workflow
o
o
o
o
o
3D, nondestructive characterization
microstructure generation / reconstruction
surface and volume meshing
crystal plasticity model formulation
simulation
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.
13
BACKUP
14
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 packing
4. cellular automaton nucleates and grows
ellipsoids
5. voxelated microstructure output
42.5μm
o reconstruction
 can be used to align, clean, and reconstruct slices
of data from serial sectioning and HEDM
* dream3d.bluequartz.net
15
Computational Specifics
• Finite Element Driver
o Finite Element All-Wheel Drive (FEAWD)
o scales to 1,024+ cores
o built on PETSc, HDF5, and FEMLib
• Computational Resource
o Ranger (XSEDE resource from Texas Advanced Computing Center)
• Performance
5
92-sec
92-sec
4.5
speedup plot
12.8-mil DOFs
Speedup
Speedup
4
3.5
113-sec
113-sec
3
2.5
2
1.5
1
394-sec
0.5
10
20
394-sec
30
40
50
60
Number of Nodes (16 cores per node)
70
Number of Nodes (16 cores per node)
16
Shape Metric
3/𝜅𝑤 𝐴𝑛 metric
𝜅𝑤 𝐴𝑛 = 𝐴𝑛 𝑊 −1
𝐴𝑛 = −1
𝑛
𝐴𝑛 𝑊 −1
−1
𝑒𝑛+1,𝑛 𝑒𝑛+2,𝑛 𝑒𝑛+3,𝑛
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
17
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
18
Material State Mapping
CP model asserts volume preserving plastic deformation.
𝐹𝑝 =
𝐹𝑝
det 𝐹𝑝
𝑚𝑎𝑝
𝑚𝑎𝑝
1/3
multiplicative decomposition
𝐹 = 𝐹𝑝 ∙ 𝐹𝑒
19
ΔCTD Criterion
𝑑𝑎
= 𝐺 Δ𝐶𝑇𝐷 − Δ𝐶𝑇𝐷𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙
𝑑𝑁
da = crack growth increment
dN = cycle increment
ΔCTD = cyclic change in crack tip displacement
ΔCTDcritical = minimum crack tip displacement required for propagation
G = 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)
20
Schmid Factor
𝜏
𝑚=
𝜎
𝜏 critical resolved shear stress
𝜎 applied stress
𝑚 = cos 𝜅 cos 𝜆
𝜅 angle between loading direction and slip plane normal
𝜆 angle between loading direction and slip direction
21
Voce-Kocks
Voce Law
𝜎 = 𝜎𝑠 − 𝜎𝑠 − 𝜎𝑜
−𝜖
exp
𝜖𝑜
σ macroscopic true stress
ε true plastic strain
σs saturation stress extrapolated to zero work-hardening rate
σo initial or threshold stress at which homogeneous plastic
deformation begins to be appreciable
Kocks
temperature and strain rate
22
The Saltykov Method to Predict
3D 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
23
Volume Meshing
• octree generation
o constructed around the grain
o refined to the element sizes of the surface mesh, and then to the largest cell size
on the boundary
• advancing front
o meshes inward from boundary, discretizing volume with quadratic tetrahedra
o facets on grain boundary unchanged, preserving conformity between adjacent
grains
• mesh cleaning
o smoothing
o back-tracking
24